It's About Time: the topology of biological time

Footnotes

¹ An obvious resolution to this problem is that a series of events occurred over a long time that resulted in the evolution of birds that laid eggs. And that noon and midnight are series of points on the line. The point here, as detailed below, is that we often regard noon and midnight as singular events: the many noons and the many midnights can each represent one thing to us. Much of what we experience is repeated in a similar fashion over time, and we think of those repeated activities as sharing a single time course.↩

² Circular reasoning, but the point is that time equals changes in neuronal activity. This is an issue that is present throughout this book, that I don't reject the word "time" even though I reject a simple interpretation of its referent. In Philosophy of Language, the usual example is Russell's "The Present King of France." We understand "time" semantically even though we don't understand what it refers to.↩

³ Parmenides was a 5th century BCE Greek philosopher who lived in Elea, on what is now the Italian coast. His work is known to us only by a set of fragments written in Epic Hexameter, the style of Homer. Parmenides' philosophy of Monism, that everything is one, challenged thinkers to falsify his propositions, as he and his pupil Zeno tried to falsify the Milesians' ideas about change. He got a lot right, including continuity. His monism could be interpreted in terms of things like the "laws of physics" being eternal.↩

⁴ Abstractions can be regarded as not having existence. Nominalism is a philosophy that rejects abstractions, so that red things exist, but red does not. Nominalism is discussed further below.

In my second year at Caltech, I was exposed to Linguistics. The professor for this course, Bozena Henisz Dostert Thompson, spent the first term in her home country of Poland, and her newlywed husband Fred Thompson took over the teaching. Thompson had diverse training and interests, having studied with logician Alfred Tarski, and worked in computer science. He developed natural language abilities in computers, which at the time were not really capable compared to what we have now. He taught us about Chomsky's theories of grammar, and I did an independent study with him that I wanted to be about Structuralism, but he made it primarily a deep dive into Nominalism (Nelson Goodman and Willard Van Orman Quine). I was slow to appreciate the good parts of Nominalism, but am now committed to avoiding categorization of people.↩

⁵ One of many beautiful phenomena in the brain, that these types of remarkably complex signals evolved, with so many accompanying specializations.↩

⁶ Thanks to Ray and Tom Magliozzi, the car guys.↩

⁷ Up the hierarchy is defined by projections from a lower area to the middle cortical layer (4) in the upper area. Projections down the hierarchy go from layer 5 to layer 2/3.↩

Phase

When we say what time it is, we provide a series of phase values for several periods. If it is 10:37 am on Tuesday 10 May 2050, we mean that it is late morning early in the week about a third of the way through the month almost halfway through the year midway in the century. These are pairs of values, a phase and a period, for the periods of a day, a week, a month, a year, and a century. Phase refers to how far around a cycle things lie. We are interested in the set of phase values over a range of periods.

Phase is a property of continuous processes. For discrete situations, in which items can be counted, we learn about doing division, where remainders are left after dividing one integer by another. For division by 4, the remainders can be 0, 1, 2, or 3. These remainders are the discrete version of phase. On a clock with 12 hours, we measure phase over a 12-hour period by saying what hour we are closest to. On a calendar page, we measure phase by what day it is over a period of a month, for which the period ranges between 28 and 31 days.

The periods we use are somewhat arbitrary, though sometimes derived from some natural phenomena (the origins of our western units of time go back primarily to the ancient Near East, developed with the recognition that 12 and 60 have many divisors). But of course we reckon time in terms of months and years that actually have varying periods. We might receive paychecks monthly and keep track of payday as a given phase, even though the distance between paydays varies on the timeline. We rely on the day of the week to organize our lives, even though weeks don't clearly correspond to natural phenomena (the number of days in a week has varied across cultures). Seconds, minutes, and hours are relatively artificial periods as well (derived from dividing larger units by factors of 360, i.e. 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360). Ptolemy divided the circle into first and second minutes using successive divisions by 60, giving us what we now term minutes and seconds.

The circle was often divided into 360 degrees, a system that persists for measuring phase. Phase is also measured relative to 2 times π, as radians, so that halfway around the circle is π and a quarter of the way around is π/2. I choose to measure phase as the fraction of the way around the circle, so halfway around is ½ and a quarter of the way around is ¼. The unit of phase is a cycle, so a half cycle and a quarter cycle are sensible quantities.

Imagine that you are the Emperor, and want to count how many soldiers you have in your army. You stand on top of a mountain looking down on your troops, and direct your generals to have the soldiers organize themselves into small groups of 3. They do so easily by shifting their positions a little bit. But one soldier is left over at the end, the remainder of your desired number after dividing by 3. You then direct them to form groups of 7. They do so, and you write down that there were 6 soldiers remaining. You do the same with groups of 17 and 29, finding remainders of 5 and 2. From those quick manipulations, you can determine the total number of troops, in this case 9253.

Figure 5: Chinese remainder theorem. Four points are plotted from arranging soldiers. The top axis shows the sizes of the arbitrarily chosen groups: 3, 7, 17, and 29. The right axis shows how many soldiers were left over after each grouping: 1, 6, 5, and 2. The bottom axis is the reciprocal of the group sizes, frequency, in units of cycles per number of soldiers. The left axis is the phase, the remainder divided by the group size, in units of cycles. The 4 points have been shifted vertically by integers until they line up. The line drawn through the points has a slope equal to the total number of soldiers, 9253.

There are several methods known for constructing solutions to the Chinese Remainder Theorem. The method of interest here is to plot the remainders divided by the periods against the reciprocal of the periods (Fig. 5). The remainders divided by periods (1/3, 6/7, 5/17, 2/29) are phase values, between 0 and 1 cycles. The reciprocals of the periods are frequencies, cycles per number of soldiers. Phase values are equivalent when they differ by any integer, since adding an integer means going around the circle to return to the original position. We therefore increment the phase values by integers until the phase vs. frequency points are collinear. In other words, there is a single line that passes through all of the points. The slope of that line, in units of cycles divided by cycles per number of soldiers, is the number of soldiers.

This result generalizes. We can do the same thing for data that aren't discrete, using continuous phase and frequency values. The slope of phase vs. temporal frequency is latency. This is best appreciated by knowing a basic property of Fourier transforms:

f(t-L) ⇔ e^2πiLω F(ω)

which says that the Fourier transform of a function f where time is shifted by some latency L is the Fourier transform of f (the function F) with phase shifted by L times the frequency ω. Even if you don't understand Fourier transforms, the point again is that the slope of phase vs. frequency is exactly latency, the pure temporal shift of any function.¹⁰

When people think of latency in the one-dimensional time domain, they generally rely on picking particular moments in time. These choices do not necessarily reflect the underlying functions (e.g., stimulus and response, neither of which is typically a "delta" function, infinitely narrow and tall, completely localized in time). The choices of moments might just be onsets or peaks of those stimulus and response functions, for example. Because of these choices, that sort of time-domain definition of latency is notoriously unreliable. In contrast, the definition used here, the slope of phase vs. frequency, is precise, accurate, and generally applicable, as it is based on the equivalence above.

The addition of integer numbers of cycles to the phase values as described above is not as ill-defined as it might seem. For our purposes, phase is measured at each of a series of temporal frequencies that are spaced closely enough together that neighboring phase values are separated by less than a cycle. The only manipulation that is needed is to ensure that the phase values are slowly increasing as a function of temporal frequency (e.g., the "unwrap" operation in Igor).

Another way to understand that latency is the slope of phase vs. frequency is to think about what latency does. If some process responds with a fixed latency to an input, when does the response occur at different frequencies? Let's take an example where the latency is 100 milliseconds (100 ms). At a frequency of 1 cycle/second (1 Hz), 100 ms is a tenth of a cycle (0.1 c). At 2 Hz, one cycle takes half a second, 500 ms, so 100 ms is 1/5 c = 0.2 c. At 5 Hz, one cycle is 200 ms, so 100 ms is 0.5 c. At 10 Hz, one cycle is 100 ms, so the phase is 1 c. If you plot these phase values (0.1, 0.2, 0.5, 1) against these frequencies (1, 2, 5, 10), you get a line where phase (in cycles) = 0.1 (s) * frequency (Hz). The slope of the line is 0.1 s = 100 ms. As frequency gets higher, the period gets shorter, and a latency takes up a longer portion of a cycle. This hopefully makes it clear that latency is exactly the slope of phase vs. frequency. This relation can be written as φ=Lω, phase is the product of latency and temporal frequency, ignoring momentarily what will be explained next.

Absolute phase

The phase at 0 Hz, what we call absolute phase (analogous to the temperature of absolute zero, which can not be reached but can be extrapolated as approximately -273°C; the term is an oxymoron, given that phase is relative), describes the shape of a function of time. Phase is by definition a relative quantity, and we use a convention where phase is measured in cycles, with 0 cycles corresponding to the peak of an even (symmetric with respect to time, after accounting for latency) function, and +0.25 cycles corresponding to an odd (antisymmetric) function with a negative phase preceding its opposing positive phase (Figure 6, black and red curves). Absolute phase values of 0.5 c and -0.25 c correspond to shifting the black and red functions by a half cycle, giving the blue and green traces. Absolute phase values just less than 0 c are termed phase leads; phase lags correspond to absolute phase values just above 0 c. These terms apply to functions with phase values around 0.5 c as well, so that an absolute phase value of 0.4 c is a phase lead. In the visual system, illustrated with examples below, neurons that respond with phase values around 0 c are called ON-center, and those with phase values around 0.5 c are called OFF-center. This is a somewhat unfortunate terminology originating with early studies where only bright stimuli could be easily generated, so cells that are excited by dark stimuli were tested by turning off the bright stimulus.

Figure 6: Even and odd functions. The black and blue traces illustrate even functions, symmetric about their peak or trough. Absolute phase is 0 c for the black function and 0.5 c for the blue function. The red and green traces are odd functions, anti-symmetric, with negative and and positive phases. Absolute phases are 0.25 c for the red and -0.25 c for the green function.

A simple example of absolute phase in the world is rhythm. Stressed beats can occur at different points in a musical or poetic line with many beats. In some musical forms, the first beat is stressed, but often alternative beats are emphasized. James Brown emphasized the "1" in his music, in contrast to the emphasis on "2" and "4" (the backbeat) in much of rhythm and blues. Poetic rhythms are often established by different kinds of "feet". The common unstressed/stressed iamb foot ("I love") contrasts with the trochee stressed/unstressed foot ("Love me"). The positions of stressed beats determines the absolute phase of the musical or poetic phrase, its shape.

Our goal is to characterize how the brain transforms external sensory stimuli into neuronal activity. These characterizations are simplified by thinking in terms of frequency, amplitude, and phase. In the frequency domain, any stimulus and any response (at least, that we can generate and measure! - formally, members of the ℒ² function space, "square-integrable" functions) can be characterized in terms of amplitude and phase as functions of frequency. The system (e.g., the brain) that turns the stimulus into the response is analyzed by correlating the stimulus and the response. In the time domain, that correlation is a slightly complicated process termed "convolution" (it's convoluted!). In the frequency domain, it is simply division of the response by the stimulus. Formally, this is division of complex numbers, but in terms of the real components of amplitude and phase, it is division of amplitudes and subtraction of phases. The result of this analysis is called a transfer function, or a kernel, among many names. We will refer to kernels, in both time and frequency domains. The phase of the kernel is thus the difference of response minus stimulus phases. The brain or other system changes the stimulus phase into the response phase, and the kernel describes this change. We model this process in terms of the response R arising from the stimulus S via the kernel K:

r(t)=k(t)⊗s(t) in the time domain, convolution (convoluted);
R(ω)=K(ω)S(ω) in the frequency domain, multiplication (much simpler!).

So time is two-dimensional, with one dimension being temporal frequency, and the other phase. A standard way to think about the two-dimensional space of time is as a disk, with the center point missing (Figure 14). The distance out from the center is frequency or period, and the distance around the disk along any fixed frequency is phase. This representation of time is clearly different from the time line!

We are quite familiar with such representations of time. Our clocks tell us what time it is by giving us phase values for some small number of periods. Analog clocks are versions of the disk, usually showing the hour (the phase over a 12 or 24 hour period) with a short hand, and the minute (phase over a 1 hour period) with a longer hand, farther out on the disk - so the opposite direction from the center compared with Fig. 14, where longer periods are farther out.

On our calendars, weeks are laid out, containing 7 days. Some of our activities are centered on certain days, and thus can be considered as having different absolute phase values. Maybe we tend to eat tacos on Tuesdays, and worship on Fridays, Saturdays, or Sundays, structuring our behaviors around different positions in the week. We will see that absolute phase in a neuron corresponds to the timing relative to some phase of the process that affects the neuron, so that the neuron anticipates, goes along with, or follows the timing of the process.

Phase in the real world, especially the brain

That is, what matters is that firing rates vary over time. Consider a neuron that cares about light inputs to the retina. If light reaching the eye varies in its luminance over time, the neuron's activity is likely to be modulated in time by the stimulus. An example is illustrated by Movie 2. The stimulus is a small spot, and the luminance changes over time in a 4-part sequence, from dark to background to bright to background and back to dark. The movie shows several cycles of the stimulus, and the audio lets you hear when spikes occur during the stimulus presentation. Firing rate increases around the times when luminance increases, either from the dark to bright transition or from the background to bright change.

Movie 2: 4-part flashing spot. The spot is centered on the neuron's receptive field, and modulated through a series of steps: dark, background, bright, and background. The cell fires spikes that are audible here, with the timing being ON sustained, the spikes occurring after the spot luminance increases. In the experiments, 25-50 cycles of the modulation are typically used to compile the average response over the cycle.

Neurophysiologists at least unconsciously believe that when this 4-part sequence is repeated, the stimulus is the same from one repetition to the next, and from one neuron to another. So the presumption is that time consists of what's happening during the process, even if that isn't explicit for them.

Averaging across many cycles of the stimulus, we show how firing rate (in impulses per second, "ips", the same as spikes per second) depends on the stimulus (Figure 7). Three neurons illustrate how there is a range of timing. The transient nonlagged cell (1) at the top responded primarily just at the luminance increases, after 0.5 and 1.5 seconds. The sustained nonlagged cell (2) also fired at those points, but continued to fire as long as the spot was bright. The transient lagged cell (3) did not fire at the stimulus onsets, but was active during the period when the spot was bright, then fired strongly at offset (2.5 s). This response lags the stimulus, therefore.

Figure 7: Transient/sustained/lagged. Examples of 3 cells in the kitten lateral geniculate nucleus illustrate fundamental aspects of timing. Responses are shown for 3 neurons to the stepping stimulus at the bottom and in Movie 2. The stimulus is a small spot that changes from dark to background to bright to background and back to dark over a 4 second cycle that repeats. The cells fire action potentials (spikes) at certain times during the stimulus cycles, and these times are shown by the histograms that are averages over all the cycles, with amplitudes in units of impulses (spikes) per second (ips). Cell 1 (transient nonlagged) fires almost solely when the spot increases in brightness, after 0.5 and 1.5 s. Cell 2 (sustained nonlagged) fires after the increases in luminance, but continues to fire while the spot stays bright between 1.5 and 2.5 s, and stops firing immediately after the spot returns to background. Cell 3 (transient lagged) stops firing when luminance increases, but then fires while the spot is bright, until at the offset of the bright spot it gives an anomalous offset discharge.

That the differences between these 3 cells is a matter of timing might not be entirely clear. The stepping stimulus used here is somewhat complicated. We also tested these neurons with the same spot, but modulated its luminance in time smoothly (Figure 8). The responses of these cells are shown to this smoothly modulated stimulus that has a sinusoidal shape (Movie 3). This stimulus greatly simplifies characterizing these cells. The responses are also somewhat sinusoidal (but can not go negative), with the dominant frequency of their modulations being the stimulus frequency (here, 1 Hz or 1 cycle/second). The phase of the responses at 1 Hz is easily measured, and corresponds with the absolute phase values (about -0.15 c, 0 c, and 0.15 c; compare to traces in Fig. 6, where the absolute phases are 0, 0.25, 0.5, and -0.25), except that those absolute phase values are actually what would occur at 0 Hz. So the difference between the phases at 1 Hz and at 0 Hz is the latency multiplied by 1 Hz. The progression of timing in the 3 cells in Fig. 8 can be appreciated by following these sinusoidal responses, going from a strong phase lead, before 0.5 s, to a strong phase lag, well after 0.5 s.

Figure 8: Sinusoidal stimulation. Responses of the cells from Figure 6 are shown for a stimulus that is modulated smoothly (sinusoidally) in time between dark and bright phases (Movie 3). The 3 neurons fire at different phases of the stimulus cycle.

Movie 3: Sinusoidal modulation. The spot changes between bright and dark smoothly in time. The responses of this cell are sustained nonlagged ON. More cycles than this are typically presented, and the temporal frequency is changed from one trial to the next. In this example the temporal frequency is 1 Hz.

The differences between these cells will be provided in more detail below. A key generalization to note is that these differences in timing rely on synaptic inhibition. Transient nonlagged cells are subject to late (meaning after excitation) inhibition, and lagged cells to early (before excitation) inhibition. Many neuroscientists think of inhibition as keeping neurons from firing. But of course, the neurons do fire sometimes. Inhibition should be thought of as determining timing. Inhibition stops transient cells from firing after their initial activity. Inhibition keeps lagged cells from firing initially.

The terms transient and sustained have been applied to certain visual neurons since the early 1970s (Cleland et al. 1971). Related terms like phasic and tonic are also used. The term lagged, on the other hand, originates in this context from Mastronarde 1987. The most important note is that these terms refer to temporal differences. That fact has often not been appreciated.

Keep in mind that just as many OFF cells exist that fire a half-cycle away from these ON responses, showing similar responses as these cells but to a stimulus that turns dark rather than bright. The distinction between ON and OFF cells becomes slightly less important when thinking of two-dimensional time, but is biologically rooted in the retinal mechanisms that produce them. Different cells respond at all of the phases around the cycle (Fig. 9). ON cells have absolute phase near 0 c, between -0.25 c and +0.25 c, OFF cells near 0.5 c, between +0.25 c and +0.75 c (note again that adding an integer number of cycles to a phase value does not change it, so -0.75 c to -0.25 c is an equivalent range for OFF cells). Cells in the quadrants between 0 and 0.25 c or between 0.5 and 0.75 c are called lagged cells, and those in the other two quadrants are called nonlagged cells, roughly speaking. As noted above, transient cells have absolute phase approaching ±0.25 c, and sustained cells have absolute phase near 0 or 0.5 c.

Figure 9: Phases. Three dichotomies are shown according to the regions of absolute phase where they occur. The ON/OFF dichotomy corresponds to phase values near 0 or near 0.5, respectively. Sustained cells have phase values near either 0 or 0.5 cycles, and transient cells have phases near ±0.25 cycles. Lagged cells have phases between 0 and 0.25 or 0.5 and 0.75 cycles, and nonlagged cells between either -0.25 and 0 or 0.25 and 0.5 cycles. Addition of an integer to a phase value does not change it. These correspondences should be regarded as approximations.

These characterizations in terms of phase are simple and useful. Absolute phase describes the shape of the kernel mentioned above. Shape is the fundamental quantity that describes a system's behavior. One can expand or contract the system without changing its shape. You can shift all of the system's elements by the same amount and recognize it as the same system. These invariances are facts of geometry. We will generalize this later from geometry to topology.

An important concept, really the primary concept, in sensory physiology, is the "receptive field" of a neuron. The receptive field can be defined broadly to be the set of stimuli that modulate the neuron's activity. Often, the term is applied to the spatial receptive field, the region in space where stimulation can affect the neuron's firing. But we will focus largely on the temporal receptive field. This concept is less familiar. It can be considered to refer to how the neuron turns stimulus timing into response timing (the temporal kernel).

The time domain temporal kernels, also known as impulse response functions, are shown in Fig. 10 for the 3 cells in Figs. 7 and 8. These are like the traces in Fig. 6 (green, black, and red). This representation of temporal receptive fields is conventional. Unfortunately, quantifying the differences between these three examples in the time domain is challenging. However, they are easily described in terms of phase values. The Fourier transforms of the functions shown in Fig. 10 provide amplitude and phase vs. frequency from which one obtains absolute phase and latency. More examples of how absolute phase corresponds to response timing are available (e.g., Figure 11 in Saul and Humphrey 1990; Figure 9 in Saul 2008; and Figures 2 and 3 in Saul et al 2005).

Figure 10: Impulse response functions. The plots show the time courses of the responses derived from random flashes of dark and bright bars at different spatial positions. The insets are the resulting space-time maps, with the bottom axes space and the left axis time. Red areas are "ON" (bright excitatory) responses and blue are "OFF" (dark excitatory) responses. Taking profiles through the spatial location with the strongest responses gives the impulse response functions. Cell 1 has a positive mode followed by a negative mode, which is typical of transient nonlagged cells. Cell 2 has a positive but not a negative mode, which is a sustained response. Cell 3 has a negative mode followed by a positive mode, characteristic of lagged responses, here a transient lagged response.

Latency and absolute phase measurements can be obtained most simply from experiments that measure responses to sinusoidal stimuli of varying frequencies, as described below. However, they can also be derived in many other ways. The impulse response functions in Fig. 10 were computed from experiments that used noise stimuli, random flashes of bright and dark spots (Movie 12F). But one could use the responses to the 4-part flashing spots. That stimulus contains a set of frequencies that are odd multiples of the lowest frequency (0.25 Hz for the 4 s cycles in Fig. 7). The drawback for this stimulus is that the strength of those components decreases with frequency, so that the component at 0.75 Hz is a third of that at 0.25 Hz, and at 1.25 Hz only a fifth. But the timing of the responses to the 4-part flashes reflects the latency and absolute phase values well (Fig. 12 in Saul & Humphrey, 1990).

Figure 11 shows the temporal receptive fields of two kitten LGN neurons. The cell on the left is transient nonlagged, with an absolute phase of about -0.2 c. Its temporal receptive field therefore lies about 80% of the way to the downward axis at low frequencies. The transient lagged cell on the right has an absolute phase of about 0.2 c, so starts about 80% of the way to the upward axis. They are about a quarter cycle apart at 0.5-4 Hz. By 6-8 Hz (portions within the 4 Hz circle), they have similar phase values. The latencies differ considerably, with the nonlagged cell completing more than 1 c while the lagged cell doesn't pass through a single cycle over about 8 Hz, as can be seen in the temporal receptive field plots.

Figure 11: Temporal receptive fields. A nonlagged and a lagged cell recorded in kitten LGN are compared. The 4-part flash responses are both transient, but peak at bright onset for the nonlagged cell and at dark offset for the lagged cell. The amplitude and phase vs. frequency plots at the bottom, obtained from stimuli as in Fig. 8, were used to plot the temporal receptive fields in two-dimensional time. The receptive fields lie over the phase and frequency points in the phase plots at the bottom, and are given pseudocolored scaling based on the amplitudes.

Turning to a less technical discussion of phase, recall that the essence of phase is that it is a relative quantity. It tells you how one thing differs from another. Many of our judgments clearly fall into this category: softness, lyricism, beauty, pain, how busy we are, etc. These are continuous quantities, though without simple quantifications. Our neurons quantify their magnitudes in their firing rates. People frequently act as if everything can be categorized, and that such judgments are not really continuous. Some things are certainly discrete, such as the whole numbers, but most categorical judgments can fairly easily be made continuous, typically by adding noise. For instance, rating a league of 10 soccer teams is generally done on a scale of 1 to 10. But if they play each other team 5 times, their records will create a more continuous distribution.

People often ignore what they consider to be messiness in order to supposedly simplify their thinking, for instance by considering race to be meaningful, or by deciding which sports team or athlete is #1. We tend to impose artificial thresholds in order to create discontinuities where there are none: if team A beats team B, A is thought to be clearly superior to B, even though if they played multiple times they would each win some and lose some, and in reality are probably similar in capabilities. Whether this tendency to categorize corresponds to the way the brain works innately, perhaps a property of associative memory, or comes about through cultural conditioning, remains unclear. It may be an adaptation for making decisions rapidly, similar to beliefs. We are nonetheless quite capable of relying on continuous, fuzzy, phase-based information and actions.

Consider the very important topic of moods. We sometimes pretend that sadness and happiness are discrete, but we know that our sadness can vary, from the sorts of devastation we feel when we lose a loved one, to when we hear about the death of somebody we don't know, to when we lose a game. Our moods obviously change over time, often at a very slow pace, but occasionally rapidly. These changes in mood motivate much of our behavior. Our goal is to understand some of the neural mechanisms underlying these functions.

Wang and colleagues 2011 recorded from neurons in the mouse amygdala, a brain region that is important for emotional processing. They found that many neurons fired in relation to the anxiety of the mice. Figure 12 illustrates some of their results.

Figure 12: Slow changes in activity. A) Mice were put in two boxes, one an open field and the other having dividers and objects to check out. In the open box, mice stayed mostly in the periphery, whereas in the enriched box they explored. The graphs show that the mice had similar amounts of movement in the two boxes, but showed more behaviors thought to be related to something like anxiety in the open field. B) Example of firing from an amygdala neuron in the two boxes. Strong activity occurred in the open field but not in the enriched box, and ramped up slowly over the half hour of recording. C) The increase in amygdala activity over 30 minutes is shown for the two boxes. These neuronal changes correlate with the anxious behaviors.

When mice are placed in an "open field", an empty box, they tend to spend most of their time near the walls, seldom venturing into the middle. This behavior is presumed to reflect their anxiety about predators being able to reach them in an open field. Wang et al. divided such a box into several regions with walls, and placed objects in some of the regions to give the mice things to explore. They compared behavior and electrophysiology in the two boxes. Panel A (top far left and far right) shows that the mice stayed mostly near the outside in the open box but moved nearer to the center in the enriched box. The graphs in panel A show that the mice moved similar distances in the two boxes, but displayed anxious behaviors (defecation and escape attempts) in the open box more than in the enriched box. Certain amygdala neurons had activity that paralleled the behavior, firing in the open field but not in the enriched box (panel B). What is remarkable is that the activity in the open box ramps up over about 20 minutes. That this slowly increasing activity mirrors the emotional time course is suggested by the differences in defecation (boli) and escape attempts (jumps) between the two boxes that appeared late in the time the mice spent in the boxes (Figure 12C).

This is direct evidence that single neurons can show slowly modulated activity, with phase values that correspond to highly sustained and lagged patterns. This must be true if you accept that single neurons underlie our behaviors, but it could be that only activity across populations of neurons would show this.

The transient, sustained, and lagged kitten LGN neurons illustrated above provide an example of how certain human behaviors might be created. Tallal 1980 studied a common problem seen in 5-10% of children entering primary school. These children had difficulties that people thought were due to defects in how they processed language. Tallal and colleagues demonstrated that the defect appears to lie at a low level, and is specific to temporal processing. These children have deficiencies in processing rapid changes in any modality (auditory, visual, tactile, motor, ...). I would characterize this as a deficit in transient neural mechanisms, those that respond best as things are changing rapidly, with a phase that leads the input by about a quarter cycle. Our speculation is that the neural substrate lies in thalamus, a part of the brain through which almost all inputs to cortex pass, covering all modalities. The main visual thalamic nucleus is the LGN. Neurons in thalamus vary in their timing, as described above and discussed further below. Loss of transient timing in thalamus could produce the problems seen in the children studied by Tallal.

Paula Tallal tells a story about when she would go to children's homes to try to help them. The project used synthesized speech that slowed down the rapid changes in sounds that distinguish consonants like /ba/ and /da/ (Figure 35). The children would put on headphones to listen to a story generated with the synthesized speech, while playing a video game related to the story. The affected children would enjoy the story, being able to finally understand what was being said. The unaffected siblings, on the other hand, would hate listening to the odd speech and would take off the headphones in disgust.

Another large group of children (as well as adults) have attention deficit hyperactivity disorder (ADHD). This disorder may correspond to a deficit in sustained and lagged mechanisms in thalamus, with phase values near and greater than 0. Those neurons might be important in sustaining attention, for instance. They could also play an important role in reinforced learning, as detailed below.

Besides these disorders, processing of time varies across human populations without being considered pathological. We know some people who are always late. If they need to be somewhere by noon, they only start getting ready at noon. We speculate that they lack transient neurons that respond ahead of stimuli, with phase leads. Other people tend to do things ahead of time, getting prepared in advance, getting to appointments early, etc. They may have an abundance of transient neurons. It may be difficult to understand this in terms of points on a line, but easy when considering deficits in sustained and transient phase processing.

A misconception that is worth addressing is that many people think analysis in the frequency domain somehow applies only to periodic functions. This is far from the truth. As noted above, any reasonable function can be examined in the frequency domain, technically the "square integrable" functions that are mainly characterized by having non-zero amplitudes that don't extend out to infinity in time, so the amplitude goes to zero at low temporal frequencies. For our purposes, there is nothing about the work described here that entails periodic changes in any variable. Keep in mind that the description of a function in the frequency domain consists of amplitude and phase across the entire range of frequencies. Periodic functions are simply those where the amplitude is zero over all but a finite subset of those frequencies.

Absolute phase and latency are weakly correlated with the temporal frequency tuning of amplitude. Lagged cells tend to be tuned to slightly lower frequencies than at least transient nonlagged cells. The underlying reason might be that cells with longer latencies (Figure 31) can not respond well at high frequencies simply because their latencies approach and exceed the stimulus periods at those frequencies. A neuron with a latency of 100 ms can not respond well above 10 Hz (period of 100 ms).

Readers will probably think that much of brain function happens in moments, rather than extended periods of time. Changes in the external world happen mostly over long time scales. Our internal thoughts and emotions are similarly slow. Most experiments demand fairly isolated processing. In the real world, we must deal with numerous distractors. We often need to search through activity that is filtered out as irrelevant before we get to the task at hand. Because most things change slowly in time, we are less aware of all the things going on at low temporal frequencies, to which we slowly adapt (see below).

Many authors continue to disregard phase, and write that different latencies across a population of neurons could be the basis of direction selectivity. We will discuss this issue at length, emphasizing that direction selectivity varies with temporal frequency due to the interaction of absolute phase and latency.

We concentrate on time here, but must mention that space should be conceived in an analogous fashion. We don't say where we are in GPS terms. We describe our location in relative terms. We live on the south side of town, and at the current time are upstairs in the middle of the study at the front of the house. The moon is low in the sky above the tree at the edge of the yard. The brain similarly uses phase to locate us in space. The hippocampal place cells mentioned above are created from two-dimensional spatial maps in entorhinal cortex. The entorhinal neurons were termed grid cells. There are grid cells with different spatial frequencies and phases, mapped across entorhinal cortex. Elsewhere in the brain, there may be many other spatial mapping strategies, but it's a solid bet that they rely on frequency and phase, just like time.

Figure 13: Topology. Open and closed sets are illustrated schematically. Open sets do not have end points, whereas closed sets do. The union of the two open sets on the left might produce a set that looks like the open set in the middle if they were shifted closer together.

For the set of points on the line, the usual open sets are those that don't have "end points" included in them. So the set of points between 0 and 1, not including 0 and 1, is open. For any point in an open set, there are arbitrarily nearby points that are also in the open set. This clearly does not hold for discrete subsets of the line like {0, 1, 2, ...} because there are no points closer to 0 than 1 in the set. The subsets that consist of the points not in a given open set are closed.

Our point of view is that moments, corresponding to points on the line, do not exist. Instead, the basic objects of time have extension, and correspond to open sets in two-dimensional time. Things happen over these open sets, and processes have durations.

Thus, thinking about the topology of time, we ask what points in time are near each other. Many people would say that times that are separated by any arbitrary short length of time are near each other, thereby thinking in terms of the natural topology on the line. The notion of continuity will arise below, but we won't go into its technicalities.

The key idea is that we think of time in terms of our activities. If we say that it's time to eat lunch (dinner for some people), sets of times that might be roughly repeated every day are thought of as near each other. These points form an open set on the line, but it is natural to think of them on the disk, where a period of a day is a circle that lies at a certain distance from the origin, and lunchtime takes up a small segment of the day.

The disk of time (Figure 14) consists of the set of all periods and phases, with periods positive (greater than 0) real numbers, and phases having values between 0 and 1, with 0 and 1 being the same point (so phase takes values on a circle). The disk is the product of the open interval of real values greater than 0, times the circle. The periods should be plotted logarithmically, so that the distance between a second and a minute (a factor of 60) is the same distance as between a minute and an hour. One can also think of this as a cone, which may give a better picture of increasing periods with distance from the apex of the cone.

Figure 14: Disk of time. At the top, the time line is plotted to show the times when somebody might eat lunch. At the bottom, those times are plotted on the disk, falling at many phases and frequencies, but predominantly over a period of one day and around noon. The legend at the top right shows the pseudocolor coding of amplitude.

What counts to us is not really when one thing happens, but instead that different things happen at related times. We wake up, and a bit later we might eat breakfast and go to work, for instance. These relationships are spread across the time line, but grouped appropriately on the disk (Fig. 14, events a day apart on the line vs. red and yellow area near the black circle at the period of a day). These daily events occur at nearby phases on the cycle of a day. The disk provides more continuity than the line.

When we walk, we move one leg forward, then bring the other forward, no matter how fast we walk. This makes sense when thought of in terms of the phases at which different behaviors occur over some periods. We don't move the right leg forward at a fixed number of seconds after the left leg: that depends on how fast we're walking. But we generally move our legs in fixed phase relationships. When dancing with a partner, this is usually necessary!

We can map points on the line to the disk, and vice versa. It should be obvious, however, that these representations of time are different from each other. For instance, the disk and the cone are equivalent topological spaces, as are the subset of the line between 0 and 1 and the entire line. Intuitively, two spaces are equivalent if they can be continuously deformed into each other. The line can not be continuously deformed into the disk.

The most obvious difference between the disk and the line is their dimensionality. As mentioned above, that time is two-dimensional might not seem apparent to people who are heavily tied to the one-dimensional view, but all of our normal time-keeping is done with two-dimensional displays: calendars showing several weeks and days in each week, clocks with hands showing the phase for at least two periods, and even digital displays showing the hour, followed typically by a colon to separate that phase from the phase given as the number of minutes in the hour. Note that a sundial is really one-dimensional, with the topology of the circle: only a single period is displayed, so that the period dimension collapses to a point, a zero-dimensional space.

A single point on the line projects to a set of points on the disk (Movie 4) that form a spiral. These points all have phases near 0 for long enough periods (periods much longer than the value of the time). With decreasing period, phase increases. The relation between period, phase and time is t=𝜑⋅pd, where t is the time, 𝜑 is the phase, and pd is the period. This relation can also be written as 𝜑 = ωt, where ω is temporal frequency, the reciprocal of period, as pointed out above, where we discussed that latency (the variable t here) is the slope of phase vs. frequency.



Movie 4: Point to spiral. A moment in time projects to the time disk as a spiral. Here, noon becomes a spiral passing through noon (period of 24 hours, phase of ½ c) on the disk.

A single point on the disk, on the other hand, projects to a series of equally spaced points on the line (Figure 15). Given that each of those points on the line projects to a spiral, and since that set of spirals is not the same as the original single point on the disk we started with, we see that these maps from the disk to the line and back to the disk do not end up with the identity map. This demonstrates that the disk and the line do not have the same topology. On the other hand, all of the spirals converge near the original source point on the disk, showing that these representations of time are closely related. The composition of maps from the line to the disk and then to the line is illustrated in Figure 16. As in Figure 14, the first map projects strongly to lunchtime on the disk, but also to all of the points in the many spirals. Those points in turn project to many points on the line outside of the original points that came about 24 hours apart. The point, again, is that the disk and the line have different topologies.

Figure 15: Disk point to line. A single point on the disk represents a period and phase. This point therefore projects to a series of points on the line separated by the period (24 hours here), and occurring at the phase (about 22 hours through the day).

Figure 16: Composition of maps. A series of events that occur at a period of about 24 hours (top) is first mapped to the disk (middle), as in fig. 11. The disk is then mapped back to the line (bottom). The resulting timeline does not match the original line at the top, because the line and the disk differ topologically.

There is, however, a natural mapping between functions on the line and on the plane: the Fourier transform. We are in fact primarily interested in such functions (rather than points), that describe how something changes over time. As an example (Figure 17), say that you have a meeting coming up at 9:00 am tomorrow. If we could measure how much you think about this meeting, you might spend a variable amount of time today, perhaps peaking before you fall asleep at midnight, then falling suddenly to 0 until you awaken at 7:00 in the morning. You might then show increasing attention to the meeting leading up to 9:00 am. This would define a function of time over the 30 hours or so before and after the meeting, with peaks around midnight and 9:00 am, and very low values between midnight and 7:00 am. Although we can describe this function in these kinds of terms, it can be difficult to provide a quantitative description in terms of time.

Figure 17: Time in the frequency domain. On the left is an imaginary plot of how much attention is paid to a meeting before and just after it occurs. This time domain function is transformed to the frequency domain on the right, with components of amplitude and phase vs. frequency. The phase vs. frequency plot at the bottom shows how phase varies fairly linearly with frequency, with a slope that is latency. Note that amplitude is plotted against log frequency, whereas phase is plotted against linear frequency. The deviations from linearity occur at high frequencies, above about 1/hour, where amplitudes are vanishingly small.

We can, fortunately, transform this function to the frequency domain, obtaining two real-valued functions of temporal frequency (the reciprocal of period). One of these functions describes how strongly you paid attention to the meeting at each temporal frequency, and the other tells when you were thinking about the meeting. Because of the peaks separated by about 12 hours, there would be strong amplitudes at periods near 12 hours. More importantly, the phase graph has a slope of about 23 hours. Because the meeting occurs at 24 hours from the start of the data, this corresponds to a sort of average time of thinking about the meeting coming about an hour ahead, despite the concentration on the meeting after its start.

The power of the frequency domain view of time lies in putting together different processes to see how they might be related. If you want to see how the attention paid to the meeting is related to some other variable, such as release of a stress hormone, alertness, sympathetic activation, or dopamine release, you could convolve the rather complicated time-domain function above with a similarly complicated time domain function for your other variable. Such comparisons on the time line are difficult to quantify. In the frequency domain, on the other hand, it is mostly a matter of dividing the amplitudes and subtracting the phases between the two functions. The phase difference tells you how your variables are related: what direction and timing any causal relationship might have. The sign of the latency gives the direction of causality. An absolute phase difference near 0 means that your variables are synchronized, but other absolute phase values give insights into the transformations going on between them.

A function in the frequency domain defines a contour above the disk, with the height being the amplitude, typically lying along a spiral (𝜑 = Lω + 𝜑₀). Over the range of frequencies where two functions have strong amplitudes, the differences in the slopes and in the phases at low frequencies determine the relationship between them. Although the amplitudes contain information that can be important, the key information lies on the disk, in these differences between the phase vs. temporal frequency information. That is, the kernel or transfer function that relates the two functions is largely defined as a set of points on the disk where amplitudes are significant. It is therefore worthwhile to study the category of functions on the disk.

An example of looking at both the time domain and frequency domain

Absolute phase increases during the period from about 100 ms before to 100 ms after the saccades (Fig. 18B; this is the same time course during which amplitudes decline). This makes responses more sustained. Latency shortens remarkably before the saccades, most strongly about 70 ms beforehand (Fig. 18C).

Figure 18: Time reversal. Data from a study by Binda and colleagues is shown on the left. Examples of results from 4 subjects are shown. The pairs of points in the ovals illustrate that when bars were flashed about 75 ms before the subject made a saccade, the subjects judged the order of the flashes incorrectly. On the right are our data replotted and including the latency changes around saccades. These latency changes have a strong narrow tuning, with the largest changes occurring about 75 ms before saccades. The relative timing of the two bars would depend on the derivative of these latency changes, so that the earlier bar would have a longer latency than the later bar.

The key idea is that the activity of these LGN cells is passed along to visual cortical neurons. The cortical neurons put together inputs from different LGN cells to create novel response properties, such as direction selectivity, as detailed below. For now, consider two problems that this situation can solve. One is that cortical cells will only respond if a stimulus is in its receptive field long enough. If a stimulus is moving across the receptive field when a saccade intervenes and moves the stimulus off the receptive field, the neuron won't respond. However, the timing changes we discovered in the LGN would compensate for the spatial shift caused by the saccade, permitting the cortical cell to respond despite the saccade. This could signal that the stimulus' movement was continuous across space.

Figure 19: Compensating for saccades. A simplified illustration is presented for how the temporal receptive field changes around saccades could compensate for the stimulus moving outside the receptive field because of the eye movement. The movies on the left show a ball rolling down an incline as a metaphor for how stimuli move across receptive fields. In the situation at the top, no saccade occurs, and the ball rolls smoothly down the slope, or the stimulus moves smoothly across the receptive field, generating a large response. In the bottom two scenarios, a saccade occurs at 0.5 seconds, shifting the ball to the right. With no compensation (middle row), the stimulus ball leaves the receptive field incline, and a lower response is created. At the bottom, the receptive field compensates for the saccade by shifting its temporal profile to match the spatial shift. The response is greater as the ball stays on the ramp longer. The spatiotemporal orientation of the receptive field is opposite that of the stimulus, because time 0 for the receptive field corresponds to when spikes occur, whereas for the stimulus time 0 is when the stimulus starts to move; this is one of the complications when thinking about convolutions.

Figure 19 presents a simplified model of how this works. Imagine that a stimulus moving across a neuron's receptive field is like a ball (the stimulus) rolling down an incline (the receptive field). At the top, there is no saccade, and the ball rolls smoothly down the incline, creating a steady response. Both the stimulus and the receptive field are smooth, and their convolution gives the strong response amplitude. In the middle row, a saccade occurs at 0.5 seconds, shifting the stimulus ball off of the receptive field incline. The neuron would stop responding when the stimulus was outside the receptive field. At the bottom, the receptive field changes around the saccade. Here, the change is a simple shift at the time of the saccade, for simplicity. What is illustrated is that the receptive field now compensates for the stimulus shift caused by the saccade. The ball stays on the incline longer and the response is improved. This shift is what we observed as the increase in absolute phase before fixational saccades.

Another problem involves integrating stimulus locations before and after saccades. In the absence of a saccade, if a stimulus moves abruptly from one spatial location to another, we notice the change. If the same change in retinal location is created by a saccade, however, we do not see it as movement of the stimulus in the world. In the first of these situations, only the stimulus moves, and the background of the world stays put on the retina, whereas in the second case, both the stimulus and background move. Note, however, that the changes in neuronal receptive field properties occur before the saccade, so do not depend solely on the relative movement of figure and ground. The solution could be that the timing changes in the LGN that make neurons more sustained permit cells that were responding to the stimulus prior to the saccade to continue their activity after the saccade, when other cells that have receptive fields at the new location after the saccade are firing. The overlapping activity in these cells with different spatial receptive fields on the retina could signal that these locations are equivalent in visual space.

Morrone and colleagues performed related psychophysical experiments on humans (Figure 18). They flashed a pair of bars at different intervals of time between the flashes, and in either order (bar A then bar B, or bar B then bar A). They also had the subjects make saccades at different times around the flashes. In one of their experiments, the subject reported which bar came first. They obtained a compelling result: when the flashes came fairly close together and about 70 ms before a saccade, the subjects reliably reported the order wrongly. That is, people consistently say that bar B flashed before bar A when in fact the reverse was true. The interpretation is that time reverses for us just before we make saccades.

In the Saul study, we noted a sudden, short duration decrease in latency for the LGN cells at 70 ms before the saccades (Fig. 18C). We speculate that the Morrone results could be interpreted in terms of this change in latency: if the first bar flashes 100 ms before the saccade, it would evoke responses at a longer latency than a second bar that flashes at 70 ms before the saccade, where latencies are greatly shortened. The response to the second bar would then occur before the response to the first bar.

Remember that the responses measured from the neurons in this study were not localized in time. The analyses needed to be performed in the frequency domain, based on phase in particular. Working in the time domain alone would not enable these results to be observed.

⁸ I was teaching calculus to engineering students at West Virginia University, and noticed that a book had been left behind by the previous teacher. After class, I went to return the book to that professor, but first scanned its content. It was The Geometry of Biological Time, by Art Winfree, a beautiful book in every way. I found the owner, Ken Showalter, in the Chemistry Department. Ken is a physical chemist, and we wound up talking about the book and many other things. And wound up collaborating, studying time. I got to spend some wonderful time with the polymath Winfree in the June snow of Utah.

At the time, Ken was working on a chemical reaction that involved the reagents sodium iodate and arsenous acid. If you put these chemicals in a jar at the beginning of a lecture, they will sit as a transparent liquid. At some point during the lecture, the fluid in the undisturbed beaker will suddenly turn black. This is an example of a clock reaction.
Video of clock reaction

Inorganic chemicals can undergo reactions at regular or irregular times. Ken had been looking at how the reaction in a tube of this solution can be triggered at one point in space, creating a wave that traveled down the tube. We worked on analyzing these propagating waves. I was living on a piece of land we called "Out Yonder" an hour's drive from Morgantown at the time, working in the woods, cooling off in the pond, then trying to figure out how the waves worked. Ken and I sent notes (a few years later there would have been email) back and forth about our progress. He was sorting out the chemistry, and I was playing with the math. We each came up with the answers from our two perspectives at the same time. I had seen that the waves could be described in a pretty (closed-form, meaning easy to write down and read) function if the reaction was a cubic function of iodide concentration. Ken had seen that the reaction was in fact a cubic. We nailed down how well the predictions fit the data, determining the speed at which the waves traveled, and published a paper with Adel Hanna.

That got me started on thinking about time. I learned many things from Ken, especially about the wonders of collaboration. Some people don't recognize that Science is about people. But humans are social animals, and we thrive on our interactions. The pleasure of doing Science is largely spending time with people you enjoy and appreciate. Every once in a while you learn something, providing enormous pleasure. ↩

⁹ My friend Robert Tajima and I took Max Delbrück's course that covered a variety of biological and philosophical topics, including the Chinese Remainder Theorem. I got to know Max on annual trips to Joshua Tree, with his wife Manny and their children (Tobi works on electronic chips that process visual stimuli, including direction selective elements that analyze moving images, unfortunately using 1-dimensional time; Ludina Delbrück Sallam is an Arabic translator; there were two older children I didn't know), English professors David and Annette Smith's family, and several other undergraduates. Max kept star charts like the ancients. My last year, he told me how his retinal detachments kept him from being able to make out the stars well enough to continue this work, and asked if I'd take over his charts. I stupidly declined, fearing I wouldn't do it justice. Max viewed Science not as building a cathedral, but instead people piling up stones. I found his attitudes refreshing and on point. Max deserved every honor possible, just a giant of a man.↩

¹⁰ The right hand side is the Fourier transform of the left hand side here (and the left hand side is the inverse Fourier transform of the right hand side). The factor that multiplies F(ω) simply produces a phase shift of Lω, so that the latency shift L on the left hand side results in a phase shift of Lω. The relation between the phase shift and ω is therefore linear with a slope of L.↩

¹¹ Phase is usually not exactly linearly related to frequency, because of linear and nonlinear filtering. We ignore those complications for simplicity. For most processes, a linear relation between phase and frequency dominates. This is especially true if amplitude is taken into account, so that weaker amplitudes are discounted when considering the phase vs. frequency relation.↩

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Stitching time

Local-to-global transformations

The example of integrating stimuli that occur before and after saccades was discussed above. This is one of many examples of how local processing over distinct open sets on the disk (that is, times before and after saccades) can be integrated into new structures (assembling the responses from neurons that respond to distinct visual field locations into a percept that sees objects remaining at one position). The brain does this on massive scales in parallel as well as serially, giving us global experiences from the numerous local bits that involve many millions of single neurons. We can dissociate local and global processing by looking at examples like that in Fig. 20, where the local elements (the 3 corners) make sense, but the global percept doesn't.

This is one of the biggest problems in neuroscience, as well as elsewhere, the local-global transformation. Something happens at one time, and something else happens at another time. How do we know whether or not these things are related, and how do we put them together into a properly ordered sequence? When we listen to music, we need to integrate hundreds of elements over time, and we can manage to extract meaning from the total that is clearly not present in any of the parts (Hasson et al. 2008; Carvalho & Kriegeskorte 2008). Uri Hasson has explored these issues extensively in several modalities. Note that those studies argue for hierarchical processing.

Almost all functional studies of neurons have considered their local properties, those confined to their limited receptive or movement fields. For example, visual and somatosensory neurons have spatial receptive fields, the region of visual space or skin where stimulation affects the neuron, but our perceptions consist of objects that we see or feel. Motor neurons typically cause a small group of muscle fibers to contract, but the brain develops networks that plan and execute complex behaviors that employ those simpler motor neurons.

To reiterate the idea presented above, sensory neurons can be thought of as having temporal receptive fields in the sense of responding at certain phases in the variation of some response property over some range of temporal frequencies. Different neurons with different temporal phases and different response properties provide inputs to other neurons, that integrate those inputs to create an emergent response property not present in the inputs.

Motor neurons have movement fields, defined as the movements that the body makes when they are activated. Single motor neurons are typically local units, but are assembled into more global units in order to execute useful behaviors. Over the disk of time, a behavior consists of many motor neuron activations that have different phases and combine together to achieve a smooth, accurate movement.

The convergence of these inputs can be thought of as a stitch between them, creating a result that is less local and more global. One can think of patches of fabric floating above the disk of time, and the brain sews the patches together to produce a larger quilt. One imagines that there are advantages in creating a smooth, continuous quilt rather than a messy, torn quilt. Mechanisms for resolving discontinuities are therefore called for, and studying continuity is worthwhile.

These somewhat vague ideas can be formalized and made more rigorous, via the mathematical field of sheaf theory. A sheaf is a mathematical object defined as a set of structures called stalks, and a map from the stalks to an underlying topological space. That map simply says where each stalk is located in the topological space. The stalks can be any sort of object that has values of any kind, with any structure. A section consists of a set of values from the stalks over an open set in the space. A sheaf has the "gluing" property, whereby if two sections are the same over the intersection of their underlying open sets, then there is a section over open sets that include those two open sets with values equal to those of the original two sections over shared points in the topological space. A sheaf means that a global section (a smooth quilt) can be sewn across those sections.

Sheaves are natural models for the way the brain works. The brain stitches together activity across open sets in time and space to form a continuous experience. Mountcastle (1957) described the structure and function of somatosensory cortex in terms of columns, which are like stalks over the tangential organization (that is, the surface parallel to the layers of cortex). For the somatosensory cortex, the tangential dimensions map the body surface, that is, space. For each point on the body, a number of different sensory submodalities can be found, including fine touch, pain, temperature, vibration, and the state of that part of the body. The stalks consist of values of these properties, so the range of temperatures, the frequency of vibration, and so on. If an extended object touches our skin, individual neurons respond to the local properties, and the brain puts that information together and creates activity that we perceive as a shape, or we recognize the object. Similarly, if our skin is contacted by some object in some way that changes over time, we can tell whether it is moving in a particular direction and how its properties are changing, leading to knowledge about what is going on (e.g., a drop of water is rolling down our shoulder).

Looking back at the sort of sheaf examined above in Fig. 14, we consider the sheaves of breakfast and lunch (Figure 21). The active stalks at these two times are about a quarter cycle apart around the period of a day. This quarter cycle phase difference is the key arrangement for combining local information into a novel, more global property, as will be described in detail below.

Figure 21: Breakfast and lunch sheaves. The time domain and frequency domain representations of when things occur are shown for two processes, when we eat breakfast and when we eat lunch. In the time domain, each of these processes occurs at disconnected points on the line. In the frequency domain, each occurs at nearby points. The relationship between the timings of breakfast and lunch are especially clear this way, as they occur a quarter-cycle apart (north for breakfast and west for lunch).

Hubel and Wiesel defined an even better example of a sheaf (Figure 22). They described visual cortical organization in terms of visual space being the base space, corresponding to the tangential organization of cortex, derived from the two-dimensional sheet created by the retina. They focused on two properties of visual cortical neurons: orientation and ocular dominance. These neurons (in cats) are all sensitive to the orientation of contours, some responding best to vertical, others to horizontal, and others to oblique orientations, over the full range of 180°. This is a good example of neuronal response properties forming a circle, with phase describing their values. The other property, ocular dominance, was a measure of the relative response amplitude of cells to stimulation through the two eyes. Some cells only responded to the contralateral eye (for a cell in the left hemisphere, this would be the right eye), other cells responded to both eyes to different degrees, and some cells only responded to the ipsilateral eye.

Figure 22: Hypercolumn. For each small patch of V1, Hubel and Wiesel defined what they called a hypercolumn, consisting of neurons that have receptive fields overlapping in one point in visual space. They showed that two visual response properties, ocular dominance and orientation, were represented completely within the hypercolumn. That is, a full range of ocular dominance and all orientations could be found. These properties are mapped fairly continuously across V1.

Hubel and Wiesel had noticed that each of these response properties is mapped across the tangential dimensions, so that neighboring neurons tended to have similar orientation preference and ocular dominance. The maps were locally continuous (note that orientation can not be mapped in a globally continuous fashion across cortex, according to the Hairy Ball Theorem of topology). Hubel and Wiesel proposed that these two maps had to be locally orthogonal, so that, ideally, neighboring cells would not have identical values in both ocular dominance and orientation. The notion of columns was extended to the idea of hypercolumns, where all values of ocular dominance and orientation are found in a small patch of cortex that has receptive fields in a small neighborhood of visual space, with all of the spatial receptive fields intersecting in at least one point. The hypercolumn is thus a stalk in a sheaf whose base space is visual space: over each small region (open set) in space, one can pick values of ocular dominance and orientation.

Ocular dominance is a less salient variable than orientation for most people. Other simple variables important in V1 include contrast, color, spatial frequency (how rapidly contrast changes over space), preferred direction, and temporal frequency (how rapidly contrast changes across time). A visual scene can be thought of in terms of choices of these features over each open set in space, corresponding to the activity of neurons in visual cortex.

Figure 23 attempts to illustrate this sort of sheaf. The color image is an actual representation of how orientation is mapped onto primary visual cortex. Each color corresponds to a particular orientation, as shown at the right edge, so this is a pseudocolor map of orientation preferences across a patch of cortex. The map is derived from an experiment where the cortex is imaged while gratings of different orientations are presented. A patch of activated cells tends to reflect slightly less light than neighboring inactive cells, so capturing images at a series of grating orientations permits this map to be assembled. You can see the effect of the hairy ball theorem at several places in the image, where the different colors come together. This area was termed a pinwheel, because the various colors rotate around it. A question arises from knowing that cortex must contain pinwheels: how do cells in the pinwheel respond to different orientations? They could respond to all orientations, but such cells are rarely observed. Alternatively, the pinwheel region could comprise a mixture of separate neurons that preferred different orientations from each other. Reports have suggested that the orientation tuning of the neurons in pinwheels sharpens at low contrasts, perhaps reflecting recruitment of longer range inputs. Evidence has been found that single neurons in the centers of pinwheels have strong orientation selectivity, and are segregated as in areas away from the pinwheel centers.

Figure 23: Sheaf of colors/orientations. An orientation map of cortex was derived from experiments using optical recording of cortical activity. Each orientation was assigned a color (as indicated at right) and regions of cortex preferring an orientation were colored as shown, with red horizontal and green vertical.

Movie 5: Sheaf in 3D. The map in Fig 23 is given a 3-dimensional representation to make clear its structure as a sheaf. Stalks that contain the full range of colors shown in A are positioned across space, and the stalks are cut off at the point where the appropriate color occurs. Rotating the sheaf eventually brings it to a viewpoint that matches the scene in Fig. 23.

The sheaf version of the hypercolumnn in Movie 5 provides a clearer illustration. At each point in space, there is a stalk that has differing colors along its length. Again, these correspond to different orientations. However, we can also just think of these stalks as containing preferences for different colors, as might be seen in another part of the brain where color processing is performed. All of the stalks are the same, except that the color corresponding to the same pixel in the image in Fig. 23 is highlighted, and the stalk is truncated at that point by making the top of the stalk much less visible. This enables a view from the top that mimics the original image in Fig. 19, but adds a depth component when viewed at other angles that reveal the stalks.

This example of a sheaf corresponds to a few hypercolumns, containing millions of neurons. We are primarily interested in single neurons, and will treat a single neuron's receptive field as a sheaf. An analogous sheaf to the hypercolumn above can be devised with the base space of time. Just as the spatial sheaf will vary over time, the temporal sheaf varies over space. This can lead to using a topological space uniting space and time, but for now we'll try to simplify things by restricting our base space to just time, as if we're only treating a single neighborhood in space with its spatial stalk.

Hubel and Wiesel described two main types of cells in cat primary visual cortex, that they called "simple" and "complex". Simple cells have spatially separated regions that respond to either increases or decreases in luminance (Fig. 44). Complex cells respond to both increases and decreases in luminance throughout their receptive fields. They theorized that complex cells were created by convergent inputs from simple cells with overlapping receptive fields.

David Hubel had a mathematics background, and saw that this operation of creating complex cells by putting together inputs from simple cells could be generalized. He termed the abstraction of this process "complexification" (see p. 3406). He saw this operation as appearing throughout cortex, building increasingly complex response properties in neurons with increasingly large receptive fields. It turns out that complex cells are usually created directly from LGN inputs rather than from cortical simple cells, but Hubel's insight that cortical processing might occur through such a general mechanism has considerable merit.

The single neuron version of the hypercolumn sheaf defined in primary visual cortex can be complexified into another sheaf by starting with two sheaves (Fig. 24). Say that one sheaf (RF2) has a value of orientation that represents vertical over an open set in visual space, and a null value outside that open set. The other sheaf (RF1) has an orientation just off vertical over an open set that is shifted in space relative to RF2. Together, the two neurons responding to these orientations over slightly different portions of space could provide input to another neuron (in white) that would thus respond to a curved contour that bends away from vertical in the lower part of its receptive field. Hubel and Wiesel labeled such further complexified cells "hypercomplex", but they are known as "end-stopped" because they respond better to short contours than to longer contours.

Figure 24: Complexification. This toy example takes two neurons with slightly different spatial receptive fields and slightly different orientations, and projects them onto a third neuron. That new neuron (white receptive field) might then respond to curved contours. Curvature is a novel response property that does not exist in RF1 or RF2.

Thus, such neurons have stitched together different spatial properties. We are most interested in how neurons stitch together different temporal structures, associating them because of their consistent temporal relationship with each other. On the line, breakfast appears to consist of separate events separated by days, but in the phase domain, one can see that they are synchronous, overlying a common neighborhood on the disk (Fig. 21). These observations are the essential notions in our theory. Processes are characterized in terms of phase, rather than being events that are localized in one-dimensional time. Different processes are united by neurons receiving convergent input from neurons whose phase values differ. This creates emergent properties.

The binding problem

Movie 6: Ramachandran on Capgras Syndrome.

This is why we resist thinking of time in ways that we didn't grow up with. Our sense of time is invested with tremendous emotional content, and we feel attached to, comfortable with, and have a certain fondness for the timeline. We don't want to give up our sense of free will, or of moving along the railroad track of time.

We need to note, however, that one kind of evidence that free will doesn't exist is based on the experiments mentioned above, where neural activity that drives our choices occurs earlier in time than the conscious belief that the choice has been made. What does earlier in time mean? The underlying assumption is that the neural activity and the sense of having chosen are well localized in time. It seems more likely that these processes are both extended in time, and overlap. The appropriate analysis compares phase vs. temporal frequency for the two processes, not arbitrary moments in time. Sapolsky does not rely on such experiments for his thesis about free will, however.

Sapolsky discusses chaos. The butterfly effect is a popular concept, where the fluttering of a butterfly's wings in Brazil leads eventually to a hurricane in the Caribbean. The idea is that weather is sensitive to starting conditions, so that even a tiny perturbation in the wind can lead to much larger effects. Unfortunately, Sapolsky oversimplifies the basis of chaos, claiming it arises from nonlinearities. That is not a great explanation. A really clear example of how to produce chaos was described by Mitch Feigenbaum, a certified genius who pioneered many aspects of chaos theory.

Figure 25 illustrates Feigenbaum's example where x_n+1 = 4λ x_n(1-x_n). That is, you start with x₀, compute x₁ based on this rule, and keep doing that for increasing n. The parameter λ is key here, as will be explained. The starting value x₀ should be between 0 and 1. If it starts at either 0 or 1, then all subsequent values are 0. But if it starts anywhere else, after many iterations the values tend to approach a fixed point. For example, if λ=0.5, and we start with x₀=0.1, the rule gives us the series 0.1, 0.18, 0.2952, 0.416114, 0.485926, 0.499604, 0.5, and then 0.5 forever (green region). If λ=0.6, the series is 0.1, 0.216, 0.406426, 0.578985, 0.585027, 0.582649, 0.583606, 0.583224, 0.583377, 0.583316, 0.58334, 0.583331, 0.58334, 0.58333, and then it oscillates around about 0.58333 asymptotically. An interesting thing happens as λ increases. At λ=0.75, the series oscillates between two points, one near 0.65918 and another near 0.67398. Then, as λ continues to increase, the series oscillates between 4 points. For instance, at λ=0.87, the 4 points are about 0.869426, 0.395064, 0.831680, 0.487159. At λ=0.89, 8 points, at λ=0.892, 16 points, and as λ increases by decreasing amounts, the oscillations keep increasing by a factor of 2. The number of points in the oscillation is the period, so the period doubles as λ increases. But these doublings occur at smaller and smaller increases in λ. At a finite λ, around 0.8925, there have been an infinite number of period doublings, meaning the period is infinite. That means that one can not predict what the value of x_n is as n gets large. The process is chaotic. Feigenbaum's process is termed the period doubling route to chaos.

Figure 25: Period doubling route to chaos. Feigenbaum's example of how changing a parameter can produce infinite periods while the parameter remains finite.

So chaos can be achieved by a deterministic process, not just by randomness. This can be demonstrated physically, for example with chemical reactions (BZ reaction, with a citation of Art Winfree; and a Ken Showalter presentation). These chemical systems demonstrate emergent properties, like many biological systems as well as the physical systems noted above.

When people claim that nonlinearities are responsible for something, like chaos, they aren't saying much. The category of nonlinearities is simply the complement of linearities. It includes practically everything (except linear things). Some nonlinearities do one thing, others another thing. As seen above, the behavior of a nonlinear process depends on parameters. When we're told something is nonlinear, we should always ask, "What kind of nonlinearity?" It's like wanting to know where somebody lives, and being told "Not in Gaborone."

Most systems have large linear components, while also showing nonlinearities, so one should keep in mind that linear and nonlinear components are combined. Purity is rare. Oversimplification is common. When analyzing systems, the initial focus is often on the linear components, because they can be analyzed more fully, and generally account for more characteristics of the system. Nonlinear components typically require more complicated analyses. One can use the linear analyses to predict how much of the behavior of the system is in fact linear, and one gains considerable insight this way.

Sapolsky describes how the brain works in his Chapter 13 in terms of combining differing inputs to generalize across their similarities. For instance, he puts together Gandhi, King, and the Mirabal sisters to create the concept of dying for one's beliefs. This is probably one of the ways the brain works. But more important might be the structure outlined here, where inputs that differ are combined to create new properties, most importantly direction.

The brain tends to signal errors. Decisions consist largely of trying to avoid errors. For instance, when we're walking around a house, our brain is making lots of decisions about how to move in order to not walk into walls. A way this works might be illuminating. Many of the pathways carrying information about motor plans pass through the cerebellar nuclei (Fig. 26). The motor plans are relayed to the places where the plans are carried out. But they are also sent into cerebellar cortex to be processed. The processing there can be characterized as looking for errors. When all looks good, the inhibitory output of cerebellar cortex (Purkinje cell activity) decreases, permitting the cerebellar nucleus cells to go ahead and relay the plans. If something seems wrong, Purkinje cell activity increases, inhibiting their targets in the cerebellar nuclei, stopping the motor plans from being executed at that time. A concrete example might be that when playing the piano, if a finger is about to hit a key but the cerebellum detects that doing so would be inappropriate, the movement is stopped. In Sapolsky's context of making moral decisions, the brain has set up values, and when a decision is about to be made that goes against those values, the error is detected and the decision is suppressed.

Figure 26: Cerebellar circuitry. The inputs to the cerebellum have projections to the deep cerebellar nuclei (DCN), which projects to various areas that create actions. The Purkinje cell output of cerebellar cortex is inhibitory, and keeps the DCN cells from firing, unless the Purkinje cell activity is decreased. One can therefore think of Purkinje cells as carrying error signals, preventing actions at the wrong times.

But ethics and morals tend to be fluid because of the complexity of the world, and the brain must rely on the direction in which a decision might push us, rather than the binary choices described in the previous paragraph. My favorite example is a skilled soccer player who has the ball at her feet. She puts together an enormous amount of information about where her teammates and opponents are located, and figures out the best moves to make with the ball. That processing might occur largely in prefrontal cortex and cerebellum, which have the inputs and mechanisms to deal with such complex situations. Sometimes she will make a bad decision, and her brain will learn from it. Sometimes she will be about to make a bad decision, but her brain will prevent her from going ahead with it. Sometimes she will make a better decision, and her teammate will get the ball and subsequently pass it to the opposition, providing another learning experience. And much of the time she will make an amazing pass with great results, or dribble past the opponents and cross the ball to a teammate for a goal. Great training for making good moral decisions, doing the right thing and heading in the best direction.

Movie 7: Model for direction selectivity. Two inputs project to a third neuron (triangle) that will be direction selective. The inputs are separated in space by a distance Δx, and the left input is delayed by a time Δt. The left input is assumed to be inhibitory and the right input excitatory. When a bar moves from left to right, it first activates the left input. After a delay, this input suppresses firing of the DS cell, despite the excitation arising from the right input once the bar has traveled the distance Δx. In the right-to-left direction, the right input is activated immediately, and the inhibition arrives much too late to prevent the DS cell from firing. The scheme on the right shows the exact same model, but Δx and Δt are replaced with spatial and temporal phase differences.

We need to combine these inputs onto a downstream cell that we want to be direction selective. How we combine them has been a subject of much discussion, as noted above, but from our point of view is not the issue, since almost any way of combining them works! We choose, for convenience, to make the left-hand, delayed input inhibitory and the right-hand input excitatory. Then, for a stimulus that moves from left to right, the left input is excited, but can not inhibit its target immediately because of its delay. That inhibition arises later, at the time when the stimulus has moved toward the right far enough to excite the right-hand input. The inhibition blocks the excitation that the right-hand input provides, and the cell does not respond.

For the opposite direction, the right-hand input is stimulated first, and does excite the direction selective cell. The stimulus only arrives on the left later, and the delay adds more time before the inhibition is passed to the direction selective cell, too late to affect it.

Hopefully this is simple to understand. This is all that is involved, in a sense. In one direction, the inputs arrive onto the direction selective cell at the same time (and in this way of combining the inputs, the inhibition blocks the excitation). In the other direction, the inputs arrive at different times (so the inhibition is ineffective). The key to this model is that the difference in direction of motion is turned into a difference in timing: in one direction we get the SAME time, in the other direction DIFFERENT times.

There are complications in this scheme, arising from the fact that we're working in the time domain. The inputs arrive at the same time only if Δx/v -Δt = 0 where v is the stimulus speed. The delay Δt between the inputs has to match how long it takes for the stimulus to travel between the inputs (Δx/v), so the speed v has to be Δx/Δt. For the rightward direction, the time difference between the inputs is Δx/v+Δt.

If the stimulus moves too fast from left to right, it will get to the right-hand input before the inhibition arrives, and thereby excite the downstream cell. If it moves too slowly, the inhibition will have died off before it gets to the right-hand input, and the excitation will similarly get through. This is a real problem, that this model only works for a limited range of speeds around Δx/Δt. However, the problem is easily solved, by working in the frequency domain.

The diagram on the right is exactly the same model, but uses a different language. Instead of the spatial and temporal differences Δx and Δt, we now label them ψ and 𝜑. These are spatial and temporal phase differences. Just by changing our language, we gain a lot. The model works exactly the same as above, but now we can say exactly what it means for the signals to arrive at the same or at different times. In one direction, we subtract the spatial and temporal differences, and in the other direction we add them. For the rightward direction, we subtract them, as above, and want the difference to be zero. For the leftward direction, we add them, and want the difference to be non-zero. Using the language of phase, no difference is 0 c, in phase, and a maximally different phase is ½ c, out of phase. So we can now specify exactly what we want. In one direction the difference of the spatial and temporal phases is 0 c, and in the opposite direction their sum is ½ c:

ψ - 𝜑 = 0 c
ψ + 𝜑 = ½ c

These two equations are easily solved for ψ and 𝜑. They are both true when ψ=¼ c and 𝜑=¼ c. That is the spatiotemporal quadrature model, spatial and temporal phase differences of a quarter cycle. Given those phase differences, you get direction selectivity.

We emphasize that the key here is to turn the difference in direction into a difference in timing. We would generalize that to most aspects of brain function, where differences in function result from differences in timing. As they say, timing is everything!

With this description in terms of phase, the model now works at any speed. That trick comes about because we insist on having the quarter cycle phase differences no matter what the stimulus might be. In reality, if the stimulus can vary, we might have a hard time creating those phase differences across all of the stimulus variations. Watson and Ahumada (1985) termed this model an ideal motion detector.

We will discuss this issue of how real neurons deviate from quadrature as the stimulus changes, but first note that real neurons do not need to be ideal motion detectors. They do not have to satisfy the equations above exactly, but instead just need to have somewhat different responses in the two directions. They may not achieve a response of 0 in the non-preferred direction, nor attain the maximal possible response in the preferred direction. If the phase difference in the non-preferred direction is 0.05 c and in the preferred direction it is 0.45 c (e.g., spatial and temporal phase differences of 0.25 c and 0.2 c), the response difference would still be significant. This is what is observed in many neurons. What is important is not so much quadrature, but that there are spatial and temporal differences. Those differences can not approach 0 c or 0.5 c, but must be somewhat near quarter cycles. We need approximate spatiotemporal quadrature.

Phase is only part of the description of functions in the frequency domain. Amplitude also plays a role. Implicit in spatiotemporal quadrature is that the amplitudes of the two inputs are the same. Otherwise, even though the phases subtract to 0 c and add to 0.5, subtraction and addition of the complex numbers comprising amplitude as well as phase would not lead to a maximal response in one direction and no response in the other direction. However, this requirement to have the same amplitudes applies only to the ideal. In real neurons, the amplitudes of the inputs can be somewhat different. The way the inputs are combined plays an additional role. For instance, a weak inhibition might be able to veto a strong excitation. In addition, we need to keep in mind that neurons are not completely linear, and they receive many inputs, not just two. The situation quickly gets more complicated than the ideal model, which we think of as giving us ideas about what to measure: theoretically-derived heuristics.

Before getting to more technical matters, another simple way to understand the importance of quarter cycles is to ask how one would move around a clock to go from 12 to 6. You can go either clockwise or counter-clockwise equally well. In order to specify which direction to take, one needs to add something between the half-cycle, saying to go from 12 to 6 through 3. Then it is clear that one should go clockwise. This is what quadrature buys, a phase difference that is between zero and a half-cycle is needed to specify direction. Winter to summer can go either forward or backward in time, but adding spring tells you to go forward. All that is really needed is some relatively small phase change, say from Monday to Tuesday for the period of a week. Note that as the change approaches either zero or a half-cycle, however, the directionality becomes less compelling. What really counts for direction selectivity is how temporal phase varies across space. If phase just jumps by a half-cycle (i.e., from 12 to 6), one can go in either direction. Ideally, temporal phase varies continuously and monotonically across spatial phase. If it forms a nice ramp for stimuli to roll down, you get an ideal motion mechanism with constant slope (Movie 8).

Movie 8: Ball on ramp. A direction selective receptive field is like an incline that the stimulus rolls down easily in one direction, but doesn't climb well in the opposite direction.

Many advertising signs use flashing lights to simulate motion. For example, light bulbs might light up sequentially in time from left to right to draw attention to text on the right side. A similar sequential use of lights occurs on certain one-way streets, where the traffic lights are programmed to turn green to allow vehicles to keep moving. Such timed lights must change between red and green dependent on their phase in the sequence. On a two-way street, the phases should be half-cycles apart.

When working in the space-time domain, phenomena can be observed that seem counter-intuitive. The best-known example is called reverse-phi. Phi motion is apparent motion, created by a series of flashed stimuli at different locations, as in the signs mentioned above. If you see a flash toward the right followed in time by a flash toward the left, you perceive leftward motion. However, if a bright stimulus is flashed on the right, followed by a dark stimulus on the left, you can perceive rightward motion even though the flash on the right preceded the flash on the left in time. Hopefully you have already considered this stimulus in terms of phase. By reversing the contrast of the flash from bright to dark, one imposes a half-cycle shift in phase (Movie 9). So now instead of the stimulus moving from 12 to 3, it seems to move from 12 to 9, that is in the opposite direction. The quarter cycle shift in space has the half cycle added to it, giving ¾ cycles, which is equivalent to -¼ cycles.

Movie 9: Reverse phi. For the first half of the movie, normal phi motion is shown, and the grating seems to move toward the left. Then, the contrast is reversed, and the direction reverses.

Shortly after the appearance of the JOSA papers, Thom Carney and Mike Shadlen produced nice demonstrations of how spatiotemporal quadrature gives us direction. They took advantage of the fact that we have two eyes that can be shown different stimuli, but that the signals from the two eyes come together in the back of the brain, in visual cortex. They showed one eye a stimulus, and the other eye the same stimulus but shifted in phase by a quarter cycle in space and in time. For either eye alone, the stimulus did not have motion in the sense that no single direction was apparent. However, when the signals from the two eyes are added in visual cortex, the motion derived from the quadrature pair is obvious. They called this cyclopean motion (Cyclops had one central eye) because the motion is only seen when the two eyes are combined into one, in analogy with cyclopean depth, where combining information from the two eyes via stereopsis produces a perception of depth. The example in Movie 10 tries to illustrate this in a simple form. On the left, a grating is modulated in time sinusoidally. On the right, a similar grating is also modulated sinusoidally. Neither grating has any clear direction up or down, though one might perceive alternating up and down motion. The two gratings are in spatiotemporal quadrature: they are offset by a quarter cycle in space, and are modulated with a quarter cycle offset in time. If you can cross or uncross your eyes so that the two plus signs merge into one, you should get a clear direction of motion (downwards if you cross your eyes and upwards if you uncross them). These demonstrations are beautiful on a number of levels. For one, they are a physiological proof of a trigonometric identity. The gratings can be written as sin(x)sin(t) and cos(x)cos(t), products of their spatial and temporal modulations, with the quarter cycle differences given by the difference between cosine and sine functions. When they are added up in visual cortex, you see the sum sin(x)sin(t)+cos(x)cos(t), which we learn in trigonometry is cos(x-t), and describes a moving grating.

Movie 10: Cyclopean motion. The two gratings are modulated in time, in spatiotemporal quadrature. Each grating has no direction, but fusing them so that your visual cortex adds them together reveals that their sum has a clear direction.

The process of detecting the direction of motion in the world involves breaking that motion up into its spatiotemporal quadrature components, the sin(x)sin(t) and cos(x)cos(t) parts that do not have motion by themselves. That decomposition is performed in retina and LGN. These components are then reassembled in visual cortex to produce the sense of direction, as demonstrated by cyclopean motion. The reason it has to work this way is that neuronal processing in the retina is local. Retinal neurons respond to local changes in brightness, signaling by their activity the timing of those changes, and by their location on the retina, the location of the stimulus in space. The local information from each of many retinal neurons is eventually put together in cortex to provide the more global and emergent response property of direction of motion.

Direction selectivity outside of visual cortex

Figure 27: Thalamic triad. In the thalamus, many neurons receive their input through a triadic arrangement. For the visual thalamic nucleus, the LGN, the input comes from a retinal ganglion cell. This input makes two excitatory synapses. One is onto a dendrite of a relay cell that projects to visual cortex. The other synapse is onto a dendrite of an inhibitory interneuron. That interneuron dendrite in turn makes an inhibitory synapse on the relay cell dendrite. The transmitters are glutamate and GABA, and the receptors for these molecules are labeled.

The excitatory input to a lagged cell also provides a strong feed-forward inhibition via these synaptic triads. A spike in the retinal axon releases the neurotransmitter glutamate, that binds to NMDA receptors on the relay cell, but also binds to AMPA receptors on the interneuron dendrite.¹⁴ This depolarizes (excites) the interneuron dendrite and leads to release of its neurotransmitter, GABA, onto the relay cell dendrite. GABA binds to GABA_A receptors and tends to reduce the depolarization of the relay cell, inhibiting its spiking activity.

Mastronarde provided a vast amount of data detailing the origins of lagged cell responses and contrasting them with nonlagged cells (Mastronarde 1987a, Mastronarde 1987b). First, he confirmed that all lagged and most nonlagged cells receive a single dominant input from one retinal ganglion cell. He noted that he only recorded lagged cells with certain electrodes, and that the parameter that best distinguished lagged and nonlagged cells was their "antidromic latency" from cortex. Antidromic latency is how long it takes for a spike generated in cortex (by experimentally providing electrical stimulation there) to travel to LGN, in the reverse direction along the axon. Lagged cells often had extraordinarily long antidromic latencies. The inference is that their axons have small diameters, which slows the conduction speed of spikes. Neurons with fine axons have small cell bodies. The difficulty in recording lagged cells with many electrodes arises because of these small cell bodies, since most electrodes will only "hear" large cells that swamp the many smaller cells in their vicinity, like listening to a crowd where a few people are shouting and most are whispering. An electrode needs to get very close to a small cell to hear it, putting its ear up to the cell's mouth, to extend the analogy. Hubel and Wiesel recognized this problem, though they didn't try to solve it: "Other types of receptive fields may yet be found in the cortex, since the sampling (45 units) was small, and may well be biased by the micro-electrode techniques. We may, for example, have failed to record from smaller cells, or from units which, lacking a maintained activity, would tend not to be detected. We have therefore emphasized the common features and the variety of receptive fields, but have not attempted to classify them into separate groups."

Confirmation of many of Mastronarde's results was obtained (Humphrey and Weller, 1988a), and extended by filling neurons intracellularly with the dye horseradish peroxidase (HRP) after recording their visual responses (Humphrey and Weller, 1988b). The HRP provides a way to visualize the structure of the neuron after sacrificing the animal, fixing the brain tissue with aldehydes, and cutting and preparing sections of the LGN to be viewed under a microscope. They demonstrated that all lagged cells have small cell bodies and fine axons. They also showed that lagged cells have a particular morphology, showing dendritic appendages known as grape clusters (Fig. 28). These appendages are the sites of triadic synapses. Some LGN relay neurons have many appendages, others have fewer, and some have none. Nonlagged cells can have dendritic appendages or not, and typically have larger cell bodies than lagged cells.

Figure 28: Lagged cell morphology. A micrograph of an HRP-filled neuron show the dendritic appendages (grape clusters) at dendritic branch points.

Mastronarde, Humphrey & Weller, and Heggelund & Hartveit all estimated that lagged cells make up about 40% of the LGN cells projecting to cortex. Previous studies of the geniculocortical projection had attributed a number of differences to the X/Y dichotomy, two classes of cells distinguished first in retina by Enroth-Cugell and Robson 1966.¹⁵ We have argued that some of these findings can be better interpreted in terms of the lagged/nonlagged dichotomy. For example, geniculocortical latencies vary with depth in the layers of cortex (Mitzdorf and Singer 1978), and in retrospect match the antidromic latencies measured by Mastronarde, implying that lagged cells project to the deeper parts of layer IV and upper layer V, whereas nonlagged cells project throughout the depth of layer IV (figure 15 in Saul and Humphrey 1992b).

The triadic synapses transform the retinal input. The inhibition inverts the sign of the input, providing a half-cycle shift in phase. Then, the excitatory transmission through the NMDA receptors creates firing in the lagged cell when the inhibition declines. This response to the change in inhibitory input provides a quarter-cycle shift. That is, the response does not occur when the retinal cell is simply not firing (as would be the case for a half-cycle shift), but when the retinal cell's firing changes by having its rate go down (this is like the negative of the derivative, producing a quarter cycle phase lag). For the step stimulus, the retinal cell gives a transient rise in firing at onset, and therefore a strong inhibition in the lagged cell (Fig. 6; figures 3 and 8 in Saul 2008). As that retinal transient goes away, the lagged cell starts firing. The retinal cell then continues to decline in firing rate during the step, so that the lagged cell continues to fire. At stimulus offset, the retinal cell suddenly stops firing and the inhibition ends while the NMDA-mediated excitation persists, leading to the strong anomalous offset discharge in the lagged cell. The inversion and differentiation combine to produce a quarter cycle phase lag between the retinal input and lagged cell output. Nonlagged cells relay their retinal input with relatively little change, though both lagged and nonlagged cells vary widely.¹⁶

Lagged cells can be difficult to record, because they don't fire immediately when the stimulus changes, and many of them do not fire at high rates. One needs to be patient and gentle with them. Whereas most visual neurons announce themselves loudly, lagged cells sometimes only reveal themselves with murmurs. After recording from many lagged cells, one learns what they sound like (the electrical signal recorded from neurons is fed to a loudspeaker, and spikes are heard as sharp pops, as in Movies 2 and 3). The first lagged cell recorded in an alert monkey was being tested by Elsie Spingath and Yamei Tang, who had never heard a lagged cell. I was sitting in the lab and got tremendously excited by hearing the familiar tune. We wound up testing that cell extensively (just three of the dozens of tests that were run were illustrated in figures 2, 3, and 4 in Saul 2008), until we had to leave for a seminar. Once you peel back the curtain on them, they can provide hours of fun as you test them with various stimuli. They are a bit like babies who can't talk much but nevertheless have a lot to say.

Lagged and nonlagged cell responses are separated by about a quarter cycle in time because of these mechanisms. Recall, however, that phase is just one dimension of time, the other being temporal frequency (or its reciprocal, period). At what temporal frequencies are lagged and nonlagged cells in quadrature? One might hope that they would remain a quarter cycle apart across all frequencies, as in the ideal motion detector. Keep in mind that low temporal frequencies are particularly interesting, for one because most of our behavior takes place over long periods/low frequencies. Can a lagged cell provide a quarter cycle phase lag over many seconds? If the input firing changes very slowly in time, will the triadic inhibition be competent to delay output firing for many seconds?

We performed simple experiments in cats to measure timing in lagged and nonlagged LGN cells. The small spot used for the step stimulus had its luminance modulated sinusoidally in time at a series of temporal frequencies (Movie 3), and we measured the phase of the response relative to the stimulus. Figure 29 shows responses to a spot modulated at 1 Hz for an OFF-center nonlagged cell, an ON-center lagged cell, and an ON-center nonlagged cell. The nonlagged cells fire a half-cycle apart, as luminance is either falling or rising. The lagged cell fires halfway between the two nonlagged cells, and thus can be seen to be about a quarter cycle away from either of the nonlagged cells. To be quantitative, we measure the phase relative to the stimulus luminance peak. The OFF-center nonlagged cell has a phase value just less than 0.5 c; the ON-center nonlagged cell is just less than 0 c; and the lagged cell is just less than 0.25 c.

Figure 29: Quarter-cycle timing differences between lagged and nonlagged cells. Responses to a sinusoidally modulated spot (Movie 3) are shown for an ON-center nonlagged cell, an ON-center lagged cell, and an OFF-center nonlagged cell. The nonlagged cells fire a half-cycle apart. The lagged cell fires halfway in between, and thus a quarter cycle from each of the nonlagged cells.

When I first obtained this result, I was thrilled. Lagged and nonlagged cells were in quadrature, and could implement the spatiotemporal quadrature model of direction selectivity. As just mentioned, however, the result in Fig. 29 is for a single temporal frequency, 1 Hz, and it matters a lot whether quadrature holds across other frequencies. We therefore looked over a range of frequencies.

Figure 30 illustrates what happens at 10 frequencies, from 0.25 to 16 Hz. With increasing frequency, you can see that the responses occur later in the stimulus cycle. Again, this is due to latency. To repeat the explanation given above in the context of this concrete example, if it takes 100 ms for a cell to respond to a stimulus, then, at low frequencies/long periods, that delay is a relatively small part of the stimulus cycle. At 1 Hz, 100 ms is 0.1 c. But as frequency increases and period decreases, the fixed latency is an increasingly large portion of the cycle. At 4 Hz, 100 ms is 0.4 c. Both the lagged and nonlagged cell show this increase in phase with frequency, but the lagged cell is clearly getting later at a faster rate. The quarter cycle difference at 1 Hz increases to about a half cycle by 4 Hz. The lagged cell has a longer latency than the nonlagged cell.

The response phase values over a series of frequencies are plotted against frequency for these two cells (Fig. 30c). The points for each cell fall approximately on a line. The best fit lines are parametrized by a slope (latency) and an intercept (absolute phase). The lagged cell has a longer latency, and an absolute phase lag relative to the nonlagged cell. The difference in latencies is about 60 ms (0.06 s). Therefore, between 0 and 4 Hz, the phase difference increases by about 0.24 c (0.06 s x 4 c/s). The quarter cycle difference at low frequencies increases to about a half cycle at 4 Hz.

Figure 30: Phase vs. temporal frequency. Responses to sinusoidally modulated spots are shown for 10 temporal frequencies in a nonlagged and a lagged cell. The response phases at each frequency are plotted in panel c. The lagged cell had phase values about a quarter-cycle later than the nonlagged cell at low frequencies, but had a longer latency, so that their phases were about a half-cycle apart at 4 Hz.

This was typical across a population of lagged and nonlagged cells. Latency is plotted against absolute phase in Fig. 31. Neurons with absolute phase leads (mostly nonlagged cells) tended to have shorter latencies than those with absolute phase lags (lagged cells). The distribution is continuous, with cells covering the entire range of absolute phase values. Note that OFF-center cells have had their absolute phase values normalized by adding or subtracting a half cycle.

Figure 31: Latency vs. absolute phase. For a population of cat LGN cells, latencies are plotted against absolute phase in the left panel. Nonlagged cells (filled symbols) typically had absolute phases leads and latencies less than 100 ms. Lagged cells (open symbols) tended to have absolute phase lags and latencies greater than 100 ms. On the right, latency and absolute phase are plotted for subzones of V1 simple cells in cats. These data fell largely into lagged-like and nonlagged-like regions, with longer latencies compared to LGN.¹⁷

We wondered above whether the triadic inhibition would be able to provide quadrature at low temporal frequencies. This remains a key open question, because we have not explored other thalamic nuclei, such as medial dorsal thalamus, where the inputs respond at much lower frequencies than in the visual system. Do the slowly ramping cells in amygdala recorded by Wang et al. (Fig. 12) project to lagged and nonlagged cells in medial dorsal thalamus, and do those medial dorsal thalamic relay cells project to cortex to provide information about how anxiety is changing in time? For the LGN, we do know that quadrature occurs down to at least 0.125 Hz, that is, 8 second periods. This is a remarkably long time in terms of how most neuroscientists think. As seen above, and below (Fig. 33), quadrature at low frequencies must be generated by mechanisms that produce phase shifts, and that seems to be how these triads work. The experiments need to be done to test this hypothesis.

Temporal frequency tuning of direction selectivity

Figure 32: Temporal frequency tuning of direction selectivity. Responses in the preferred (solid line and square symbols) and the nonpreferred direction (dashed line and circles) are plotted against temporal frequency for a V1 simple cell. The cell is direction selective at low frequencies, but loses direction selectivity at about 4 Hz.

Across the population of visual cortical neurons in cat, a bias was seen for direction selectivity to be more common at lower (below 4 Hz) than at higher (above 4 Hz) frequencies (Fig. 33, adult results). This bias matches a tendency for timing to change gradually across the receptive field at lower but not at higher frequencies (Reid, 1988, Direction selectivity and the spatiotemporal structure of the receptive fields of simple cells in cat striate cortex, PhD Thesis Rockefeller University, p. 72; Saul & Humphrey, 1992b; Murthy et al., 1998), and of course it matches the relative phase difference between lagged and nonlagged cells, pointing to that difference as the likely source of cortical direction selectivity.

Figure 33: Temporal frequency tuning of direction selectivity in kittens and cats. The top graph plots the average direction selectivity across populations of V1 cells, and the bottom plots the proportions of those populations that had direction selectivities greater than 0.3.

Over many years, I encountered cells like the one illustrated in Fig. 32, and never lost my amazement at them. Perhaps this is because they represent the rare prediction that is validated, but more likely the strong feelings they evoke are due to my doubts about that prediction. When the world allays our doubts, we are led to believe most strongly. That is one reason why it pays to be skeptical. Most of the time we don't doubt our beliefs, and force our reality to correspond to those beliefs, rather than letting reality counter our doubts.

Note that these findings depend heavily on analyses in terms of phase. It would be difficult to understand the temporal frequency tuning of direction selectivity and its basis in timing differences without relying on phase measurements. The fact that latency is the slope of phase vs. frequency showed us that lagged and nonlagged cells are in quadrature at low but not higher frequencies. Since direction selectivity is related to quadrature phase, we could have predicted what we later were surprised to find.

A key observation that arose from these findings is that temporal quadrature is difficult to obtain at low temporal frequencies. Figure 34 illustrates the relationship between frequency and time delays that provide quadrature. At low temporal frequencies, long time delays are required, and they vary over a wide range of delays for a narrow range of frequencies. From 1 Hz down to 0.25 Hz, the delays go from 250 ms to 1000 ms. It is therefore hard to imagine obtaining quadrature at low frequencies from mechanisms that operate in the time domain.

Figure 34: Quarter cycles as a function of temporal frequency. The amount of temporal delay that provides a quarter-cycle phase difference is plotted vs. linear temporal frequency. At low frequencies (blue arm), long delays that vary rapidly with frequency are required. At high frequencies (green arm), delays are only tens of milliseconds and could be generated by numerous mechanisms.

On the other hand, at high frequencies, the time delays do not vary as much, and are shorter. From 5 Hz to 20 Hz, the delays go from 50 ms to 12.5 ms. Those sorts of delays could be obtained from cells with differing latencies.

The result discussed above, what I call the temporal frequency tuning of direction selectivity, is hypothesized to arise from the temporal frequency tuning of the phase differences in the geniculate inputs to visual cortex. The similarity of these two sets of tunings in adult cat provides some evidence, and we eventually found more evidence by examining two other cases of geniculocortical projections. Several other findings supported this hypothesis. Comparing the geniculate and cortical distributions of latency and absolute phase in Fig. 31 shows their similarities.

More evidence

Figure 35: Typical temporal frequency tuning of direction selectivity in monkeys and cats. Whereas cats have many V1 cells that are direction selective only at low frequencies, monkeys show direction selectivity at a wide range of frequencies.

In summary, visual cortical direction selectivity appears to be generated by sets of geniculate inputs that differ in timing. This is only a small part of a big story, though, so let's pursue some of the other questions.

Movie 11: Drifting grating. A sinusoidal grating drifts upward. A receptive field is shown to illustrate that such a cell would respond as the dark bars passed over the OFF zone (triangle) and the bright bars passed over the ON zone (plus sign).

Drifting gratings are a commonly employed class of visual stimuli. The example in Movie 11 has a spatial profile that's a sinusoid, and the grating moves continuously from bottom to top. Although it clearly moves upwards, it never gets anywhere. The points at the top edge disappear, while new points appear at the bottom edge. This stimulus thus dissociates directed motion from displacement in space. One should argue that the bars or edges of the grating are clearly displaced.

An alternative stimulus consists of random dots, some of which move toward the left and others just appear and disappear randomly (Movie 12). In this case, the visual system has a difficult time tracking any features. Nonetheless, a clear direction of motion is perceived.

Movie 12: Random dot motion. A stimulus developed by Curtis Baker has been used widely. The red dots move somewhat randomly, but a percentage of them move coherently to the left, enough to enable an observer to ignore the incoherent dots and reliably guess the direction of motion of the field.

The sinusoidal drifting grating can be described by sin(fy-ωt), where f is the spatial frequency and ω the temporal frequency (y is the vertical axis and t is time). The random dot stimulus is clearly more complicated, but in the frequency domain has components that are gratings drifting to the left. It also has components that drift in other directions. One can quantify the extent to which the noise moves leftward, and match the strength of that direction to the contrast, the difference in luminance between the dark and bright bars, of the grating.

Bill Newsome and associates used these random dot stimuli to demonstrate a connection between neuronal activity and behavior. Such a connection is termed a "linking hypothesis", and has been a challenge to demonstrate. In Newsome's experiments, macaques were trained to report the direction of motion in these random dot displays by making an eye movement in the perceived direction. Simultaneously, recordings were made from single cells in visual cortical area MT, where almost all of the cells are direction selective. When the percentage of dots moving randomly was low, so that the motion was highly coherent, the monkeys responded correctly on almost all trials, and the neuron's response was strong when the dots moved in the preferred direction and weak in the nonpreferred direction. When the dot coherence was lower, the monkey's decisions and the neuron's firing became less reliable. They computed psychometric functions describing the monkey's behavior and neurometric functions for the neuron's firing, and showed that they were closely related. They estimated that the monkey's decisions could be determined based on the activity of about 20 neurons.

These moving dot stimuli are useful for adaptation experiments (see below) as well, at least when one isn't concerned so much about frequency specificity. Such adaptation experiments don't seem to have been applied to experiments like Newsome's, however.

Numerous other stimuli are used to understand vision. One of the great things about studying vision is that we can produce so many stimuli to probe the system. We primates are highly visual animals, and have enormous visual abilities.

Movie 12 presents several of the stimuli that we used to obtain data. A series of bars are shown at different positions across the receptive field. Each bar's luminance is modulated over time. These 6 examples employ a variety of modulation patterns. These include binary white noise; Gaussian white noise; naturally modulated noise; sinusoidal modulation at different frequencies; spatially correlated natural noise; and sparse noise. The term "white" here means uncorrelated, whereas "natural" implies correlations over time exist.



Movie 12: Examples of random bar stimuli. These 6 movies illustrate a few of the ways cortical neurons are tested. Because these neurons tend to have oriented receptive fields, bars are placed at the preferred orientation, and modulated in time. A) Binary white noise, where on each frame each bar is either dark or bright; B) Gaussian white noise, where on each frame the luminance of each bar is chosen from a Gaussian distribution of luminance values; C) Natural noise, like B but the luminance values for each bar are correlated across time; D) Sinusoidal modulation at different frequencies for each bar, so that over many trials a range of frequencies is applied to each bar; E) Spatially correlated natural noise, like C but also correlated across space; and F) Sparse noise, where one dark or bright bar at a time is presented at a random location.

By adding together two drifting gratings that differ in orientation, one obtains a plaid pattern that moves in a direction in between those of the two gratings (Movie 13). Direction selective neurons early in the visual pathway respond to one of the two gratings that make up the plaid, but not to the combination. In visual cortical area MT, some cells respond to the plaid, rather than to the component gratings (higher order motion, as touched on below). These findings have prompted numerous studies.

Movie 13: Plaid. A vertical grating moving rightward is combined with a horizontal grating moving up. The sum is a plaid moving diagonally.

Investigators who work in the visual system tend to be visual chauvinists, thinking we are immersed in the richest sensory system, if not the only worthwhile sort of scientific effort (😜). We admit, nonetheless, that other domains exist. In fact, much of what we do has analogues elsewhere. The auditory system is sort of worthwhile, and one of the analogues is Shepard tones, which are auditory gratings. They consist of a set of tones separated by an octave, with all of the tones shifted together either up or down in log frequency (Sound 1). The motion is clear, but it never gets anywhere. Like direction selective cells in the visual system, the auditory system has neurons that are selective for the direction of a sweep, either rising tones or falling tones.

Sound 1: Shepard tone

The tritone paradox is made from a pair of Shepard tones that are a tritone (half an octave, or a flatted fifth) apart. They are the analogue of going from 12 to 6, or 1 to 7, etc., on the clock. Again, jumping a half cycle does not give a clear sense of direction.

Sound 2: Shepard Tritone

Deutsch' tritone paradox is that some listeners hear the tritone change as moving up and others as moving down, and one listener will hear one change as up and another as down. See for more discussion.

Sound 3: Tritone paradox

Music and language are wonderful places to study time. Although also true for vision, it seems to be clearer to most people that hearing depends deeply on timing. In most music performances, structures are created at a series of time scales. The form is revealed over relatively long periods, there might be a repeated chorus of 32 measures, during which repeated 8 bar sections occur, and each bar has a structure of notes related to beats. Musicians play with these temporal structures, varying the phases at which substructures occur.

Similarly, spoken language unfolds over time, based on semantics, syntax, phonology, tone, and numerous other ways of varying what we say. Languages differ greatly in the timings of their spoken discourses. Tallal's work on children who have trouble in primary school demonstrated that the affected children don't process the rapid tonal changes that make up the phonemes for certain consonants. The sounds of 'ba' and 'da' differ only during a period of about 40 ms, where 'da' has a descending frequency component (Figure 36). The affected children can't hear this difference, and so when told "This is a 'b', and this is a 'd'", they can't make the association. Because those ba and da sounds are also associated with other sounds, bar and dark are not processed correctly, and the phrases and sentences and paragraphs and discourse that contain them can be misunderstood. These children can learn to compensate, however.

Figure 36: Difference between /ba/ and /da/. Audiograms are shown. A. The spoken sentence "I love this bar of dark chocolate." B. Excerpts from the audiogram in A are shown for the syllables /ba/ and /da/. The difference between these is in the 40 ms consonants, where /da/ has power at high frequencies that get lower over time.

The chemical senses, taste, smell, and chemical pain, tend to operate on longer time scales than vision, touch, and hearing. We respond to many foods in complicated fashions as their components provide stimulation at different times. Studies of wine perceptions indicate that bitterness precedes astringency, for example. Persistence of flavors, aromas, and burning sensations indicate that sustained neurons are important in these modalities. The primary pathway to cortex for olfactory and gustatory signals does not follow the vision/touch/hearing rule of projecting to thalamus on the way to cortex, but there is a strong projection through medial dorsal thalamus, where slow changes might undergo phase transformations. Trigeminal pathways to lateral thalamus might be amenable to testing for temporal processing.

Vestibular sensations occur over a range of time scales. One of the many important roles for this modality is the vestibulo-ocular reflex (VOR). The direction of gaze can be centered on an object of interest despite head movements, for instance while walking. The VOR compensates for head movements at high frequencies (0.5 to 2 Hz), and the optokinetic reflex handles lower frequencies (0.05 to 0.5 Hz).

Figure 37: McCollough effect. Adapting to the green and magenta gratings at the top for several minutes induces tinges of colors to the black-and-white gratings at the bottom.

Timing aftereffect

Figure 38: Timing aftereffects. Responses to a drifting grating are shown for control and adapted conditions.

This behavior was seen in all simple cells. It can be seen in a brief paper by Movshon and Lennie (put Fig. 1b on top of a or c) but was not noted there. It can also be seen in many experiments where cells were stimulated by gratings drifting for many seconds: comparing the responses during the first few seconds to the responses after 4 or more seconds reveals this effect. We observed timing aftereffects in alert monkey visual cortex as well as in anesthetized monkeys and cats. Timing aftereffects had not been previously described. This phenomenon is the coolest thing I learned from years of doing experiments.

Models

Figure 39: Mutual inhibition model. Positing inhibition between two cells, along with potentiation of the inhibition by adapting, predicts timing aftereffects.

We performed additional experiments in which we mapped receptive fields of simple cells under control and adapted conditions. The interpretation of the results from this study provides a more detailed explanation for the model above, and more importantly for how direction selectivity is generated. Figure 40 shows the fuller model. The structure is outlined in A, with inputs to the two cortical simple cells from a set of LGN cells. These inputs are in spatiotemporal quadrature: the left simple cell gets ON and OFF inputs from nonlagged cells, and the right simple cell from lagged cells. Temporal quadrature arises from the timing differences between the lagged and nonlagged cells. Spatial quadrature is set up by offsetting the receptive fields in space, interdigitating the lagged receptive fields a quarter cycle in space from the nonlagged fields (recall that ON and OFF responses are a half-cycle apart).

Panel B shows the responses that the structure in A produces. The space-time maps show that the left and right simple cells receive spatiotemporal quadrature inputs from LGN. Adding the mutual inhibition at the bottom of panel B has the effect seen above where the onset of the response is reduced, but here only at certain spatial positions because of the quadrature. The receptive fields are given an orientation in space-time, which is the same as being direction selective. The cells share the same space-time orientation, with responses coming earlier in time moving from top to bottom in space.

Figure 40: Spatiotemporal model. A) Nonlagged and lagged cells project to two V1 simple cells in spatiotemporal quadrature. B) Responses in space and time are shown before and after implementing the mutual inhibition. C) Both cells fire strongly in the preferred direction (from top to bottom in B) but at different times. They fire at the same time in the nonpreferred direction and therefore have lower amplitudes.

Panel C shows how responses to drifting gratings come out of this spatiotemporal organization. In the preferred direction (top toward bottom), the two cells respond well, a half cycle apart, out of phase (as in Movie 6). In the nonpreferred direction (bottom toward top), responses are weaker because the responses are in phase and thus the inhibition is in phase with the excitation (as in Movie 6), reducing its effect.

This model is based on spatiotemporal quadrature, so it is unsurprising at its core. However, people have the intuition that a direction selective cell must be inhibited by cells that prefer the nonpreferred direction, so a cell that is selective for downward motion is inhibited by cells that respond to upward motion. There is little evidence for that arrangement, and it doesn't make sense for simple cells because the timing would not match systematically for cells with opposite space-time orientations. Timing must be taken into account when trying to understand neuronal processing!

In addition, the model uses feedback, which tends to be difficult for people to process intuitively. Electrical engineers are highly conversant with feedback, which is fundamental to the way that amplifiers work, for example. Population biologists know that ecological systems depend on feedback between species occupying the same niche. Much of the richness of the brain arises from the massive feedback in its circuits.

Evidence for the model in Figure 39 comes from several excellent studies that measured space-time receptive fields while deriving the contributions of excitatory and inhibitory inputs. Murthy and Humphrey showed that inhibition is tuned in the same direction as excitation (see their Figure 12). Priebe and Ferster provided strong support by measuring excitatory and inhibitory receptive fields, showing that they were in quadrature (their Figure 6). Monier at al. wrote that "a majority of cells showed both excitation and inhibition tuned to the preferred direction". Znamenskiy et al. demonstrated that excitatory and inhibitory neurons in mouse visual cortex with similar response properties are connected. Lien and Scanziani also provided results on geniculate and cortical responses in mice consistent with the model proposed above. Kuo and Wu found a related organization for direction selective auditory neurons in the inferior colliculus.

Our model is broadly supported, although objections are sometimes made based on ignorance of phase. Livingstone claimed that lagged cells are never biphasic (transient), meaning they don't have absolute phase values near 0.25 c. This is of course not true, as shown by the examples above and the scatterplots of absolute phase values. Ralph Freeman claimed that lagged and nonlagged cells differed in phase by a half-cycle. The claim seems to be based on the distributions of latencies of lagged and nonlagged cells. As with Livingstone, they apparently thought that lagged cells were all monophasic (sustained; see the biphasic transient lagged cells above or Fig. 11 G,H in Saul & Humphrey, 1990). Alan Freeman argued that the relative delay between lagged and nonlagged responses would not suffice to produce direction selectivity, completely ignoring the phase differences.

Chariker et al. (2021) modeled direction selectivity in macaque visual cortical neurons based on temporal response differences between ON and OFF cells. They claimed that lagged and nonlagged input can not explain macaque direction selectivity. They argued that the differences between the temporal frequency dependence of direction selectivity in monkey and cat invalidates the dependence on lagged and nonlagged cells. As described above, the opposite is true: the difference in temporal frequency tuning of direction selectivity is matched by the differences in LGN absolute phase and latency. Furthermore, they claimed that "lagged cells have not been found in macaque LGN", ignoring Saul (2008).

Priebe and Ferster showed the clearest evidence for our model, but denied it because they refused to think in terms of phase, and didn't understand why excitation and inhibition are tuned in the same direction. They wrote: "In this paper, we provide evidence that the excitatory and inhibitory synaptic inputs to simple cells prefer motion in the same direction, which is the direction that evokes the most spikes. While excitation and inhibition were tuned to the same direction, the two components were evoked asynchronously by moving stimuli. This difference in the timing of excitation and inhibition appeared in responses to simple gratings as a 180° phase difference between the excitatory and inhibitory inputs." That last sentence applies of course to motion in the preferred direction. They did not consider the relationship between excitation and inhibition for motion in the nonpreferred direction: "Our results suggest that inhibition plays little role in shaping direction selectivity in simple cells. It is unclear, however, why inhibition should come from the same direction as excitation". They too were stuck on latency: "Most models of direction selectivity are based on input that differs in latency for different spatial positions".

I had repeatedly attempted to show David Ferster how direction selectivity works in our model, but he never accepted my reasoning. I attribute this to a prior belief that inhibition is not important, and a difficulty in understanding phase. This is common among the many people I've talked with about direction selectivity, and a prime reason why I wanted to lay this out here.

Figure 41: STDP. The change in synaptic strength was measured in an output (post) cell for stimulation of excitatory input (pre) cells. The relative timing of input and output activation determined the degree and sign of the change in EPSC amplitude.

The time course of STDP is an example of a kernel, as described above, a function that describes how input and output are correlated. This kernel, however, describes how the plastic change depends on the relative timing of input and output spikes, rather than how the timing of output activity depends on input activity. The plastic change is a function of the relative phase between input and output timing. This function has a phase approaching -0.25 c, meaning potentiation depends transiently on the input/output timing: input must precede output, and not overlap extensively. For depression, input spikes must lag output spikes. This view, that generalizes the single spike concept to the more realistic world where continuous spike trains are involved, is crucial to formulating and interpreting experiments. The view that is more common limits itself to single spikes, missing the richness of the more continuous view. Buonomano concludes that "STDP cannot capture the relationship between events separated by seconds and beyond." It does just that, in fact, because those events evoke many spikes in both input and output neurons.

Postsynaptic resonance

Direction selective cells need to integrate inputs from other neurons that differ in space and time, as described above. As described in detail above, these inputs will fire at the same time (in phase) when stimuli move in one direction, but will not be synchronous (out of phase) for the opposite direction. On average, assuming that the two directions of motion occur about equally often, the inputs should not be associated with each other, since their net correlation (in phase = 1, out of phase = -1, an average of 0) would be small. However, as with STDP, synaptic changes typically depend in a particular way on activity in the postsynaptic neuron, the output. Synaptic changes are "gated" by the postsynaptic activity. This was termed postsynaptic resonance by Rauschecker and Singer (1979). In the nonpreferred direction, the direction selective neuron is not activated, and so no synaptic change occurs. This permits the association because of the synchrony in the preferred direction (ARVO poster; Feidler et al. 1997).

Experience dependent modification

If kittens grow up in an environment that is lit only by brief flashes 8 times a second generated by a strobe light, they don't experience smooth motion.¹⁸ Several labs showed that cats raised that way did not have as many direction selective neurons in visual cortex as cats reared with continuous lighting. We showed that the spatiotemporal receptive fields of strobe-reared cats did not have the space-time oriented receptive fields associated with direction selectivity (figure 3 in Humphrey & Saul 1998). Instead, the receptive fields were of the classic Hubel & Wiesel variety (figure 4 in Humphrey & Saul 1988), with different parts of the receptive field separated by half cycles.

Cortical simple cells in strobe-reared cats had regions with lagged or with nonlagged timing, but not in the same neuron. Allen Humphrey provided an explanation for why this is the case. In a continuously lit environment, inputs with different timings can be associated onto a cortical neuron because moving stimuli cause those inputs to be active at the same time as stimuli move across the receptive field. But with very brief strobe flashes, lagged and nonlagged neurons do not fire synchronously, because the lagged cells have longer latencies, and by the time they fire, the nonlagged cells are silent following their brief responses (figure 10 in Humphrey et al 1998). This prevents neurons from having both lagged and nonlagged inputs.

We speculate that this is a general paradigm for how the brain sets itself up to behave adaptively. The relative timing of different inputs creates potential associations that depend on the timing relationships. Associations are made when timing is right in a statistical sense, when the in-phase timings occur often enough and lead to some sort of reinforcement, generally meaning postsynaptic activity. The result is that the brain is full of neurons that are selectively active for processes that occur in one direction in time but not the other. Muscles contract when we need to move in one direction, and relax when we move in the opposite direction, based on signals from neurons that fire in this direction selective pattern.

The world is not symmetric with respect to many of these processes. Things tend to fall toward the earth, in our experience. We might therefore expect that we have more neurons that respond to things falling down than up. Experiments have been performed to suggest this is the case: raising kittens in an environment where visual movement is predominantly in one direction biases the visual cortical population toward responding to the experienced direction. Below, we will rely on this idea to suggest why we think that time itself has a direction.

Motion blindness

It's possible to lose direction, as well. A rare case of motion blindness was studied by several investigators. A woman known in the literature as LM had seemingly intact function after suffering a hemorrhage, but could not see visual motion. As Josef Zihl, the scientist who studied her most, reported in a review paper (Zihl & Heywood 2015):

"People, dogs, and cars appear restless, are suddenly here and then there, but disappear in between. Very often I don't even know where they have left, because they move too fast, so I lose them quite often." Fluids appeared frozen, like a glacier, which caused great difficulty, for example, with pouring tea or coffee into a cup; filling a glass with water became impossible. Most events were much too fast for her and she needed a considerable time to perform even simple routine activities, such as cutting bread or using the vacuum cleaner. She could no longer use the tube, bus or tram, which severely restricted her mobility. She also found it very irritating to meet friends and have a chat with them because she could not respond in time to their handshake and because she found their moving hand disturbing. In addition, the experience of talking to them was very unpleasant because she had to avoid watching their (changing) facial expressions while speaking, in particular, their lips seem to "jump rapidly up and down, and I am very often unable to listen to what they were saying." In contrast, when people, faces, objects and cars were stationary, she had no difficulty in seeing them "clearly" and could recognize them immediately and accurately. The perception of colors had not changed, and she reported no difficulty with perceiving the position of objects and judging correctly both how far away they were and the distances between them. She reported that reading took more time than before, writing had become somehow difficult. Psychiatric examination revealed no psychopathological symptoms, in particular depression, anxiety or agoraphobia.

Figure 42: LGN spatial receptive fields. An OFF-center and an ON-center cell are illustrated. These cells have radial symmetry, the same amplitudes at any given distance from the center. They can therefore be replotted below as 1-dimensional functions of radial distance.

One can interpret the receptive field in at least two ways. It can be thought of as the region of space and time in which stimuli affect the cell's activity. But it can equivalently be considered as the cell's internal way of changing stimuli into responses. This second notion is philosophically appealing, as it is causal. Note that it is difficult to think about the receptive field over time in Fig. 43 as a region of time. Its time course runs from about 0 to 0.2 s. Where is that 200 ms interval on the time line? One thinks of these time domain functions as kernels that are convolved with stimuli to produce neuronal activity, not as regions of time. We don't have 1960s cells.

For the frequency domain representations, the kernels are simply multiplied by the stimulus to produce the response, and it is easy to view them as the filters, the actual regions of spatial and temporal frequencies where the neuron responds. Both interpretations are easily made when frequency and phase are the dimensions.

Figure 43: Spatiotemporal receptive fields in spatiotemporal and in spatiotemporal frequency domains. LGN cells can be modeled as spatial Gabor functions and impulse response functions (spatial and temporal kernels). In the frequency domain, Gabors are Gaussians in amplitude. The complicated impulse response functions are also simply Gaussian functions of amplitude vs. frequency. Phase is linear with frequency, with absolute phase determining the shape of the space or time receptive field and the slope giving the spatial offset or latency.

In the frequency domain, there are two dimensions for each space dimension (amplitude and phase as functions of spatial frequency). But again, we can restrict space to the single radial dimension, and thus radial spatial frequency. The Fourier transform of a Gabor function has amplitude that is a Gaussian function of spatial frequency peaked at the center frequency of the Gabor, and phase that is linear with a slope equal to the offset of the Gabor's center (fig. 43). For temporal frequency, we described above how the temporal kernels (impulse response functions) generally have low-pass amplitudes and linear phase with absolute phases and latencies that vary across geniculate neurons.

Note that the space and time domain descriptions have fairly complex shapes that may be easy to look at and seemingly understand, but are difficult to describe in words or numbers. The frequency domain descriptions are simpler, with Gaussian amplitudes and linear phase, with the slope of the phase vs. frequency line giving spatial and temporal offsets, and absolute phase providing a single number that tells us about the shape of the spatial profile and temporal kernel.

These receptive field models are sheaves, with the base space being time and space relative to when and where the stimulus occurs, and the stalks being the set of real numbers that correspond to amplitudes, as in Fig. 43. For a particular LGN receptive field, the stalks take values corresponding to the Gabor function in space and the impulse response in time, or to just amplitude at each frequency and phase (Fig. 11). Note that most stalks in the frequency domain have values near 0, because non-trivial amplitudes exist only at the frequency-phase values determined by their quasilinear relationship.

We now move to visual cortical (V1, primary visual cortex) receptive fields. These are constructed by neuronal connections from LGN to cortex as well as by intracortical connections. A worthwhile way to begin is to take a V1 neuron that receives inputs from just two LGN neurons. These are excitatory inputs, and our first order approximation is that the V1 neuron just adds the contributions from the LGN inputs. Having two spatial dimensions complicates matters, but there is again a way to reduce this dimensionality. V1 cells are orientation selective, meaning they respond better at a certain orientation in two-dimensional space than at neighboring orientations. This orientation, for our simplified two-input example, would be perpendicular to the axis through the offset centers of the inputs. Note that motion of a contour is always perpendicular to its orientation, so parallel to the axis between the inputs, as in Movie 7.

Then, the space-time receptive field of the V1 neuron is the weighted combination of the impulse response functions (IRFs) in time, where the weights are given by the spatial Gabors:

V1 RF(r,t) = Gabor1(r)*IRF1(t) + Gabor2(r)*IRF2(t).

A typical V1 receptive field, as documented by Hubel and Wiesel, looks like Fig. 44 (top), with offset ON and OFF zones. The ON and OFF polarities can be incorporated into either the Gabors or the IRFs.


Figure 44: V1 simple cells. Two LGN inputs are shown as red and blue functions of time and space. At the top, the inputs have the same timing but opposite contrasts, and are shifted in space. They create a classic kind of non-direction selective simple cell with separated OFF (black) and ON (yellow) zones. At the bottom, the inputs differ in time as well as space, and combine to make a direction selective simple cell.

If we shift one of the LGN inputs in time, so that the inputs are offset in time as well as in space, the cortical simple cell becomes somewhat direction selective, slightly oriented in space-time, as in Figure 44 (bottom).

LGN cells do not tend to differ much in spatial phase. Simple cells do, on the other hand, as in the example in Fig. 45, where one simple cell (in red) has 3 subzones (OFF/ON/OFF) and the other (in blue) has 2 (ON/OFF). Figure 45 shows receptive fields in both spatial and temporal quadrature, as illustrated in Fig. 40.

Figure 45: Direction selectivity from spatiotemporal quadrature. Multiple LGN inputs create two simple cell inputs that differ in space and time by quarter cycles. They generate a highly direction selective receptive field.

The frequency domain representation is the same as above, since the Fourier transform of a sum is the sum of the Fourier transforms:

V1 RF(f,ω) = Gaussian1(f)e^{2πi(D1f+ψ1₀)}∗A1(ω)e^{2πi(L1ω+φ1₀)} + Gaussian2(f)e^{2πi(D2f+ψ2₀)}∗A2(ω)e^{2πi(L2ω+φ2₀)}

where each input has factors of gaussian amplitude and linear phase as functions of spatial frequency (f), and factors of amplitude and linear phase as functions of temporal frequency (ω). For a classic Hubel & Wiesel simple cell, the Gaussians would be shifted in location, that is, the slope of spatial phase vs frequency (D1-D2), and the temporal absolute phases shifted (φ1₀-φ2₀) by a half-cycle (or the Gaussians would have opposite polarity). For a direction selective simple cell, the spatial and temporal phases would each be shifted by about a quarter cycle (Figure 45).

The two inputs each have two factors, a spatial receptive field and a temporal receptive field. These factors are separable, there is no dependence of one factor on the other. Their sum, the direction selective V1 receptive field, on the other hand, does show a dependence on the temporal and spatial structures that can no longer be separated. It is inseparable in space and time: the timing differs across the spatial receptive field (or vice versa). Inseparability is another way of viewing the emergent property of direction selectivity.

In terms of the sheaf, again, the two inputs have amplitude values above the frequencies/periods and phases (Figure 46 top). The red and blue inputs here are in spatiotemporal quadrature. The figure only shows the temporal sheaf. Almost one cycle of phase is included, from 100 ms (10 Hz) to 4 s (0.25 Hz). The relative phases are a quarter cycle apart (Fig. 46 middle). That quarter cycle is apparent at low frequencies (absolute phase) and the inputs have identical latencies (100 ms), so the quarter cycle is maintained across frequencies. At low and high frequencies, the blue input has phase values that approach 0 c (to the right). The red input approaches 0.25 c (up) at low and high frequencies. The cortical cell combines these inputs, linking the stalks over distinct open sets that differ by a quarter cycle. Although the frequency domain is less familiar, it is easier to interpret than the space-time plot in Fig. 45, just showing the quarter cycle relationship that defines direction selectivity, keeping in mind that there is a similar spatial frequency sheaf. Note also that the frequency domain makes it clear that there are important things going on at low temporal frequencies - these are hidden in the spacetime plots.


Figure 46: Direction selective sheaf. Quadrature inputs are shown over 2-dimensional time. The movie illustrates the quarter cycle difference shown in the graph.

The quarter cycle shifts of the sheaf define a "morphism" on the sheaf that characterizes this emergent property. The morphism here consists of rotating the sheaf by a quarter cycle (movie at the bottom of Fig. 46). This is an example of a linear transformation. As described above, this is the Hilbert Transform. The Hilbert Transform simply shifts phase by a quarter cycle at all frequencies, with no change in amplitude. Note that this happens locally in V1 and elsewhere (such as for creating lagged cells in the thalamus) in the early visual pathway. The collection of these morphisms forms a sheaf that is represented by other brain structures, and the process generalizes to produce increasingly complex emergent properties.

The neurons whose temporal properties are shown above also have spatial receptive fields. We made the two neurons to have quarter-cycle differences in space as well, and that plays into the morphism, making responses direction selective.

The illustrations here are of receptive fields in the early visual pathway. These are local processing elements, with only a small portion of space and time activating the neurons. As the complexification process is repeated many times across the tens of billions of neurons in the brain, global concepts are created.

We perceive motion even when stimuli do not contain the cues we've discussed so far (that is, first-order motion). Second-order motion does not contain quarter-cycle differences in luminance. Movie 14 is an example of such a stimulus. Motion here must be processed by neurons that have inputs that complexify the luminance signals to obtain temporal contrast across space. These contrast differences are in spatiotemporal quadrature. The motion is often the cue that lets us see form: motion breaks camouflage. In visual cortical area MT, almost all neurons are direction selective, and many respond well to higher-order motion as well as to first-order motion.

Movie 14: Second order motion. Three columns have their randomly chosen dark and bright pixels changed at each frame.

Motion is a key factor in breaking camouflage. Movie 15 is a typical example. Our brains respond extremely well to motion and its directions. Notice how well the whole spider becomes visible when its parts move. The brain integrates all of these separated moving elements into the complex object that is the spider.

Movie 15: Breaking camouflage. We see things that move even when we don't see them when stationary.

Recall that we make eye movements several times a second, even when we are fixating. This means that the sheaf of a scene shifts several times a second, and the brain knits these shifting sheaves repeatedly. It helps that the brain knows ahead of time what eye movements are about to occur, so, as discussed above, neurons change their receptive fields to knit the local information together into a sensible view of the world.

What the brain does to produce global concepts is thus a matter of the sequences of the morphisms of receptive field sheaves. Analyzing the neural pathways in this way might allow a systematic view of the processing that makes the brain so smart. At each point in the processing chains, some properties of the neurons are ignored, and others are created by convergence of inputs with slightly different properties. It is not necessary that this is hierarchical, and is certainly not purely serial or totally ordered. One expects that aspects are somewhat hierarchical, in that the processes are likely to lead to increasing globalization. In many cases, spatial location is ignored in order to recognize objects: this type of processing occurs in the ventral streams of sensory cortex. But in the dorsal streams, those spatial properties are preserved and extended, and the object recognition is relatively ignored.

Providing a concrete example of these abstract concepts, we can go back to what was described above. Let's start with retinal ganglion cells, with spatial and temporal receptive fields like those in Fig. 43. These neurons respond to small flashed spots with either transient or sustained firing. They project to LGN cells. Those cells can have receptive fields that are similar to their inputs, but some transform their retinal input to generate approximately quarter-cycle phase lags (the Hilbert transform). Putting together lagged and nonlagged projections to V1 simple cells (which are often postsynaptic to other simple cells as in Fig. 40) creates direction selectivity. Simple cells with a range of spatial phases can then result in complex cells (which are often postsynaptic to geniculate inputs, at least partly bypassing simple cells). Direction selective V1 neurons with differing spatial phases project to area MT to generate strongly direction selective cells with large spatial receptive fields, and that are sometimes direction selective for plaids, a second-order property. We thus have a sequence:

RGC → X_L ⊕ X_N → ∑DS S → ∑DS C → MT

where the retinal ganglion cells project to lagged and nonlagged cells in LGN, which project to V1 to help create direction selective simple cells, and then direction selective complex cells (which also receive inputs directly from LGN), and those V1 cells project to MT to make direction selective neurons with large receptive fields and plaid responses. Each of the arrows represents a projection between populations of neurons, and mathematically a set of morphisms between sheaves. Each transformation leads to novel response properties, while ignoring properties that were present beforehand.

By considering each step in these chains, we obtain a more precise characterization of the mechanisms at work in the brain that create emergent properties. Timing is altered in the lagged cells at the first morphism. Direction selectivity emerges next. Spatial generalization happens over the next two stages.

The examples here have often been greatly simplified to employ two inputs to one postsynaptic cell that appears to be above the inputs on a hierarchy. In fact, the brain is a complex network. Each neuron typically receives many inputs. Simple and complex cells receive both geniculate and cortical inputs. We could model the real situation as a sheaf that is a sum of all of these inputs.

In addition, as modeled in Fig. 40, neurons feed back on each other. The network is not simply feed-forward up a hierarchy. In sheaf theory, many diagrams are feedforward, but feedback is also studied. Feedback often tests our intuition, as might be apparent from Fig. 40.

Movie 16: Self-motion. Note how different parts of the scene move with respect to each other. Some motion occurs that is not due to self-motion: birds flying, water flowing. When we move forward or backward, we experience optic flow.

When we move, our heads tend to move. For example, when we walk, our heads bob up and down, as well as sideways and forwards and backwards. If we're trying to keep our eyes fixed on something, we need to compensate for these movements. We have a vestibular system that is activated by head movements, and the vestibular neurons project to the oculomotor neurons to perform this compensation. This is the vestibular-ocular reflex (VOR). Together with the optokinetic reflex (OKR), we can largely keep our eyes steady on our targets while moving.

The receptive fields of accessory optic neurons are very large. They are presumably built from the projections of many direction selective retinal ganglion cells originating from widespread retinal locations and having large receptive fields themselves. As described above for the V1 to MT projection, spatial localization is discarded in the retina to accessory optic system projection. These transformations permit behaviors that are influenced by the entire visual field, global in the sense of something that is happening everywhere in space.

Smooth pursuit eye movements are made when we track a small object that moves through the visual field. For example, we may keep our eyes on a bird flying across the sky. The visual stimulus in this case is the object that we want to keep centered on our eyes plus a background that moves across the retina. It has both a global component and a local component. Area MT is one of several cortical regions that are important for mediating the behavior that lets us track the object. The brain uses error signals for most sensorimotor processes, and here the error signal is called retinal slip. When the object being tracked moves away from the fovea, a velocity signal is provided to the motor system by direction selective cells. That signal enables eye movements to reduce that velocity toward 0. Most studies have emphasized how the brain generates the motor behavior that controls eye movements. The sensory perception depends on both discarding localization to produce the optokinetic reflex, and direction selective cells with small receptive fields that monitor how well the object stays on the fovea, the center of the retina.

MT

An important type of spatial transformation works in a different way. In parietal cortex, where MT resides, spatial processing is emphasized. Visual processing in the early visual pathway is retinotopic, meaning that receptive fields are localized on the retina. In AOS, this localization is lost. In MT, it is diminished. In other parietal areas, such as LIP, retinotopy is replaced by other mapping strategies.

LIP

Our perception is clearly not retinotopic: when our eyes move, the world does not move with the retina. The results presented earlier, about how the LGN shifts neuronal timing starting before saccades are executed, is an example of how the brain breaks away from retinotopy. In LIP, a remapping of space seems to occur around saccades. Before a saccade is executed, some LIP neurons shift their receptive field to the location from where a stimulus was before the saccade to the location where the stimulus will be after the saccade occurs. That allows these neurons to respond more continuously to the stimulus across the saccade, as suggested by the timing changes we saw in LGN.

One might guess that the entire map of the visual field is shifted. That may not happen. LIP is an important part of a large network that mediates attention, and it is possible that only what we are attending to gets remapped. We don't care so much about the rest of space, we only care about where we are looking. Thus, what LIP does is a great example of the sheaf morphisms we've discussed. A lot of information is thrown away, with a gain of information about what we care about. This is a kind of rejection of globalization.

In temporal cortex, down by our ears, spatial localization tends to be discarded in a really important manner. Cells in this part of the brain can have amazing stimulus specificity. Faces are an example. There are neurons that respond specifically to a familiar face (and sometimes to other representations of the bearer of the face, like their name). These neurons will respond to their preferred face without regard to how the face is oriented or where it is seen in visual space. The size of the face doesn't matter. A lot of information is nulled out in order to process what does matter.

These examples of globalization all point to analyses in terms of phase as the way to understand what's going on. As what counts is how local elements are related to each other, as opposed to where they appear on the retina or other sensory peripheral organs, that relativity is captured by phase. Faces are a great example, with elements like eyes that are found relative to other elements like each other and the outline of the face. We give directions this way, sometimes indicating that an object is at 2 o'clock, or that we need to go up and to the left.

We take ideas from numerous sources, and put them together to generate novel ideas. A chef might combine disparate ingredients from several geographic areas to create new dishes that meld related and unrelated flavors. An architect collects building designs and finds new ways to surround spaces. Inventors assemble ideas into devices and other products that can change millions of lives. Populations arise from two cells.

Figure 47: Learning timing in cerebellum. A simulation from Medina & Mauk (their figure 3) shows a variety of timing in granule cells in a. The cells project to Purkinje cells with strengths that are modified by experience (b and c). LTD and LTP create appropriate timing of the eyeblink conditioning reflex. LTD alone in panel C fails to refine the timing. CS: conditioned stimulus=sound; US: unconditioned stimulus=airpuff.

Medina and Mauk theorized that this cerebellar function depends on having a range of timing in cerebellar granule cells (these neurons make up about half of the total number of neurons in the whole brain!). Repeated conditioning leads to appropriate timing through LTD and LTP of the synapses from the different input granule cells onto the cerebellar output neurons, called Purkinje cells. These mechanisms generalize to other behaviors, and the cerebellum is also subject to developmental differences in environment and experience.

Note that the Purkinje cells are inhibitory, and have a high maintained firing rate (spikes shown in b and c). The conditioning leads to a pause in Purkinje cell firing prior to the airpuff, permitting their targets in the cerebellar nucleus to respond. LTP increases Purkinje cell firing rates, and thereby suppresses activity in the nucleus cells. LTD has the opposite effects.

Another good example of how the brain modifies our behaviors involves how we maintain our gaze on an object when we're moving. This was discussed above. When we walk, our heads move around. This would be expected to make it hard to see what we're looking at. But several mechanisms stabilize our vision. The vestibulo-ocular reflex (VOR) takes signals about how our head is moving, and feeds that to our eye muscles to compensate for the head motion. If your head turns toward the right, your eyes need to turn to the left to keep looking where you want. We also use the signals from the accessory optic system that arise from "retinal slip", the visual signal that the eyes are not stabilized. When your head turns to the right, the world moves to the left. The visual system computes the speed of that motion and tells the eye muscles to move the eyes at that speed to match the retinal slip. This is the optokinetic reflex (OKR).

Ideally, these mechanisms would compensate perfectly for our head movements. That compensation is measured by "gain". Gain is the ratio of eye movement output to head movement (for VOR) or retinal slip (for OKR) input, as in audio amplifiers (turning the volume knob of an amplifier changes the gain, controlling how loud the output sound is). Perfect compensation in our case means a gain of 1, so the head movements are exactly compensated by eye movements. Neither the VOR nor the OKR has a gain of 1, however. The gains of these mechanisms depend on the temporal frequencies of the head movements. The OKR approaches a gain of 1 at low frequencies, whereas the VOR has a low gain at low frequencies and a higher gain at high frequencies (Figure 48). Together, their gains add up to something near 1 across frequency (Schweigart et al, 1997).

Figure 48: Gaze compensation. VOR and OKR gains and phases are plotted against head rotation frequency. OKR gain is close to 1 at low frequencies. A weighted sum of VOR and OKR gains provides good compensation.

The brain has another trick up its axons, one that is actually ubiquitous. The cerebellum does many many many things, and one of them is to detect errors and fix them. All of those tens of billions of granule cells receive inputs from across the brain, and a set of circuits that seem like the same machine repeated many millions of times processes those inputs and produces an inhibitory output from Purkinje cells (Figure 26). For the VOR, the head motion signals from the vestibular nuclei project to the oculomotor nuclei that drive eye movements. A set of Purkinje cells also projects to the vestibular nuclei. These Purkinje cells inhibit the vestibular neurons that would drive eye movements in the same direction as the head movements, thereby helping to compensate. They learn how to fix errors by modifying the strengths of synapses when errors occur. A general lesson is that much of what the brain does involves monitoring for errors, generating error signals, and using them to improve function. Neuronal activity commonly consists of error signals.

What counts, of course, is timing. The numerous granule cells are activated at different phases of the head movements. LTD adjusts the strength of the synapses from the granule cells onto the Purkinje cells dependent on the phases. The cerebellum generates exquisite modulations of the basic VOR by manipulating the timing of the inhibition onto the vestibular nucleus cells.

The OKR is similarly modulated by the cerebellum. Visual retinal slip signals activate different granule cells, and LTD tunes Purkinje cells to supervise the gain of the reflex. Together, VOR, OKR, and their cerebellar controls enable us to maintain visual function and not experience the kind of disruptions we get from unstabilized moving cameras.

Most of our skilled behaviors rely on learning those skills based on cerebellar mechanisms. Playing music, sewing, speaking, processing emotional issues, solving puzzles, and all the other amazing things we do, are generated by processing in various parts of the brain, but cerebral cortex has a large responsibility, and cerebellar cortex massages our actions to make them beautiful.

Another lesson from the workings of cerebellum is that the cerebellar nucleus neurons that receive the inhibition from Purkinje cells can be thought of as having been programmed to generate some behavior. They can't execute this behavior, however, until the inhibition goes away. This is a common strategy. Certain inhibitory neurons have high firing rates, and keep their targets from firing. Only when the inhibitory neurons pause their firing are their targets released, and a behavior is executed.

The best example of this comes from saccade generation. The superior colliculus in the midbrain programs saccades, determining their direction and amplitude. Inhibitory omnipause neurons in the brainstem keep the saccade from starting until they pause, at which time the eyes move in a partly ballistic fashion, meaning without further guidance, until the eyes reach their target. The inhibition sets the timing of these behaviors.

My therapist told me a story about a young patient who was having serious behavior problems. They decided to equip the child with a smart watch that reported his heart rate. His foster mother monitored him this way, and had him jump on a trampoline at a certain time when he needed to exert himself. She also delivered shoulder rubs at other times. Keeping track of his behavior over time led to improvements in his behavior. This is a great example of what time is: it's when we do things, feel things, and think things. When we're thirsty, we drink. When we're in a bad mood, we act out. When we're tired, we sleep. When we're thinking about time, it's about time.

Figure 49: Looking at the brain at different scales. Note how disciplines tend to cross these scales.

Biologists often cross the borders between molecular biology and cellular function, and between neurophysiology and behavior, for instance. Physicists often manage to bridge these divides as well, showing how superconductivity arises from quantum interactions in solids with large scale structures that encourage electrons to travel long distances. Reductionism fails to account for emergent properties.

Conceptualizing time as points on a line tends to be reductionist, and thinking in the frequency domain can be more wholistic. The frequency domain by definition encompasses all scales, whereas the time domain usually focuses on the smallest scales, at least in neuroscience.

We mentioned above that nothing can go faster than the speed of light. Why is that the case? It is not intuitive that there should be a maximum speed, and why should it be the particular speed at which light travels? This is a fascinating fact of nature. The speed of light has a finite value, but behaves asymptotically. That is, it can be approached but not reached by anything that has mass. To accelerate mass, it takes energy, and as the speed approaches the speed of light, it would take an infinite amount of energy to reach the speed of light. In the sense of attaining it, the speed of light is infinite. Massless particles (photons) move at the speed of light.

If the speed of light were actually infinite, on the other hand, everything would happen at once. There would be no time! We would see the whole universe at once, nothing would appear to change. Lorentz showed that time dilates, that is gets longer, as the speed of something approaches the speed of light. At the speed of light, time becomes infinite. That is, periods are infinite, and the frequencies are all 0: nothing changes (hen ta panta, in Parmenides' Greek).

The finite speed of light is analogous to the finite temperature of absolute zero. At that temperature (about -273°C), nothing changes. That is, no molecules move. Heat is exactly the motion of molecules. Entropy is minimized, there is no disorder. Absolute zero can only be approached asymptotically, and not reached. Somehow our intuition is violated less by absolute zero than by the speed of light. And we have even better intuition about zero frequency, which similarly can not be reached, but is routinely characterized as DC, direct current, in electronics.

Note that the speed of light does not change in different "reference frames". That is, no matter how, where, and when photons are generated, they alway move through space and time at the same speed (until they are absorbed by matter). Whereas if you throw a ball from a moving car it will have the sum of the velocities of the car and the throw, the light coming out of the headlights of a moving car moves at the same speed as when the car is not moving. This is true even if the car was moving close to the speed of light. This deviates from our intuition, but makes sense from the relativistic viewpoint.

The color of the light coming out of the headlights of the moving car is affected by the car's velocity, however. We're more familiar with how pitches change when a siren is moving toward or away from us. If an ambulance is coming toward us, the pitch rises, and if it moves away, the pitch has lower frequencies. Because galaxies are moving away from each other, their light is shifted to lower frequencies, corresponding to longer wavelengths. That makes their visual spectrum redder, so this phenomenon is called the red shift. The photons emitted by the galaxies have less energy. If the galaxies were moving relative to each other near the speed of light, the red shift would make the frequencies of emitted light fall to near zero, so the wavelengths of the light would be extremely long.

Early on, we mentioned Nominalism, a philosophic view that rejects abstractions. In college, I had been exposed to Linguistics. The professor for this course, Bozena Henisz Dostert, spent the first term in her home country of Poland, and her newlywed husband Fred Thompson took over the teaching. Thompson had diverse training and interests, having studied with Alfred Tarski, and worked in computer science. He developed natural language abilities in computers, which at the time were not really capable compared to what we have now. He taught Noam Chomsky's theories on syntax and philosophy, and I later did an independent study with him that I wanted to be about structuralism, but he made it primarily a deep dive into Nominalism (Nelson Goodman and Willard Van Orman Quine). Some of Chomsky's ideas came out of Goodman and Quine.

I was slow to appreciate the good parts of Nominalism, but am now committed to avoiding categorization of people. In any large population of humans, there will be a large variance in almost any characteristic, usually comparable to the variance across different populations. This means that saying that one population is different from another is wrong. For instance, mean IQ might differ slightly between men and women, but is not significant because of the large variance in IQ across both groups. Randomly picking a woman and a man is not going to result in any consistent difference in IQ.

Rejecting abstractions can be difficult in general. Our brains generalize from our experiences, providing us with the means to make categorical decisions. These are beliefs that we should try to question whenever possible, however. Abstractions give us a lot of power to understand the world, but often delude us, because they don't correspond to any physical objects. We can't say that "red" exists on its own, it only corresponds to perceptions about red things.

Buonomano discusses the term "mental time travel" at length. This is the idea that the past and future are imagined for adaptive purposes. It presents problems when viewed from the perspective of the timeline and its focus on the supposed present. But knowing that we have nonlagged and lagged neurons makes this transparent. The brain anticipates future behaviors by having phase leads, and responds to the past by having phase lags. Current activities retain a link to the future through the phase lags, so that future reinforcements can be effective. Temporal discounting, the fact that future reinforcements lose effectiveness with distance from the behavior that produces them, simply depends on the decreasing activity of lagged cells, as illustrated by Sagvolden. Neuronal activity is ongoing across time, and our brains therefore have information about the past and future built into that activity.

Nevertheless, we have great difficulty understanding very low frequencies. The distant future and past remain challenges to our cognitive capabilities. Most neurons adapt to things that don't change quickly, and stop responding. I wonder whether there are neurons that maintain activity across currently unimaginable periods. Our oldest memories represent an example of neuronal activity spanning many years.

Cham and Whiteson give some examples of definitions of time. One of them approximates the view here: "Time is what tells us when things happen." Closer would have been "when things are happening." Whereas they characterize space as "goo", malleable across its extents, they hardly recognize that time is gooey across its extents. Things happen over periods of time, and what counts are the phases over the periods/frequencies. These authors also discuss time in terms of the filmstrip model, although they emphasize continuity afterwards. But whereas they describe a sheaf-type model for space, they fail to do so for time. They see time as linked snapshots, rather than the two-dimensional topological space upon which everything changes. The open sets of time are clearly interconnected locations when viewed as the disk, but not as the line.

Time, like most things we believe exist, is a delusion. A delusion is a belief that is not true. Our beliefs come out of our experience, and we think they are justified by evidence. We are sure that our beliefs are true, but they are not justified by evidence, and so are not true.

Our beliefs originate in the workings of our brains. The adaptive value of having beliefs is presumably that we need to make decisions without having all the evidence. If we are driving and need to make a turn against traffic, we scan the roads and try to determine if it is safe to turn. But we also take into account our prior experience. After a variable amount of time, we decide that it is likely to be safe to make the turn. Every once in a while, our belief that it is safe turns out to be false.

In other words, mistakes are always cases where our beliefs are false. Accumulating more evidence makes it less likely that we will make a mistake, but there are no guarantees.

The prior experience is a particularly important part of beliefs not being true. Most of the time, we don't have the time to accumulate enough evidence, and must make a guess. Educated guesses are enabled by having lots of past experience that can guide present decisions. Through experience, we develop intuition, and often rely heavily on that intuition. That intuition tends to be useful and correct, but sometimes fails miserably.²⁰

Being deluded about everything constitutes our normal state of being. This is neither good nor bad. We think we feel time flowing by, which is seldom harmful. Belief is adaptive, a very good thing. But wherever possible, we should check our beliefs, and be skeptical, rather than blindly accepting what we think we know. Common sense remains a great thing, but is not necessarily grounded in the realities of our extremely complicated world. Cham and Whiteson make this point repeatedly in the context of the weirdness of physics. The ways our brains work are at least as weird, creating our behaviors, emotions, movements, perceptions, and thoughts from mechanisms that arose through hundreds of millions of years of adapting to varying environments. We encounter novel environments, however, and often sense and do the wrong things.

This is especially clear in science. Scientists are continually coming up with ideas about what might be true. Almost all the time, they are deluded. One of millions of examples is how Einstein held for years to his belief that the universe was static, when in fact it is expanding. A good scientist always tries to show that her beliefs are not valid. Most actual scientists unfortunately don't work that way. We champion our ideas, often at the expense of revealing the truth.

Time is not alone as a basic physical concept that we don't understand (see Cham & Whiteson, Rovelli, Feynman, ...). Space is also different from what we normally think about. Properties of elementary particles like charge, mass, and spin do not correspond to anything satisfying. Gravity is a mystery. Physicists continue to look for interpretations of quantum theory. The universe is full of mysteries like dark matter and dark energy. So we should not feel frustrated about being deluded. Everything we know is wrong, or flawed.

On the other hand, Rovelli frames a cogent analysis in terms of quantum gravity. His conclusion involves seeing time in terms of heat. This is related to the idea that time arises from entropy increasing, but is more nuanced. The sense in which time gets cast in this analysis consists of the timeline, and the flow of time from past to future. Timing does not play a role. The idea that breakfast is a time for those who eat it doesn't occur. Timing may not depend on heat for its general application.

A key concept since the early 20th century is spacetime, the topological space underlying everything. Einstein argued successfully that the way to understand gravity is to view it as distorting spacetime. This might sound familiar, as direction selective receptive fields are analogous distortions of space and time. Distorting a straightforward separable receptive field into an oriented, inseparable version of space and time changes the way neurons respond to moving stimuli, just as a black hole warping spacetime changes the way light moves around it. And spacetime is nothing other than gravity, just as a spacetime receptive field is nothing other than a neuron's response properties.

From the point of view of biology, many of these physical questions don't matter much. We can be easily deluded about quantum effects at small scales that we don't experience. What counts is what we experience, which is what matters to the brain. Biology and geology, what is commonly called evolution, are based on how things change in time. But we don't directly experience the evolution of stars, unless they are popular musicians.

A recurring question is when did time begin? The usual answer is that time began at the Big Bang. The universe may have been vanishingly small, with no size at all, but expanded dramatically. This expansion had several phases, but they occurred within 10^-36 seconds (ten million Planck times). Since time had begun only that long ago, the time disk only had periods less than that. In terms of frequency, the reciprocal of period, only enormously high frequencies were present. Thus, anything that was present changed rather quickly. It seems that space was one of those things that was present, and it expanded at a predictably enormous rate. This may be true even when taking into account that only high spatial frequencies were present, since the Planck length is relatively large. On the other hand, the Planck volume is relatively miniscule! So the speed at which space expanded in three dimensions might not have been so fast.

If one fails to appreciate that time consists of frequencies and phases, recognizing these interactions becomes nearly impossible.

Movie 17: Clarinet vs. oboe timbres.

The timbres of different sounds can thus be fruitfully illustrated as sheaves with amplitudes over the frequency and phase space of two-dimensional time. However, the frequencies we're talking about here (20-20000 Hz) are much higher than the frequencies we consider when most things, including musical sounds, change in time (less than 20 Hz).

Sound texture is usually described in terms that don't explicitly depend on time. Usually the number of voices is emphasized: homophony, polyphony, monophony, heterophony. For me this is another example of a bias against the primacy of time. All of the other components of music make up the substrate of texture: whether few or many pitches are employed, whether differing timbres are heard, how elaborately melodic elements occur, the thickness of the harmonies, the dynamic range, the complexity of the forms, variety of tempi, metrical changes and oddities, and the intricacies of the rhythmic approaches. These textural elements can occur synchronously but are typically shifted over time. Polymetric music is synchronous and asynchronous at the same time, so to speak! A fugue has notably rich textures because of the counterpoint, simultaneous melodies. Listen and watch this visualization of a fugue (Movie 18), noting how it depends on temporal phase.

Movie 18: Bach fugue. The "Little" fugue in G minor, BWV 578.

Melodies reveal themselves over time. They have a duration, usually in terms of a number of measures or beats. They have a certain density of notes over that duration. The rhythmic and metrical structure, as well as the tempo and dynamical changes, define the temporal characteristics of melodies. And the way that pitches change over time is the essence of a melody.

Harmony is viewed as the vertical dimension of music, as opposed to melody being horizontal. Vertical refers to synchrony, but harmonies change over time predominantly. Chord sequences are the horizontal dimension of harmonies in many types of music. Popular songs are known by their chord changes, so that blues changes (based on I, IV, and V, tonic, fourth, and dominant chords) form a category of songs. Rhythm changes, derived from "I Got Rhythm" by George Gershwin, use I-vi-ii-V chords. Musicians compose a wide range of melodies that follow a single set of changes, often known as contrafacts. A list of many of these is at List_of_jazz_contrafacts. A good example is Charlie Parker's "Koko", a contrafact on the chord changes of "Cherokee", played at a terrifically fast tempo with enormous virtuosity.

Many pieces of music have a tonal center, given by the key signature. That tonal center can change during the piece, however, via a modulation to a new key. Modulations have directions. Sometimes they move downwards or upwards in pitch. Often they move between minor and major modes in either direction. Other aspects of harmony can change during a piece, such as modes, extensions, resolutions, and voicings.

In many forms of music, the harmonic sequences resolve. A dramatic tone is produced by invoking tension with less euphonic harmony that resolves to something more euphonic and reassuring. A typical set of chord changes that resolves is a minor chord followed by a cadence on the fifth degree, resolving to the tonic. When a phrase does not resolve, the tension remains, providing the listener the sense of things to come over time.

Music is usually not symmetric over time: playing a melody or a chord sequence or a rhythm in reverse order gives a different experience. One of the common manipulations in serial music is to do these sorts of reversals, for instance playing a melody followed by playing it in the reverse, "retrograde" direction. The intervals between consecutive pitches might also be reversed ("inverted") in pitch space.

Dynamics, the loudness of musical sounds, depend completely on time. Musicians use dynamics to express emotions and to draw pictures. Ravel's Bolero increases monotonically in volume and textural density throughout its duration, parallel to the Michael Snow film discussed below. Amplification became important in order to provide greater dynamic ranges to certain instruments that can't sound loud acoustically. Usually, music has crescendi and decrescendi passages where loudness increases and decreases. There are also accents on certain beats that emphasize parts of the rhythmic and melodic structures.

The form of a piece of music comprises longer durations. For example, a popular song might have a chorus of 32 bars, with the first, second and fourth sets of 8 bars having roughly the same structure, and the third set varying that structure. This is an AABA form, with the B section sometimes called the bridge. Typically, the chorus is repeated, often twice at the beginning of the performance and once or twice at the end.

Sonata form has three main parts: exposition, development, and recapitulation. This form became so popular during the 19th century that it developed numerous variations. The main idea is that a theme (or set of themes) is presented, developed by changing aspects of it, followed by coming back to the original theme. This form usually depends on modulations of the tonal center, timbral, rhythmic and harmonic variations, and different textural presentations.

The elements of form most often have durations of many seconds. How does the brain know where it lies in the form? Neurons that are sensitive to these elements might be active over many seconds. However, it is possible that neurons are active at different phases of the more extended elements as well as being specific to those elements.

Another obvious temporal component of music is tempo. Tempi can be denoted in terms of how many beats (e.g., quarter notes) should be played per minute. This is clearly a frequency. However, it is important to keep in mind that the primary element of timing is phase. A piece of music is often performed at different tempi, with the elements occurring at the same phase each time, but at different times on the timeline. When learning a piece of music, it is common to slow down the tempo so that one can find the notes more easily. That does not interfere with future performance because what counts is phase, not the points on the timeline where notes occur.

Meter refers to how many beats occur in each measure, where the music has been divided into a series of units (measures) that are assembled into larger units (e.g., phrases, choruses, movements). This concept is derived from early poetry. In ancient Greece, an important metrical structure was Epic Hexameter. This probably originated in oral traditions for singing the stories that are now associated with Homer. When analyzing the Iliad and Odyssey, each line is divided into six feet, and each foot has 2 or 3 syllables, with the stress varying slightly across those syllables. Not only the metrical structure, but also the structure of the phrases in epic poetry is characterized by parataxis, the use of shorter phrases without subjunctive clauses. Analogous structures appear in music.

I don't know much about tala, an analog of Western music's meter for Indian classical music. It is a more flexible construct, sometimes carrying information about longer sequences of notes. It depends less on accented beats, though a variety of accents are used. It can be altered by the improvising performer. That is, a tala opens the musician to creative expressions in time. As in Western music, the tempo can be varied: the tala is a matter of phase (Movie 19).

Movie 19: Tala. Explanation of a tala and improvised performance based on it.

By structuring oral traditions that presented texts in a musical form, the texts may have been more likely to be preserved, as the temporal structures prevented alterations of the words to fit. We are generally aware of the phenomenon where we can remember the words when we know the melody to which they are set.

Finally, we can hardly dissociate rhythm and time. Cardiac rhythms, walking rhythms, working rhythms, daily rhythms, monthly rhythms, seasonal rhythms, and the rhythms of our lifespans have all been reflected in musical compositions. Inversely, musical rhythms affect our physiological rhythms. We change our musical preferences over the various phases of our lives. And music changes dramatically over a typical human lifespan.

Rhythm is mostly about timing (as opposed to duration, for example). Although a simple rhythm might consist of a series of beats at different phases of a cycle, more complex rhythms involve multiple individual rhythms superimposed over one of more cycles. A trained percussionist can play a trap drum set (a collection of drums and cymbals and other devices) with two hands and two feet, creating four distinct rhythms, "four limb independence". This is an extension of the exercise where you pat your head and rub your tummy. One hand is playing beats on a drum while the other is hitting a cymbal at different times, one foot hitting the bass drum for emphatic beats and the other foot driving the high-hat cymbal in its own patterns.

These sorts of polyrhythms are typical of certain composers. Elliot Carter used them extensively. So did Don Vliet (Captain Beefheart). Ba'aka music, as well as much of West African music, is fundamentally polyrhythmic. Scriabin, Brahms, Boulez, Stravinsky, Ligeti, Reich (note that the shift is a latency difference between the two parts), Jo Jones and Max Roach on Booker Little's Cliff Walk, and many others used numerous varieties of multiple rhythms overlapping. Drum circles generally involve many players interacting with distinct patterns that create a complex groove.

Music presents these elements in orders that are highly structured in time. Serial music plays with these ordering concepts, for example by playing a twelve-tone row forwards and backwards. Pierre Boulez serialized aspects of music beyond pitch, ordering dynamics, durations, attacks (the approaches to sounding notes), and timbre. Ordering in time elsewhere in music is based on gradual, consistent changes that are described by phase values at low frequencies.

Composer Karlheinz Stockhausen wrote a thesis he called "How time passes by" in which he details the structures involved in serialization, with reference especially to his composition Zeitmasse (Time-measures). He refers to phase repeatedly, but in an idiosyncratic usage.

Movie 20: Zeitmasse. Note how the conductor signals the phase values in each measure.

The most common unit of music is the note. A note might have a pitch, a duration, an attack and release, a phase relative to the meter, an intensity, and a timbre. Mostly, however, a note's sound depends on the other notes. This relative quality tells us that it is the note's phases relative to the rest of the music that dominates how it is heard. The directions of changes in pitch, harmony, dynamics, and the other parameters are what matter.

Finally, here is an account of listening to music by somebody who could not hear, Helen Keller:

Helen Keller wrote the following letter to the New York Symphony Orchestra in March 1924. Here's how she describes listening to Beethoven's "Ninth Symphony" over the radio:

"Dear Friends: I have the joy of being able to tell you that, though deaf and blind, I spent a glorious hour last night listening over the radio to Beethoven's "Ninth Symphony." I do not mean to say that I "heard" the music in the sense that other people heard it; and I do not know whether I can make you understand how it was possible for me to derive pleasure from the symphony. It was a great surprise to myself. I had been reading in my magazine for the blind of the happiness that the radio was bringing to the sightless everywhere. I was delighted to know that the blind had gained a new source of enjoyment; but I did not dream that I could have any part in their joy.

Last night, when the family was listening to your wonderful rendering of the immortal symphony someone suggested that I put my hand on the receiver and see if I could get any of the vibrations.

He unscrewed the cap, and I lightly touched the sensitive diaphragm. What was my amazement to discover that I could feel, not only the vibration, but also the impassioned rhythm, the throb and the urge of the music!

The intertwined and intermingling vibrations from different instruments enchanted me. I could actually distinguish the cornets, the roil of the drums, deep-toned violas and violins singing in exquisite unison. How the lovely speech of the violins flowed and plowed over the deepest tones of the other instruments!

When the human voices leaped up thrilling from the surge of harmony, I recognized them instantly as voices more ecstatic, upcurving swift and flame-like, until my heart almost stood still. The women's voices seemed an embodiment of all the angelic voices rushing in a harmonious flood of beautiful and inspiring sound.

The great chorus throbbed against my fingers with poignant pause and flow. Then all the instruments and voices together burst forth — an ocean of heavenly vibration — and died away like winds when the atom is spent, ending in a delicate shower of sweet notes.

Of course this was not "hearing," but I do know that the tones and harmonies conveyed to me moods of great beauty and majesty. I also sense, or thought I did, the tender sounds of nature that sing into my hand-swaying reeds and winds and the murmur of streams. I have never been so enraptured before by a multitude of tone-vibrations.

As I listened, with darkness and melody, shadow and sound filling all the room, I could not help remembering that the great composer who poured forth such a flood of sweetness into the world was deaf like myself. I marveled at the power of his quenchless spirit by which out of his pain he wrought such joy for others — and there I sat, feeling with my hand the magnificent symphony which broke like a sea upon the silent shores of his soul and mine."

The Auricle, Vol. II, No. 6, March 1924. American Foundation for the Blind, Helen Keller Archives.

Movie 21: Yasujiro Ozu's Late Spring. A classic example of Ozu's use of time.

Movie 22: Michael Snow's Wavelength. Probably the best known example of slow cinema.

Movie 23: Andy Warhol's Empire. A clip of a much much longer film.

Cinema uses movement through time in many creative ways. Time is compressed by cutting out (the boring) parts, just including the framing elements that surround the boring parts. Our brains fill in those skipped scenes. Frank Borzage introduced the calendar flipping visual in "Three Comrades" to let the viewer know that years had passed between the end of World War I and the comrades' new difficult lives in post-war Germany (Movie 24). Another great temporally fascinating scene from Borzage is in "Little Man, What Now?" (Movie 25), where Margaret Sullavan's character goes around a merry-go-round, telling Douglas Montgomery's character something each time she comes around to him. A great dependence on phase.

Movie 24: Clip from Frank Borzage's "Three Comrades". Franchot Tone, Robert Young, and Robert Taylor play veterans of WWI who transcend their circumstances with the help of Margaret Sullavan. Borzage pioneered this way of showing how time has passed.

Movie 25: Clip from Frank Borzage's "Little Man, What Now?".

Often, filmmakers use flashforwards and flashbacks to reveal more information. Traditionally, these sequences were marked to reduce confusion, but sometimes a degree of sophistication is presumed so that the film rejects a consistent flow of time. Richard Friedman's "Christmas Crime Story" works backwards and forwards to tell a story that forks to bad and good endings. In "Groundhog Day", Harold Ramis loops February 2 with variations as Bill Murray's character gradually and non-monotonically changes from miserable to heroic and fulfilled.

Hollis Frampton made my favorite example of how film uses time. A mathematical proposition called Zorn's Lemma examines an object called a partial order. A partial order is a relation where elements can be ranked as in a total order, but some pairs of elements can not be ranked that way. It is almost always depicted as a tree, with branches extending upwards. Two elements on the same branch can be ordered, but elements on separate branches can not. This is a spatial realization of the relation. Frampton's film "Zorns Lemma" (Movie 26; reading the linked article's Production section might be illuminating) instead bases the partial order on time. He takes the letters in the Roman alphabet (put into 24 classes by interchanging I and J, and U and V), and displays them, usually in a shot of a sign, for one second at a time, running through the 24 letters before cycling back. Eventually each letter is replaced by a non-alphabetic image, representing the top of that branch. What the mathematical Zorn's Lemma states is that in any partially ordered set where each branch is entirely below some height, the entire tree is below some height. That sounds completely reasonable, as if it must be true. However, the statement is taken to apply to infinite as well as finite sets, and those might not satisfy the statement (since an infinite series of finite values can be infinite, like the whole numbers). In fact, Zorn's Lemma can not be proven based on the axioms of Set Theory. However, it can be proven if one also assumes the Axiom of Choice or equivalent statements that aren't provable by Set Theory. The Axiom of Choice says that given a collection of non-empty sets, one can form a new set containing one element from each of the original sets. Sounds reasonable, but hard to say for infinite sets. Frampton substituted time for the usual spatial dimension in which partial orders are presented. Frampton's manipulations in time could have been done by panning across space, but he went to great lengths to construct a phase-based view of time, "the illusion of space or substance", that resonates with what I've described in this book.

Movie 26: Zorns Lemma. Hollis Frampton's masterpiece illuminating partial orders over the space of time.

Consider also how film and video create the illusion of motion. Frames are presented at a high enough rate that our brains knit together these individual frames, giving the sense of motion when the images on each frame in a sequence change slightly. The familiarity of this technology leads people to think that the brain sees motion in a similar way. Analogous errors are made when brains are said to function like computers. As we've seen, spatiotemporal quadrature is used to perceive motion, not comparisons between frames occurring at discrete times.²⁴

Varying the rate at which frames of a film are captured vs. projected permits slow- or fast-motion. Stan Brakhage filmed towels falling down in slow motion to make money in a fabric softener commercial. Another of his commercial films was a collaboration with the ingenious physicist/author George Gamow (inventor of the Big Bang), "Mr. Tompkins Inside Himself". I haven't found a way to view that film, but another Gamow film (without Brakhage) is "Mr. Tompkins in Wonderland" (Movie 27). That film lacks Brakhage's artistic touch, but explores many of the physical and cinematic concepts we have discussed here.

Movie 27: George Gamow's Mr. Tompkins in Wonderland. A review of some of the Physics discussed above.

Many thousands of films have played with time. Readers will have favorites, and I would be delighted to be enlightened about them. I can be reached at alan@adale.org.

Fig. A1: The PSTH. A stimulus is presented, and whenever spikes occur, a histogram is incremented to reflect where in time during the stimulus responses occur. Here, the stimulus is a spot placed in a neuron's receptive field and its luminance is modulated in time.

Figure A1 shows an example of two PSTHs derived from an experiment like that in the left panels of Fig. 7. The stimulus was a small spot that changed from bright to background to dark and back to background (Movie 2), with that cycle repeated many times. On each cycle, cells generate spikes at slightly different times, but tend to show the same pattern, as revealed by the averages. The nonlagged cell in Fig. A1a fired just after the 1 second time, then nearly stopped firing. It fired strongly again after 2 seconds, and continued to fire until the dark spot went off at 3 seconds. It was silent for about 200 ms, then produced occasional spikes again until the bright spot reappeared. The lagged cell in Fig. A1b fired spikes throughout the stimulus cycle, with the exception of right after the dark spot came on at 2 seconds. This neuron fired strongly after the dark spot went off at 3 s. From these kinds of data one can measure parameters that can be compared across groups of cells.

The visual pathway originates in the retina. The output of the retina consists of the axons of ganglion cells, which project to numerous parts of the brain. One of the targets of the retinal ganglion cells is the lateral geniculate nucleus (LGN). The LGN has several divisions and cell types, but the main structural feature is that many LGN neurons have a dominant input from a single retinal ganglion cell. These LGN neurons relay the signal from the retina on to area V1 (first visual area) in cortex. Cortex is divided into 6 layers. The input from subcortical regions like the LGN is primarily targeted to the middle layer IV, with different LGN cells targeting slightly different parts of layer IV and nearby layers (a substantial input is found in layer VI as well). Connections within and between cortical layers provide the substrates for processing information, which is then passed on to cortical and subcortical regions. Area V1 sends a massive input back to LGN, and feeds visual information to cortical areas V1, V3, V4, MT, and elsewhere (Fig. 2).

A function of time (or space) corresponds to another function of temporal (or spatial) frequency. If we take a function f(t) in the time domain, it corresponds to a function F(ω) in the frequency domain, where ω is temporal frequency. The frequency domain function is two-dimensional, often described as taking values in the complex plane. The dimensions are amplitude, the distance from the origin in the complex plane, and phase, the direction from the origin in the complex plane. So we can write F(ω) = A(ω) ⋅ e^2πiφ(ω). The functions A(ω) and φ(ω) are the amplitude and phase components. The complex exponential e^2πiφ(ω) has values on the unit circle (the circle defined by the points lying at a distance of one unit from the origin) in the complex plane. A phase of φ=0 gives a value of 1, a phase of φ=¼ gives a value of i=√-1, a value of φ=½ gives a value of -1, and a value of φ=¾ gives a value of -i=-√-1. The conventional arrangement of the complex plane has 1 to the right, i upwards, -1 leftwards, and -i downwards.

Many interpretations of the mathematical objects like the complex plane exist. For our purposes, a function of temporal frequency changes amplitude across temporal frequencies. Usually, amplitudes are small at low and high frequencies. That means that little responses are present for slow and fast changes. Phase changes systematically with frequency, increasing with increasing frequency. The change is approximately linear. Phase represents WHEN responses occur, so the linearly increasing phase with frequency means that responses occur later in the cycle as frequency increases. This is obvious, because the length of a cycle decreases with increasing frequency. Responses only occur after some latency, and that latency is a greater portion of the cycle as frequency increases.

Afterword

We have speculated that huge questions about emergent properties and the power of our brains might be posed in the context of the example of direction selectivity. Timing is key to that example, and we expect that would carry over in other domains.

An emergent property that is broadly familiar is childbirth. Living things produce more of their kind, and in the case of sexual reproduction, two individuals who have both commonalities and differences produce a new individual by combining themselves. Generations of this process over millennia under evolutionary pressures create unimaginable novelty.

The numerous speculations above are not likely to be borne out by experiments. But perhaps some of them will be useful at stimulating thinking about time. I wonder why more people don't appreciate the frequency domain. It may just be that it seems foreign, and I hope that the explanations here might make people more comfortable with it, since it seems so simple and provides so much more explanatory power.

As the late phases of this work approach, I have entered into the late phases of my life. A diagnosis of Primary Progressive Aphasia and Alzheimer's disease means that I am likely to lose language, among other things. I am quite curious about how that will feel, as language is one of the most compelling of our temporal faculties. Will I continue to enjoy music and film, as well as emotional contact with my loved ones? One of my all-time favorite books is Douglas Hofstadter's Le Ton beau de Marot: In Praise of the Music of Language. I have had the experience of communicating with people without a common language, and am interested in how well I'll be able to do that. Will my creativity decline? Will I still enjoy music? Lots to learn!

Some biographic information. My path through time

Interactions with people who influenced my thinking

I started my undergraduate work in Physics. I wanted to work with Murray Gell-Mann on particle physics. The only time I saw him, however, was when I spotted him hunting mushrooms. In high school, I had won $50 in a science fair for my presentation on A Hypothetical Mechanical Model of the Photon (in 1971, that was a lot of money for me). It was not soundly based on Physics, but was entertaining, partly by depending on an integral that nobody I asked could evaluate, including brilliant friends like Andy Zangwill, David Sze, and Brad Osgood, as well as our teachers, until somebody at Carnegie-Mellon University identified it as an elliptic integral. The project hypothesized faster-than-light behavior. In college, one of the greatest physicists, Richard Feynman, known at Caltech as God, did an occasional session with first and second year students. One day I asked him about tachyons, particles that travel faster than light. He quickly dismissed the notion of tachyons, citing the fact that they would emit Cerenkov radiation and continuously speed up to infinite speeds (Steven Kenneth Kauffmann, Cerenkov Effects in Tachyon Theory, 1970). He then went back to answering the sorts of questions he loved, such as how sugar dissolves. My appreciation of that attitude took awhile.

My first year Physics advisor was George Zweig. He was a prodigy, who obtained a faculty position when he was in his mid-20s. He came up with the idea of quarks, unusual particles that are the building blocks of protons and neutrons, independently of the guy in the next office, Gell-Mann, who got the credit (Zweig called them "Aces"; Gell-Mann got the word quark from James Joyce' Finnegan's Wake). By the time I knew Zweig, he had switched his research to Neuroscience, in particular auditory physiology. He told me about that and gave me some fantastic reading, Georg von Békésy's studies on how the cochlea works. I had no use for any domain other than Physics at the time, and Zweig's enthusiasm started to change my perspective. Physics lasted through half of my second year, with my last course Feynman's class on Quantum Mechanics.

Robert Sharp was a huge influence on me, through his excellent lectures and the trips we took to the Grand Canyon and Death Valley. I went ahead of the group to the Canyon, hiking down Hermit Trail on my own before going with Professor Sharp the next day. He pointed out the massive fossil bed near the top that I had completely missed. He knew practically every rock in Death Valley, telling us what was around each bend. Professor Sharp told stories about the discovery of seafloor spreading, how various unconformities emerged, and how sand dunes stood and fell. Now there's a mountain on Mars with his name, an appropriate honor. I spent much of my college years in the mountains and the desert, with numerous trips to the Canyon. I often wished I could get a degree in Nature.

I had been exposed to Linguistics. The professor for this course, Bozena Henisz Dostert Thompson, spent the first term in her home country of Poland, and her newlywed husband Fred Thompson took over the teaching. Thompson had diverse training and interests, having studied with logician Alfred Tarski, and worked in computer science. He developed natural language abilities in computers, which at the time were not really capable compared to what we have now. He taught us about Chomsky's theories of grammar, and I did an independent study with him that I wanted to be about structuralism, but he made it primarily a deep dive into Nominalism (Nelson Goodman and Willard Van Orman Quine). I was slow to appreciate the good parts of Nominalism, but am now committed to avoiding categorization of people.

So I started to think Biology might have some worthwhile ideas. That was reinforced by doing some experiments for the Psycholinguistics term, in which I performed experiments with my biologist and soccer friend John Lehmann. We borrowed a tachistoscope, a slide projector with a shutter, that can flash images very briefly, from Dahlia and Eran Zaidel, who were working in Roger Sperry's lab. Sperry had pioneered studies of the two hemispheres of the human brain, thanks to observations on a set of patients who had their hemispheres separated surgically to alleviate symptoms of epilepsy. John and I examined how stimuli were lateralized, processed better on one side of the brain. John's stimuli worked far better than mine.

My best friend, Robert Tajima (Taj), helped me enormously to move further toward Biology. I accompanied him to Sperry's course that ranged from planes of section (pure basic anatomy) to consciousness. Even more revealing was the Neurophysiology course, two terms of which were taught primarily by Jack Pettigrew. His thrilling lectures, talk sessions, and personal guidance were life-changing.

Jack was an Australian maverick. He was a free spirit, possessed the kind of one-in-a-billion intellect that captivated any audience, and always overflowed with ideas. He didn't always chase down the evidence, but in some cases worked quite hard to do so, in particular with his claim that big bats are primates. His lab was decorated inside and out with art made by his wife Rona, and both his lab and his home were populated with varied experimental animals. His most memorable lecture was "Love is a plastic state," in which he argued that our brains become more susceptible to change when we're in love (or on certain drugs or under other conditions). Jack gave me a paper by William C. Hoffman about the "Lie algebra of visual perception" that proved instrumental in seeing how mathematical ideas can be applied to neuroscience. Although I didn't make the connection, Hoffman proposed a form of the idea I treat in this book, that sheaf theory applies to the brain. Jack similarly gave me a wonderful paper by Max Cynader and David Regan that proposed a simple arithmetic solution to how we determine whether a moving object is going to hit us in the head. Jack suggested that Max might be a good person to work with. He was right.

Jack is known for many discoveries and theories. He showed how neurons in visual cortex help to create stereopsis to let us see depth. He proposed and tested a number of molecules that might regulate cortical plasticity. By comparing visual mechanisms in different species, he argued for convergent evolution that produces binocular vision with similar algorithms despite varying structures. He famously campaigned for the idea that big bats are flying primates. Jack took on projects involving baobab trees and rock art. And he measured switching times for the bistable perception in binocular rivalry, correlating those times with bipolar disease and schizophrenia.

Jack collaborated with dozens of other great minds, including the Dalai Lama. He studied an equally broad collection of non-human creatures. And his home was the whole world, despite his love for Queensland, Pasadena, Tasmania, and the several other places he lived.

Taj and I also took Max Delbrück's course that covered a variety of biological and philosophical topics. I got to know Max on annual trips to Joshua Tree, with his wife Manny and their children (Tobi works on electronic chips that process visual stimuli, including direction selective elements that analyze moving images; Ludina Delbrück Sallam is an Arabic translator; there were two older children I didn't know), Humanities professors David and Annette Smith's family, and several undergraduates. Max kept star charts like the ancients. My last year, he told me how his retinal detachments kept him from being able to make out the stars well enough to continue this work, and asked if I'd take over his charts. I stupidly declined, fearing I wouldn't do it justice. Max viewed Science not as building a cathedral, but instead people piling up stones (also Beckett). I found his attitudes refreshing and on point. Max deserved every honor possible, just a giant of a man.

David Smith gave me a special gift in bringing musician Charles Lloyd to campus on a series of Wednesday evenings over a span of many months. Charles talked mostly about transcendental meditation and mucusless diet, but did discuss music. He taught me many things from his expert perspective. Notably, when I brought up Booker Little, he immediately said "the greatest trumpet player who ever lived" and proceeded to regale me about how Booker put him up when he first got to New York City, and how they had been close friends from childhood. He asked us to bring in a record one week, and I brought in Coltrane's Village Vanguard album. He proceeded to play the masterpiece Chasin' the Trane, picking up the tone arm at the end and putting it back at the beginning 3 times. Charles brought along his filmmaker friend Eric Sherman, who screened a series of great films by Frank Borzage, Fritz Lang, Josef Sternberg, Kenji Mizoguchi, Orson Welles, John Ford, Peter Bogdanovich, and others. This sparked my interest in film, which, along with music, has occupied much of my time since.

After college came two years in Central African Republic as a Peace Corps Volunteer. Three months of training in MBaiki and Bangui gave me a start on speaking Sango, the National Language. I learned how to fix my mobylette, kobé nguinza na gozo (eat manioc with manioc), and communicate with people without a common language. Halfway through training, we had a week to travel. I walked into the rain forest, hoping to see how the Ba'aka live. The water filters had not been turned on at the agricultural university where we stayed, and everybody got sick that week. I learned some Sango on my walk, but when I got to a tiny village, I asked for water as I was feeling terrible. They didn't want to give me their rather dirty water, so they gave me mbako, manioc whiskey. I tried to slake my thirst with the mbako, which was a bad idea. They fed me caterpillars that were delicious, and put me to sleep in a tobacco-drying hut. I woke up at 3 am and wandered around trying to get some water, checking all the leaves for any dew. Heading back to the university in the morning, I learned about everybody being sick. After recovering, I went back out and made my way to Mongoumba, hoping to take the President's boat to Bangui. The town's gendarme confronted me about what I was doing there. He was suspicious until I dug out a card we'd be given that said I was cooperant. Suddenly we were best friends. He took me to his house, taught me to play kissoro, a stones in holes game. He had his wife cook me the best meal I've ever had, wild mushrooms. After several days, I gave up on the President's boat and found rides to Bangui.

I lived up north in Ndélé for the rest of my time there, with a month back in Pittsburgh with my father, who had suffered his fourth heart attack. He told me to go back to CAR, and he'd see me get my PhD, an idea I hadn't ever considered. I hung out with lots of people who I never would have known otherwise. My neighbors Jacques, Ali, and Malik, commerçant brothers, were wonderful friends and teachers. A student, Lopère Jean-Nestor, became a close friend, along with his compatriots from the villages of Digba and Kottisako, who I referred to as the Ouadda Boys. They often joined me for dinner, which was probably their only meal during the week. They played music and sang, and we talked about all sorts of things out of school.

Amara, a musician who played the double reed keita, came to town from Tchad. He could support himself because Ndélé had a Sultan with a court. Amara used circular breathing, had cheeks like Dizzy Gillespie, and a tremendously big sound. We got to be friends, talking without a common language about music, and drinking douma, honey beer. I had three pieces of technology with me: a Walkman cassette recorder, a Fuji Single-8 movie camera, and my Leblanc clarinet. I recorded Amara and the Ouadda Boys and various other music on cassettes, and listend to a set of tapes I'd brought along. I shot rolls of film that were sent by mail to Fuji and then to people I designated. Amazingly, I recovered every roll of film when I got back to the states. I played the clarinet with Amara, but mostly sitting on the rocks above town. The Black Kites would swoop past, grazing my head.

When I returned from visiting my father, I got to work on a construction project that the Ambassador had funded, a new building at the school where I taught. I borrowed a motorcycle and rode around Bangui putting together all the supplies we needed, and a truck to take them to Ndélé. I rode with the truck, getting to be friends with the driver. When we got to the Bangoran River 60 km south of Ndélé, the driver refused to cross, afraid his engine would flood. I pleaded with him for a couple hours, then decided I would walk ahead, hoping that he would come along after the crazy white guy. The plan worked. I knew that Digba and Kotissako were not far from the river, but he didn't. I bought the whole crew douma when we got to Kotissako. A group of newly arrived construction volunteers arrived unexpectedly. I found them places to stay and hired another cook to help. I had already hired locals to help with the demolition of the old building, and now the construction, and we got to work. I didn't deal well having other Americans around, partly because most of them had no interest in the community. Another truck had come up without me to bring cement for our needed concrete work. The cement wound up being stolen by the Préfét, David Zoumandé. I met with him regularly, and would ask him about the cement. I finally got fed up with him, and told him explicitly that he had stolen the cement. He kicked me out of his office and told me not to come back. That was fine with me, since I hated speaking French with him and didn't need him for anything. That truck also left our load of reinforcing rods, rebar, at the other side of the Bangoran, which we later went down and walked across the river, at the end of Ramadan. The Ambassador had also funded a project to bring water from a spring into the town. I was in charge of the money for that project as well, but the work was supervised by the chef-de-genie-rural, a wonderful man named Thadée Ozzenguet. We traded formboards and rebar, and became close and helpful to each other. Many years later, I corresponded with his son who was in Canada. At the end of the rainy season, two more Americans arrived, Tony Nathe and Mark Rand, who stayed and taught for the next year. The construction volunteers left, though two came back to replace the subroofing a little later.

We had great colleagues at the school, Eleguendo Joel, Yamtiga Albert, and others. Joel taught history, and got up early every morning to listen to the news just in case a student asked about it. He fought the system, and had spent time in prison because of that. I was a math teacher, but taught Biology and English as well, because faculty tended not to show up in distant Ndélé. The school closed for a month in 1977 so that the students could practice marching in anticipation of the coronation of Bokassa I. I also had a class kicked out of school for a term because they were misbehaving. Students had a difficult road for many reasons. Most of them came from villages that were too hard to walk home to during the week, and they only really ate on weekends. Some didn't have comfortable places to sleep at night. They were under tremendous pressure from family, townspeople, and teachers. The girls had to deal with men soliciting them for sex, and the risk of getting pregnant. There weren't books, only little notebooks they had to buy and copy into from the board.

The last thing I did in CAR was a trip to the most beautiful place in the world, Koumbala, with the other teachers and the superintendent of primary schools, whose wife was the director of our school, and who had a car. I bought a sheep, and we drove to Koumbala in his Land Rover with the sheep in the back. At Koumbala, we walked to the top of the waterfall, slaughtered the sheep in the river, cleaned the tapeworms out of the intenstines, stuffed them with the organ meat, and ate all the parts. I was a vegetarian, but tried everything and saw why people love the meat that most Americans never venture to eat. After a wonderful weekend, we headed back, but ran out of gas a few kilometers before town. We walked the rest of the way. When I got to Tony and Mark's house, Mark told me that my father had died. I sat and cried for hours. Mark accompanied me on the trip to Bangui atop a truck. I didn't have many chances to say goodbye to friends, most notably Jean-Nestor. After a long trip to Pittsburgh, I was quite ill, and my mother sent me to my father's doctor. It took another week to recover.

While in college, I pined to get away from academia and live away from cities. I succeeded in that by living in Ndélé. Back in the states, I fell in love with the piece of woods we called Out Yonder, and bought it from my brother-in-law, who had purchased it with money he got from doing a commercial for the Post Office. Peace Corps gives you a "readjustment allowance", and I used it for the land. Otherwise I didn't need much money thanks to a pretty spartan lifestyle, and a teaching job at my cousin June Himelstein's alternative high school. But eventually I ran out of money, and looked to get a job at the nearby West Virginia University. They paid me $3000/year to teach algebra and calculus, and I got a Master's degree. I got sucked back into academia.

The Clock Reaction

I was teaching calculus to engineering students at West Virginia University, and noticed that a book had been left behind by the previous teacher. After class, I went to return the book to that professor, but first scanned its content. It was The Geometry of Biological Time, by Art Winfree, a beautiful book in every way. I found the owner, Ken Showalter, in the Chemistry Department. Ken is a physical chemist, and we wound up talking about the book and many other things.

At the time, Ken was working on a chemical reaction that involved the reagents sodium iodate and arsenous acid. If you put these chemicals in a jar at the beginning of a lecture, they will sit as a transparent liquid. At some point during the lecture, the fluid in the undisturbed beaker will suddenly turn black. This is an example of a clock reaction.
Video of clock reaction

Inorganic chemicals can undergo reactions at regular or irregular times. Ken had been looking at how the reaction in a tube of this solution can be triggered at one point in space, creating a wave that traveled down the tube. We worked on analyzing these propagating waves. I was living Out Yonder at the time, an hour's drive from Morgantown, working in the woods, cooling off in the pond, then trying to figure out how the waves worked. Ken and I sent notes (a few years later there would have been email) back and forth about our progress. He was sorting out the chemistry, and I was playing with the math. We each came up with the answers from our two perspectives at the same time. I had seen that the waves could be described in a pretty (closed-form, meaning easy to write down and read) function if the reaction was a cubic function of iodide concentration. Ken had seen that the reaction was in fact a cubic. We nailed down how well the predictions fit the data, determining the speed at which the waves traveled, and published a paper with Adel Hanna.

That got me started on thinking about time. I learned many things from Ken, especially about the wonders of collaboration. Some people don't recognize that Science is about people. But humans are social animals, and we thrive on our interactions. The pleasure of doing Science is largely spending time with people you enjoy and appreciate. Every once in a while you learn something, providing enormous pleasure.

This book will hopefully provide such pleasure, or at least be stimulating. It has roots in Winfree's book, and steals its subtitle from his title. I got to spend some time in June 1980 with Winfree in Salt Lake City at a meeting about Mathematical Biology. We went up to the mountains, still covered by snow, and slid down on trash bags. Winfree was driven by creativity. He liked to push the limits. Yet his work was accessible and applicable.

Carl Irwin made it possible for me to attend that meeting. Carl was a math professor at WVU who for some reason took an interest in me despite my never taking a class from him. He had earlier gotten me to a meeting in Philadelphia organized by Stephen Grossberg, where I met George Sperling, Stuart Geman, and others who would reappear in my life. Carl, and Paul Brown, a neurophysiologist at WVU in whose lab I worked, taught me that helping other people without expectation of getting anything back is a prime goal. Paying it forward is a beautiful philosophy. I have an expensive electric bike that Sohail Khan gave me, saying "Pay it forward" and have learned that the point of life is to help other people.

Goro Kato spent a year at WVU as a visiting professor, and asked me who I was at a Math Department meeting. Goro has an infectious, cheery personality. He got me studying abstract algebra again, insisting on rigor that I have a hard time achieving. Our friendship persists, and he has been my collaborator on the sheaf theory presented here, though my poor understanding leaves the presentation here lacking.

The Math Department treated me well, and I enjoyed most of the course work, particularly Topology with Sam Nadler. But I was more interested in Neurophysiology, and searched out somebody on the other campus to work with, finding Paul Brown. Paul studied somatosensory processing in the spinal cord of cats. I participated in some experiments, but mostly came in at night to use his computer. Paul had gone to an NIH-funded program to give computers to neurophysiologists. These computers were built by Ted Kehl, and used his BASIL language. I ran the computer to simulate the development of kitten visual cortex, based on ideas I'd formulated thanks to Jack Pettigrew. Paul was on sabbatical in Scotland much of the time, and we corresponded by mail about papers I was reading. The training I received from Paul stands as one of the several large pillars I've stood upon.

I wound up at Brown University because it seemed relatively close to my Out Yonder land, and because I had met Stuart Geman and knew Jerry Daniels from Jack Pettigrew's lab. My 4 years in Providence involved a lot of suffering around my lack of money and a lot of hard work, sometimes with difficult people. But there were plenty of great times, including my eldest two daughters, listening to music with Phil Stallworth, learning to think critically with Jim McIlwain, and especially knowing Philip Fiagbedzi and his wife Christina. Philip got me through my comprehensive exams, although my Probability professor asked me about martingales, which I had missed learning about while at a Neuroscience meeting. Jerry Daniels and Leon Cooper, my advisers, helped me through a lot of tough times, but also created them for me. My graduate colleagues Paul Munro, Mike Paradiso, Mark Bear, Brad Seebach, Eugene Clothiaux, Doug Reilly, and Chris Scofield helped immensely to get me through my difficulties.

Mike Paradiso introduced me to Mike Shadlen, who told me about spatiotemporal quadrature. His son Josh Shadlen happens to have become a sheaf theorist, although Josh doubts its applicability to anything.

As my PhD program was coming to an end, I contacted Jenny Lund, an expert visual cortical anatomist, who had just moved to Pittsburgh with her husband Ray, an equally expert developmental neuroscientist. Jenny was enthusiastic about my coming to do a postdoc, but steered me to their new hire, Allen Humphrey, who I knew from a job interview he had made at Brown. We agreed that I would join them. However, at the vision meeting in Florida that year, I ran into Max Cynader at the Myakka River state park. I had applied to do my graduate work with him several years back (but decided against it because I thought Nova Scotia was too far away from Out Yonder), and he suggested I come to Halifax now as a postdoc. So I wrote to Jenny and Al to decline their offer with regret, mentioning that I would love to come work with them later. In a very encouraging way, they asked me to postpone our plans until after I was done with Max.

Working with Max Cynader was a revelation about collaborations. Max and I finished each other's sentences. We were quite similar in scientific temperament, being fascinated with ideas. I continued to do experiments by myself, but got inspiration and support from Max. He decided to move to Vancouver, so I had a highly productive year with him before going to Pittsburgh. My work on adaptation aftereffects in single neurons had started in Providence, but took off in Nova Scotia. I got to look at temporal processing.

During my time at Dalhousie, Al Humphrey suggested that I write a grant to fund my postdoctoral work with him. Al and Rosalyn Weller had done experiments in Murray Sherman's lab in Stony Brook, looking at cat lateral geniculate neurons by filling their axons with the enzyme horseradish peroxidase (HRP) and visualizing the cell bodies, later progressing to experiments where they filled the cell bodies directly. They became interested in cells reported by David Mastronarde, and Al sent me Mastronarde's and their yet unpublished manuscripts on lagged cells. I wondered whether lagged and nonlagged cells might be in spatiotemporal quadrature, and wrote a draft of the grant that proposed to examine that question and whether they might have something to do with cortical direction selectivity. Al rewrote the grant almost completely to focus on the experiments he had been doing. That was wise, because we were awarded the funding. When I got to Pittsburgh, Al gave me Peter Bishop's papers to read, and we started trying to fill cells with HRP after initial extracellular recordings of their visual responses. The anatomical work went very slowly, only recovering a few cells in a year. Meanwhile, the receptive field recordings went beautifully, and soon I could see that lagged and nonlagged cells were in quadrature.

We published the LGN study in 1990, after sending a draft of the manuscript to Murray Sherman. Sherman was critical, but did not keep us from publishing. Subsequently, he tried to sabotage our work. He and Dan Ulrich put out a short paper arguing that lagged cells were an epiphenomenon of anesthesia. Paul Heggelund and Espen Hartveit, who had been recording from lagged cells as well, along with Al and me, responded with evidence that refuted Sherman's claims. But Sherman was influential in making people skeptical of our findings. This is despite how clearly Mastronarde had initially laid out the story, and the replications by Heggelund and Hartveit and ourselves. The business of Science, of course, is a social sport.

I was anxious to get back to recording in visual cortex, and that work soon proved fruitful as well. I was also funded through the 1990s, and performed studies on adaptation and direction selectivity, and on development of timing in kittens. Jordan Feidler, a graduate student who had been doing neural modeling, became the major participant in the research, along with Adi Murthy. Jordan left before finishing his PhD because he needed to support his family, but has served the US well at Mitre, Raytheon, and The Aerospace Corporation. Adi continues to do research involving eye movements. Along with Paul Baker, our programmer, we had great times in and out of the lab in Pittsburgh. Peter Carras joined Al and me to work on replicating and extending some of our findings in monkeys. Unlike Max Cynader, Al Humphrey and I had different strengths and weaknesses. That of course can make for excellent collaborations, and we put together a set of experiments that revealed exciting novel concepts and results about visual processing. Al writes really well and understands what reviewers need, and made it possible for me to be funded. I wish I had incorporated more of his strengths than I did.

After dozens of unsuccessful job interviews, including for positions outside neuroscience research after I'd given up on that, I heard from Max Snodderly. I knew Max from discussions about cortical processing. His wife, Kristen Harris, interviewed for the department chair position in Pittsburgh but accepted a job at the Medical College of Georgia. Max was given a position in the Ophthalmology Department. I interviewed with Max in Boston in November 2002, walking through snowy Boston with my broken leg in an external fixator, and he hired me to start in March 2003. Then I heard nothing from him. I forced the issue by showing up in May, and accompanying him to a meeting with the Ophthalmology Chair, Julian Nussbaum. Julian had me go through the processing to start in June. Unfortunately, the university did to me what I found out they did to many people, and didn't pay me for my first month, which led to a grave financial crisis for me that lasted many years. Julian was my savior during most of my 18 years in Augusta, but he was fired at the end, adding another straw that led to my retirement at the end of 2021.

I was fortunate to have Jay Hegdé come to the university. He was a constant source of friendship, knowledge and critical thinking. Unfortunately, the university screwed him horribly. The administrators (including the head lawyer) who committed these crimes earned my hatred.

The final straws that made me want to retire were interference in the way I ran the graduate neuroscience course, and the new vet and Animal Care Committee chair telling me I couldn't have music in my lab. I'd always had music in my labs, and most of the scientists I'd admired had the same. I was happy to retire for many reasons, but getting away from those people was important.

One of the other reasons is that I wanted to write this book. I started it years ago, but it's impossible to do all the work that's needed for a book while still working 60-80 hours a week. My retirement plans were altered by getting married to my wonderful wife Sarah. Retirement has been busy, but I'm finding time for this work. I hope readers will appreciate some of the ideas herein.

At a point where this book was well along, my friend Tim Meier turned me on to his former mentor Robert Sapolsky (Tim joined his lab because there was a piano there!), who had recently written a book about how free will is a delusion (my term). This book has been a great inspiration in many ways, for its content as well as for its style. I have been helped by so many people that I dare not try to mention all of them. But like the Sapolsky book, I got a lot out of Andy Zangwill's recent biography of physicist Phillip Anderson.

Numerous other people are cited in the text. I want to clarify an important point that might be missed by most readers. Biologists are trained to be critical. We read scientific papers to find out what's wrong with them, not so much to learn about their content. In this book, I therefore criticize many colleagues. These criticisms are by no means personal. They reflect different ways of thinking, and that is kind of the point of this book, to distinguish my thinking from that of others. I'm doubtless wrong about many things, especially because I've gone way beyond the evidence to speculate wildly. That seems appropriate in this book that is meant for generalists as well as specialists. The goal is to make people think, and when you find out places where I'm wrong, that's great. I'm glad to hear from people, and will alter this book as I learn where it can be improved.

I've attempted to not write a scholarly book, in the sense that statements are always supported in the text. Many more citations would have been nice to have for many of us, but detract from readability for most readers. This is a choice that was difficult. I originally organized a split stream that included far more details that the reader could ignore by sticking to the streamlined version. That was unwieldy, unfortunately. Perhaps a later edition will reinstate that, but I suspect that a general audience book might be welcomed even for the scholars among us. However, tell me where something important is missing, and I'll put it in here.

An appendix is included to provide some basics that might be useful to some readers. It provides some information about how the experimental results are analyzed and interpreted. A brief tutorial about neurons is followed by some relevant math ideas.