#24. The Pictures in Your Head [neuroscience]

NE

Red, theory; black, fact.

My post on the thalamus suggests that in thinking about the brain, we should maintain a sharp distinction between temporal information (signals most usefully plotted against time) and spatial information (signals most usefully plotted against space). Remember that the theory of General Relativity, which posits a unified space-time, applies only to energy and distance scales far from the quotidian.

In the thalamus post, I theorized about how the brain could tremendously data-compress temporal information using the Laplace transform, by which a continuous time function, classically containing an infinite number of points, can be re-represented as a mere handful of summarizing points called poles and zeroes, scattered on a two-dimensional plot called the complex frequency plane. Infinity down to a handful. Pretty good data compression, I'd say. The brain will tend to evolve data-compression schemes if these reduce the number of neurons needed for processing (I hereby assume that they always do), because neurons are metabolically expensive to maintain and evolution favors parsimony in the use of metabolic energy.

Ultimately, the efficiency of the Laplace transform seems to come from the fact that naturally-occurring time functions tend to be pretty stereotyped and repetitious: a branch nodding in the wind, leaves on it oscillating independently and more rapidly, the whole performance decaying exponentially to stillness with each calming of the wind; an iceberg calving discontinuously into the sea; astronomical cycles of perfect regularity; and a bacterial population growing exponentially, then shifting gears to a regime of ever-slowing growth as resources become limiting, the whole sequence following what is called a logistic curve.

Nature is very often described by differential equations, such as Maxwell's equations, those of General Relativity, and Schrodinger's Equation, the three greats. Other differential equations describe growth and decay processes, oscillations, diffusion, and passive but non-chemically energy-storing electrical and mechanical systems. A differential equation is one that contains at least one symbol representing the rate of change of a first variable versus a second variable. Moreover, differential equations seem to be relatively easy to derive from theories. The challenge is to solve the equation, not for a single number, but for a whole function that gives the actual value of the first variable versus the second variable, for purposes of making quantitative, testable predictions, thereby allowing testing of the theory itself. The Laplace transform greatly facilitates the solution of many of science's temporal differential equations, and these solutions are remarkably few and stereotyped: oscillations, growth/decay curves, and simple sums, magnifications, and/or products of these. Clearly, the complexity of the world comes not from its temporal information, but from it's spatial information. However, spatial regularities that might be exploited for spatial data compression are weaker than in the temporal case.

The main regularity in the spatial domain seems to be hierarchical clustering. For an example of this, let's return to the nodding branch. Petioles, veins, and teeth cluster to form a leaf. Leaves and twigs cluster to form a branch. Branches and trunk cluster to form a tree. Trees cluster to form a forest. This spatially clustered aspect of reality is being exploited currently in an approach to machine intelligence called "deep learning," where the successive stages in the hierarchy of the data are learned by successive hidden layers of simulated neurons in a neural net. Data is processed as it passes through the stack of layers, with successive layers learning to recognize successively larger clusters, representing these to the next layer as symbols simplified to aid further cluster recognition. This technology is based on discoveries about how the mammalian visual system operates. (For the seminal paper in the latter field, see Hubel and Wiesel, Journal of Physiology, 1959, 148[3], pp 574-591.)

Visual information passes successively through visual areas Brodmann 17, 18, and 19, with receptive fields becoming progressively larger and more complex, as would be expected from a hierarchical process of cluster recognition. The latter two areas, 18 and 19, are classed as association cortex, of which humans have the greatest amount of any primate. However, cluster recognition requires the use of neuron specialist sub-types, each looking for a very particular stimulus. To even cover most of the cluster-type possibilities, a large number of different specialists must be trained up. This does not seem like very good data compression from the standpoint of metabolic cost savings. Thus, the evolution of better ability with spatial information should require many more new neurons than with the case of temporal information.

My hypothesis here is that what is conferred by the comparatively large human cerebral cortex, especially the association cortices, is not general intelligence, but facility with using spatial information. We take it on and disgorge it like water-bombers. Think of a rock-climber sizing up a cliff face. Think of an architect, engineer, tool-and-die maker, or trades person reading a blueprint. Now look around you. Do we not have all these nice buildings to live and work in? Can any other animal claim as much? My hypothesis seems obvious when you look at it this way.

Mere possession of a well developed sense of vision will not necessarily confer such ability with spatial information. The eyes of a predatory bird, for instance, could simply be gathering mainly temporal information modulated onto light, and used as a servo error for dynamically homing in on prey. To make a difference, the spatial information has to have someplace to go when it reaches the higher brain. Conversely, our sense of hearing is far from useless in providing spatial information. We possess an elaborate network of brain-stem auditory centers for accomplishing exactly this. Clearly, the spatial/temporal issue is largely dissociable from the issue of sensory modality.

You may argue that the uniquely human power of language suggests that our cortical advantage is used for processing temporal information, because speech is a spaceless phenomenon that unfolds only in time. However, the leading theory of speech seems to be the Wittgenstein picture theory of meaning, which postulates that a statement shows its meaning by its logical structure. Bottom line: language as currently understood is entirely consistent with my hypothesis that humans are specialized for processing spatial information.

Since fossil and comparative evidence suggests that our large brain is our most recently evolved attribute, it is safe to suppose that it may be evolving still, for all we know. There may still be a huge existential premium on possession of improved spatial ability. For example, Napoleon's strategy for winning the decisive Battle of Austerlitz while badly outnumbered seems to have involved a lot of visualization. The cultural face of the zeitgeist may reflect this in shows and movies where the hero prevails as a result of superior use of spatial information. (e.g., Star Wars, Back to the Future, and many Warner Bros. cartoons). Many if not most of our competitive games take place on fields, courts, or boards, showing that they test the spatial abilities of the contestants. By now, the enterprising reader will be thinking, "All I have to do is emphasize the spatial [whatever that means], and I'll be a winner! What a great take-home!"

Let me know how it goes, because all this is just theory.

#13. The Neural Code, Part II: the Thalamus [neuroscience, engineering]

A hypothetical scheme of the thalamus, a central part of your brain.

EN     NE     

Red, theory; black, fact.

Thalamic processing as Laplace transform

More in Deprecated, Part 1. I postulate that the thalamus performs a Laplace transform (LT). All the connections shown are established anatomical facts, and are based on the summary diagram of lateral geniculate nucleus circuitry of Steriade et al. (Steriade, M., Jones E. G. and McCormick, D. A. (1997) Thalamus, 2 Vols. Amsterdam: Elsevier).  What I have added is feedback from cortex as a context-sensitive enabling signal for the analytical process. I originally guessed that the triadic synapses are differentiators, but now I think that they are function multipliers.

Thalamic electrophysiology

The thalamic low-threshold spike (LTS) is a slow calcium spike that triggers further spiking that appears in extracellular recordings as a distinctive cluster of four or five sodium spikes. The thalamus also has an alternative response mode consisting of fast single spikes, which is observed at relatively depolarized membrane potentials.

The thalamic low-threshold spike as triggered by a hyperpolarization due to an electric current pulse injected into the neuron through the recording electrode. ACSF, normal conditions; TTX, sodium spikes deleted pharmacologically. From my thesis, page 167.

Network relationships of the thalamus

Depolarizing input to thalamus from cortex is conjectured to be a further requirement for the LTS-burst complex. This depolarization is conjectured to take the form of a pattern of spots, each representing a mask to detect a specific pole of the stimulus that the attentional system is looking for in that context.

The complex frequency plane is where LTs are graphed, usually as a collection of points. Some of these are "poles," where gain goes to infinity, and others are "zeroes," where gain goes to zero. I assume that the cerebral cortex-thalamus system takes care of the poles, while the superior and inferior colliculi take care of the zeroes. 

If this stimulus is found, the pattern of poles must still be recognized. This may be accomplished through a cortical AND-element wired up on Hebbian principles among cortical neurons. These neurons synapse on each other by extensive recurrent collaterals, which might be the anatomical substrate of the conjectured AND-elements. Explosive activation of the AND network would then be the signal that the expected stimulus has been recognized, as Hebb proposed long ago, and the signal would then be sent forward in the brain via white matter tracts to the motor cortex, which would output a collection of excitation spots representing the LT of the desired response.

Presumably, a reverse LT is then applied, possibly by the spinal grey, which I have long considered theoretically underemployed in light of its considerable volume. If we assume that the cerebral cortex is highly specialized for representing LTs, then motor outputs from cerebellum and internal globus pallidus would also have to be transformed to enable the cortex to represent them. In agreement with this, the motor cortex is innervated by prominent motor thalami, the ventrolateral (for cerebellar inputs) and the ventroanterior (for pallidal inputs).

Brain representation of Laplace transforms

The difficulty is to see how a two-dimensional complex plane can be represented on a two-dimensional cerebral cortex without contradicting the results of receptive field studies, which clearly show that the two long dimensions of the cortex represent space in egocentric coordinates. This just leaves the depth dimension for representing the two dimensions of complex frequency.

03-01-2020:
A simple solution is that the complex frequency plane is tiled by the catchment basins of discrete, canonical poles, and all poles in a catchment basin are represented approximately by the nearest canonical pole. It then becomes possible to distinguish the canonical poles in the cerebral cortex by the labelled-line mechanism (i.e., by employing different cell-surface adhesion molecules to control synapse formation.)

Recalling that layer 1 of cortex is mostly processes, this leaves us with five cortical cell layers not yet assigned to functions. Four of them might correspond to the four quadrants of  the complex frequency plane, which differ qualitatively in the motions they represent. The two granule-cell layers 2 and 4 are interleaved with the two pyramidal-cell layers 3 and 5. The two granule layers might be the top and bottom halves of the left half-plane, which represents decaying, stabilized motions. The two pyramidal layers might represent the top and bottom halves of the right half-plane, which represents dangerously growing, unstable motions. Since the latter represent emergency conditions, the signal must be processed especially fast, requiring fast, large-diameter axons. Producing and maintaining such axons requires correspondingly large cell bodies. This is why I assign the relatively large pyramidal cells to the right half-plane.

Intra-thalamic operations

It is beginning to look like the thalamus computes the Laplace transform just the way it is defined: the integral of the product of the input time-domain function and an exponentially decaying or growing sinusoid (eigenfunction). A pole would be recognized after a finite integration time as the integrand rising above a threshold. This thresholding is plausibly done in cortical layer 4, against a background of elevated inhibition controlled by the recurrent layer-6 collaterals that blocks intermediate calculation results from propagating further into the cortex. The direct projections from layer 6 down to thalamus would serve to trigger the analysis and rescale eigenfunction tempo to compensate for changes in behavioral tempo. Reverberation of LTS-bursting activity between thalamic reticular neurons and thalamic principal neurons would be the basis of the oscillatory activity involved in implementing the eigenfunctions. This is precedented by the spindling mechanism and the phenomenon of Parkinsonian tremor cells.

Mutual inhibition of reticular thalamic neurons would be the basis of the integrator and multiplication of functions would be done by silent inhibition in the triadic synapses (here no longer considered to be differentiators) via the known disinhibitory pathway from the reticular thalamus. 

A negative feedback system will be necessary to dynamically rejig the thalamus so that the same pole maps to the same spot despite changes in tempo. Some of the corticothalamic cells (layer 6) could be part of this system (layer 6 cells are of two quite different types), as well as the prominent cholinergic projections to the RT.

Consequences for object recognition

The foregoing system could be used to extract objects from the incoming data by in effect assuming that the elements or features of an object always share the same motion and therefore will be represented by the same set of poles. An automatic process of object extraction may therefore be implemented as a tendency for Hebbian plasticity to involve the same canonical pole at two different cortical locations that are connected by recurrent axon collaterals.

#3. AviApics 101 [population, engineering, evolutionary psychology]

PO     EN     EP     

Red, theory; black, fact.

Here, I go into detail about the human population controller introduced in the previous post.

I assume that, like everything in the natural (i.e., evolved) world, it is a masterful piece of engineering, as Leonardo Da Vinci declared.

The way to build an ideal controller is the inverse plant method, where the controller contains the mathematical inverse of a mathematical model of the system to be controlled.  To derive the model, you take the Laplace transform of the system's impulse response function. For populations, a suitable impulse would be the instantaneous introduction of the smallest viable breeding population into an ideal habitat.

What happens then is well known, as least in microbial life forms too simple to already have a controller: unrestrained, exponential population growth as per Malthus, with no end in sight.

This exponential curve is then the impulse response function we need, and its Laplace transform is simple: 1/(S - r), where S is complex frequency and r is the Malthusian constant, that is, percent population growth rate per year. The mathematical inverse is even simpler: S - r, which is calculated as set point X minus controlled variable Y. The result is summed with perturbation P and made equal to Y. The result is usually simplified to permit predictions about controller performance, but that is not needed in this discussion.

The control effort is E(S - r), which can be multiplied out as ES - Er. Remember that everything has been Laplace transformed in these expressions, and that ES becomes the time differential of e when transformed back into the real world. Multiplication by a constant such as r stays multiplication, however. Control effort in the real world is then rate of change of e minus r times e. (Lowercase variables are the un-transformed versions.) Since e = x - y, and since x is constant, x becomes zero when differentiated, and drops out of the expression. Control effort is then -dy/dt - er. <Corrected 5 Jun '16.>

I theorize that women calculate -dy/dt, and men calculate er. When they get together, the complete population control effort is exerted, resulting in stability, which the world rewards. However, on average, the men and the women will be pulling in opposite directions exactly 50% of the time, if we model population variation as a sine wave centered on the set point.

A prediction is that women unconsciously react to evidence of increased birth rate or decreased death rate by wanting fewer children. Men react to excess absolute population relative to set point by violence, and to breathing room under the set point by partying.

That negative sign in front of the male contribution was puzzling at first, until I realized that it must derive from the married state itself, and not from the base male response to population error. This could be the origin of statements such as: "Marriage is the exact opposite of the way you think it will be." 

The level of the noise produced so copiously by small children is probably the signal that women unconsciously integrate to estimate birth rate, and the wailing and long faces following a death probably serve the same purpose for estimating death rate, aided by reading the tabloids. [My (married) older brother once showed me the developmental time course of child noise in the air with his hand, and it looked like an EPSP, the response of a neuron to an incoming action potential. The EPSP is the convolution kernel by which a neuron decodes a rate code.] The men have to calculate absolutes, not rates, however. The male proprietary instinct causes them to divvy up the limiting resource for breeding (jobs in our present society) into quanta that can be paired off with people like pairs of beads on adjacent wires of an abacus. Excess people left over at the end of this operation spells trouble. Politicians are right to worry about jobless rates.

#2. The Iatrogenic Conflicts of the Twentieth Century [population, engineering]

The Edwardian era (1901-1911) in small-town Ontario, and La Belle Epoque will soon be over. (From a photo owned by Constance M. Mooney of Ottawa, Canada)

PO     EN     

Red, theory; black, fact.

Medical advances during a turbulent century

In 1911, the anti-syphilis drug salvarsan, invented by Paul Ehrlich, became widely available to the public, at a time when this disease was cutting a wide swath of morbidity and mortality. Three years later, World War I broke out.

In 1937, sulfa drugs, the first effective treatment for tuberculosis, became available to the public. Two years later, World War II broke out.

In 1945, both penicillin and streptomycin became available to the public, followed in short order by the first mass vaccinations, notably against smallpox. In that decade (1945-1955), the Cold War between the United States and the Soviet Union began. That one nearly finished us in 1962, the year of the Cuba Missile Crisis, when a nuclear WW III was narrowly averted.

My conclusions

In the human brain, there is a wholly unconscious controller for population density with a feedback delay of some two to four years, that answers every sudden downtick in the death rate with a brutal, reflexive uptick. Recently, these downticks in the death rate have been due to advances in medicine, hence my title for this post. "Iatrogenic" means roughly "caused by doctors."

Moreover, last year I noted that the headlines were all about ISIS, an unusually disruptive phenomenon of the Muslim world. I then checked to see what the main preoccupation of the headline-writers had been exactly four years previously. This seemed to be the Arab Spring, when many old governments in the Arab world were being thrown off. I concluded that these regimes had somehow been suppressing population growth.

An engineering model

I began to reason thus: if this controller is real, it should be just as analyzable as Watt's steam-engine governor, using standard engineering approaches. If it has a significant feedback delay, then a perturbation sufficiently rich in high-frequency harmonics (i.e., sufficiently sudden) should drive it briefly into a damped oscillation.

Evidence for the engineering model

In support of these conclusions, I present the US Census Bureau statistics on the percent growth rate of the human population for the 20th century, international yearly figures, aggregated to "World," and extended back to 1900 with decade-wise World data from the historical estimates table. At roughly the end of WW II, we see a huge jump in the growth rate followed by a sharp drop bottoming at 1960, followed by another sharp peak at 1962, followed by a leveling off superimposed on a gradual decline, the latter possibly due to increasing absolute numbers. This time series could be construed as showing a damped oscillation. See below.

The historical global population growth rate scaled to population.

11-07-2018

My surmise that the post-1964 decline in the plot would disappear if corrected for changing absolute numbers is confirmed by calculation based on US Census Bureau data. See below. Furthermore, the plot shown below appears to level off at 78 million new people per year, which is probably the upper trigger level for the controller. There is probably no formal lower trigger level, making this controller asymmetric. Oscillation begins well before this level is reached, however, reflecting the presence of a differential control term, as discussed in the next post. The sharp upstroke in growth rate that occurs at 1980 may be due to the eradication of smallpox over the decade 1967-1977. The downturn after 1988 was probably due to the AIDS pandemic. The data are coarse-grained before 1950 and do not show the upstrokes in 1911-1914 and 1937-1939 that I would have predicted from the two world wars.

World population growth rates in persons per year with no scaling. Note the reaction in 1960.

Center: a centrifugal speed governor familiar in 1914. The Steam Museum, Kingston, Canada, 2012.