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Edited by: Ramon Guevara Erra, Laboratoire Psychologie de la Perception, CNRS, France

Reviewed by: Paul Rapp, Uniformed Services University of the Health Sciences, USA; Paolo Massobrio, University of Genoa, Italy

*Correspondence: James J. Wright

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper furthers our attempts to resolve two major controversies—whether gamma synchrony plays a role in cognition, and whether cortical columns are functionally important. We have previously argued that the configuration of cortical cells that emerges in development is that which maximizes the magnitude of synchronous oscillation and minimizes metabolic cost. Here we analyze the separate effects in development of minimization of axonal lengths, and of early Hebbian learning and selective distribution of resources to growing synapses, by showing in simulations that these effects are partially antagonistic, but their interaction during development produces accurate anatomical and functional properties for both columnar and non-columnar cortex. The resulting embryonic anatomical order can provide a cortex-wide scaffold for postnatal learning that is dimensionally consistent with the representation of moving sensory objects, and, as learning progressively overwrites the embryonic order, further associations also occur in a dimensionally consistent framework. The role ascribed to cortical synchrony does not demand specific frequency, amplitude or phase variation of pulses to mediate “feature linking.” Instead, the concerted interactions of pulse synchrony with short-term synaptic dynamics, and synaptic resource competition can further explain cortical information processing in analogy to Hopfield networks and quantum computation.

Two long-standing controversies impeding the development of brain function theory concern, respectively, the functional significance of cortical synchrony, and the significance of columnar cortical organization, or its lack. It seems possible that the difficulty these problems pose singly might be overcome by finding a mutually consistent resolution, since both involve uncertainties about the way neurons code, store, and transfer information.

Synchrony is a favored mechanism for “binding” and cognition (Eckhorn et al.,

For synchrony to mediate complicated cognitive processes, it is commonly supposed synchronous oscillation involves detailed information exchange between the linked cells, via coded pulses. Yet, simulations and analysis show that synchrony acts only to extract the in-phase component in exchanged traveling waves, rejecting out-of-phase components (Chapman et al.,

It has been recently argued that synchrony may be simply a measure of level of activation (Merker,

However, if the position and connections of each of the cells specifies their spatial and temporal inputs and outputs, rather than any property intrinsic to the cells, co-ordination of their output pulses could represent associations of spatial and temporal information

The problem of synchrony, so formulated, becomes a problem of anatomical order, and synaptic dynamics.

Uncertainties concerning the functional significance of cortical anatomical order, and its embryonic development, have persisted despite dramatic early advance (Hubel,

From early research in the field (Hubel and Wiesel,

Although the surface organization of OP in V1 in some species shows columnar order with hexagonal rotational periodicity (Muir et al.,

Theories for the emergence of cortical connections at millimetric scale have focused on V1's columnar structure and its associated response maps. Following the work of Hubel and Wiesel (

A related puzzle concerns the superficial patch system, composed of relatively long-range, largely excitatory (Hirsch and Gilbert,

In an attempt to account for the above findings, we (Wright and Bourke,

We were able to explain the emergence of macrocolumns and OP response maps in V1 before eye-opening, their singularities, and continuity at column margins (e.g., Bosking et al.,

An important property of our account is its explanation of the variation of apparent OP with stimulus orientation, angle of movement relative to orientation, object velocity, and object length found by Basole et al. (

Our account remained deficient in a number of ways. We did not describe the developmental sequence. We did not show that the twin cell selection pressures upon which our theory depended—ultra-small-world connectivity and maximum synchrony—were mutually compatible. We did not fully explain the periodic “skipping” of patch connections, and outside animals with columnar V1, did not explain the nature of non-columnar cortex, beyond suggesting how the columnar structure would break down beyond certain limits. However, somewhat to our surprise, in line with the findings of occasional order in non-columnar cortex described above, later findings in the somatosensory cortex (S1) (Wright et al.,

We now attempt to correct these deficiencies, and propose a modular functional structure in all cortical areas, whether or not the area is columnar. In the Conclusion we argue that since this developmental account of structure depends on neither atomistic “feature” responses of cells, nor requires frequency coded signal exchanges, these properties are not necessarily essential for cognitive function at maturity. Instead, we generalize the basic assumptions of our hypotheses to fast time-scales and extensive range of metabolic competition, and show that a role for synchronous binding via dynamic synapses with potentially powerful application to cortical computation is implied.

* During early cell division cortical cells fire synchronously and organize themselves into small world configurations (Downes et al.,

* Minimization of metabolic demand for axons implies ultra-small world (Cohen and Havlin,

* To account for survival dependence on synchrony, we assume synchrony maximizes uptake, or one or more unspecified essential resources, and that adjacent synapses cooperate in attracting resources, but compete with each other for its consumption, thus affecting synaptic dispositions. We assume synaptic adaptations, including short term depression and facilitation (Markram and Tsodyks,

* The restricted size of dendritic trees relative to axonal trees enforces network sparsity (Liley and Wright,

The neural dynamics necessary for the generation of synchronous oscillation are captured in the following simplified neural field equations, as we have previously demonstrated (Robinson et al.,

Subscript

_{p}(

_{p}(

_{Σ}(_{q}(

_{p}(

An overall power function pre-synaptic density/distance relationship for excitatory pyramidal cells can be approximated, for simplicity, by two cell populations, each with axonal trees characterized by different exponential density/distance relations.

We term these alpha and beta cells. Both types are excitatory, and it is to be understood that alpha cells are approximate to superficial patch cells, and beta cells to short-axon intracortical cells. Inhibitory cells are considered as strictly local, and are not explicitly considered other than as enabling generation of oscillation. Thus, average axonal density as a function of distance from pyramidal somas is given by

ρ is average probability of synaptic connection between any two excitatory cells.

Subscripts α, β indicate whether the cells are of alpha or beta type.

_{α} and _{β} are fractions of the selected cell population composed of alpha and beta cells.

λ_{α} and λ_{β} are inverse length constants of the axonal trees of alpha and beta cells, respectively, and these vary with cortical area and species. Units of inverse length are arbitrary, so only relative axonal lengths are considered.

^{qr} for alpha cells is represented as

It can be shown that the emergence of synchronous steady-states depends upon local excitatory/inhibitory oscillation and on the exchange of excitatory pulses at longer range (Wright,

This sum can be decomposed into subsets of connections between pairs of alpha cells, pairs of beta cells, and pairs of alpha and beta cells.

where the subscript βα means connection from an alpha cell to a beta cell, etc.

Under the added assumption that values of ε^{qr} are initially random, geometrical solutions for positions of cell bodies creating maximum co-resonance, and thus maximizing each subset of

In the simulations to follow we explore, firstly, the effects on development of ultra-small world organization as if it were the sole influence in determining the emergent cell body arrangements, next the effect of maximization of synchrony as it were the sole influence, then their combined, staged, influences, and finally, based on the simulation results for cell body positions, we consider the late optimization of antenatal synaptic distributions.

Since relative lengths of intracortical axons vary widely with species and cortical area we seek to relate this variation to differences in anatomical order. For given λ_{α} and λ_{β} the values of _{α} and _{β} = 1 − _{α} meeting the power function pre-synaptic density/distance relation, can be found via Equation (4) by minimization of

At a distance

The values of _{β} = 1 − _{α} and _{α} and λ_{β}. It can be seen that _{β} increases to high values for alpha cells with long axons (patch cell surrogates, small values of λ_{α}), and beta cells with short axons (short axon intracortical excitatory cells, high values of λ_{β}). Values of _{β} ≈ λ_{α}, on the diagonal, and tend to small value when both λ_{α} and λ_{β} are large.

_{α} and λ_{β}, to a power function representing the ideal ultra-small world average density vs. distance relation_{β} = 1 − _{α} of cells with axons of inverse length λ_{β} required to obtain best fit.

At distances of cell separation greater than

The surplus of alpha axonal tree density over beta cell density as a function of range, and vice-versa, is given by

Equations (6a,b) can be regarded as selection forces operating over the range of the growing axons, whereby alpha and beta cells respectively influence the probability of survival of all other cells as the network develops. By analogy, Equations (6a,b) can be used in simulation as forces pulling or pushing all other surviving cells into positions in which the forces are in equilibrium. In opposition to these forces of cell selection operating at all ranges, cell differentiation and growth can be conveniently modeled as an expansive force,

In simulation, 8000 points were randomly distributed on a plane with toroidal bounds, and assigned randomly to alpha and beta categories with weighted likelihood in the ratio required for power function approximation at a particular pair of axonal inverse lengths. Each point was intended to represent a small group of cells. At the initiation of the simulation, the forces of selection (Equations 6a,b) were applied until the 2000th time-step, after which they were linearly scaled down until they reached zero at the 4000th time-step. The simulation was then allowed to run until the 20,000th time-step, at which time it had been verified that all cell positions were static.

In the early stages of this process it was as if, at each time-step, cells divided and either survived or died at their prior position, the surviving cells appearing at positions more consistent with an optimum distribution. As cell positions approached the optimal, and the selection forces fell to zero, the growth force (Equation 6c) enlarged the surface area covered by the points, as if cells at positions now adequate for optimum survival were now increasing in numbers.

The smoothing parameter,

Simulations were performed with differing initial random distribution of cells, degree of smoothing of the local repulsive force, scale of the long-range forces, and variation of the relative scale of the selection and growth forces.

Figure _{α} and λ_{β}, displayed in relation to the axes of Figure _{β} and _{β} and

To establish the robustness of this finding the following further comparative simulations were run. The linear scale-down of the forces of selection was varied to between 1000 and 2000, and 4000–6000. The selective forces were doubled in total gain, and halved in total gain. The short-range growth force was varied separately, by halving the force and doubling the range, and by doubling the force and halving the range. None of these variations had any significant effect on the emerging patterns except the last one, in which the simulation concluded without closure together of all the points, as expected, given the choice of parameters for Equation (6c).

In contrast to the requirement to minimize axonal lengths, the maximization of synchrony in a state of equilibrum requires that signal exchange between all cells be symmetrical (whether by direct connections or by intermediate connections in the field)—and assuming a uniform average pulse rate over the field, this requires all reciprocal couplings to be equal.

Connections between alpha cells and alpha cells, and beta cells and beta cells, are symmetric

It is seen that in all outcomes alpha cells surround beta cells in roughly hexagonal or other polygonal forms, with few breaks of their continuity. In contrast to the preceding simulation no blurring of the pattern develops near the diagonal, and the size of the sides of elements of the closed meshworks vary as the value of _{α} and λ_{β} pair.

To check the robustness of these findings, the same simulation duration variations as in the preceding cases were run, and shown to produce no significant influence on the outcome. As an additional check, two further simulations were run in which the total gain of both _{βα} and _{αβ} were either doubled or halved—also without change in outcome.

These results reveal that the two criteria of development we applied (maximum synchrony and ultra-small-world axonal connections) are not wholly compatible.

Since the two selection criteria for development did not generate the same result, we next tried their combination, reasoning that the selections for ultra-small-world axonal lengths must precede the Hebbian stabilization of synchronous equilibrium.

To achieve this effect, we started the simulations from time-step 0 using Equations (6a,b), beginning the linear scale-down of the selective forces at time-step 2000 as usual, but then changing the selection Equations to (7a–d) from 3000 until the usual conclusion. This led to the outcome seen in Figure

It can be seen that the outcome is a mixture of those shown in Figures _{β} and _{β} and

In comparative simulations we found, as expected, that the earlier the (Equations 7a–d) equations were initiated, the more they tended to overwhelm the earlier influence of the Equations (6a,b) equations.

Having shown that realistic cell body positions can arise in the simulations, we next considered consequent effects whereby distribution of synaptic resources and growth would further maximize synchrony. As previously mentioned, consistent with the power-law average presynaptic densities, the preponderance of pre-synapses made by alpha cells linking to other alpha cells must be at distance

This distribution of alpha to alpha connections will maximize the term _{αα} in Equation (5b) and we will show in subsequent sections that the same periodicity of synaptic resources and synapses between alpha and beta cells leads to realistic outcomes maximizing the terms _{αβ} and _{βα} to Equation (5b). Beta to beta synaptic distribution follows without introduction of periodicity. Provisional synaptic density functions, including the prior Hebbian-induced symmetry, are therefore:

We introduced the positive-valued parts of Equations (8a–d) as long-range forces in a third stage in simulation, in analogy to the two prior stages resulting in Figure

There is little difference from Figure

We also ran a comparative simulation set using Equations (8a–d) as the only selection forces. The results did not look anatomically realistic at all. The networks of alpha cells were broken, various geometrically regular and irregular patterns appeared with no orderly change toward the diagonal. In short, the result was pathological, as might have been expected, since deployment of synaptic resources cannot begin until cell body positions are established.

In this section we show diagrammatically how the periodicity of resources and pre-synapses required to maximize _{αα} (Equation 5b) leads concurrently, via further selection of the disposition of synapses, to maximization of _{αβ} and _{βα}, and maximization of _{ββ}. That is, along with the formation of interpatch connections, goes “like to like” connections, and local connections within macrocolumns. Finally we extend the scheme to non-columnar cortex.

Figure

_{α} = 0.25, λ_{β} = 1.50, from Figure 5

Each alpha cell generates a field of potential connections about itself at short range, and since we can take the single cell in the simulation to represent a cluster of cells, these proximal connections are those within a patch. Each alpha cell (or patch) selected in the figure falls in the zone of synaptic connection at distance

It can also be seen that the rings of potential synaptic contact at distance

Figure

The inset shows that if the beta cells themselves form into a re-entrant loop of connected cells, analogous to a three dimensional representation of a Möbius strip (for reasons next given) then the formation of alpha to beta synapses and the reciprocal beta to alpha synapses, in the red, blue, and green colored domains, form a connection system maximizing the synchronous resonance between the beta cells and the surrounding alpha network—i.e., maximizing _{αβ} and _{βα} —since they form a map of nearest-connected-neighbor alphas onto nearest-connected-neighbor betas. This map of connections can be represented as

Arrangement of the beta-to-beta connections in the Möbius-strip-like manner is that which will maximize _{ββ}, since, all connections being sparse, unconnected cells can be closely situated to each other in space, while established connections are between cells packed as closely as possible—yet this mapping is one between cell connection systems with topological identity, as required for maximum co-resonance. Thus, the beta cells can form a tight network, maximizing _{ββ} and concurrently permitting the maximization of _{αβ} and _{βα}.

Figure

The rings of potential connection outward from the alpha cell intersect some other alpha cells in a regular, periodic, manner. The development of synapses on these intersected alpha cells, but not on all nearby alpha cells, accounts for the spatially periodic nature of patch cell connections, while where the rings of potential connection pass through beta cells in areas of similar “OP” in different maps, preferential synapse formation forms “like to like” connections, shown here as black stars. Although not indicated in this figure, the repetition of mirror reversal in adjacent maps in one direction can account for the prolongation of patch connections mainly in one direction, as is seen in experimental data (Bosking et al.,

It may be mentioned in passing that, as the simulations show, the most efficient packing of cells is achieved by hexagonal arrays of alpha cells surrounding beta cells. An exception would occur when the ratio of beta cells to alpha cells is approximately π/4. Then a uniform square network of alpha cells would be possible and the adjacent OP maps would be able to be arranged exactly in mirror reflection, producing a stable map with the properties of OD columns.

The schema for columnar cortex can be readily extended to non-columnar cortex, since sparse synaptic networks architectures can be interwoven with each other. Figure

_{α} = 2.25, λ_{β} = 3.00 from Figure

A small portion of a simulation in Figure

In both the main figure and the inset, alpha cells and their connections have been highlighted in red to show systems of closed connections with distances between nearest neighbors equal to

Thus, as a consequence of the sparsity of neural connections, similar networks can exist in either columnar or non-columnar cortex, the only difference being the differing balance between optimizing ultra-small-world axonal connections vs. Hebbian-symmetric equilibrium. Because of the intertwining, OP would appear to be “pepper and salt” in the non-columnar case, and relatively orderly in the columnar case. At some places in the non-columnar cortex (e.g., near the top right of the image) there are occasional areas approaching the orderliness of columnar cortex.

These simulation outcomes support our prior hypotheses on organization of columnar cortex. They explain the development of cortex in sequential, partially antagonistic but ultimately synergistic steps, and extend the scope to non-columnar cortex. The differences between columnar and non-columnar cortex are shown to be attributable to variable compromise between ultra-small-world axonal length optimization, and maximization of synchrony, depending upon the relative lengths of patch axons and short-axon cells in the cortical area. It appears the entire cortex may be a matrix of overlapping and intertwined elements, in which each element is a folded central topographic map with patch projections to and from the surrounding cortex—an arrangement within which metabolic efficiency and speed of communication reach optimum.

The results also explain how non-columnar cortex can exhibit small areas with properties like columnar cortex, against a predominantly apparently random background pattern (Van Hooser et al.,

With the notable exception of the Möbius configuration of synapses within each element's short-axon cells, all the synaptic connections in the theoretical account have already been demonstrated by direct anatomical means. The most rigorous anatomical test of the model therefore depends on whether or not the closure of synaptic connections into the Möbius configuration, shown in Figure

A simpler test is to relate the ratio of lengths of patch cell axons and cells with short axons to the degree of columnar organization in different cortical areas and species. To the best of the authors' knowledge there are not readily available and sufficiently exhaustive data, cortex and species-wide, to make this test. A complication arises since we have simplified the many classes of excitatory cortical neurons into two classes. Since, realistically, a larger variety of cell types must be considered, the ratio of lengths might be converted to a measure of skewness of axonal length distribution, to enable the test.

Further tests of hypothesis are possible along the lines of our earlier reported experiment (Wright et al.,

If this model is found to be consistent with further anatomical findings, then it will become imperative to undertake extensive modeling of likely processes underlying the assumed metabolic uptake effect of synchrony, and of the biochemistry of the assumed metabolic competitions. We have avoided speculation on these issues for present purposes, although we are aware of the large and incomplete body of relevant data available. However, regardless of the biochemical detail, the time-constants and range of the metabolic exchanges are of importance to the range of application of the theory. If the time-constants are slow, and the range of metabolic exchanges short, then the theory's application would be confined to slower, developmental, processes. However, if these processes are fast, and occur over long ranges, then there are important implications for the role of synchrony in information processing generally, as argued further below. We consider first the implications for later cortical organization on the time-scale of long-term learning.

The late antenatal configuration of each of our hypothetical elements provides a ground state, with maximum potential information storage capacity, which is both an initial organizational framework for learning, and a default state for forgetting. As postnatal learning begins, the antenatal organization would be progressively overwritten. The organization of the connection overwriting can be inferred from our explanation of object representation and feature tuning in V1, as previously cited in the Introduction (Wright and Bourke, _{P}, is transmitted with axonal delays to each cortical column surrounding an OP singularity as a mapped transformed pattern, _{p}
_{x} is the velocity of the stimulus pattern moving over the visual cortex along axis _{x}, _{y}, _{x}_{y} are dominant spatial frequencies of the visual object's projection and mapped response respectively, ω is the temporal frequency of response, ν is the speed of electrocortical wave transmission, and

Thus, the four dimensional world of moving objects becomes represented as stored learning within the four dimensional space defined by the complex vector pairs,

There are wider implications still, if metabolic competition between synapses is fast, on the time-scale of fast synaptic dynamics, and operates by rapid diffusion over ranges large compared to a synapse. As mentioned in the Introduction, there is no need for the exchange of frequency-coded information in pulses, outside the time frame set by dynamic synapses. The assumption was made in Equation (5a) that average pulse activity is uniform and synaptic connectivity symmetrical. However, synchronous equilibria are permitted more generally, when for all pairs of cells _{ij} measures the degree to which the _{i}, _{ij}} are in steady-states of synaptic adaptation and pulse exchange. That is, the set of all possible synchronous states is the set of ways synaptic resources and cell pulses can be aligned to steady state in concert. Equation (11) has analogous form to the energy function

Since capacity for computation does not imply the learning of behaviorally useful computations, we must further suppose supervision of the cortex by limbic system interventions on activation and neuromodulation, as initially proposed in the “Triune Brain” concept (Maclean, _{ij}} are consolidated if they facilitate goal-directed limbic activity.

Synchronous states could thus mediate the storage of information and its release when suitably activated, and lead to evolution of individual, novel—essentially “creative”—behavioral adaptations.

JW and PB are jointly responsible for this work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was supported by use of the NeCTAR Research Cloud and Intersect Australia Limited. The NeCTAR Research Cloud is a collaborative Australian research platform supported by the National Collaborative Research Infrastructure Strategy. The first author wishes to acknowledge the essential contribution of Adrienne Edith Wright.