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Conceived and designed the experiments: BE DJS. Performed the experiments: PWL. Analyzed the data: BE DJS PWL. Wrote the paper: BE DJS.

The authors have declared that no competing interests exist.

Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In the rodent somatosensory cortex, groups of neurons called “barrels” correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. Here we demonstrate that similar patterns can be simulated by means of local growth-promoting and growth-retarding interactions within the circular domains of single barrels. The model correctly predicts that larger barrels contain more spatially complex subbarrel patterns, suggesting that the development of barrels and of the patterns within them may be understood in terms of some relatively simple dynamic processes. We also simulate the full nonlinear equations to demonstrate the predictive value of our linear analysis. Finally, we show that the pattern formation is robust with respect to the geometry of the barrel by simulating patterns on a realistically shaped barrel domain. This work shows how simple pattern forming mechanisms can explain neural wiring both qualitatively and quantitatively even in complex and irregular domains.

Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In rodent somatosensory cortex, groups of neurons called “barrels” correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. We suggest that these so-called subbarrel patterns arise spontaneously during development through a pattern-forming instability. We use a simple chemotaxis and branching model to explain the patterns and their dependence on the size of the barrel.

Mechanisms underlying the attainment of the central nervous system's highly structured organization are varied and numerous. The identification of developmentally regulated molecular signals are critically important for understanding neural function as well as fundamental processes of disease and repair. The complexity of the details notwithstanding, it is likely that many aspects of neural development can be understood in terms of relatively simple operational principles that govern the specific interactions among neurons and/or other elements, e.g., glia. Spatial patterns such as coat markings in animal skin and colors and textures in seashells are ubiquitous in biology, and theoretical studies have been able to account for a remarkable variety of them using models based on dynamical interactions among surprisingly small numbers of factors

The rodent somatosensory cortex contains striking spatial patterns of neuronal cell bodies and processes wherein discrete anatomical structures in layer IV called “barrels” correspond functionally with the representation of well-defined body surfaces

Cytochrome oxidase staining of individual whisker barrels reveals that there are patterns in the innervation of thalamic axons and that these patterns belong to only a few different classes.

See also

There are many plausible models for pattern formation during neural development. By way of illustration and to show the underlying concepts, we will use a variant of the Keller-Segel

The main idea of spontaneous pattern formation is to show that spatially homogeneous activity in a model is unstable to perturbations that have a characteristic wave-length but stable to other perturbations. Thus, those in the unstable regime will grow and produce a

A) Spatial interactions of the “surround inhibition”- or “Mexican hat” type (Ai) and its Fourier transform (Aii); note peaks at nonzero values of k. B) Interactions destabilize the uniform state. (Bi) small inhomogeneities (solid arrow) are amplified (Bii) by local positive feedback (solid arrow) while neighboring regions are depressed (dashed arrows). In Biii, because of the depression, neighboring regions are amplified (solid arrows) and their outer neighbors are in turn depressed (dashed arrows). C) Final patterned state. D) The complexity of the pattern is determined by the size of the domain. Di) there is a minimal length scale for creating a pattern; Dii) as domain size increases, the pattern expands to fill it; Diii) if the domain is large enough, a repeat of the pattern is inserted.

We assume that in equation (1),

So far, the description of instability has been general in that we have not made use of the shape or size of the domain (the barrels). In this section, we state our main results which describe the patterns one expects to form spontaneously as we decrease the diffusivity of the chemoattractant. Recall from the previous section, that the spatial form of the patterns is determined by

For a disk-shaped domain, it is convenient to write the eigenvalue problem in polar coordinates,

A) Plots of the first 4 (denoted by

The patterns in

Dark regions correspond to highest density of thalamocortical axons. White areas correspond to density less than background. Blue labels are the named patterns seen in the data. All barrels are drawn at the same diameter. Numbers in parentheses,

Throughout this discussion, we used no-flux boundary conditions to obtain the patterns. A similar sequence occurs with fixed boundary conditions. In fact, it follows from the general theory of second order linear differential equations

The above analysis suggests the types of patterns that are possible for the full non-linear system for parameters near the loss of stability of the constant state. In this section, we numerically solve equations (1) and (2) on a fixed radius disk and vary the values of chemotaxis and diffusion.

For mathematical simplicity, we have treated the barrels as disks, but real barrels have less regular shapes. A natural question is whether the qualitative shapes of the patterns are robust to irregularities in the actual barrel domains. In order to examine this, we chose a specific barrel with a very clear mercedes pattern (see

(A) Image of the actual barrel showing the piecewise linear approximation of the boundary (red). (B) Two eigenfunctions with nearby eigenvalues. (C) Superposition of two patterns in (B).

Here we demonstrate that appropriate and complex anatomical patterns can be understood in the context of general pattern forming mechanisms in a circular domain. Emergence of spatial patterning is common in development, and a number of such processes have been modeled as dynamical systems. In a classic paper on morphogenesis, Turing

There are many possible mechanisms for pattern formation. The present model is based on chemotaxis and diffusion, though other processes, employing chemorepulsion and/or additionally involving activity-dependent competition for resources, are also plausible. Here we use growing thalamocortical axons as the fundamental, interacting elements, as virtually all empirical studies support a central role of these afferent fibers in establishing the basic pattern of barrels within the face area of the primary somatosensory cortex (e.g.

The mechanism(s) by which thalamocortical axons interact with each other and with the cortical neurons themselves are largely unknown, and our model makes no explicit assumption – or prediction – regarding the detailed processes underlying subbarrel formation. Indeed, our model employs only two key variables – thalamocortical axon density and chemoattractant concentration, though numerous morphogenetic factors involving the growth and elaboration of axons, dendrites and synapses are almost certainly involved in establishing the organization of cell bodies and neuropil within each barrel. A number of molecules thought to be important for barrel formation are themselves regulated by neuronal activity, though at present the role of activity in the formation of barrels or subbarrels remains unclear

One question we have not addressed in this paper is why there are sub-barrel structures at all. The

For the biological portion of this study we reanalyzed 113 rat barrels whose subbarrel patterns were described previously (Land and Erickson, 2005). These specimens were derived from layer 4 of the somatosensory cortex in young rats ranging in age from postnatal day 10 (P-10) through P-16. Cortices were prepared as tangential, in vitro slices. Slices were prepared by standard methods. Live slices subsequently were fixed in 4% paraformaldehyde, sectioned at 80 µm and stained histochemically for cytochrome oxidase (CO) (Land and Simons, 1985). Each of the barrels chosen for the current analyses contained one of four basic subbarrel arrangements that are recognized based upon the pattern of CO-dark and CO-light zones. We acquired images of CO-stained barrels with a SPOT RT digital camera (Diagnostic Instruments, Sterling Heights, MI) using a Kodak 47B Wratten filter and imported them into Photoshop (Adobe Systems Incorporated, San Jose, CA). To further enhance the contrast between CO-dark and CO-light regions, the original RGB color images first were converted to grayscale. We then applied the Equalize command, which finds the brightest and darkest values in the composite image and remaps them so that the brightest value is depicted as white and the darkest value as black. Resulting equalized images were analyzed using Scion Image (Scion Corporation, Frederick, MD). We used the Freehand Selection tool to outline the perimeter of CO-stained barrels and then exported the area data into a spreadsheet (Excel, Microsoft Corporation, Redmond, WA) We organized the data into groups of barrels that exhibited a particular subbarrel pattern (i.e., cb, me, etc.) and determined the mean and standard deviation of barrel areas associated with each pattern.

The nonlinear partial differential equations models were solved on a

Matlab code to obtain eigenfunctions for a realistic barrel shape.

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This posthumously published paper is dedicated to the late Dr. Peter Land. We thank Susan Erickson, Justin Crowley and James McCasland for their insights and suggestions.