Abstract
This paper challenges and extends earlier seminal work. We consider the problem of describing mathematically the spontaneous activity of V1 by combining several important experimental observations including (1) the organization of the visual cortex into a spatially periodic network of hypercolumns structured around pinwheels, (2) the difference between short-range and long-range intracortical connections, the first ones being rather isotropic and producing naturally doubly periodic patterns by Turing mechanisms, the second one being patchy, and (3) the fact that the Turing patterns spontaneously produced by the short-range connections and the network of pinwheels have similar periods. By analyzing the PO maps, we are able to classify all possible singular points (the pinwheels) as having symmetries described by a small subset of the wallpaper groups. We then propose a description of the spontaneous activity of V1 using a classical voltage-based neural field model that features isotropic short-range connectivities modulated by non-isotropic long-range connectivities. A key observation is that, with only short-range connections and because the problem has full translational invariance in this case, a spontaneous doubly periodic pattern generates a 2-torus in a suitable functional space which persists as a flow-invariant manifold under small perturbations, for example when turning on the long-range connections. Through a complete analysis of the symmetries of the resulting neural field equation and motivated by a numerical investigation of the bifurcations of their solutions, we conclude that the branches of solutions which are stable over an extended range of parameters are those that correspond to patterns with an hexagonal (or nearly hexagonal) symmetry. The question of which patterns persist when turning on the long-range connections is answered by (1) analyzing the remaining symmetries on the perturbed torus and (2) combining this information with the Poincaré–Hopf theorem. We have developed a numerical implementation of the theory that has allowed us to produce the predicted patterns of activities, the planforms. In particular we generalize the contoured and non-contoured planforms predicted by previous authors.
Highlights
The primary area (V1) of the visual cortex is one of the first locations targeted by connections from the thalamus which relays inputs from the retina
We focus on the processing of the local orientation of visual stimuli which is reflected by the ability of some neurons to fire only when a drifting grating of specific orientation is presented in their receptive field
Direct numerical simulations of the dynamics on the perturbed torus allow us to fix the actual dynamics which is induced on the invariant tori when switching on the long-range connections
Summary
The primary area (V1) of the visual cortex is one of the first locations targeted by connections from the thalamus which relays (and processes) inputs from the retina. The problem of the effect of long-range connections with discrete translation invariance on the Turing patterns which bifurcate when these connections are inactive has been addressed by Bressloff [12] (see Baker and Cowan [13]) in the following context These authors assumed that the periods of the patterns and of the lattice of pinwheels were not correlated. Our primary motivation here is to understand how the long-range connections with discrete lattice symmetry affect the dynamics of the network This is made possible by using the fact that these connections act as a perturbation [19] of the dynamics generated by the local connections.
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