Traveling fronts and wave propagation failure in an inhomogeneous neural network

Abstract

We use averaging and homogenization theory to study the propagation of traveling wavefronts in an inhomogeneous excitable neural medium. Motivated by the functional architecture of primary visual cortex, we model the inhomogeneity as a periodic modulation in the long-range neuronal connections. We derive an expression for the effective wavespeed and show that propagation failure can occur if the speed is too slow or the degree of inhomogeneity is too large. We find that there are major qualitative differences in the wavespeed for different choices of the homogenized weight distribution.

Introduction

A common starting point for analyzing the large-scale dynamics of cortex is to treat nerve tissue as a continuous two-dimensional medium and to describe the dynamics in terms of Wilson–Cowan equations of the form [1], [2], [3]

$$\text{τ}{}_0\text{∂u(x,t)}\text{∂t}\text{=−u(x,t)+}\text{∫}\text{−∞}\text{∞}\text{w}\text{̂}\text{(x,x′)f(u(x′,t))}\text{d}\text{x′+I(x,t).}$$

The scalar field u(x,t) represents the local activity of a population of excitatory neurons at cortical position $\text{x∈}\text{R}{}^2$ at time t, τ₀ is a time constant (associated with membrane leakage currents or synaptic currents), I(x,t) is an external input, and the positive distribution $\text{w}\text{̂}\text{(x,x′)}$ is the strength of connections from neurons at x′ to neurons at x. (Populations of inhibitory neurons can be incorporated into Eq. (1.1) by taking u to be a vector-valued field and replacing w by a matrix of distributions with both positive and negative components.) The nonlinearity f is often assumed to be a smooth monotonically increasing function of the form

$$\text{f(u)=}\text{1}\text{1+}\text{e}{}^{\mathrm{-\beta (u-\alpha )}}\text{,}$$

where β is the gain parameter and α the threshold. This becomes a step function in the high-gain limit β→∞, i.e., f(u)=Θ(u−α), where Θ(x)=1, if x>0 and is zero otherwise.

A major simplification concerning the structure of cortex is to assume that it is homogeneous and isotropic, with the strength of connections depending on the (Euclidean) distance between presynaptic and postsynaptic neurons. More specifically, $\text{w}\text{̂}\text{(x,x′)=w(∣x−x′∣)}$ with w(s) a decreasing function of separation s. Under this approximation the weight distribution is invariant with respect to the Euclidean group of rigid body motions in the plane. Euclidean symmetry plays a key role in determining the types of solutions that can be generated spontaneously in such networks [4], [5].

Another important property of homogeneous networks is that they support traveling waves in the form of propagating fronts separating regions of high and low activity [6], [7]. This is of interest since a number of experimental studies have observed waves of excitation propagating in cortical slices (typically from rat somatosensory cortex) when stimulated appropriately [8], [9], [10]. The propagation velocity of such waves is of order $\text{0.06}\text{ms}{}^{\mathrm{-1}}$, which is much slower than the typical speed of $\text{0.5}\text{ms}{}^{\mathrm{-1}}$ found for action potential propagation along axons. It is possible that the conditions for wave propagation in the intact cortex are different since much of the long-range cortical connectivity is lost during slice preparation. On the other hand, traveling waves of electrical activity have also been observed in vivo in the somatosensory cortex of behaving rats [11], turtle and mollusk olfactory bulbs [12], [13], turtle cortex [14] and visuomotor cortices in cat [15]. Often these traveling waves are found to occur during periods without sensory stimulation with the subsequent presentation of a stimulus inducing a switch to synchronous oscillatory behavior [16]. Hence, determining the conditions under which cortical wave propagation occurs could be important for understanding the processing of sensory stimuli as well as more pathological forms of behavior such as epileptic seizures and migraines.

The dependence of wave velocity on the underlying pattern of synaptic connectivity has been analyzed in the case of homogeneous networks [7], [17], [18]. However, certain care must be taken in interpreting these results, since traveling wave solutions are sensitive to the degree of homogeneity in the connectivity pattern, particularly in the case of slowly moving waves. This is a consequence of the fact that traveling wavefronts are not structurally stable solutions, i.e., they correspond to heteroclinic orbits of an equivalent dynamical system.¹ Keener [19], [20] has recently explored this issue within the context of reaction–diffusion equations modeling excitable chemical media. Using a careful application of averaging or homogenization theory, he has shown that if the level of non-uniformity is sufficiently large or the wave velocity is sufficiently small then propagation failure can occur.

In this paper, we extend the analysis of Keener [19], [20] to the case of wave propagation in inhomogeneous neural networks. Our work is motivated by the fact that although the assumption of cortical homogeneity is reasonable at the macroscopic level, there are additional structures at the microscopic level that could modify this simple picture (see Section 2.) For example, in the case of the primary visual cortex (V1) there are a number of functional maps superimposed on the underlying retinotopic representation of the visual field [21], [22]. These maps, which correspond to additional cortical labels such as ocular dominance and orientation preference, introduce an approximate periodic tiling of the cortical plane whose wavelength is comparable to a single hyperecolumn spacing of around 1 mm (in cats and monkeys). It is likely that such functional periodicity is associated with a corresponding periodicity in the anatomy. Within the framework of large-scale cortical models (1.1), the periodic microstructure of the cortex can be incorporated by taking a weight distribution²

$$\text{w}\text{̂}\text{(x,x′)=w(∣x−x′∣)h}\text{∣x−x′∣}\text{ε}\text{k}\text{x′}\text{ε}\text{,}$$

where h,k are 2π-periodic functions and ε determines the microscopic length-scale. If k=1, then the cortex still acts like a homogeneous medium with the effective connection strength a periodically modulated function of spatial separation. On the other hand, if there are inhomogeneities arising from the microscopic level then these will result in a periodically heterogeneous medium for which k≠1. Absorbing the factor h into w and substituting into Eq. (1.1) gives the inhomogeneous evolution equation (for zero external inputs)

$$\text{τ}{}_0\text{∂u(x,t)}\text{∂t}\text{=−u(x,t)+}\text{∫}\text{−∞}\text{∞}\text{w(∣x−x′∣)k}\text{x′}\text{ε}\text{f(u(x′,t))}\text{d}\text{x′.}$$

The main goal of this paper is to determine the behavior of propagating wavefronts in a one-dimensional version of Eq. (1.4) by extending the averaging method of Keener [19], [20] to the case of integro-differential equations. This generates an analytical expression for the wavespeed and waveform shape, from which it can be deduced that propagation failure occurs when the medium is sufficiently inhomogeneous (see Section 3). An explicit calculation of the wavespeed is then carried out in the case of piecewise linear dynamics (see Section 4). We show that the ε-dependence of solutions is sensitive to the choice of weight distribution w. In particular, there are major qualitative differences between exponential and Gaussian weight distributions due to the fact that in the latter case there are transcendentally small (non-perturbative) ε-dependent contributions to the wavespeed. Interestingly, such terms only occur for smooth nonlinearities in the models considered by Keener [19], [20].