Traveling pulses in a stochastic neural field model of direction selectivity

Paul C Bressloff,Jeremy Wilkerson

doi:10.3389/fncom.2012.00090

Abstract

We analyze the effects of extrinsic noise on traveling pulses in a neural field model of direction selectivity. The model consists of a one-dimensional scalar neural field with an asymmetric weight distribution consisting of an offset Mexican hat function. We first show how, in the absence of any noise, the system supports spontaneously propagating traveling pulses that can lock to externally moving stimuli. Using a separation of time-scales and perturbation methods previously developed for stochastic reaction-diffusion equations, we then show how extrinsic noise in the activity variables leads to a diffusive-like displacement (wandering) of the wave from its uniformly translating position at long time-scales, and fluctuations in the wave profile around its instantaneous position at short time-scales. In the case of freely propagating pulses, the wandering is characterized by pure Brownian motion, whereas in the case of stimulus-locked pulses, it is given by an Ornstein–Uhlenbeck process. This establishes that stimulus-locked pulses are more robust to noise.

Highlights

Continuum neural field models represent the large-scale dynamics of spatially structured networks of neurons in terms of nonlinear integro-differential equations, whose associated integral kernels represent the spatial distribution of neuronal synaptic connections (Wilson and Cowan, 1972, 1973; Amari, 1977)
As in the case of non-linear partial differential equation (PDE) models of diffusively coupled excitable systems (Keener, 1981; Kuramoto, 1984), non-local neural fields can exhibit a diverse range of spatiotemporal dynamics, including solitary traveling fronts and pulses, stationary pulses, and spatially localized oscillations, spiral waves, and Turing-like patterns
Neural fields have been used to model a wide range of neurobiological phenomena, including wave propagation in cortical slices (Pinto and Ermentrout, 2001; Richardson et al, 2005) and in vivo (Huang et al, 2004), geometric visual hallucinations (Ermentrout and Cowan, 1979; Bressloff et al, 2001), EEG rhythms (Nunez, 1995; Robinson et al, 2001; Liley et al, 2002; Steyn-Ross et al, 2003), orientation tuning in primary visual cortex (V1) (Ben-Yishai et al, 1995; Somers et al, 1995), short term working memory (Camperi and Wang, 1998; Laing and Chow, 2002), control of head direction (Zhang, 1996), direction selectivity (Xie and Giese, 2002), motion perception (Giese, 1999), and binocular rivalry waves (Bressloff and Webber, 2012a)

Summary

Introduction

Continuum neural field models represent the large-scale dynamics of spatially structured networks of neurons in terms of nonlinear integro-differential equations, whose associated integral kernels represent the spatial distribution of neuronal synaptic connections (Wilson and Cowan, 1972, 1973; Amari, 1977). We have recently shown how perturbation methods for studying fluctuating fronts in reaction-diffusion PDEs (Schimansky-Geier et al, 1983; de Pasquale et al, 1992; Armero et al, 1998; Sagues et al, 2007) can be extended to the problem of front propagation in stochastic neural fields (Bressloff and Webber, 2012b), and have used this to investigate the effects of noise on binocular rivalry waves (Webber and Bressloff, submitted). Such methods exploit a separation of time-scales in which there is a diffusive-like displacement (wandering) of the front from its uniformly translating position at long time-scales, and fluctuations in the front profile around its instantaneous position at short time-scales

Methods

Results

Conclusion