Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

Paul C. Bressloff, Zachary P. Kilpatrick
2011 SIAM Journal on Applied Mathematics  
We analyze radially symmetric bumps in a two-dimensional piecewise-smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases
more » ... in activity. We show that in the case of a Mexican hat weight distribution, sufficiently strong synaptic depression can destabilize a stationary bump solution that would be stable in the absence of depression. Numerically it is found that the resulting instability leads to the formation of a traveling spot. The local stability of a bump is determined by solutions to a system of pseudolinear equations that take into account the sign of perturbations around the circular bump boundary.
doi:10.1137/100799423 fatcat:xbheqmc7sfcqtfvsj3hrrptryi