Spiral waves in nonlocal equations

We present a numerical study of rotating spiral waves in a partial integro- differential equation defined on a circular domain. This type of equation has been previously studied as a model for large scale pattern formation in the cortex and involves spatially nonlocal interactions through a convolution. The main results involve numerical continuation of spiral waves that are stationary in a rotating reference frame as various parameters are varied. We. find that parameters controlling the strength of the nonlinear drive, the strength of local inhibitory feedback, and the steepness and threshold of the nonlinearity must all lie within particular intervals for stable spiral waves to exist. Beyond the ends of these intervals, either the whole domain becomes active or the whole domain becomes quiescent. An unexpected result is that the boundaries seem to play a much more significant role in determining stability and rotation speed of spirals, as compared with reaction- diffusion systems having only local interactions.