Modeling collective excitations in cortical tissue

We study a two-dimensional system of spiking neurons with local interactions depending on distance. The interactions between the neurons decrease as the distance between them increases and can be either excitatory or inhibitory. Depending on the mix of excitation and inhibition, this kind of system exhibits a rich repertoire of collective excitations such as traveling waves, expanding rings, and rotating spirals. We present a continuum approximation that allows an analytic treatment of plane waves and circular rings. We calculate the dispersion relation for plane waves and perform a linear stability analysis. Only waves that have a speed of propagation below a certain critical velocity, are stable. For target patterns, we derive an integro-differential equation that describes the evolution of a circular excitation. Its asymptotic behavior is handled exactly. We illustrate the analytic results by parallel-computer simulations of a network of 10(6) neurons. In so doing, we exhibit a novel type of local excitation, a so-called 'paternoster'. Copyright (C) 1998 Elsevier Science B.V. All rights reserved.