Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses

We consider the existence and stability of standing pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain. Assuming an arrangement of synaptic connections described by lateral inhibition, previous formal arguments have demonstrated the existence of both stable and unstable standing pulses [S. Amari, Biol. Cybern., 27 (1977), pp. 77-87]. These results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions. Implicit in the lateral inhibition arrangement, however, is the assumption that the dynamics of inhibition are instantaneous. Here we present two arguments demonstrating the loss of stability of standing pulse solutions through a Hopf bifurcation when more realistic inhibitory dynamics are considered. The rst argument parallels Amari's formal presentation, while the second provides a rigorous analysis of the linearized system. Additionally, we extend the existence of solutions to include a broader range of conditions by constructing a standing pulse using singular perturbation analysis.