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 sy stem of integro-differential equations used to describe the activity of syn aptically 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 exis tence of both stable and unstable standing pulses [S. Amari, Biol. Cybern., 27 (1977), pp. 77-87]. These results have formed the basis for several rec ent hypotheses regarding the generation of sustained activity patterns in p refrontal cortex and other brain regions. Implicit in the lateral inhibitio n arrangement, however, is the assumption that the dynamics of inhibition a re instantaneous. Here we present two arguments demonstrating the loss of s tability of standing pulse solutions through a Hopf bifurcation when more r ealistic inhibitory dynamics are considered. The rst argument parallels Ama ri's formal presentation, while the second provides a rigorous analysis of the linearized system. Additionally, we extend the existence of solutions t o include a broader range of conditions by constructing a standing pulse us ing singular perturbation analysis.