SYNCHRONIZATION IN MODEL NETWORKS OF CLASS I NEURONS

Abstract. We study a modification of the Hoppensteadt-Izhikevich canonical model for networks of class I neurons, in which the ‘pulse’ emitted by a neuron is smooth rather than a δ-function. We prove two types of results about synchronization and desynchronization of such networks, the first type pertaining to ‘pulse ’ functions which are symmetric, and the other type in the regime in which each neuron is connected to many other neurons. 1. introduction In the work of Hoppensteadt and Izhikevich [3, 4] a model for networks of class I neurons was derived, which is canonical in the sense that a huge class of detailed and biologically plausible models of weakly-connected neurons near a ‘saddle-node on limit-cycle ’ bifurcation can be reduced to it by a piecewise-continuous change of variables. We refer to [4] for the precise conditions under which this reduction is valid, and to [3], chapter 8, for a details of the derivation. The Hoppensteadt-Izhikevich canonical model is described by the equations (1) θ′i = h(ri; θi) + n∑ j=1 w(sij; θi)δ(θj − pi), 1 ≤ i ≤ n, where θi is the phase of the i-th neuron, and in which h(r;α) = (1 − cos(α)) + (1 + cos(α))r describes the internal dynamics of the i-th neuron, which is excitable if r < 0 and oscillatory if r> 0, w(s;α), which describes the response of neuron i to a pulse from neuron j is the 2pi-periodic extension of the function defined for α ∈ [−pi, pi] by (2) w(s;α) = 2 arctan tan