Switching Activity in an Ensemble of Excitable Neurons

Abstract

In [1], a stable heteroclinic cycle was proposed as a mathematical image of switching activity. Due to the stability of the heteroclinic cycle, the sequential activity of the elements of such a network is not limited in time. In this paper, it is proposed to use an unstable heteroclinic cycle as a mathematical image of switching activity. We propose two dynamical systems based on the generalized Lotka – Volterra model of three excitable elements interacting through excitatory couplings. It is shown that in the space of coupling parameters there is a region such that, when coupling parameters in this region are chosen, the phase space of systems contains unstable heteroclinic cycles containing three or six saddles and heteroclinic trajectories connecting them. Depending on the initial conditions, the phase trajectory will sequentially visit the neighborhood of saddle equilibria (possibly more than once). The described behavior is proposed to be used to simulate time-limited switching activity in neural ensembles. Different transients are determined by different initial conditions. The passage of the phase point of the system near the saddle equilibria included in the heteroclinic cycle is proposed to be interpreted as activation of the corresponding element.