A dynamical analysis of collective behavior in a multi-population network with infinite theta neurons

Abstract

Interactions among many populations are common in both biological and engineering networks. This paper investigates the dynamics of a multi-population network composed of identical theta neurons, where each population has an infinite number of neurons. These neural oscillators are globally interconnected by pulse-like synapses whose sensitivity is adjustable. In this paper, the analytical technique developed by Ott–Antonsen is employed to streamline the dynamics of a large-scale network into a small set of variables and parameters, thereby representing the network’s overall state. The investigation indicates that our network can display either symmetric or asymmetric states. Meanwhile, an analysis of bifurcations with codimension-1 and -2 is conducted to examine the origins of the network’s multistability, oscillations, and hysteresis. Particular attention is paid to the influence of various network components, such as coupling patterns and population size. The analysis results reveal a strong correlation between multistability and the presence of a supercritical Hopf bifurcation with an attractive manifold. The evaluation procedure demonstrates the important role of balanced coupling in regulating the overall macroscopic dynamics of the network. Additionally, extensive testing suggests that networks with instantaneous synapses can exhibit asymmetric states even with homogeneous inter-population coupling, and this type of synapse removes Hopf bifurcations in two-population bifurcation scenarios. In three-population setups, there are subcritical Hopf bifurcations with an attractive manifold, leading to oscillations within specific parameter ranges. Our study provides new insights into the collective dynamics of neuronal nuclei in similar basal ganglia structures.