Complex dynamics induced by multiple timescales in a Wilson-Cowan model with homeostatic plasticity

Homeostatic adjustment of synaptic weights is important to prevent runaway excitation in neural populations, and it occurs on a slower timescale than the timescale of neural activity. Homeostatic plasticity in neural mass models can induce complex dynamical behaviors, including mixed-mode oscillations (MMOs) and chaos. In this paper, we investigate dynamical mechanisms by studying a single-node Wilson-Cowan model with homeostatic plasticity, which has three timescales associated with the activities of the excitatory/inhibitory (E/I) populations and the homeostatic connection weight, WI, from the I to the E population. We study how the relative timescale separations induce various dynamical behaviors. Analysis in two-timescale settings unveils two typical mechanisms underlying transitions. (1) Considering E as fast and the other variables as slow, canard-induced MMOs due to the presence of a folded node are observed. Bifurcations of folded singularities, including type II folded saddle-node and degenerate folded node, explain the dynamical transitions. (2) Considering WI as slow and E/I as fast, period-doubling cascades and canard explosion, induced by a degenerate folded point, explain the dynamical transitions. Extending to a three-timescale framework introduces interactions between singularities defined in the two-timescale settings and enables a more detailed description of the dynamics. Folded singularities in the three-timescale setting determine the structure of singular orbits. The degenerate folded point in the 2F/1S case determines the transition between MMOs and relaxation oscillation. This paper provides a comprehensive understanding of the role of three timescales in this single-node system and highlights how the connection weights between populations induce complex dynamical behaviors.