Analysis of a two-timescale neuronal ring model with voltage-dependent,piecewise smooth inhibitory coupling

We present an analysis of activity patterns in a neuronal network that consists of three mutually inhibitory neurons with voltage-sensitive piecewise smooth coupling. One of the observed propagating solutions appears to be contrary to the network architecture and is characterized by a sudden “turn-around” of trajectories during fast transitions between quasi-stable states. Standard fast-slow analysis fails to describe the mechanism underlying this activity pattern due to the voltage-sensitive nature of the coupling. We exploit the piecewise smooth nature of the coupling and consider a sequence of fast subsystems defined in a piecewise way.   Our analysis shows that there are three possible scenarios during fast jumps, which may depend on both the fast dynamics and the slow dynamics. First, the fast dynamics may succeed to equilibrate at (or near) a critical manifold, after which the slow dynamics relaxes to its own fixed point. Second, while the fast dynamics tries to equilibrate to a critical manifold, the slow dynamics may push the fast system through a bifurcation, which forces a second fast jump to a new critical manifold. Third, the presumed critical manifold may be lost prior to fast subsystem equilibration, through effects that may relate to either the slow dynamics or the fast dynamics, in which case the fast dynamics is forced to approach a new critical manifold directly. In the second and third cases, we observe the sudden “turn-around” during fast jumps.