Activity Patterns in a Two-Timescale Neural Ring

I will present an analysis of activity patterns in a neuronal network that consists of three mutually inhibitory neurons with voltage-sensitive piecewise smooth coupling and directionally asymmetric weights. This network model is motivated by the respiratory neuronal network in the mammalian brainstem and is able to exhibit various activity patterns including bistability of relaxation oscillation solutions in which activation propagates around the ring in opposite directions. 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 provides the set of fast subsystem fixed points and transition surfaces parametrized by slow variables, but due to the voltage-sensitive nature of the coupling it fails to describe the mechanism underlying the sudden “turn-around” during fast jumps. I will discuss analysis of a linear, reduced form of the model system that preserves the solution structure, which reveals novel adaptive escape and adaptive release phase transition mechanisms. To determine where the fast jumps actually go, we exploit the piecewise smooth nature of the coupling to 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.  This is joint work with Choongseok Park from NC A&T University.