| Abstrakt: |
We present an analysis of activity patterns in a neuronal network that consists of three mutually inhibitory neurons with voltage-sensitive piecewise smooth coupling. 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. By considering a linear, reduced form of the model system that preserves the solution structure, we are able to perform a thorough analysis of the oscillations, 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. First, the fast dynamics may succeed to equilibrate at (or near) a critical manifold branch, after which the slow dynamics relaxes to its own fixed point, pulling the slaved fast variables along the critical manifold. 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 component, after which the slow relaxation follows. Third, the critical manifold component expected to be attracting may be lost prior to fast subsystem equilibration or may be inaccessible due to separatrix geometry, 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. [ABSTRACT FROM AUTHOR] |
