Discipline: Mathematics & Statistics
Subcategory: STEM Research
Choongseok Park - North Carolina A&T State University
Co-Author(s): Jonathan Rubin, University of Pittsburgh, Pittsburgh, PA
In this study, we present an analysis of activity patterns in a neuronal network, which consists of three mutually inhibitory cells with voltage- sensitive piecewise smooth coupling. This network model was motivated by recent development of respiratory neuronal network model in the mammalian brainstem and is able to exhibit various activity patterns including the bistable relaxation oscillation solutions with different order of activations. A reversed order of activations, which is observed in one of the bistable solutions, is contrary to the network architecture and characterized by a sudden ‘turn-around’ during transitions (fast jumps) between states. Standard fast-slow analysis provides the set of fixed points of fast subsystem and transition surfaces parametrized by slow variables but due to the voltage-sensitive nature of coupling it fails to describe the mechanism underlying the sudden ‘turn-around’ during fast jumps. To determine where fast jumps actually go, we consider the structure of fast subsystem which is modulated by slow dynamics as well as fast dynamics. Piecewise smoothness of coupling enables us to consider a sequence of fast subsystems in piecewise way. Our analysis shows that there are three possible scenarios during fast jumps, which incorporate the fast dynamics and slow dynamics. First, the fast dynamics succeeds to equilibrate at (or near) a presumed fixed point manifold and then the slow dynamics relaxes to its own fixed point, pulling the slaved, fast variables along the fixed point manifold. Second, while the fast dynamics tries to equilibrate at a fixed point manifold, the slow dynamics pushes the fast system through a bifurcation, which forces a second fast jump to a new fixed point manifold and then the slow relaxation follows. Third, the presumed fixed point manifold is either already lost by the slow dynamics or blocked by the structure of the fast subsystem, thus the fast dynamics is forced to approach a new fixed point manifold directly. In the second and third cases, we observe the sudden ‘turn-around’ during fast jumps.
Funder Acknowledgement(s): NSF 1700199
Faculty Advisor: None Listed,
NSF Affiliation: HBCU-UP