Coarse-Graining and Simplification of the Dynamics Seen in Bursting Neurons

Abstract

© 2016 Society for Industrial and Applied Mathematics. A system of coupled bursting neurons often exhibits transitions from a quiescent state (no action potentials) to a bursting state (where there are several action potentials in each burst and periods of no action potentials between bursts) to a spiking state (continuous train of action potentials). However, in certain applications, the exact details of the complex dynamics that underlies these transitions may not be important and one would like to use a simplified model that captures the dynamics only roughly. We have developed a numerical method, based on an equation-free approach, that enables us to rationalize the type of bifurcations the simplified model ought to exhibit. The method maps between the variables of a bursting neural network (for which the equations are known) and the variables of a simplified model (for which the equations are in principle unknown). By moving back and forth between the variables of the detailed system and the variables of the simplified system using restriction and lifting operators, and simulating the detailed system for short periods of time, we can calculate the stationary solutions of the simplified model and their effective stabilities and create bifurcation diagrams for the simplified model. We illustrate our approach on a model of a single neuron that consists of three ordinary differential equations and show that the system could be simplified using two ordinary differential equations. We then show how our method can be applied to a network of several neurons. Our work is motivated by studies of the neural control of breathing.