Nonlinear Dynamics

, Volume 68, Issue 3, pp 305–328

Complete dynamical analysis of a neuron model

Authors

    • Neuroscience Institute, Department of Mathematics and StatisticsGeorgia State University
Original Paper

DOI: 10.1007/s11071-011-0046-y

Cite this article as:
Shilnikov, A. Nonlinear Dyn (2012) 68: 305. doi:10.1007/s11071-011-0046-y

Abstract

In-depth understanding of the generic mechanisms of transitions between distinct patterns of the activity in realistic models of individual neurons presents a fundamental challenge for the theory of applied dynamical systems. The knowledge about likely mechanisms would give valuable insights and predictions for determining basic principles of the functioning of neurons both isolated and networked. We demonstrate a computational suite of the developed tools based on the qualitative theory of differential equations that is specifically tailored for slow–fast neuron models. The toolkit includes the parameter continuation technique for localizing slow-motion manifolds in a model without need of dissection, the averaging technique for localizing periodic orbits and determining their stability and bifurcations, as well as a reduction apparatus for deriving a family of Poincaré return mappings for a voltage interval. Such return mappings allow for detailed examinations of not only stable fixed points but also unstable limit solutions of the system, including periodic, homoclinic and heteroclinic orbits. Using interval mappings we can compute various quantitative characteristics such as topological entropy and kneading invariants for examinations of global bifurcations in the neuron model.

Keywords

Neuron modelBifurcationBurstingBistabilityPoincaré mappingParameter continuationComplex dynamics

Copyright information

© Springer Science+Business Media B.V. 2011