Complete dynamical analysis of a neuron model |
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Authors: | Andrey Shilnikov |
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Institution: | 1.Neuroscience Institute, Department of Mathematics and Statistics,Georgia State University,Atlanta,USA |
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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. |
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