Analysis of dynamics of a map-based neuron model via Lorenz maps

What is it about?

The work undertakes the study of the discrete neuron model introduced by Courbage, Nekorkin and Vdovin in 2007. In this model the membrane voltage dynamics was captured by the iterates of the piecewise linear discontinuous map, coupled with the linear equation for the recovery variable. However, in some later version of the model the piecewise linear function was replaced by a cubic polynomial with a discontinuity point, following the prominent FitzHugh-Nagumo model. We have augmented the system with an additional parameter responsible for varying the slope of the cubic function. Showing that on a large subset of the multidimensional parameter space the return map of the voltage dynamics is an expanding Lorenz map, we analyze both chaotic and periodic behaviour of the system and describe complexity of spiking patterns fired by a neuron. This is achieved by using and extending some results from the theory of Lorenz-like and expanding Lorenz mappings.

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Photo by Hans-Peter Gauster on Unsplash

Photo by Hans-Peter Gauster on Unsplash

Why is it important?

Map-based models form an important class of models describing the dynamics of a single nerve cell, which effectively complement earlier ODE-based ones. Although these models might seem abstract from the biological point of view, their electrophysiological relevance is in many cases satisfactory. Moreover, their relative low computational complexity enables to employ them successfully in larger scale simulations of neuronal circuits motivated by biological or clinical issues. However, prior to examination of large ensembles of coupled neurons, it is desirable to understand the dynamical mechanisms behind the phenomena observed in the chosen single neuron model.

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