FitzHugh-Nagumo equations with generalized diffusive coupling

Anna Cattani

doi:10.3934/mbe.2014.11.203

Article Contents

2014, Volume 11, Issue 2: 203-215. Doi: 10.3934/mbe.2014.11.203 open access

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FitzHugh-Nagumo equations with generalized diffusive coupling

Anna Cattani^1,

1.
Department of Mathematical Sciences, Corso Duca degli Abruzzi 24, 10129 Torino

Received: August 31, 2012

Revised: February 28, 2013

Published: September 2013

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Abstract

The aim of this work is to investigate the dynamics of a neural network, in which neurons, individually described by the FitzHugh-Nagumo model, are coupled by a generalized diffusive term. The formulation we are going to exploit is based on the general framework of graph theory. With the aim of defining the connection structure among the excitable elements, the discrete Laplacian matrix plays a fundamental role. In fact, it allows us to model the instantaneous propagation of signals between neurons, which need not be physically close to each other.
This approach enables us to address three fundamental issues. Firstly, each neuron is described using the well-known FitzHugh-Nagumo model which might allow to differentiate their individual behaviour. Furthermore, exploiting the Laplacian matrix, a well defined connection structure is formalized. Finally, random networks and an ensemble of excitatory and inhibitory synapses are considered.
Several simulations are performed to graphically present how dynamics within a network evolve. Thanks to an appropriate initial stimulus a wave is created: it propagates in a self-sustained way through the whole set of neurons. A novel graphical representation of the dynamics is shown.

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Mathematics Subject Classification: Primary: 92C42; Secondary: 68R10.

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References

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