FitzHugh-Nagumo equations with generalized diffusive coupling

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2014, 11(2): 203-215. doi: 10.3934/mbe.2014.11.203

FitzHugh-Nagumo equations with generalized diffusive coupling

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

Received September 2012 Revised March 2013 Published October 2013

Abstract
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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.

Keywords: stability analysis, FitzHugh-Nagumo model, Biological neural network, travelling pulses., diffusive coupling.

Mathematics Subject Classification: Primary: 92C42; Secondary: 68R1.

Citation: Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203

References:

[1]	R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs,, Linear Algebra Appl., 436 (2012), 99. doi: 10.1016/j.laa.2011.06.035.
[2]	N. Burić and D. Todorović, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222.
[3]	A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, ().
[4]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6.
[5]	A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.
[6]	J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002).
[7]	Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHugh-Nagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389.
[8]	A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.

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References:

[1]	R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs,, Linear Algebra Appl., 436 (2012), 99. doi: 10.1016/j.laa.2011.06.035.
[2]	N. Burić and D. Todorović, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222.
[3]	A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, ().
[4]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6.
[5]	A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.
[6]	J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002).
[7]	Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHugh-Nagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389.
[8]	A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.

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