# American Institute of Mathematical Sciences

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

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.
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.  Google Scholar [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.  Google Scholar [3] A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [6] J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002).   Google Scholar [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.  Google Scholar [8] A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.   Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [6] J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002).   Google Scholar [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.  Google Scholar [8] A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.   Google Scholar
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