# American Institute of Mathematical Sciences

March  2014, 9(1): 111-133. doi: 10.3934/nhm.2014.9.111

## The derivation of continuum limits of neuronal networks with gap-junction couplings

 1 Department of Mathematical Sciences, Corso Duca degli Abruzzi 29, 10129 Torino, Italy, Italy

Received  April 2013 Revised  March 2014 Published  April 2014

We consider an idealized network, formed by $N$ neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for $N \to \infty$ of the resulting discrete model is thoroughly investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies for passing to the limit are analysed: i) a more conventional approach, based on a fixed nearest-neighbour connection topology accompanied by a suitable scaling of the diffusion coefficients; ii) a new approach, in which the number of connections to any given neuron varies with $N$ according to a precise law, which simultaneously guarantees the non-triviality of the limit and the locality of neuronal interactions. Both approaches yield in the limit a pde-based model, in which the distribution of action potential obeys a nonlinear reaction-convection-diffusion equation; convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are discussed, both theoretically and numerically.
Citation: Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks and Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111
##### References:
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##### References:
 [1] R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, Linear Algebra Appl., 436 (2012), 99-111. doi: 10.1016/j.laa.2011.06.035. [2] A. Cattani, "Multispecies'' Models to Describe Large Neuronal Networks, Ph.D. Thesis Polytechnic University of Turin, 2014. [3] P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progress in Nonlinear Differential Equations and their Applications, 50, Birkhäuser, Basel, 2002, 49-78. [4] G. B. Ermentrout and D. H Terman, Mathematical Foundations of Neuroscience, 1st edition, Springer, New York, 2010. doi: 10.1007/978-0-387-87708-2. [5] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [6] R. FitzHugh, Motion picture of nerve impulse propagation using computer animation, J. Appl. Physiol., 25 (1968), 628-630. [7] M. Galarreta and S. Hestrin, Electrical synapses between Gaba-Releasing interneurons, Nature Reviews Neuroscience, 2 (2001), 425-433. doi: 10.1038/35077566. [8] 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), 500-544. doi: 10.1016/S0092-8240(05)80004-7. [9] J. Keener and J. Sneyd, Mathematical Physiology, 1st edition, Springer-Verlag, New York, 1998. [10] E. Marder, Electrical synapses: rectification demystified, Current Biology: CB, 19 (2009), R34-R35. doi: 10.1016/j.cub.2008.11.008. [11] J. D. Murray, Mathematical Biology I, An Introduction, 3rd edition, Springer-Verlag, New York, 2002. [12] S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology, Numer. Methods Partial Differential Equations, 18 (2002), 218-240. doi: 10.1002/num.1000. [13] A. C. Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487-533. [14] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [15] P. Wallisch, M. Lusignan, M. Benayoun, T. I. Baker, A. S. Dickey and N. G. Hatsopoulos, Matlab for Neuroscientists, Elsevier/Academic Press, Amsterdam, 2009. [16] Y. C. Yu, S. He, S. Chen, Y. Fu, K. N. Brown, X.-H. Yao, J. Ma, K. P. Gao, G. E. Sosinsky, K. Huang and S. H. Shi, Preferential electrical coupling regulates neocortical lineage-dependent microcircuit assembly, Nature, 486 (2012), 113-117. doi: 10.1038/nature10958.
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