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 & Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111
References:
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A. Cattani, "Multispecies'' Models to Describe Large Neuronal Networks,, Ph.D. Thesis Polytechnic University of Turin, (2014).

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G. B. Ermentrout and D. H Terman, Mathematical Foundations of Neuroscience,, 1st edition, (2010). doi: 10.1007/978-0-387-87708-2.

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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.

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R. FitzHugh, Motion picture of nerve impulse propagation using computer animation,, J. Appl. Physiol., 25 (1968), 628.

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M. Galarreta and S. Hestrin, Electrical synapses between Gaba-Releasing interneurons,, Nature Reviews Neuroscience, 2 (2001), 425. doi: 10.1038/35077566.

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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. doi: 10.1016/S0092-8240(05)80004-7.

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J. Keener and J. Sneyd, Mathematical Physiology,, 1st edition, (1998).

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E. Marder, Electrical synapses: rectification demystified,, Current Biology: CB, 19 (2009). doi: 10.1016/j.cub.2008.11.008.

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S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology,, Numer. Methods Partial Differential Equations, 18 (2002), 218. doi: 10.1002/num.1000.

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A. C. Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.

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J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (1994). doi: 10.1007/978-1-4612-0873-0.

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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. doi: 10.1038/nature10958.

show all references

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]

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, (2000), 49.

[4]

G. B. Ermentrout and D. H Terman, Mathematical Foundations of Neuroscience,, 1st edition, (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. 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.

[7]

M. Galarreta and S. Hestrin, Electrical synapses between Gaba-Releasing interneurons,, Nature Reviews Neuroscience, 2 (2001), 425. 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. doi: 10.1016/S0092-8240(05)80004-7.

[9]

J. Keener and J. Sneyd, Mathematical Physiology,, 1st edition, (1998).

[10]

E. Marder, Electrical synapses: rectification demystified,, Current Biology: CB, 19 (2009). doi: 10.1016/j.cub.2008.11.008.

[11]

J. D. Murray, Mathematical Biology I, An Introduction,, 3rd edition, (2002).

[12]

S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology,, Numer. Methods Partial Differential Equations, 18 (2002), 218. doi: 10.1002/num.1000.

[13]

A. C. Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487.

[14]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (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, (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. doi: 10.1038/nature10958.

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