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:
[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.  Google Scholar

[2]

A. Cattani, "Multispecies'' Models to Describe Large Neuronal Networks, Ph.D. Thesis Polytechnic University of Turin, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. FitzHugh, Motion picture of nerve impulse propagation using computer animation, J. Appl. Physiol., 25 (1968), 628-630. Google Scholar

[7]

M. Galarreta and S. Hestrin, Electrical synapses between Gaba-Releasing interneurons, Nature Reviews Neuroscience, 2 (2001), 425-433. doi: 10.1038/35077566.  Google Scholar

[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.  Google Scholar

[9]

J. Keener and J. Sneyd, Mathematical Physiology, 1st edition, Springer-Verlag, New York, 1998.  Google Scholar

[10]

E. Marder, Electrical synapses: rectification demystified, Current Biology: CB, 19 (2009), R34-R35. doi: 10.1016/j.cub.2008.11.008.  Google Scholar

[11]

J. D. Murray, Mathematical Biology I, An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[13]

A. C. Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487-533. Google Scholar

[14]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. Cattani, "Multispecies'' Models to Describe Large Neuronal Networks, Ph.D. Thesis Polytechnic University of Turin, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. FitzHugh, Motion picture of nerve impulse propagation using computer animation, J. Appl. Physiol., 25 (1968), 628-630. Google Scholar

[7]

M. Galarreta and S. Hestrin, Electrical synapses between Gaba-Releasing interneurons, Nature Reviews Neuroscience, 2 (2001), 425-433. doi: 10.1038/35077566.  Google Scholar

[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.  Google Scholar

[9]

J. Keener and J. Sneyd, Mathematical Physiology, 1st edition, Springer-Verlag, New York, 1998.  Google Scholar

[10]

E. Marder, Electrical synapses: rectification demystified, Current Biology: CB, 19 (2009), R34-R35. doi: 10.1016/j.cub.2008.11.008.  Google Scholar

[11]

J. D. Murray, Mathematical Biology I, An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[13]

A. C. Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487-533. Google Scholar

[14]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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