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Initial boundary value problems for coupled nerve fibres

Published online by Cambridge University Press:  20 January 2009

P. Grindrod
P. Grindrod
Affiliation:
Department of Mathematical Sciences, The UniversityDundee
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In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the form

Here W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 2 , June 1985 , pp. 249 - 269
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1. Bell, J., Modelling parallel unmyelinated axons: Pulse trapping and ephaptic transmission, SIAM Journal of Appl. Math. 41 (1981).CrossRefGoogle Scholar
2. Courant, R. and Hilbert, D., Methods of mathematical physics, Vol. 1 (Interscience, N.Y., 1966).Google Scholar
3. Fitzhugh, R., Mathematical models of excitation and propagation in nerve fibres, Biological Engineering, Ed. Schwan, H. P. (McGraw-Hill, N.Y., 1969).Google Scholar
4. Grindrod, P. and Sleeman, B. D., Qualitative analysis of reaction-diffusion systems modelling coupled unmyelinated nerve axons, IMA J. Math. Appl. in Med. and Biol. 1 (1984).Google ScholarPubMed
5. Grindrod, P. and Sleeman, B. D., Comparison Principles in the Analysis of Reaction-Diffusion Systems Modelling Unmyelinated Nerve Fibres (Dundee University Applied Analysis Report AA843, 1984).Google Scholar
6. Rauch, J. and Smoller, J. A., Qualitative theory of the FitzHugh-Nagumo equations, Adv. in Math. 27 (1978).Google Scholar
7. Schonbeck, M. E., Boundary value problems for the FitzHugh-Nagumo equations, J. Diff. Eqns. 30 (1978).Google Scholar