Synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo type with nonlinear coupling
Article Sidebar
Full Text: 
PDF
Main Article Content
Abstract
The synchronization in complete network consisting of nodes is studied in this paper. Each node is connected to all other ones by nonlinear coupling and is represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the sufficient condition on the coupling strength to achieve the synchronization is found. The result shows that the networks with bigger in-degrees of nodes synchronize more easily. The paper also presents the numerical simulations for theoretical result and shows a compromise between the theoretical and numerical results.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
Ambrosio, B., & Aziz-Alaoui, M. A. (2012). Synchronization and control of coupled reaction-diffusion systems of the FitzHugh-Nagumo-type. Computers and Mathematics with Applications, 64, 934-943.
Ambrosio, B., & Aziz-Alaoui, M. A. (2013). Synchronization and control of a network of coupled reaction-diffusion systems of generalized FitzHugh-Nagumo type. ESAIM, 39, 15-24.
Aziz-Alaoui, M. A. (2006). Synchronization of Chaos. Encyclopedia of Mathematical Physics, Elsevier, 5, 213-226.
Belykh, I., De Lange, E., & Hasler, M. (2005). Synchronization of bursting neurons: What matters in the network topology. Phys. Rev. Lett, 94, 188101.
Corson, N. (2009). Dynamique d'un modèle neuronal, synchronisation et complexité (doctoral dissertation). University of Le Havre, France.
Corson, N., & Aziz-Alaoui, M. A. (2009). Complex emergent properties in synchronized neuronal oscillations. In From system complexity to emergent properties (pp. 243-259). Springer, Berlin, Heidelberg.
Ermentrout, G. B., & Terman, D. H. (2009). Mathematical Foundations of Neurosciences. Springer.
Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol, 117, 500-544.
Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Terrence J. Sejnowski and Tomaso A. Poggio The MIT Press, Cambridge, Massachusetts, London, England.
Keener, J. P., & Sneyd, J. (2009). Mathematical Physiology: Systems Physiology, Second Edition. Antman S.S., Marsden J.E., and Sirovich L. Springer.
Murray, J. D. (2002). Mathematical Biology. I. An Introduction, Third Edition. Springer.
Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proc. IRE, 50, 2061-2070.
Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization, A Universal Concept in Nonlinear Science. Cambridge: Cambridge University Press, England.