1. Instability Analysis of Nerve Cell Dynamics in the FitzHugh-Nagumo Model Nasrin Sultana*, Sampad Das and M. Osman Gani** Department of Mathematics, Jahangirnagar University, Savar, Dhaka.
Abstract
We study the two-variable FitzHugh-Nagumo reaction-diffusion system for neuron excitation. The periodic action potentials of the nerve cells can be treated as the periodic traveling waves in one dimension. That motivates us to study the existence and the stability of periodic traveling waves in a one-parameter family of solutions. It is observed that periodic traveling waves change their stability by a stability change of Eckhaus type in a two-dimensional parameter plane. We determine the stability boundary between stable and unstable periodic traveling waves. We also calculate essential spectra of the periodic traveling waves.
Methodology
We used the periodic traveling wave (PTW) continuation package WAVETRAIN [2] to calculate the existence and stability of PTW solutions in one dimension in a two-dimensional parameter plane by the method of continuation.
Firstly, we formulate the four-dimensional Ordinary Differential Equations (ODEs) by using a traveling wave co-ordinate.
Secondly, we linearized partial differential equations(PDEs) and
Finally, using these linearized PDEs, we derive the eigenvalue problem with necessary boundary conditions using Floquet theory and Bloch transformation. In order to understand the stability of PTWs, we calculate the essential spectra of the PTWs. 1 3
Bifurcation Diagrams 4
Mathematical Model 5
Numerical Results 2
(1)
u: fast activator (excitable) variable
v: slow inhibitor (recovery) variable
Nullclines of model (1)
Left to right: Existence and stability of PTW solution of (1), the essential spectra of two PTW solutions of (1), first one is a stable PTW and the second one is an unstable PTW (Eckhaus type).
Conclusion
We have studied the existence and stability of the periodic traveling wave solutions in the standard FitzHugh-Nagumo model. Our results showed the stability change of Eckhaus type of the PTWs. The instability occurs in the waves having sufficiently large periods. This is our main finding in this work. We explained these phenomena by calculating the dispersion curves and essential spectra of the PTWs.
References
[1]. R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical journal, vol. 1, no. 6, p. 445, 1961.
[2]. J. A. Sherratt, “Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations,” Applied Mathematics & Computation, vol. 218, pp –4694, 2012.
[3]. J. D. M. Rademacher, “Homoclinic bifurcation from heteroclinic cycles with periodic orbits and tracefiring of pulses,” Ph.D. dissertation, University of Minnesota, 2004.
[4]. M. O. Gani and T. Ogawa, “Instability of periodic traveling wave solutions in a modified fitzhugh–nagumo model for excitable media,” Applied Mathematics and Computation, vol. 256, pp. 968–984, 2015.
[5]. J. D. M. Rademacher, B. Sandstede, and A. Scheel, “Computing absolute and essential spectra using continuation,” Physica D, vol. 229, pp. 166–183, 2007.
[6]. J. A. Sherratt, “Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations,” Advances in Computational Mathematics, vol. 39, no. 1, pp. 175–192, 2013.