Bifurcation analysis of the travelling waveform of FitzHugh–Nagumo nerve conduction model equation

Abstract

The FitzHugh-Nagumo model for travelling wave type neuron excitation is studied in detail. Carrying out a linear stability analysis near the equilibrium point, we bring out various interesting bifurcations which the system admits when a specific Z(2) symmetry is present and when it is not. Based on a center manifold reduction and normal form analysis, the Hopf normal form is deduced. The condition for the onset of limit cycle oscillations is found to agree well with the numerical results. We further demonstrate numerically that the system admits a period doubling route to chaos both in the presence as well as in the absence of constant external stimuli. (c) 1997 American Institute of Physics.

Hopf Bifurcation in a Chaotic Associative Memory

“…6 (k) − 0.00048698x 3 6 (k) x 6 (k + 1) = 0.543931x 5 (k) + 0.84625x 6 (k)− 0.00412636x 3 5 (k) − 0.010156x 2 5 (k)x 6 (k)− 0.00833215x 5 (k)x 2 6 (k) − 0.00227861x 3 6 (k) (38) and the Hopf Bifurcation is stable, since Re(a(μ 0 )) < 0 (see Table 2). This result was expected, since the norm of all eigenvalues in Case 2 is smaller than 1, except the complex pair, which is approximately one.…”

“…The center manifold reduction and the normal forms to describe Hopf Bifurcation, based on Crawford's view [33], were used by many authors [35,36,37,38,39]. We present below, these two methodologies for discrete systems.…”

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

“…Neurons receive incoming sensory inputs, encode them into various biophysical variables and produce relevant outputs [2,3]. Theoretical modeling and numerical simulations are prominent tools in understanding various functional mechanisms in neural computations [4,5,6]. Hindmarsh-Rose (HR) [7], Izhikevich [2,3], FitzHugh-Nagumo (FHN) [8] and FitzHugh-Rinzel (FHR) [9,10] systems are such types of biophysical models that produce diverse firing properties.…”

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

“…Neurons receive incoma) Electronic mail: arghamondalb1@gmail.com b) Electronic mail: aziz.alaoui@univ-lehavre.fr ing sensory inputs, encode them into various biophysical variables and produce relevant outputs 2,3 . Theoretical modeling and numerical simulations are prominent tools in understanding various functional mechanisms in neural computations [4][5][6] . Hindmarsh-Rose (HR) 7 , Izhikevich 2,3 , FitzHugh-Nagumo (FHN) 8 and FitzHugh-Rinzel (FHR) 9,10 systems are such types of biophysical models that produce diverse firing properties.…”