Analytical Properties and Solutions of the FitzHugh – Rinzel Model

Abstract

The FitzHugh-Rinzel model is considered, which differs from the famous FitzHugh-Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh-Rinzel model.

On Integrability of the FitzHugh – Rinzel Model

“…In this note we have considered the FitzHugh -Rinzel model for description of the propagation from one neuron to another. This topic was studied in the paper [15], and this note has been written under the influence of this article.…”

“…However, there is the expansion of the general solution of Eq. (1.2) in the Puiseux series with three arbitrary constants t 0 , a 3 and a 5 in the form [15] Let us note that we have a critical movable pole of the general solution and the three arbitrary constants t 0 , a 3 and a 5 for the expansion of the general solution in the Puiseux series. These arbitrary constants may correspond to first integrals with the leading terms [22,23].…”

“…The solution of the FitzHugh -Rinzel system of equations (3.9) has two arbitrary constants. The authors of [15] considered the same dependence of t for functions y(t) and w(t) in the form y = w − σ β . (3.12) In this case there are two equations in the form…”

“…In the recent paper [15], the authors considered the analytical properties of the so-called FitzHugh -Rinzel model and built some simple solutions of the system of equations used to describe the process of neuron propagation. They showed that the general solution of a thirdorder equation is represented by a Puiseux series with three arbitrary constants.…”

On solutions to a FitzHugh–Rinzel type model

“…The FitzHugh-Rinzel (FHR) system [1][2][3][4][5] is a three dimensional model deriving from the FitzHugh-Nagumo (FHN) model [6][7][8][9][10][11][12][13][14][15] to incorporate bursting phenomena of nerve cells. Generally, in many cell types, bursting oscillations are characterized by a variable of the system that changes periodically from an active phase of rapid spike oscillations to a silent phase during which the membrane potential only changes slowly [7].…”