Three Scenarios for Changing of Stability in the Dynamic Model of Nerve Conduction

Abstract

Abstract. The paper deals with the specific cases of the changing of stability of slow integral manifold of singularly perturbed systems of ODE via the dynamic model of nerve conduction. It is shown that the proper choice of the additional parameters of the system allows us to construct the slow integral manifold with multiple change of its stability.Keywords: singular perturbations, canards, delaying of the loss of stability, critical phenomena, Hindmarsh-Rose model. Citation: Shchepakina EA. Three scenarios for changing of stability in the dynamic model of nerve conduction.

Canards Existence in the Hindmarsh–Rose model

“…This result confirms the genericity of the condition (σ 2 < 0) that we have highlighted and provides a simple and efficient tool for testing the occurrence of "canard solutions" in any three or four-dimensional singularly perturbed systems with one or two fast variables. Applications of this method to the famous coupled Hindmarsh-Rose model has enabled to confirm the existence of "canard solutions" in such systems as already stated by Shchepakina [28]. However, in this paper, only the case of pseudo singular points or folded singularities of saddle-type has been analyzed.…”

“…The other parameters (a, b, c, d, I, s, α and r) reflect the physical features of the neurons and the dot indicates derivative with respect to the time t. We notice that the parameter r 1. Existence of canard solutions in such system (4.1) has been originally suspected by Shilnikov et al ([29], p. 2149) and highlighted by Shchepakina [28]. Thus, by posing x → y 2 , y → y 1 , z → x 1 and t → εt with ε = r and according to the previous definitions, the Hindmarsh-Rose model may be written as a three-dimensional singularly perturbed system with k = 1 slow variable and m = 2 fast variables:…”

“…(b − d) (c + I + sα)According to equations (3.16) we find that:p = σ 1 = T r [J] = s, q = σ 2 = −2 (b − d) (c + I + sα) .Thus, according to Proposition 3.4, the pseudo singular point is of saddle-type if:− 2 (b − d) (c + I + sα) < 0. (4.10)In her work Shchepakina[28] used the following parameter set: a = 1, b = 3, c = 1, d = 0.275255, I = 2.7 and α = −1.2. She found a canard without head (seeFig.…”

“…Singular canard solution of the Hindmarsh-Rose (4.1) model in the (x 3 , y) plane phase, its critical manifold (in green) and the second-order approximation in ε of the slow invariant manifold (in blue) with the following parameter set: a = 1, b = 3, c = 1, d = 0.275255, I = 2.7, α = −1.2 and for the "duck parameter" value s = 3.0810445478558141214.According to equations (3.16) we find that:p = σ 1 = T r [J] = s, q = σ 2 = −2 (b − d) [s (ỹ 2 − α) −x 1 ] .Thus, according to Proposition 3.4, the pseudo singular point is of saddle-type if and only if:− 2 (b − d) [s (z − α) −x] < 0. (4.13)In her work Shchepakina[28] used the following parameter set: a = 1, b = 3, c = 1, d = 0.275255, I = 2.7 and α = −1.2. According to equation (4.13) and with such a parameter set, i.e.…”

Canards Existence in the Hindmarsh–Rose Model

“…In her work Shchepakina [28] used the following parameter set: a = 1, b = 3, c = 1, d = 0.275255, I = 2.7 and α = −1.2. According to equation (4.13) and with such a parameter set, i.e.…”