Hopf bifurcations in dynamics of excitable systems
Monica De Angelis
TL;DR
This work analyzes Hopf bifurcations in the three-variable FitzHugh-Rinzel neuronal model by performing a linear stability analysis of fixed points via a spectral equation and the Lienard-Chipart criterion, augmented with the instability coefficient power (ICP) method. A detailed expression for the linear invariants $A_1,A_2,A_3$ is derived in terms of the quadratic form $\Gamma$ and bifurcation parameters, enabling the classification of Hopf onset driven by the admissible critical point $\bar{U}$ and by the parameter $-\eta = -\varepsilon\beta$. The authors show that, by introducing a positive bifurcation parameter $R$ (related to $\Gamma$), Hopf bifurcations can be either simple oscillations or steady-plus-oscillatory, with explicit conditions and frequency characterizations (through $\varphi^2 = A_2(R)$). They further extend the analysis to the direction $-\eta = -\varepsilon\beta$, illustrating similar bifurcation structures and highlighting the broader applicability of the results, including connections to Josephson junction dynamics and potential generalizations to additional model coefficients.
Abstract
A general FitzHugh-Rinzel model, able to describe several neuronal phenomena, is considered. Linear stability and Hopf bifurcations are investigated by means of the spectral equation for the ternary autonomous dynamical system and the analysis is driven by both an admissible critical point and a parameter which characterizes the system.