Chaotic Dynamics and Bifurcation Analysis of the Hindmarsh-Rose Neuron Model with Blue-Sky Catastrophe under Magnetic Field Influence

TL;DR

This study shows that adding a magnetic-flux feedback to the Hindmarsh–Rose neuron model with a blue-sky catastrophe reshapes its bifurcation landscape and firing patterns. By exploring two forms of nonlinear magnetic coupling and employing ISI bifurcation diagrams, phase-space analysis, Poincaré sections, and the largest Lyapunov exponent $\lambda_{\max}$, the work reveals a nonmonotonic influence of coupling strength: weak coupling preserves intrinsic spiking–bursting transitions, intermediate coupling induces chaotic bursting, and strong coupling yields structured irregular dynamics. The findings demonstrate robust qualitative behavior across coupling forms and highlight electromagnetic feedback as a tunable mechanism to control instability and chaos in slow–fast neuronal systems. This has potential implications for understanding EM field effects in neural excitability and for designing neuromodulation strategies.

Abstract

We investigate the impact of magnetic-field-induced feedback on the dynamics of a Hindmarsh-Rose neuron model exhibiting a blue-sky catastrophe. By introducing a magnetic flux variable that couples nonlinearly to the membrane potential, we demonstrate that electromagnetic effects profoundly reshape neuronal firing patterns and bifurcation structure. Interspike-interval bifurcation analysis reveals a nonmonotonic dependence on the magnetic coupling strength, with weak coupling preserving regular spiking and bursting, intermediate coupling promoting chaotic bursting, and strong coupling yielding structured irregular dynamics. These transitions are quantitatively characterized using the largest Lyapunov exponent computed via the Wolf algorithm and supported by Poincaré sections and time-series analysis. Our results establish electromagnetic feedback as a robust and tunable mechanism for controlling instability and chaos in slow-fast neuronal systems.

Figures (5)

• Figure 1: ISI bifurcation diagrams of the Hindmarsh–Rose neuron model with blue-sky catastrophe under polynomial magnetic coupling $W(\phi)=\alpha+3\beta\phi^{2}$ for different values of the magnetic coupling strength $k_1$. The external current $I_\mathrm{ext}$ is used as the bifurcation parameter. Insets show representative membrane potential time series corresponding to selected parameter values (arrows). Increasing magnetic coupling induces fragmentation of periodic branches, intermittent dynamics, and chaotic bursting, followed by suppression of complexity at strong coupling.
• Figure 2: ISI bifurcation diagrams of the Hindmarsh–Rose neuron model with blue-sky catastrophe under polynomial magnetic coupling $W(\phi)=-\tanh(\phi)$ for different values of the magnetic coupling strength $k_1$. The external current $I_\mathrm{ext}$ is used as the bifurcation parameter. Weak coupling preserves the intrinsic blue-sky catastrophe and bursting dynamics, moderate coupling stabilizes periodic spiking, while strong coupling reintroduces irregular and chaotic firing.
• Figure 3: Dynamics of the Hindmarsh--Rose neuron model with magnetic-field coupling at $k_1 = 1.13$. (a) Interspike interval (ISI) bifurcation diagram as a function of the external current $I_{\mathrm{ext}}$, showing banded and irregular firing regimes. (b) Time series of the state variables $x$, $y$, $z$, and $\phi$ at $I_{\mathrm{ext}} = 3.5$, illustrating amplitude-modulated spiking and slow adaptation dynamics. (c) Poincaré section constructed from successive intersections of the trajectory, revealing a continuous curved set characteristic of quasiperiodic or weakly chaotic motion. (d) Zoomed time series of the membrane potential $x(t)$ highlighting clustered spiking and variable interspike intervals.
• Figure 4: Dynamics of the Hindmarsh--Rose neuron model with magnetic-field coupling at $k_1 = 0.95$. (a) Interspike interval (ISI) bifurcation diagram as a function of the external current $I_{\mathrm{ext}}$, showing dense and fragmented ISI distributions characteristic of chaotic bursting. (b) Time series of the state variables $x$, $y$, $z$, and $\phi$ at $I_{\mathrm{ext}} = 3.2$, illustrating irregular bursting and slow modulation. (c) Poincaré section constructed from successive intersections of the trajectory, revealing a scattered set associated with a chaotic attractor. (d) Zoomed time series of the membrane potential $x(t)$ highlighting irregular interspike intervals within bursts.
• Figure 5: Correlation between ISI bifurcation structure and Lyapunov instability in the Hindmarsh--Rose neuron model with magnetic-field coupling. Top row: Interspike interval (ISI) bifurcation diagrams as a function of the external current $I_{\mathrm{ext}}$, with points colored according to the sign of the largest Lyapunov exponent computed using the Wolf method (blue: $\lambda_{\max} \le 0$, red: $\lambda_{\max} > 0$). Bottom row: Corresponding largest Lyapunov exponent $\lambda_{\max}$ evaluated over the same parameter range. From left to right, the panels correspond to increasing magnetic coupling strength $k_1$. Weak coupling yields predominantly regular dynamics with isolated chaotic windows, intermediate coupling produces extended chaotic regimes, and strong coupling leads to structured irregular dynamics with reduced but persistent instability.