Stability analysis of the Hindmarsh-Rose neuron under electromagnetic induction

TL;DR

This work extends the Hindmarsh-Rose neuron model by incorporating electromagnetic induction through a memristive coupling, yielding a four-dimensional system. The authors perform a thorough dynamical analysis, deriving the equilibrium structure and employing a linear stability (Routh–Hurwitz) framework to map how the magnetic coupling parameter k reshapes the fixed-point landscape. They show that increasing k reduces the number of equilibria from three to one and stabilizes fixed points, effectively regularizing chaotic dynamics into periodic or damped behavior. The findings illuminate how magnetic flux can modulate neuronal excitability and bursting, with potential implications for neural networks and neurocomputational processing under external magnetic influences.

Abstract

We consider the Hindmarsh-Rose neuron model modified by taking into account the effect of electromagnetic induction on membrane potential. We study the impact of the magnetic flux on the neuron dynamics, through the analysis of the stability of fixed points. Increasing magnetic flux reduces the number of equilibrium points and favors their stability. Therefore, electromagnetic induction tends to regularize chaotic regimes and to affect regular and quasi-regular ones by reducing the number of spikes or even destroying the oscillations.

Figures (10)

• Figure 1: Regions characterized by the number of equilibrium points in the $s-I_{ext}$ plane, for different values of the magnetic-flux coupling parameter: $k=0$ (i), $k=5$ (ii), $k=10$ (iii) and $k=15$ (iv). In the shadowed region (A) there are three equilibrium points, while in region (B), there is a single equilibrium point. At the border (full lines), there are two equilibrium points. The dotted line repeats the case $k=0$, for comparison. Notice that the region (A) shrinks with increasing $k$.
• Figure 2: Stability maps of the equilibrium points $E$, in the plane $s-I_{ext}$, for $k=0$ (i), $k=5$ (ii), $k=10$ (iii) and $k=15$ (iv). The colored regions correspond to one equilibrium point [unstable (dark-gray), stable (light-gray)] and three equilibrium points [1 unstable (green), 2 unstable (purple), all unstable (red)]. The latter only appears, within the resolution of the figure, near the vertex of region (A).
• Figure 3: Projections of the phase portrait in different planes, and corresponding time series for the membrane potential $x$ and the current $y$ vs. $t$, for (a) $k=0$, (b) $k=5$ and (c) $k=10$, with $r = 0.008$, $s=4$ and $I_{ext}=3.25$. In (a), we observe the chaotic attractor of the HR neuronal model, while the dynamics becomes regularized as $k$ increases.
• Figure 4: Heat-plots of the largest Lyapunov exponent $L_{max}$ as a function of $r$ and $k$, using $s=4$, $I_{ext}= 3.25$. In the right-hand-side panel we amplified the low $k$ region. We observe that for sufficiently low $k$ chaotic trajectories (red scale) can exist when $r$ is also low, but increasing $k$ leads to negative values of $L_{max}$, indicating regular behavior. The darker blue region corresponds to damped oscillations towards a fixed point.
• Figure 5: Bifurcation diagrams and the largest Lyapunov exponent as a function of $k$, for $r=0.008$, with $s=4$, $I_{ext}= 3.25$. (i) In these and following bifurcation diagrams, magenta and blue lines correspond respectively to the local minima and maxima of the timeseries $y(t)$, (ii) largest Lyapunov exponent. (iii) interspike intervals: time elapsed between maxima (blue) and between minima (magenta). After approx. $k>9$, the single value corresponds to the period of simple oscillations, which become damped near $k>11$.