Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model

We study the phenomenon of noise-induced torus bursting on the base of the three-dimensional Hindmarsh-Rose neuron model forced by additive noise. We show that in the parametric zone close to the Neimark-Sacker bifurcation, where the deterministic system exhibits rapid tonic spiking oscillations, random disturbances can turn tonic spiking into bursting, which is characterized by the formation of a peculiar dynamical structure resembling that of a torus. This phenomenon is confirmed by the changes in dispersion of random trajectories as well as the power spectral density and interspike intervals statistics. In particular, we show that as noise increases, the system undergoes $P$ and $D$ bifurcations, transitioning from order to chaos. We ultimately characterize the transition from stochastic (tonic) spiking to bursting by stochastic sensitivity functions.

Figure 1

Bifurcation diagram: stable (black dash-dotted line) and unstable (black dotted line) equilibria, maximal and minimal values of $x$ ($z$) coordinates along stable (blue solid) and unstable (blue dashed line) limit cycles, and maximal and minimal values of $x$ ($z$) coordinates along invariant tori (red thick solid line).

Figure 2

Deterministic trajectories (in projection on the $xOy$ the $xOz$ planes) and corresponding time series $x\left(t\right)$ for (a) $\beta =-0.162$ and (b) $\beta =-0.15$.

Figure 3

Stochastic trajectories (in projection on the $xOy$ and $xOz$ planes) and corresponding time series $x\left(t\right)$ for $\beta =-0.159$: (a) $\varepsilon =0.0001$, (b) $\varepsilon =0.0004$.

Figure 4

Stochastic trajectories (in projection on xOy and xOz planes) and corresponding time series $x\left(t\right)$ for $\beta =-0.15$: (a) $\varepsilon =0.001$ and (b) $\varepsilon =0.005$.

Figure 5

Noise-induced torus bursting: $z$ coordinates of stochastic trajectories for (a) $\beta =-0.159$ and (b) $\beta =-0.15$.

Figure 6

Power spectral density for (a) $\beta =-0.159,\varepsilon =0$ (deterministic limit cycle); (b) $\beta =-0.162,\varepsilon =0$ (deterministic torus); and (c) $\beta =-0.159$ ($\varepsilon =0.0001$ and 0.0004).

Figure 7

ISI PDFs for (a) $\beta =-0.162$ ($\varepsilon =0$), (b) $\beta =-0.159$, and (c) $\beta =-0.15$.

Figure 8

ISI statistics: (a) mean value (overall), (b) coefficient of variation (overall), (c) mean value (in burst), and (d) mean quiescence interval, for $\varepsilon =-0.159$ (blue solid line), $\varepsilon =-0.155$ (red dashed line), and $\varepsilon =-0.15$ (green dash-dotted line).

Figure 9

ISI return maps during noise-induced bursting: (a) $\beta =-0.162$ ($\varepsilon =0$), (b) $\beta =-0.159$, and (c) $\beta =-0.15$.

Figure 10

The largest Lyapunov exponent (LLE) in dependence on the noise intensity.

Figure 11

Deterministic limit cycle (black) and phase trajectories (light gray and dark gray) starting from different initial points for $\beta =-0.15$ with corresponding time series $x\left(t\right)$.

Figure 12

Stochastic sensitivity of limit cycles: (a) nonzero eigenvalues ${\lambda }_{1,2}\left(t\right)$ of the stochastic sensitivity matrix for $\beta =-0.15$, (b) maximal eigenvalue ${\lambda }_1\left(t\right)$ (red thick solid) along the limit cycle (black thin solid) for $\beta =-0.15$, and (c) stochastic sensitivity factor in the zone of tonic spiking limit cycles.

Figure 13

Set of confidence ellipses (confidence torus) around the deterministic limit cycle (black thick solid) for $\beta =-0.15$ and $\varepsilon =0.0005$ with fiducial probability $P=0.99$: (a) in $xyz$ space, (b) in projection on the $xOy$ plane.

Figure 14

Mahalanobis distance from the cycle to the pseudoseparatrix for (a) $\beta =-0.159$ and (b) $\beta =-0.15$.

Figure 15

Point of the limit cycle (black circle), confidence ellipses (solid and dashed lines) with fiducial probability $P=0.99$, pseudoseparatrix (red dash-dotted line), point of the minimal Mahalanobis distance from cycle to pseudoseparatrix (red asterisk) for (a) $\beta =-0.159$ (for point $t=6$) and (b) $\beta =-0.15$ (for point $t=4.2$).