Noise-induced spiking-bursting transition in the neuron model with the blue sky catastrophe

We study a special variant of the noise-induced transition between spiking and bursting regimes associated with the blue sky catastrophe bifurcation in the Hindmarsh-Rose neuron model. We show that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, noise can transform the spiking oscillatory regime to the bursting one. This phenomenon is studied by means of power spectral density and interspike intervals statistics. We show that noise shifts the bifurcation value, so that bursting activity can be observed for a wider parameter range. Moreover, we reveal that the stochastic spiking-bursting transitions in this system are accompanied by the change in sign of the Lyapunov exponent. We perform a detailed quantitative analysis of these phenomena with an approach that uses a concept of the stochastic sensitivity function, the confidence domains method, and Mahalanobis metrics.

Figure 1

Minimal and maximal $z$-coordinates along limit cycles (a), period $T\left(\alpha \right)$ (b), and length $L\left(\alpha \right)$ (c) of limit cycles. The value $\overline{\alpha }$ corresponds to the blue sky catastrophe bifurcation.

Figure 2

Deterministic phase trajectories and time series for (a) $\alpha =0.004$, (b) $\alpha =0.0083$, and (c) $\alpha =0.00835$.

Figure 3

Phase trajectories and time series for $\alpha =0.00835$ with (a) $\varepsilon =0.005$ and (b) $\varepsilon =0.03$.

Figure 4

Power spectral density for $z\left(t\right)$ for $\alpha =0.00835$ with (a) $\varepsilon =0.005$, (b) $\varepsilon =0.01$, and (c) $\varepsilon =0.03$.

Figure 5

Burst duration: (a) mean value ($m$); (b) coefficient of variation $C_V$.

Figure 6

Mean value of interspike intervals in active phase ($m_s$) and mean duration of quiescent phase ($m_q$) for $\alpha =0.00835$.

Figure 7

Stochastic bifurcation diagrams: $z$ coordinates of limit cycles (a) and burst duration (b) in dependence on the parameter $\alpha$ for fixed values of the noise intensity $\varepsilon$.

Figure 8

Largest Lyapunov exponent in dependence on noise intensity.

Figure 9

Deterministic phase trajectories and time series for $\alpha =0.00835$: A character of the transient process depends on the position of the initial point for a trajectory.

Figure 10

Stochastic sensitivity of the limit cycle for $\alpha =0.00835$: (a) eigenvalues of SSF along the limit cycle; (b) Mahalanobis distance between the limit cycle and the pseudoseparatrix; (c) zones of the cycle corresponding to the maximal SSF value (circle) and the minimal Mahalanobis distance (asterisk).

Figure 11

Confidence ellipses and pseudoseparatrix for (a) $\alpha =0.00835$ and (b) $\alpha =0.0083301$ (for the point corresponding to the minimal Mahalanobis distance between cycle and pseudoseparatrix line).