Abstract
We analyze the effect of weak-noise-induced transitions on the dynamics of the FitzHugh–Nagumo neuron model in a bistable state consisting of a stable fixed point and a stable unforced limit cycle. Bifurcation and slow-fast analysis give conditions on the parameter space for the establishment of this bi-stability. In the parametric zone of bi-stability, weak-noise amplitudes may strongly inhibit the neuron’s spiking activity. Surprisingly, increasing the noise strength leads to a minimum in the spiking activity, after which the activity starts to increase monotonically with an increase in noise strength. We investigate this inhibition and modulation of neural oscillations by weak-noise amplitudes by looking at the variation of the mean number of spikes per unit time with the noise intensity. We show that this phenomenon always occurs when the initial conditions lie in the basin of attraction of the stable limit cycle. For initial conditions in the basin of attraction of the stable fixed point, the phenomenon, however, disappears, unless the timescale separation parameter of the model is bounded within some interval. We provide a theoretical explanation of this phenomenon in terms of the stochastic sensitivity functions of the attractors and their minimum Mahalanobis distances from the separatrix isolating the basins of attraction.
Highlights
Fixed points, periodic, quasiperiodic or chaotic orbits are typical solutions of deterministic nonlinear dynamical systems
Through bifurcation and slow-fast analyses, we determined the conditions on the parameter space for the establishment of a bi-stability regime consisting of a unique stable fixed point and a stable unforced limit cycle
Introducing noise to the system causes transitions from the basin of attraction of the fixed point to that of the limit cycle and as well, from the basin of attraction of the limit cycle to that of the fixed point
Summary
Periodic, quasiperiodic or chaotic orbits are typical solutions of deterministic nonlinear dynamical systems. Gutkin et al (2009) and Tuckwell et al (2009) used the Hodgkin–Huxley equations in the bistable regime (fixed point and limit cycle) with a mean input current consisting of both a deterministic and random input component, to computationally confirm the inhibitory and modulation effects of Gaussian noise on the neuron’s spiking activity. They found that there is a tuning effect of noise that has the opposite character to SR and CR, which they termed inverse stochastic resonance (ISR).
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