Abstract
Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ∼τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t → 2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.