Noise-Induced Synchronization and Desynchronization of Neural Oscillators

We study an influence of white Gaussian noise on synchrony of neural oscillators. In neurophysiology, the property of a single neuron to provide identical outputs for repeated noisy input (in terms of our problem, an array of such identical neurons driven by common noise to behave synchronously) is referred as “reliability” [1]. The first quantifier of the synchrony we use is the Lyapunov exponent (LE, measures the exponential growth rate of small deviations from the trajectory) for a single oscillator [2, 3, 4]. Within the phase approximation for limit circle oscillators, one can derive the explicit expression for the LE. For weak noise the LE is negative, what means synchronization of identical systems subject to common noise: this result appears to be general for smooth limit circle oscillators. In some systems, moderate noise can desynchronize oscillations [5]. The simplest neuron-like systems (Fitz Hugh–Nagumo and Morris–Lecar; the results for the latter are not presented in this poster) exhibit such a behavior under noisy influence close to the transition from excitable behavior to periodic spiking. Moderate noise is shown to be able to desynchronize neuron-like oscillators not only in the regime of periodic spiking but also ones in the excitable state. For characterization the approach of event synchronization [6] is also used. Phase Model and Lyapunov Exponent Within the phase approximation, one neglects deviations from the limit circle of the noiseless system and takes into account only motion along this circle. The system states on this limit circle can be parameterized by a single parameter, phase φ. With a stochastic force (in the case of N-component noise) the equation for the phase reads (in Stratonovich formulation) φ̇ = ω+ N ∑ j=1 f j(φ)ξ j(t) , (1) where 2π/ω is the period of the limit circle in the noiseless system, f j(φ) is the 2π-periodic sensitivity of the phase to the j-th component of noise, and ξ j is white Gaussian noise: 〈ξ j(t)ξk(t + t ′)〉 = 2D jδ jkδ(t ′). The stationary solution of the Fokker-Planck equation corresponding to Eq. (1) with periodic boundary conditions is the distribution