The variance of phase-resetting curves

Abstract

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.

Predicting the Responses of Repetitively Firing Neurons to Current Noise

“…Therefore the rate of charge accumulation, measured in units of charge 2 per interspike interval, is d* σ Noise 2 /ω. The coefficient of variation of interspike intervals (CV) predicted by the Ermentrout et al [12] approximation for our simulations, (and our experimental results) is: …”

“…The standard errors for the slopes obtained from the regression allow a direct measure of reliability of each point on the PRC, and the proportion of the variance in ISI that can be accounted for in the regression is obtained as R 2 . It should be noted that the standard errors obtained by this method are not a measure of the variance of the PRC, as described in Ermentrout et al [12], and they do not vary in the same way with the amplitude or slope of the phase resetting curve. They are standard errors of the estimates for the PRC obtained in the regression.…”

“…Ermentrout et al [12] derived an approximation for the evolution of the variance of the latent phase distribution for a phase model subjected to Gaussian white noise. In their approximation, the variance of the phase distribution at the time of the average interspike interval is equal to the variance of the Gaussian white noise times the integral of the square of the PRC.…”

“…We used a phase model to predict interspike intervals of STN neurons during stimulation, and adapted the approximation of Ermentrout et al [12] to predict the coefficient of variation of interspike intervals in subthalamic neurons as a function of the amplitude and duration of the noise pulses and neuronal firing rate. Our results indicate that the phase-based simplification of the STN neuron, and perhaps other repetitively firing cells, can accurately predict responses to temporally complex trains of inputs even when the perturbations in timing are large enough to obscure the oscillatory nature of the neuron's firing.…”

Origins and suppression of oscillations in a computational model of Parkinson’s disease

“…Over many stimuli, the distributions will converge to a steady-state probability distribution independent of the starting distribution. The steady-state distribution is the eigenvector, λ ( φ ), corresponding to the largest eigenvalue of the transition matrix, $p(\varphi \mid {\varphi }_{\mu },{\varphi }_{\sigma }^2)$ (Ermentrout et al, 2011). …”