Stochastic dynamics of uncoupled neural oscillators: Fokker-Planck studies with the finite element method

We have investigated the effect of the phase response curve on the dynamics of oscillators driven by noise in two limit cases that are especially relevant for neuroscience. Using the finite element method to solve the Fokker-Planck equation we have studied (i) the impact of noise on the regularity of the oscillations quantified as the coefficient of variation, (ii) stochastic synchronization of two uncoupled phase oscillators driven by correlated noise, and (iii) their cross-correlation function. We show that, in general, the limit of type II oscillators is more robust to noise and more efficient at synchronizing by correlated noise than type I.

Figure 1

(a) Phase response curves for two limit cases from neuroscience. Left, type-I excitability or neural integrator; right, type-II excitability or neural resonator. (b) Phase evolution of two oscillators (black and gray lines) and threshold crossings (i.e., spikes as black and gray dots) for different levels of input correlation $r$. The number of synchronous spikes clearly increases with $r$ and appears to be larger for type II (right) than for type I (left), as confirmed in Fig. 3. Equation parameters ${\omega }_1={\omega }_2=1$, ${\sigma }_1={\sigma }_2=1$.

Figure 2

(a) Coefficient of variation of the cycle duration as a function of the noise amplitude (lines, FEM; circles, SDE). (b) Stochastic synchronization quantified as the correlation coefficient of the phases, $R$, increases with increasing input correlation $r$, but it is consistently larger for type II than for type I (dashed lines, $R=r$ for reference). (c) Effect of the difference of the intrinsic frequencies on stochastic synchronization. As the difference increases, synchrony deteriorates quickly, but more rapidly for type I. (d) The noise amplitude $\sigma$ does not affect synchrony as long as the phase oscillator approximation is valid ($\omega =1$, larger than the average of the term with $\sigma$).

Figure 3

(a) Probability density $P({\phi }_1,{\phi }_2)$ on a gray color scale. The probability of the synchronous states, ${\phi }_1={\phi }_2$, is larger for type II (right panels) than for type I (left panels). The correlation coefficient of the stochastic inputs here is $r=0.6$ (parameters ${\omega }_1={\omega }_2=1$, ${\sigma }_1={\sigma }_2=1$), where ${\omega }_1$ means small omega with subindex 1. (b) Probability density of the phase difference, $P\left(\Delta \right)$ (thick lines)$.$The probability of being around $\Delta =0$ is larger for type II than for type I for any intermediate value of $r$ (only three representative values shown). This result is in agreement with the results obtained from numerical integration of system (2), shown as thin lines with markers. (c) Cross-correlation function $C\left(\tau \right)$ normalized in such a way that $C\left(0\right)$ gives the correlation coefficient $R$ (see text).