Time scales of spike-train correlation for neural oscillators with common drive

We examine the effect of the phase-resetting curve on the transfer of correlated input signals into correlated output spikes in a class of neural models receiving noisy superthreshold stimulation. We use linear-response theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem and contrast the results for type I vs type II models and across the different time scales over which spike correlations can be assessed. We find that, on long time scales, type I oscillators transfer correlations much more efficiently than type II oscillators. On short time scales this trend reverses, with the relative efficiency switching at a time scale that depends on the mean and standard deviation of input currents. This switch occurs over time scales that could be exploited by downstream circuits.

Figure 1

(Color online) (a) A schematic of the setup and correlation metric used in this study. Two cells receive both common and independent input, here modeled as white noise. Correlation of output spike trains is measured as the normalized covariance of spike counts over a finite time window $T$ [see Eq. (5)]. (b) The principal result of our study; that correlation transfer efficiency depends on both internal dynamics and time scale of readout, with relative efficiency between type I (red or light gray) and type II (blue or dark gray) switching as readout time scale changes. The graph shows ${\rho }_T$ for a particular set of model parameters $\left(\omega ,\sigma \right)$ vs the measurement time window $T$ ($T$ is shown on a logarithmic scale). Insets show the phase-resetting curves used for the type I and type II models.

Figure 2

Firing rate (left) and CV (right) of the theta model neuron $\left(\alpha =0\right)$ over a range of intrinsic phase speed $\omega$ and noise variance ${\sigma }^2$. Sample spike trains are illustrated for four sets of $\left(\omega ,\sigma \right)$ values (see white squares), notably mean-driven (high $\omega$, low $\sigma$-bottom-most spike train) and fluctuation-driven (low $\omega$, high $\sigma$-top-most spike train). Level sets of $\tilde{\sigma }\equiv \frac{\sigma }{\sqrt{\omega }}$ are plotted (see text in Sec. 3C).

Figure 3

(Left) Susceptibility $S\left(\omega ,\sigma \right)$ for global model parameter $\alpha =0$ (top), $\alpha =0.5$ (middle), and $\alpha =1$ (bottom) over a range of intrinsic phase speed $\omega$ and noise variance ${\sigma }^2$; lines are level sets of $\tilde{\sigma }\equiv \sigma /\sqrt{\omega }$. (Right) Spike count correlation ${\rho }_T$ vs fraction of correlated noise $c$ from Monte Carlo simulations for $\alpha =0$ (top), $\alpha =0.5$ (middle), and $\alpha =1$ (bottom). Error bars indicate the standard deviation over eight trials. The specific $\left(\omega ,\sigma \right)$ values plotted are those shown in Fig. 2. ${\rho }_T$ is measured over a time window of $T=32\text{ms}$.

Figure 4

Scatter plots of susceptibility $S\left(\omega ,\sigma \right)$ vs firing rate $\nu$ (a) and susceptibility vs coefficient of variation (CV) (b) for global model parameter $\alpha =0$ (light gray), $\alpha =0.5$ (medium gray), and $\alpha =1$ (black). $S\left(\omega ,\sigma \right)$ was computed for the ranges of intrinsic phase speed $\omega$ and noise variance ${\sigma }^2$ shown in Fig. 3.

Figure 5

Susceptibility $S\left(1,\sigma \right)$ for a continuum of models specified by $\alpha ∊\left[0,1\right]$.

Figure 6

${\text{lim}}_{\sigma \to 0^{+}}S\left(1,\sigma \right)$ for a continuum of models specified by $\alpha ∊\left[0,1\right]$ (dashed black line). This is a good approximation for small ($\sigma =0.2$; dark gray line) and moderate ($\sigma =1$; light gray line) values of $\sigma$.

Figure 7

Spike count correlation ${\rho }_T\left(\omega ,\sigma \right)$ as measured from Monte Carlo simulations with global model parameter $\alpha =0$ (top row), $\alpha =0.5$ (middle row), and $\alpha =1$ (bottom row). Measurement time intervals are, from the leftmost column, $T=1$, $T=2$, $T=8$, and $T=32\left(\text{ms}\right)$. ${\rho }_T$ is computed for a range of intrinsic phase speed $\omega$ and noise variance ${\sigma }^2$. The fraction of noise variance that is received by both oscillators is $c=0.1$; therefore, to recover the approximate susceptibility $S_T\left(\omega ,\sigma \right)\approx {\rho }_T\left(\omega ,\sigma \right)/c$, multiply by 10. As $T$ increases, $S_T$ approaches the $T\to \infty$ limit illustrated in Fig. 3.