Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability

The activity of spiking neurons is frequently described by renewal point process models that assume the statistical independence and identical distribution of the intervals between action potentials. However, the assumption of independent intervals must be questioned for many different types of neurons. We review experimental studies that reported the feature of a negative serial correlation of neighboring intervals, commonly observed in neurons in the sensory periphery as well as in central neurons, notably in the mammalian cortex. In our experiments we observed the same short-lived negative serial dependence of intervals in the spontaneous activity of mushroom body extrinsic neurons in the honeybee. To model serial interval correlations of arbitrary lags, we suggest a family of autoregressive point processes. Its marginal interval distribution is described by the generalized gamma model, which includes as special cases the log-normal and gamma distributions, which have been widely used to characterize regular spiking neurons. In numeric simulations we investigated how serial correlation affects the variance of the neural spike count. We show that the experimentally confirmed negative correlation reduces single-neuron variability, as quantified by the Fano factor, by up to 50%, which favors the transmission of a rate code. We argue that the feature of a negative serial correlation is likely to be common to the class of spike-frequency-adapting neurons and that it might have been largely overlooked in extracellular single-unit recordings due to spike sorting errors.

Figure 1

Test for weak stationarity of the counting process. Spike count of neuron No. 1 observed in successive intervals of $1\mathrm{s}$ length as a function of time. Gray parts did not deviate significantly from the null hypothesis of a nonstationary time series. Black parts significantly deviated from this null hypothesis and were assumed to be weakly stationary. See text for details on the test procedure.

Figure 2

(Color online) Empirical interspike interval distribution and conditional mean of one MB extrinsic neuron. (a) Histogram of the length of $N=1530$ ISIs (gray). Log-normal (red) and a gamma (dashed blue) model distribution for MLE parameters. (b) Conditional mean (CM) of the $(i+1)\text{st}$ interval in dependence on the length of the $i\text{th}$ interval (estimated in bins of $15\mathrm{ms}$).

Figure 3

Partial autocorrelation function for experimentally observed spike trains. Box plots describe the distribution of rank-order correlation across $N=21$ neurons for different serial lags. In 14 cases, the sequence of ISIs exhibited a significant $(P<0.01)$ negative correlation of lag 1. For higher lags $(p>1)$ the serial correlation did not significantly deviate from 0 $(P=0.01)$. Here we tested only neurons with a unimodal shape of the interval distribution.

Figure 4

(Color online) Numeric simulations of the log-normal model. The parameter of first-order serial correlation was set to either ${\beta }_1=-0.1$ (left) or ${\beta }_1=-0.5$ (right). The parameters of $E\left[\Delta \right]=50\mathrm{ms}$ and $C_v=0.5$ were fixed for both cases. (a) Model (red line) and empiric (gray histogram) interval distribution show a good agreement. (b) Interval return map. The serial dependence of the $(i+1)\text{st}$ on the $i\text{th}$ interval is difficult to see. The linear regression (dashed line) with linear correlation coefficient ${\varrho }_1$ of the interval series ${\Delta }_s$ underestimates the correlation parameter ${\beta }_1$. (c) In the interval return map of the log-transformed series $\log \left({\Delta }_s\right)$ the strong negative correlation is clearly visible (right). The empiric correlation coefficient ${\widehat{\beta }}_1$ is an unbiased estimate of the true correlation ${\beta }_1$. (d) PACFs estimated for serial lags $p⩽10$.

Figure 5

(Color online) Comparison of linear correlation coefficient (black circles) and MLE estimator for $\widehat{\beta }$ (red triangles). Each single estimate is based on a numeric simulation of 10 000 intervals with parameters of neuron No. 1 in Table . The serial correlation parameter $\beta$ has been varied in the range $[-0.99,0.99]$.

Figure 6

(Color online) Effect of serial interval correlation on spike count variability. We obtained numerical spike train realizations using our model (2). We varied the serial correlation parameter $\beta$ and adjusted the parameters $(\mu ,\sigma )$ to obtain a fixed value of $C_v$. (a) Fano factor $\left(J\right)$ as a function of $\beta$ for three different values of $C_v$ as indicated. The horizontal dotted lines indicate the expectation $J=C_v^2$ for a renewal process. The vertical dashed line represents the renewal model where $\beta =0$. (b) Ratio of $J$ and $C_v^2$. For the physiologically relevant range with negative first-order serial correlation $\beta ∊[-0.5,-0.1]$ (cf. Table ), the Fano factor is clearly smaller than the $C_v^2$ (gray shaded area). For the renewal model $({\beta }_1=0)$ we find $J∕C_v^2\approx 1$ as expected. Each data point represents 10 000 trials. Each trial comprised on average 100 spikes; this number is sufficiently large to avoid a significant bias of estimation for Fano factor and $C_v^2$ [14]. The black square reproduces the results given in [24] as the average for seven cortical neurons with significant first-order serial correlation of ISIs (see Discussion).