Variability and randomness in stationary neuronal activity

The patterns of neuronal activity can be different even if the mean firing rate is fixed. Investigating the variability of the firing may not be sufficient and we suggest to take into account the notion of randomness. The randomness is related to the entropy of the firing, which is bounded from above by the entropy of the Poisson process (given the mean interspike interval). Thus, we propose the Kullback–Leibler distance with respect to the Poisson process as a measure of randomness in a stationary neuronal activity. Under the condition of equal mean values the KL distance does not depend on the time scale and therefore can be compared to the coefficient of variation employed to measure the variability. Furthermore, this measure can be extended to account for correlated neuronal firing. Finally, we analyze the variability and randomness for three common ISI distributions in detail: gamma, lognormal and inverse Gaussian.