As these correlations can significantly improve information transmission and weak-signal detection, it is an important task to develop analytical approaches to these statistics for well-established computational models. A consequence of these two ubiquitous features is that the successive time intervals between spikes, the interspike intervals, are not independent but correlated. The generation of the action potential is a random process that can be shaped by correlated fluctuations (colored noise) and by adaptation. The elementary processing units in the central nervous system are neurons that transmit information by short electrical pulses, so called action potentials or spikes. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability. We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel’s time scale. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence.
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