Adaptation with correlated noise predicts negative interval correlations in neuronal spike trains

Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Here we predict observed ISI serial correlations from primary electrosensory afferents of weakly electric fish using an adaptive threshold model with a noisy spike threshold. We derive a general relationship between serial correlation coefficients (SCCs) and the autocorrelation function of added noise. Observed afferent spike-trains fall into two categories based on the pattern of SCCs: non-bursting units have negative SCCs which remain negative but decay to zero with increasing lags (Type I SCCs), and bursting units have oscillatory (alternating sign) SCCs which damp to zero with increasing lags (Type II SCCs). Type I SCCs are generated by low-pass filtering white noise before adding it to the spike threshold, whereas Type II SCCs are generated by high-pass filtering white noise. Thus, a single parameter (the sign of the pole of the filter) generates both types of SCCs. The filter pole (equivalently time-constant) is estimated from the observed SCCs. The predicted SCCs are in geometric progression. The theory predicts that the limiting sum of SCCs is −0.5, and this is confirmed from the expressions for the two types of filters. Observed SCCs from afferents have a limiting sum that is slightly larger at −0.475 ±0.04 (mean ±s.d.). The theoretical limit of the sum of SCCs leads to a perfect DC block in the power spectrum of the spike-train, thereby maximizing signal-to-noise ratio during signal encoding. The experimentally observed sum of SCCs is just short of this limit. We conclude by discussing the results from the perspective of optimal coding. Author summary Many neurons spontaneously emit spikes (impulses) with a random time interval between successive spikes (interspike interval or ISI). The spike generation mechanism can have memory so that successive ISIs are dependent on one another and exhibit correlations. An ISI which is longer than the mean ISI tends to be followed by an ISI which is shorter than the mean, and vice versa. Thus, adjacent ISIs are negatively correlated, and further these correlations can extend over multiple ISIs. A simple model describing negative correlations in ISIs is an adaptive threshold with noise added to the spike threshold. A neuron becomes more resistant (refractory) to spiking immediately after a spike is output, with refractoriness increasing as more spikes are spaced closer together. Refractoriness reduces as spikes are spaced further apart. We show that a neuron can generate experimentally observed patterns of correlations by relating it to the noise in the spike threshold. Two different types of filtered noise (low-pass and high-pass) generate the observed patterns of correlations. We show that the theoretical sum of the sequence of correlations has a limiting value which maximizes the information a neuron can transmit. The observed sum of correlations is close to this limit.