Serial interspike interval correlations of excitable neurons with memory

Abstract

Many theoretical studies of spiking neurons rely on thesimplifying renewal assumption, meaning that interspikeintervals (ISIs) of a spike train are statistically independ-ent. Experimental measurements of serial ISI correlationshave revealed, however, that the renewal assumption isviolated in various systems. For instance, strong but short-lived, negative correlations have been reported for P-unitsof weakly electric fish [1,2] and for pyramidal neurons inrat entorhinal cortex [3]. For a review, see [4]. Positive cor-relations that extend over many ISIs have been observedin [5]. Neurons exhibiting spike-frequency adaptationtypically display negative serial correlations, whereas slowinput variations typically induce positive serial correla-tions [6].Theoretical work has gained much insight into the benefi-cial role of serial correlations for signal transmission [7].In that case, the calculation of the serial correlation coef-ficient (SCC) could be accomplished by the use of a par-ticularly simple neuron model that operates in theoscillatory regime and features a dynamic firing threshold.In general, however, the SCC is difficult to access analyti-cally, which is due to the non-Markovian nature of non-renewal neuron models. Here, we present a novel tech-nique that allows us to calculate the SCC of a large class ofnon-renewal neurons operating in the excitable regime.Specifically, we consider neurons with discrete internalstates or discrete states of the external driving function.The analytical approach is based on a generalization ofthe discrete kinetic scheme used to investigate residencetime correlations in driven bistable systems [8,9].Having established the general method to obtain the SCCfor arbitrary lags between ISIs, we consider two specialcases that are of particular interest. The first case mimics aleaky integrate-and-fire (LIF) neuron with dynamicthreshold, which has been proposed as a model for spike-frequency adaptation. By discretizing the threshold intothree states, we find a negative SCC at lag one and vanish-ing correlations at higher lags. In the second case, we con-sider a neuron that participates in a network that switchesbetween Up and Down states. The theory reveals positiveserial correlations that decay exponentially with the lag.Interestingly, for slow two-state synaptic input the SCC atlag one is maximized at an optimal amplitude of the two-state driving. Our theoretical expressions agree well withextensive simulations of noisy LIF neurons. Finally wemention, that an analytical formula for the SCC could beuseful to model measured data by tuning the parametersof the discrete neuron model.