The resemblance of an autocorrelation function to a power spectrum density for a spike train of an auditory model

Abstract

Abstract. In this work we develop an analytical approach for calculation of the all-order interspike interval density (AOISID), show its connection with the autocorrelation function, and try to explain the discovered resemblance of AOISID to the power spectrum of the same spike train.Keywords: power spectrum, autocorrelation, inter-spike interval density PACS: 87.19.lc, 87.19.ll, 43.80.+p Introduction. As it is well-known, the Fourier transformation allows imaging a signal as a sum of sinusoidal components. In case of a spike train, it seems to be not consequent to consider a sequence of sharp pulses as a sum of smooth sinusoids. Apparently, this is one of the reasons why neurophysiologists use the histogram of interspike intervals, in particular, all-order interspike interval density (AOISID), more often, then the power spectrum density (PSD). The power spectrum of a short in time pulse inevitably contains high frequency components, which do not have anything to do with interspike intervals. So, the spectrum provides just the redundant information about a pulse shape, which does not play a role in an inter-neuron communication. Actually, in the past, before the fast Fourier transformation algorithm invention and its wide applications, the usual tool for the signal analysis was a correlometer providing the autocorrelation function (ACF), which is stated [1] to be directly connected with AOISID.In the presented work we describe a quite unexpected connection between ACF and PSD observed in the auditory system models. The connection between ACF and AOISID is also rigorously derived here.The presented study has been motivated by the discovery of the resemblance of the PSD at the output of a simple neural model [2] to the AOISID at the output of much more complex model [1] of the same auditory system of mammals with similar parameters of input signals (Fig. 1A). The problem of the analysis of this resemblance was that we had analytical expressions for the PSD of the simple model, but did not have ones for the AOISID. As per the complex model, here we had only AOISID plots and a not clear enough statement about the direct correspondence between AOISID and ACF.Model description. The mentioned PSD has been calculated for the model, which is described in details in the paper [2]. It consists of three Leaky Integrate-and-Fire neurons, two of which are the input (sensory) elements and are driven by sinusoidal signals,