Suppose that during a stationary input scenario, we observe a neuron firing a first spike at time . While the interval distribution describes the probability that the next spike occurs at time , the autocorrelation function focuses on the probability to find another spike at time – independent of whether this is the next spike of the neuron or not.
In order to make the notion of an autocorrelation function more precise, let us consider a spike train of length . The firing times might have been measured in an experiment or else generated by a neuron model. We suppose that is sufficiently long so that we can formally consider the limit . The autocorrelation function of the spike train is a measure for the probability to find two spikes at a time interval , i.e.
where denotes an average over time ,
We note that the right-hand side of Eq. (7.18) is symmetric so that holds. The calculation of the autocorrelation function for a stationary renewal process is the topic of the Section 7.5.2.
It turns out that the autocorrelation function is intimately linked to the power spectrum of a neuronal spike train, also called noise spectrum. The power spectrum (or power spectral density) of a spike train is defined as , where is the power of a segment of length of the spike train,
The power spectrum of a spike train is equal to the Fourier transform of its autocorrelation function (Wiener-Khinchin Theorem). To see this, we use the definition of the autocorrelation function
In the limit of , Eq. (7.20) becomes identical to (7.4) so that the assertion follows. The power spectral density of a spike train during spontaneous activity is called the noise spectrum of the neuron. Noise is a limiting factor to all forms of information transmission and in particular to information transmission by neurons. An important concept of the theory of signal transmission is the signal-to-noise ratio. A signal that is transmitted at a certain frequency should be stronger than (or at least of the same order of magnitude as) the noise at the same frequency. For this reason, the noise spectrum of the transmission channel is of interest. As we will see in the next section, the noise spectrum of a stationary renewal process is intimately related to the interval distribution .
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