7.4 Autocorrelation function and noise spectrum

Suppose that during a stationary input scenario, we observe a neuron $i$ firing a first spike at time $t$. While the interval distribution $P_{0}(s)$ describes the probability that the next spike occurs at time $t+s$, the autocorrelation function $C(s)$ focuses on the probability to find another spike at time $t+s$ – 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 $S_{i}(t)=\sum_{f}\delta(t-t_{i}^{(f)})$ of length $T$. The firing times $t_{i}^{(f)}$ might have been measured in an experiment or else generated by a neuron model. We suppose that $T$ is sufficiently long so that we can formally consider the limit $T\to\infty$. The autocorrelation function $C_{ii}(s)$ of the spike train is a measure for the probability to find two spikes at a time interval $s$, i.e.

 $C_{ii}(s)=\langle S_{i}(t)\,S_{i}(t+s)\rangle_{t}\,,$ (7.18)

where $\langle\cdot\rangle_{t}$ denotes an average over time $t$,

 $\langle f(t)\rangle_{t}=\lim_{T\to\infty}{1\over T}\int_{-T/2}^{T/2}f(t)\,{% \text{d}}t\,.$ (7.19)

We note that the right-hand side of Eq. (7.18) is symmetric so that $C_{ii}(-s)=C_{ii}(s)$ 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 ${\mathcal{P}}(\omega)={\rm lim}_{T\to\infty}{\mathcal{P}}_{T}(\omega)$, where ${\mathcal{P}}_{T}$ is the power of a segment of length $T$ of the spike train,

 ${\mathcal{P}}_{T}(\omega)={1\over T}\left|\int_{-T/2}^{T/2}S_{i}(t)\,e^{-i% \omega\,t}\,{\text{d}}t\right|^{2}\,.$ (7.20)

The power spectrum ${\mathcal{P}}(\omega)$ of a spike train is equal to the Fourier transform $\hat{C}_{ii}(\omega)$ of its autocorrelation function (Wiener-Khinchin Theorem). To see this, we use the definition of the autocorrelation function

 $\displaystyle\hat{C}_{ii}(\omega)$ $\displaystyle=\int_{-\infty}^{\infty}\langle S_{i}(t)\,S_{i}(t+s)\rangle\,e^{-% i\omega\,s}\,{\text{d}}s$ $\displaystyle=\lim_{T\to\infty}{1\over T}\int_{-T/2}^{T/2}\,S_{i}(t)\,\int_{-% \infty}^{\infty}\,S_{i}(t+s)\,e^{-i\omega\,s}\,{\text{d}}s\,{\text{d}}t$ $\displaystyle=\lim_{T\to\infty}{1\over T}\int_{-T/2}^{T/2}\,S_{i}(t)\,e^{+i% \omega\,t}\,{\text{d}}t\,\int_{-\infty}^{\infty}S_{i}(s^{\prime})\,e^{-i\omega% \,s^{\prime}}\,{\text{d}}s^{\prime}$ $\displaystyle=\lim_{T\to\infty}{1\over T}\left|\int_{-T/2}^{T/2}S_{i}(t)\,e^{-% i\omega\,t}\,{\text{d}}t\right|^{2}\,.$ (7.21)

In the limit of $T\to\infty$, 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 $\omega$ 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 ${\mathcal{P}}(\omega)$ 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 $P_{0}(s)$.