7 Variability of Spike Trains and Neural Codes

7.4 Autocorrelation function and noise spectrum

Suppose that during a stationary input scenario, we observe a neuron ii firing a first spike at time tt. While the interval distribution P0(s)P_{0}(s) describes the probability that the next spike occurs at time t+st+s, the autocorrelation function C(s)C(s) focuses on the probability to find another spike at time t+st+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 Si(t)=fδ(t-ti(f))S_{i}(t)=\sum_{f}\delta(t-t_{i}^{(f)}) of length TT. The firing times ti(f)t_{i}^{(f)} might have been measured in an experiment or else generated by a neuron model. We suppose that TT is sufficiently long so that we can formally consider the limit TT\to\infty. The autocorrelation function Cii(s)C_{ii}(s) of the spike train is a measure for the probability to find two spikes at a time interval ss, i.e.

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

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

f(t)t=limT1T-T/2T/2f(t)dt.\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 Cii(-s)=Cii(s)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 𝒫(ω)=limT𝒫T(ω){\mathcal{P}}(\omega)={\rm lim}_{T\to\infty}{\mathcal{P}}_{T}(\omega), where 𝒫T{\mathcal{P}}_{T} is the power of a segment of length TT of the spike train,

𝒫T(ω)=1T|-T/2T/2Si(t)e-iωtdt|2.{\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 𝒫(ω){\mathcal{P}}(\omega) of a spike train is equal to the Fourier transform C^ii(ω)\hat{C}_{ii}(\omega) of its autocorrelation function (Wiener-Khinchin Theorem). To see this, we use the definition of the autocorrelation function

C^ii(ω)\displaystyle\hat{C}_{ii}(\omega) =-Si(t)Si(t+s)e-iωsds\displaystyle=\int_{-\infty}^{\infty}\langle S_{i}(t)\,S_{i}(t+s)\rangle\,e^{-% i\omega\,s}\,{\text{d}}s
=limT1T-T/2T/2Si(t)-Si(t+s)e-iωsdsdt\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
=limT1T-T/2T/2Si(t)e+iωtdt-Si(s)e-iωsds\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}
=limT1T|-T/2T/2Si(t)e-iωtdt|2.\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 TT\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 ω\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 𝒫(ω){\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 P0(s)P_{0}(s).