Article Figures & Data
Figures
- Figure 1.
Spectral analysis of simple models of neuronal activity. A, Typical spike train generated by a single Poisson process. B, The interarrival times of spikes in the Poisson process are independent of earlier interarrival times, i.e., the Poisson process is memoryless. This implies that spikes are not correlated in time, and the autocorrelation function of the signal is zero for positive time lags. C, The power spectral density is the Fourier transform of the autocorrelation function and quantifies the amount of variance (“power”) per frequency component. For the Poisson process the power spectral density is flat. D, Three Poisson processes (#1–3) with distinct intensities (“firing rates”) contribute to the total signal (bottom). Only one process is active at each time, as determined by a hidden Markov process. The time series to be analyzed is the number of spikes in each fixed, small time interval along the time axis (interval length 2 ms, not shown). E, The autocorrelation function of the spike count time series falls off slowly for increasing time lags and indicates long-range correlations. F, The estimated power spectral density of the spike count time series seemingly follows a 1/fα power-law, as assessed by fitting a line (dashed, slope −α ≈ −1.8) in a double-logarithmic plot. Similar power law behavior is often seen in spectral densities from electrophysiological recordings. Note that both power spectral densities have been smoothed by a moving average filter for easier visualization.
Tables
Term Description 1/f noise A signal with a frequency content such that its power spectral density S(f) is proportional to a small power α of the reciprocal of the frequency, i.e., S(f) ≈ 1/fα, where the exponent α satisfies 0 ≤ α ≤ 3. Such signals are characteristic of complex processes without a preferred timescale, e.g., long-range-dependent signals. Continuous-time Markov chain A stochastic process X(t) in continuous time for which the conditional probability distribution at time t depends only on a finite number n of values X(s1), X(s2), …, X(sn) from its past, s1 < s2 < … < sn < t. Such a process is also called a process with finite memory. The special case where n = 0 is a memoryless process. Eigenvalue Principal component analysis decomposes the covariance matrix of a multivariate data set into an ordered set of eigenvectors along which the data varies the most. The corresponding eigenvalues represent the variance of the data explained by each eigenvector. Eigenvector coefficients The components of a principal mode (or eigenmode), i.e., of an eigenvector of the covariance matrix in principal component analysis. Hilbert transform A tool from signal processing that shifts the phase factors of a bandpass-filtered signal. Adding the Hilbert transform as imaginary part to the original signal results in an analytic signal, from which the instantaneous phase and the envelope of the signal can be calculated. Hurst exponent The Hurst exponent H (where 1/2 ≤ H ≤ 1) quantifies the self-similarity of a signal. A stochastic process X(t) is called self-similar if X(at) has the same probability distribution as aHX(t) for each a > 0. Long-range-dependent signals (e.g., arising from a process with a 1/f power spectral density) exhibit values of H close to 1. Long-range dependence A stochastic process is long-range dependent if its autocorrelation function R(τ) falls off slower than exponentially for large values of the lag τ. Such processes have infinite memory and typically give rise to power spectral densities with a 1/f distribution. Power spectral density The power spectral density S(f) is the Fourier transform of the autocorrelation function of a stochastic process X(t). It is closely related to the Fourier transform of the signal (which does not exist if X(t) is long-range dependent) and quantifies the frequency content of X(t), i.e., S(f) is the contribution to the variance (or ″power″) of X(t) at the frequency f. Projection weight The scalar product of the original signal with an eigenvector of the covariance matrix in principal component analysis, i.e., the coefficient the eigenvector has when the signal is written as a linear combination of all eigenvectors. This quantifies the amount the eigenvector in question contributes to the signal. Wavelet decomposition A time–frequency representation of a signal X(t) that quantifies its local frequency content, similar to windowed Fourier analysis. The signal X(t) is convolved with a family of short test signals that are rescaled and translated copies of a single ″mother″ signal (wavelet).
