Locally stationary wavelet coherence with application to neuroscience

Time series analysis is used extensively in neuroscience in order to study the interdependence between two simultaneously recorded signals (Pereda et al., 2005). Neurophysiological time series are inherently non-stationary, and the detection of changes in covariance structure is important as they reflect changes in the functional connectivity of the system. This, therefore, allows us to make inferences on how segregated areas of the brain are interacting. Our aim is to develop a method of localised coherence in order to analyse simultaneous recordings of neural activity taken from two areas of a rat’s brain: the hippocampus and the prefrontal cortex, as in the experimental set-up of Jones & Wilson (2005). While the cross-correlation function provides a natural estimate of the relationship between two series in the time domain, the cross-spectral density function can be used similarly in the spectral domain. The coherence function is derived from the normalisation of the crossspectrum by the individual spectra and, roughly speaking, measures the correlation between the signals as a function of frequency. The main problem with this approach is that it assumes stationarity. Windowed Fourier analysis allows for non-stationarity by splitting the signal into segments (Daubechies, 1992). Although this overcomes the assumption of global stationarity, it still requires stationarity within each section. Since wavelets are localised in both time and scale, they provide a natural approach to the modelling of series with time varying spectral characteristics (see Vidakovic (1999) for an introduction to wavelets). Unlike time resolved Fourier coherence which employs a constant window width for all frequencies, the wavelet transform uses shorter windows for higher frequencies, which leads to more “natural" localisation (see Daubechies (1992) for more on this topic). In this paper we propose a new measure of wavelet coherence termed ‘locally stationary wavelet coherence’. This is derived from the Locally Stationary Wavelet time series model of Nason et al. (2000). Following the work of Dahlhaus (1996), the model adopts the rescaled time principle, replacing the Fourier basis representation by a system of non-decimated wavelets. Due to the particular bias correction implied by the model, our new statistic differs significantly from wavelet coherence measures proposed previously.