Methods

The intention here is to explore the effects of temporal filtering and the spatial variation of the autocorrelations in real FMRI data. We could attempt to examine the autocorrelation, or equivalently the power spectral density, itself. However, this would give $N$ (number of scans or time points) data points for each voxel. Instead, we use:

$$S_{\rho}=N/[1+2\sum_{\tau=1}^{N-1}\rho_{xx}(\tau)]$$ (18)

This produces a single value whose variation can then be easily visualised. The value $S_{\rho }$ corresponds approximately to $1/k_{\mbox{\scriptsize {\emph{eff}}}}$ in equation 7 when performing a t-test ($c=[1]$ and $X=[1,...,1]^T$) with large $N$.

An $S_{\rho}=n$ indicates white noise and $0<S_{\rho}<n$ indicates a time series with positive autocorrelation. We examined one rest/null dataset from a normal volunteer. Two hundred echo planar images (EPI) were acquired using a 3 Tesla system with time to echo (TE) = 30ms, TR=3 secs, in-plane resolution 4mm and slice thickness 7mm. The first 4 scans were discarded to leave $N=196$ scans and the data was motion corrected using AIR (Woods et al., 1993). To calculate $S_{\rho }$ at each voxel we arbitrarily used the nonparametric PAVA autocorrelation approach to estimate the autocorrelation.