Most functional Magnetic Resonance Imaging (fMRI) studies implicitly assume that the functional connectivity (FC) between brain regions is constant over time. However, FC has been shown to be non-stationary. Studies indicate that dynamic FC (DFC) contains neurobiologically meaningful information not available in conventional static connectivity [1]. Evaluation of DFC involves successive sliding-windows, with correlation being evaluated in each of them. One of the critical issues concerning sliding-window analysis is the choice of window size, with arbitrary choice of fixed window size leading to arbitrary results [2]. Techniques based on Dynamic Conditional Correlation (DCC) [3] and timeseries stationarity [1] have been developed lately to address this issue. They search for the window length within a predefined range which satisfies certain mathematical criteria. However the choice of minimum window length (MWL) from which to start the search is an arbitrary small number. While we would like the window to be as small as possible so that maximum dynamics is captured, the main issue is that the correlation estimated for such small windows might not be reliable, that is, the estimated correlation might deviate largely from the ground-truth correlation. It is thus necessary to objectively assess the impact of window length on the reliability of estimated correlation. In this work, we propose a novel strategy for selecting MWL. Since the ground-truth can be precisely controlled in artificial data, we performed simulations. Using the error in evaluated correlation compared to ground-truth correlation as the metric, we compared our method with (i) the popular arbitrary fixed window length approach, and (ii) the DCC method [3], which is model-based and requires no ad-hoc choices. Note that while ours is a method to obtain MWL required to give reliable correlation, DCC and fixed-window approaches are DFC methods which do not consider reliability of correlations in their formulation. Hence we hypothesize that our method would result in minimum window lengths which give more reliable correlation estimates than those obtained from DCC and fixed-window methods.

The MWL required to give reliable correlations can be obtained only when the ground-truth correlation is known, which is possible through simulations. We simulated timeseries pairs with 1000 time points each, with predefined correlation between them. As in our pervious study [1], we used a multivariate vector autoregressive (MVAR) model to generate pairs of timeseries as follows:

$$Y(t)=\sum_{i=1}^pA_{i}*Y(t-i)+\epsilon$$ where $$$\epsilon$$$ represents noise vector with covariance matrix $$$C=\begin{bmatrix}1 & {v(t)} \\{v(t)} & 1 \end{bmatrix}$$$ and $$$Y(t)=\begin{bmatrix}{y_{1}(t)} & {y_{2}(t)}\end{bmatrix}$$$ denotes simulated time series. $$$A_{i}$$$ is regression coefficient matrix for delay $$$i$$$ , chosen to be zero matrices such that only zero-lag correlation (no time-lagged relationships) were considered. We simulated timeseries pairs with constant predefined correlation between them($$$v(t)=constant$$$), with correlation being varied from 0 to 1 with step-size of 0.05. Window length was varied from 5 to 100 with step-size of 1, and DFC was evaluated over the range of window lengths. We used sliding-windowed Pearson’s correlation to evaluate DFC. This procedure was iterated 100 times. Employing the widely used Dickey-Fuller (DF) test [4], we obtained MWL required to ensure reliable correlation, that is, to ensure stationarity of correlation values with 95% confidence.

Having calculated the MWL required to estimate correlation reliably, we performed simulations to validate our minimum stationary window length (MSWL) method and compared it with DCC and fixed-window methods. We considered two scenarios as follows. Case-1:$$$v(t)=0.5+0.3*\sin(0.01t)$$$ , which represents a slow and less-varying periodic change in correlation, and case-2:$$$v(t)=0.5+5*\sin(0.5t)$$$,which represents a fast and largely varying periodic change in correlation. We iterated 100 times and compared the error between estimated correlations and ground-truth correlation for all methods.

The reliability with which the estimated correlation from simulated data matched the ground-truth correlation value (in terms of estimation error in Fig.1 and correlation stationarity in Fig.2) depended on the window length and the simulated correlation value itself, with shorter windows and smaller simulated correlations leading to larger error and less reliability/stationarity. We then obtained the relationship between MWL and correlation (Fig.3), which showed that the highest value of MWL (=45 time points) was obtained with a correlation around 0. Results for case-1 (Fig.4) showed that all methods gave comparable performances when correlation was nearly constant over time. However with case-2 (Fig.5), wherein correlation changes were faster like in real data, our MSWL method performed the best with lowest error and error variability. This suggests that MSWL can automatically determine MLWs during the evaluation of DFC successfully, thus giving highly reliable connectivity estimates. These findings should be taken into consideration in the evaluation of sliding-window based dynamic connectivity.