Tracking the time-varying cortical connectivity patterns by adaptive multivariate estimators

In the latest years, the problem of the definition and estimation of brain connectivity has become a central one in Neuroscience, as a way to understand the organization and interaction of cortical areas during the execution of cognitive or motor tasks. Among various methods established during the years, the Directed Transfer Function (DTF) and the Partial Directed Coherence (PDC) are frequency-domain approaches, based on the multivariate autoregressive modeling (MVAR) of time series, and able to describe interactions between cortical areas in terms of the concept of Granger causality. Anyway, the classical estimation of these methods requires the stationarity of the signals. In this way, transient pathways of information transfer remains hidden. The objective of this study is to test a time-varying multivariate method for the estimation of rapidly changing connectivity relationships between cortical areas of the human brain, based on DTF/PDC and on the use of adaptive MVAR modeling. This approach overcomes the requirement of stationarity of the signals, thus allowing the observation of transient influences between the cortical areas during the execution of a task. Time-varying DTF and PDC were obtained by the adaptive recursive fit of an MVAR model with time-dependent parameters, by means of a generalized recursive least-square (RLS) algorithm, taking into consideration a set of EEG epochs. The multi-trial RLS algorithm involves the information of the actual past of the signal, with a weighting factor tuning how data in the distant past affect the estimation of the present sample. Simulations were performed under different levels of Signal to Noise Ratio (SNR), number of trials (TRIALS) and frequency bands (BAND), and of different values of the RLS adaptation factor adopted (factor C), and were followed by an analysis of variance (ANOVA) of the performances. The results indicated that during simulations, time-varying DTF and PDC are able to estimate correctly the imposed connectivity patterns under reasonable operative conditions of SNR ad number of trials. Moreover, the capability to follow the rapid changes in connectivity is highly increased by the number of trials at disposal, and by the right choice of the value adopted for the adaptation factor C. Fig.1 shows the results of the ANOVA on the settling time during a step variation in connectivity, for different values of the adaptation factor C and of the number of trials, for the DTF and PDC estimators. It can be noted that a convenient choice of C can assure a tradeoff between the estimation speed and variance. Fig.2 shows the results of the error on the estimation of the strengths of different arcs of the connectivity model. In Fig.3 the time-frequency distribution of the DTF related to three cortical areas is shown. The connectivity model is the same as shown in Fig.2, with a change of connectivity in arc 1->3. In conclusion, the results of the simulation study indicate that DTF and PDC computed on adaptive MVAR can be effectively used to estimate time-varying patterns of functional connectivity between cortical activations, under general conditions met in practical EEG recordings.