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In recent years, there has been a growing need to analyze the functional connectivity of the human brain. Previous studies have focused on extracting static or time-independent functional networks to describe the long-term behavior of brain activity. However, a static network is generally not sufficient to represent the long term communication patterns of the brain and is considered as an unreliable snapshot of functional connectivity. In this paper, we propose a dynamic network summarization approach to describe the time-varying evolution of connectivity patterns in functional brain activity. The proposed approach is based on first identifying key event intervals by quantifying the change in the connectivity patterns across time and then summarizing the activity in each event interval by extracting the most informative network using principal component decomposition. The proposed method is evaluated for characterizing time-varying network dynamics from event-related potential (ERP) data indexing the error-related negativity (ERN) component related to cognitive control. The statistically significant connectivity patterns for each interval are presented to illustrate the dynamic nature of functional connectivity.

The human brain is known to be one of the most complex systems and understanding its connectivity patterns for normal and disrupted brain behavior remains as a challenge. Over the last decade, there has been a growing interest in studying brain connectivity. In literature, three kinds of brain connectivity have been addressed to define interactions between different regions of the human brain: anatomical connectivity, functional connectivity, and effective connectivity [

Functional connectivity can be inferred from different neuroimaging data such as the functional magnetic resonance imaging (fMRI), electroencephalography (EEG), and magnetoencephalography (MEG) [

A network is a mathematical representation of a real-world complex system with relational information and can be represented by a graph consisting of a set of vertices (or nodes) and a set of edges (or connections) between pairs of nodes. The presence of a connection between two vertices means that there is some kind of relationship or interaction between the nodes. In order to emphasize the strength of the connectivity between nodes, one can assign weights to each of the edges and the corresponding graph is called a weighted graph. In the study of functional brain networks, nodes represent the different brain regions and the edges correspond to the functional connectivity between these nodes which are usually quantified by the magnitudes of temporal correlations in activity. Depending on the measure, functional connectivity may reflect linear interactions such as correlation or nonlinear interactions such as phase synchrony [

Currently, topological features of functional brain networks such as clustering coefficient, path length, small world parameter [

In recent years, there has been an interest in characterizing the dynamic evolution of functional brain networks. Most of the existing approaches to dynamic network analysis are either graph theory based such as direct extensions of component finding [

Recently, researchers in signal processing have addressed problems in dynamic network analysis such as detection of anomalies or distinct subgraphs in large, noisy background [

In this paper, we first construct time-varying graphs, which are needed to describe the brain activity across time, by quantifying the time-varying phase synchrony between different electrodes of the EEG data [

Phase synchrony is defined as the temporal adjustment of the rhythms of two oscillators while the amplitudes can remain uncorrelated. The first step in quantifying the phase synchrony between two signals is to estimate the instantaneous phase of the individual signals,

Two major approaches to extracting the instantaneous phase are the Hilbert transform and the complex wavelet transform. The Hilbert transform-based method obtains an analytic form of the signal and estimates the instantaneous phase from this analytic form [

It is important to note some limitations of PLV that have been investigated in recent work, specifically in the context of intertrial phase synchrony [

Rihaczek distribution is a complex time-frequency distribution that provides both a time-varying energy spectrum as well as a phase spectrum with good time-frequency localization for phase modulated signals [

For multicomponent signals, cross-terms occur at the same time- and frequency locations as the original signals and will lead to biased energy and phase estimates. In order to eliminate these cross-terms, we proposed a reduced interference version of the Rihaczek-distribution, which is referred to as RID-Rihaczek, by applying a Choi-Wiliams (CW) kernel function to filter the cross-terms in the ambiguity domain [

Let

We propose a dynamic graph summarization framework consisting of constructing time-varying graphs from pairwise phase synchrony measure, identifying event windows, revealing the most important and informative connectivity patterns to summarize each event window with a key graph and to describe the dynamic evolution of the network over time.

In order to describe the evolution of time-varying connectivity patterns in the brain network, we first need to obtain the time-varying graphs. We quantify the bivariate relationship between nodes within the network and construct the time-varying graphs by considering the average synchrony within a frequency band at a certain time as

Once the time-varying graphs are obtained, we need to identify meaningful time intervals which may account for the underlying neurophysiological events such as error-related negativity or Pe event-related potential elicited in the process of decision making. For this purpose, we propose to quantify the change in node

After determining the event intervals, we need to form key graphs which best summarize the particular intervals. For this purpose, we need to distinguish between transient (high variance) and stationary (low variance) interactions within a given time interval and obtain a key graph which captures the transient or dynamic interactions. The ideal key graph should describe dynamic behavior of the particular interval with minimal redundancy. This is analogous to finding signal components that have low and high variance in a given data set and this separation in terms of variance is usually addressed through principal component analysis (PCA). Hence, we propose to employ PCA in order to extract key graphs and summarize the dynamics of the event intervals with minimal redundancy.

Let

Let the eigenvalues of the

In order to ensure minimal redundancy, we need to project the original data set,

Since the distribution of the interactions under the null hypothesis which form a key graph for a particular interval cannot be obtained analytically, we resort to generating random networks to derive this distribution. For each key graph extracted for a given time interval, we derived an ensemble of 2000 surrogate time-varying networks by randomly reshuffling the edge weights [

To evaluate the performance of the proposed measure in summarizing the event intervals with biological data, we use a set of EEG data containing the error-related negativity (ERN). The ERN is an event-related potential that occurs following performance errors in a speeded reaction time task [

There has been longstanding interest in time-frequency representations of the ERN [

In this paper, we analyzed data from 90 subjects corresponding to the error responses. For each subject, time- and frequency dependent phase synchrony between all possible electrode pairs is computed by RID-Rihaczek-based PLV measure and time-varying graphs,

Event interval detection: 6 event intervals are identified which roughly correspond to the stimulus processing (−1000 to −102 ms), pre-ERN (−101 to 0 ms), ERN (1 to 117 ms), post-ERN (118 to 259 ms), Pe (260 to 461 ms), and intertrial (462 to 1000 ms) intervals, respectively. The subjects respond to the stimulus at time 0 ms where the response is represented by the red spike.

The detected event intervals are consistent with the speeded reaction-time task as the subjects respond to the stimulus at time 0 ms. The first interval indexes complex processing of the imperative stimulus before making a response. The Pre-ERN and Post-ERN intervals, just before and after the ERN, index activity around the incorrect motor response. Importantly, the ERN interval (117 ms time window after the response) and Pe interval (260–461 ms time window) are detected successfully by the event detection algorithm. The Pe (error-positivity) interval corresponds to a P3-like component observed subsequent to the incorrect response [

For each event interval detected from the mean time-varying graph sequence,

For each event interval, a key graph is obtained using the framework described in Section

Stimulus processing

Pre-ERN

ERN

Post-ERN

Pe

Intertrial

We also focused on the change in connectivity for FCz electrode with the remaining 61 electrodes within the key graphs for Pre-ERN, ERN, and Post-ERN intervals and compared these connectivity values to identify if FCz has stronger connectivity during the ERN interval compared to the Pre-ERN and Post-ERN intervals. We used a Welch’s

In this paper, we proposed a new framework to summarize the dynamic evolution of brain networks. The proposed approach is based on finding the event intervals and revealing the informative transient or dynamic interactions within each interval such that the key graph would summarize the particular interval with minimal redundancy. Expectable results from the application to real EEG data containing the ERN supports the effectiveness of the proposed framework in determining the event intervals of dynamic brain networks and summarizing network activity with a few number of representative networks.

Future work will concentrate on exploring different event interval detection and key graph extraction criteria such as entropy-based divergence measures and Bayesian approaches such as the one discussed in [

This work was in part supported by the National Science Foundation under Grant no. CAREER CCF-0746971.