2.1 –Connectivity maps

The motivation for the proposed connectivity analysis is best explained by the following analysis of airline traffic. Consider the air traffic from BOS to ATL and BOS to DEN. A classical connectivity analysis of the nodes ATL and DEN would examine the connection between these two cities. For the sake of argument, let us assume that all the flights of a certain airline from BOS to DEN go through ATL. We consider a more refined, high dimensional notion of connectivity by analyzing the changes in connectivity between ATL and DEN with reference to flights that emanated from other locations, such as BOS. By observing that there was a reduction in connectivity between BOS and DEN but no change in or even an increase in connectivity between BOS and ATL. In other words, we may observe that when the flights of one airline company that flies from BOS to DEN via ATL are canceled, it is possible that those passengers book alternative flights with another carrier due to the cancellations. Note that the traffic between ATL and DEN includes flights by all other carriers; thus, it would be difficult to observe the reduction in traffic due to cancellations by a single carrier. By finding a city, BOS, from which a given carrier’s flights to DEN only go through ATL, a cancellation becomes more apparent.

In Electroencephalography, such technique may reveal the same type of significant insights, while the nodes in this case are simply the electrodes placed on the scalp and a reference-based connectivity function is required to measure the relationships between them. Therefore, in the same way, we analyze the connectivity between electrodes A and B (Fig 1) by examining either the direct connectivity or information transfer between A and B (left panel) or the induced change in connectivity between A and B, as seen by the change in the information transfer between reference electrode C and A with respect to the information transfer between C and B.

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larger image TIFF original image Download: Fig 1. In addition to the direct connectivity approach (left panel), the connectivity maps approach brings features involving a reference electrode when addressing bi-electrode connectivity. Adding such an electrode expands the connectivity approach, as the relative changes in connectivity are uncovered. https://doi.org/10.1371/journal.pone.0185852.g001

Our proposed discrimination and classification mechanism is based on a novel connectivity analysis tool that we term “Connectivity Maps”. These maps are generated in a relatively simple process that consists of the following 6 steps:

Several preprocessing tasks are performed, and the raw signals are broken into relevant intervals. Technical description of the preprocessing phase is detailed below under “Experimental Setup”. The signal in several frequency bands is decomposed and reconstructed, and then time windows that maximize the correlation between electrodes are sought. A network of correlations with respect to different base or reference electrodes is constructed. Fisher-based feature extraction (See below definitions for Fisher and Relative Fisher matrices) from the set of connectivity maps obtained for the different reference electrodes is performed. An optimization analysis is conducted based on the chronological presence of stimuli. Post analysis and processing of the results is performed, including discrimination between the correlation matrix and Fisher-based features of the healthy subjects and those of the schizophrenia patients. In addition to the resulted parameters, this step is also performed under different constraints, such as specific time frames or following specific stimuli, for a broader statistical analysis.

These steps are shown in Fig 2 below.

Given a window size W, a maximum phase φ and a number of trials i ∈ {1..t} for an electrode, let us define a connectivity function F conn between two electrodes, such as a cross correlation, as

F conn can be replaced with any other metric of bi-electrode connectivity. For example, the average-over-trials function could be replaced with the corresponding max or median function.

For each electrode, two initial connectivity maps are created and presented using matrices. The correlation connectivity map for electrode A is defined by where N is the total number of subjects and W ∈ {50 ms, 100 ms, 150 ms, 200 ms}. Such a cross-correlation function with different possible window sizes alongside a small phase limit φ can be efficient in detecting synchronization of the flow of information in a system.

In a similar way, a standard deviation matrix map can be defined as follows:

CM_STD A (B) = Std subject F Conn (A,B). A healthy-schizophrenia patients Fisher-score matrix can then be created using the following equation:

Feature matrices are then created for additional features, for prediction and classification. Three characteristics were used to create these additional features: 1. Relative distances: the differences in the direct connectivity map are computed with respect to the reference electrode (see Fig 1). The relative distances are defined by

2. Feature multiplication: Multiplication elements that are extensions of the relative distances are defined. As reference electrode is shared by both elements of each multiplication matrix, these features amplify contribution of such successful, “multi-purpose” reference points. For each pair of relative elements, as defined above, a multiplication element is created, defined as follows:

3. Fisher relative score features matrix: This matrix is defined by

The feature classification procedure collects only the most statistically significant discriminating features rather than the holistic network connectivity level. The procedure is as follows:

Set the parameters K 1 , K 2 with the goal of limiting the number of electrodes involved in the process of interest. Construct a feature matrix F of size K 1 electrodes x K 2 features by letting each row F i represent the source electrode i and its derived features. Steps 2–6 were tested with K 1 chosen from {0.1N, 0.2N,.., N}, and K 2 from {5%,10%}. The elements of row i of the feature matrix are all features obtained from electrode i, as defined above: To increase the clarity of the maps, only the top 10% of features in terms of healthy-schizophrenia patients variance were retained [25]. For each row, calculate the average of the maximal K 1 elements. Choose the K 2 features that have maximal scores. Use Laplacian regularization for feature scoring. Alternatively, another feature scoring method may also be used [26]. Analyze the projected results for new segments of acquired data using a statistical model that features 2 states or models; compare the likelihood of each model (healthy individuals or schizophrenia patient) using Chi-squared tests followed by model comparison [27]. Classify the feature set according to the likelihoods of the two models.

Post-procedure validation: Perform cross-validation between patients’ data using the ‘leave one out’ method [28]. In each phase, one subject’s data is left out of the training set, and all properties are computed without using this data. Testing and predictions are then conducted with the data that was left out, treating this data as a fresh, newly acquired dataset.