S.I. Dimitriadis , Yu Sun , K. Kwok , N.A. Laskaris , A. Bezerianos

1. A Tensorial Approach to Access Cognitive Workload related to Mental Arithmetic from EEG Functional Connectivity Estimates AIIA-Lab, Informatics dept., Aristotle University of Thessaloniki, Greece SINAPSE, National University of Singapore Temasek Laboratories, National University of Singapore S.I. Dimitriadis, Yu Sun, K. Kwok, N.A. Laskaris , A. Bezerianos

2. Outline • Introduction • functional connectivity have demonstrated its potential use • for decoding various brain states • high-dimensionality of the pairwise functional connectivity • limits its usefulness in some real-time applications • Methodology • Brain is a complex network • Functional Connectivity Graphs (FCGs) were treated • as vectors • FCG can be treated as a tensor • Tensor Subspace Analysis Results Conclusions

3. ΕΕG & CONNECTOMICS FROM MULTICHANNEL RECORDINGS TO CONNECTIVITY GRAPHS FUNCTIONAL/EFFECTIVE MEASURES

4. EEG & CONNECTOMICS COMPLEXITY IN THE HUMAN BRAIN

5. Motivation and problem statement • Association of functional connectivity patterns with particular cognitive tasks has been a hot topic in neuroscience • studies of functional connectivity have demonstrated its potential use for decoding various brain states e.g. mind reading • the high-dimensionality of the pairwise functional connectivity limits its usefulness in some real-time applications • We used tensor subspace analysis (TSA) to reduce the initial high-dimensionality of the pairwise coupling in the original functional connectivity network • which • would significantly decrease the computational cost • and • 2. facilitate the differentiation of brain states

6. Outline of our methodology FCG matrix TENSOR VECTOR

7. The linear Dimensionality Reduction Problem in Tensor Space Tensor Subspace Analysis (TSA) is fundamentally based on Locality Preserving Projection • X can be thought as a 2nd order tensor (or 2-tensor) in the tensor space • The generic problem of linear dimensionality reduction in the second order space is the following: • Given a set of tensors (i.e. matrices) • find two transformation matrices U of size and V of that maps these tensors to a set of tensors such that “represents” , where The method is of particular interest in the special case where and is a nonlinear sub-manifold embedded in

8. Optimal Linear Embedding • The domain of FCGs is most probably a nonlinear sub-manifold embedded in the tensor space • Adopting TSA, we estimate geometrical and topological properties of the sub-manifold from “scattered data” lying on this unknown sub-manifold. • We will consider the particular question of finding a linear subspace approximation to the sub-manifold in the sense of local isometry • TSA is fundamentally based on Locality Preserving Projection (LPP) Given m data points sampled from the FCG sub-manifold , one can build a nearest neighbor graph Gto model the local geometrical structure of M 2 . Let S be the weight matrix of G . A possible definition of is as follows: The function is the so called heat kernel which is intimately related to the manifold structure AND |.| is the Frobenius norm of matrix.

9. Optimal Linear Embedding In the case of supervised learning (classification labels are available), the label information can be easily incorporated into the graph as follows: Let and be the transformation matrices. A reasonable transformation respecting the graph structure can be obtained by solving the following objective functions: The objective function incurs a heavy penalty if neighboring points and are mapped far apart With D be a diagonal matrix the optimization problem was restricted to the following equations Once is obtained, can be updated by solving the following generalized eigenvector problem Thus, the optimal and can be obtained by iteratively computing the generalized eigenvectors of (4) and (5)

10. Data acquisition: Mental Task (summation) 1 subject 64EEG electrodes Horizontal and Vertical EOG > 40 trials for each condition Different difficult levels e.g. 3+4 … 234 + 148

11. Preprocessing • ICA (Independent Component Analysis) • Signals were filtered within frequency range of 0.5 to 45 Hz (from δ to γ band) • the data from level 1 (addition of 1-digit numbers – 187 trials) and level 5 (addition of 3-digit numbers – 59 trials) was used for the presentation of TSA • Initial FCG Derivation and standard network analysis • Functional Connectivity Graphs (FCGs) were constructed using a phase coupling estimator called PLV • We computed local efficiency (LE) which was defined as: where in ki corresponded to the total number of spatial directed neighbors (first level neighbors) of the current node, N was the set of all nodes in the network, and dset the shortest absolute path length between every possible pair in the neighborhood of the current node

12. Brain Topographies and Statistical Differences of LE between L1 and L5 BRAIN TOPOGRAPHIES OF NODAL LE Trial–averaged nodalLEmeasurements (across various frequency bands) for PO7 and PO8 sensors

13. Trimmed FCGs • We reduced the input for tensor analysis from FCGs of dimension N x N to sub-graphs of size N’ x N’ (with N’=12sensors corresponding to the spatial neighborhood of PO7 and PO8 sensors) • The total number of connections between sensors belonging in the neighborhood is: 66/2016≈3% of the total number of possible connections in the original graph The distribution of PLV values over the parieto-occipital (PO) brain regions and the topographically thresholded functional connections

14. Machine learning validation The TSA algorithm, followed by a k-nearest-neighbor classifier (with k=3), was tested on trial-based connectivity data from all bands 10 – fold Cross – Validation scheme (90% for training and 10% for testing) The following table summarizes the average classification rates derived after applying the above cross-validation scheme 100 times • The selected options for TSA were: • Weight mode=Ηeat Kernel • Neighbor mode=1 • Supervised learning • Number of dimensions=3 Table 1 summarizes the classification performance of TSA+knn and of LDA+knn (Linear Discriminant Analysis) as a baseline validation feature extraction algorithm

15. Conclusions • We introduced the tensorial approach in brain connectivity analysis • The introduced methodological approach can find application in many other situations where brain state has to be decoded from functional connectivity structure • After extensive validation, it might prove an extremely useful tool for on-line applications of : Human-machine interaction Brain decoding Mind reading