A Tensorial Approach to Access Cognitive Workload related to Mental Arithmetic from EEG

Functional Connectivity Estimates

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

AIIA-Lab, Informatics dept., Aristotle University of Thessaloniki, GreeceSINAPSE, National University of Singapore

Temasek Laboratories, National University of Singapore

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OutlineIntroduction

• 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

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Methodology • Brain is a complex network• Functional Connectivity Graphs (FCGs) were treated as vectors • FCG can be treated as a tensor• Tensor Subspace Analysis

ResultsConclusions

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FUNCTIONAL/EFFECTIVE MEASURES

ΕΕG & CONNECTOMICS

Intro Method Results Conclusions

FROM MULTICHANNEL RECORDINGS TO CONNECTIVITY GRAPHS

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COMPLEXITY IN THE HUMAN BRAINEEG & CONNECTOMICS

Intro Method Results Conclusions

Motivation and problem statementIntro Method Results Conclusions

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• 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 1. would significantly decrease the computational cost and 2. facilitate the differentiation of brain states

Intro Method Results Conclusions

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FCG matrixTENSOR

VECTOR

Outline of our methodology

The linear Dimensionality Reduction Problem in Tensor Space

Intro Method Results Conclusions

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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:1. 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

11 ln 22 ln ),(,..., 2211

211 nlnlYY ll

m

211,..., nn

mXX

iYVXUY i

Ti iX

The method is of particular interest in the special case whereand is a nonlinear sub-manifold embedded in

MXX m ,...,1

Optimal Linear Embedding

Intro Method Results Conclusions

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• 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)

1. Given m data points sampled from the FCG sub-manifold , one can build a nearest neighbor graph G to model the local geometrical structure of M 2 . Let S be the weight matrix of G . A possible definition of is as follows:

1{ ,..., }mX X X 1 2n nM

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j i If X is among the k nearest neighbors of X (1)0 Otherwise

i jX Xt

ijeS

The function is the so called heat kernel which is intimately related to

the manifold structure AND |.| is the Frobenius norm of matrix.

2i jx x

te

Optimal Linear Embedding

Intro Method Results Conclusions

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In the case of supervised learning (classification labels are available), the label information can be easily incorporated into the graph as follows:

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j iIf X and X share the same label; (2)0 Otherwise

i jX Xt

ijeS

2

,min (3)T T

i j ijU V ij

U X V U X V S

ii ijjD S

U U U(D - S )V D V (4)

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 apartWith 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

V V V(D - S )V λD V (5)

Thus, the optimal and can be obtained by iteratively computing the generalized eigenvectors of (4) and (5)

Data acquisition: Mental Task (summation)

1 subject64 EEG electrodesHorizontal and Vertical EOG> 40 trials for each condition

5 Different difficult levelse.g. 3+4 … 234 + 148

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Intro Method Results Conclusions

Preprocessing

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Intro Method Results Conclusions

• 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:

, ,

1,

( )1 (6)

( 1)i j h i

jhj h G

i N i i

dLE

N k k

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 d set the shortest absolute path length between every possible pair in the neighborhood of the current node

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Intro Method Results Conclusions

Brain Topographies and Statistical Differences of LE between L1 and L5

BRAIN TOPOGRAPHIES OF NODAL LE

Trial–averaged nodalLE measurements (across various frequency bands) for PO7 and PO8 sensors

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Intro Method Results Conclusions

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

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Machine learning validation

Intro Method Results Conclusions

1. The TSA algorithm, followed by a k-nearest-neighbor classifier (with k=3), was tested on trial-based connectivity data from all bands

2. 10 – fold Cross – Validation scheme (90% for training and 10% for testing)3. The following table summarizes the average classification rates derived after applying

the above cross-validation scheme 100 times

BILATERALLY PO Left PO Right PO

TSA LDA TSA LDA TSA LDA 93.59 83.28 87.81 80.39 91.68 79.31 96.70 85.65 89.86 81.29 90.40 77.411 96.13 86.42 91.13 82.14 90.27 80.282 95.00 84.12 90.31 79.41 87.63 79.311 94.40 83.49 87.63 78.39 86.54 76.402 97.59 87.21 83.86 76.25 85.86 76.05 95.86 87.34 84.86 75.83 87.54 76.91

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

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Conclusions

Intro Method Results 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