Blind Source Separation of Brain Signals

Blind source separation(BSS) is a well known problem which has implemantation in many different fields, like audio signals.

Abstract

Blind source seperation(BSS) is a well known problem which has implemantation in many diffrent fields, like audio signals. In the BSS problem we have unknown sources that are linearly mixed with an unknown mixing matrix, and need to be extracted by using only the mixtures. In this project we have looked for an algorithm that will give a novel solution for this problem for EEG signals.

The problem
The recording of the electroencephalogram signals from electrodes on the scalp is a noninvasive method for measuring the electrical activity of the brain. Two types of activity is recorded from the brain: background activity which includes all the activity which includes all the activity in the brain, and electrical activity evoked in response to an even(Event-Related Potentials, ERP). In our project we focused in seperating the signals in order to find the ERP parts of the signals.

The solution
BSS Problem and Solution In the BSS Problem we need to find the original signals s, that were mixed using the mixing matrix A, by using only the combinations x that were created.
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since we don’t know the distribution function of the sources we use an approximation for it:
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this approximation is known as the Quasi-Maximum Log Likelihood. The solution to this problem is to aproximate the function W which is the inverse of A. In order to achieve the best approximation possible we find W by minimizing the Maximum Log Likelihood function:
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In order to achieve better and faster results we used sparse represantation of the combination signals. We created the sparse reprasentation by using the diff function on artificial signals we created ourselfs, and the wavelet transform on the EEG signals.

Tools
Fminunc Method In this early stage we used the built in Matlab “fminunc” function in order to find the minimum of the QML function.
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As we can see the results were relatively fine, as we can see the reconstructed signals are very similar to the original signals. The problem is that the gradient of the QML function wasn’t close to zero, meaning that the true minimum was not found! Gradient Descent Algorithm Here we used an optimization method known as gradient decent, and the iterative update of W was:
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As we can see, the results using this algorithm were also good. The problem with this method was that the algorithm did not converge many times, or converged only after many iterations. Equivariant Adaptive Separation via Independence (EASI) This is a set of algorithms, based on serial updating of the calculated matrix W, and the update we implemented:
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In order to meet stability and convergence conditions g() should be a non linear function and we used the same function we used to approximate the distribution of the sources in the QML function. The advantage of this algorithm is that its’ performance does not depend on the mixing matrix A, the convergence and stability only depend on the distribution function of the sources.
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In this case we can see that the results were not so good since the reconstructed signals differ substantially from the originals. Here the artificial signals are longer than the ones we worked with in previous algorithms, Because the results depend on the length of the signal, and bigger signals are needed for convergence. Relative Newton Algorithm The algorithm includes the following steps until convergence is achieved: 1. Compute U(k) by using: 2. Find V(k) by minimizing L(V;U(k)) 3. Update the separation matrix W(k+1)=V(k+1)W(k) separation results:
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As we can see, the separation of the artificial signals was excellent. The computation time of this algorithm was very short. The matrix:
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is a very good approximation of the identity matrix. BSS of EEG Signals Using Relative Newton Algorithm From reasons described above we chose to separate real EEG signals by using the Relative Newton Algorithm. We first checked if the algorithm can separate an artificial signal which we added to the EEG signals, in the image below we can see the separated signals after adding a sinus and square wave to the mixed signals.
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Then we separated the EEG signals that were collected in the Evoked Potentials Lab in the Technion the sources:
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and the separated signals:
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The separated signals which we received look a lot like EEG signals, but with a small addition of noise. The peaks that can be seen in the separated signals can be attributed to a response to an event (such as blinking). However the separation of the artaficially added sinus and square wave did not go well, and although we can see that the shape of the added signal was reconstructed, the result signal contains to much noise.

Conclusions
We found that the Relative Newton algorithm gives rather good results in separating EEG signals. The algorithm is fast and therefore can be used by researchers for applications that use EEG. Future research can be done on pre-processing which will reduce the noise levels of the separated signals.

Acknowledgment
We are grateful to our project supervisor Michael Bronstein for his help and guidance throughout this work, we’d also like to thank Johanan Erez and all the lab staff, and finally we’d like to thank the evoked potential laboratory in the technion who gave us the EEG signals.
We are also grateful to the Ollendorf Minerva Center Fund for supporting this project.