# SOS-MUSIC, a subspace approach for EEG source imaging promoting Sparsity On active Sources

Joud C.¹,², Albera L¹, Merlet I¹, Coloigner J²

1. University of Rennes, Inserm, LTSI-UMR 1099, Rennes, France

2. University of Rennes, CNRS, INRIA, Inserm, IRISA, UMR 6074, EMPENN - ERL U 1228, Rennes, France

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## INTRODUCTION

### Context

* Localizing brain epileptic sources for patients with **treatment-resistant** epilepsy (30% of cases).

### Solving the EEG inverse problem

<figure><img src="image_3.png" /><figcaption>Source Signals</figcaption></figure>

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Where:

* **X**: the EEG data matrix of size (N x T)

* **S**: the signal matrix of size (Dx T)

* **G**: the **lead field** matrix of size (N x D)

with

N: The number of EEG electrodes

D: The number of source dipoles

T: The number of time samples

### MUSIC (Multiple Signal Classification)[1]: a subspace approach

* Decomposition of the observed signal into **signal** and **noise orthogonal** subspaces.

**2-MUSIC metric:**

$$ \Upsilon(g(\theta)) = \frac{g(\theta)^T E_{\nu} E_{\nu}^T g(\theta)}{g(\theta)^T g(\theta)} $$

Where:

* $g(\theta)$ is the column of $G$ of the $\theta$ source;

* $E_{\nu}$: the noise subspace;

* The $\theta$ minimizers of $\Upsilon$ give the epileptic sources.

## RESULTS

### Scenario 1.a

<figure><img src="image_6.png" /><figcaption>Fig 1. Localization error in mm for one surface epileptic source</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig 2. CPU time in seconds for one surface epileptic source</figcaption></figure>

### Scenario 1.b

<figure><img src="image_8.png" /><figcaption>Fig 3. Localization error in mm for one deep epileptic source</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig 4. CPU time in seconds for one deep epileptic source</figcaption></figure>

### Scenario 2

<table><caption>TABLE 1. Localization error and CPU time in the presence of two synchronous epileptic sources</caption><thead><tr><td>Methods</td><td>Euclidean distance [mm]</td><td>CPU Times [s]</td></tr></thead><tbody><tr><td>2-MUSIC</td><td>80.68</td><td>1.8</td></tr><tr><td>4-MUSIC</td><td>71.2</td><td>231.7</td></tr><tr><td>MCE</td><td>72.7</td><td>2540</td></tr><tr><td>SOS-MUSIC</td><td>0</td><td>710.7</td></tr></tbody></table>

## References

* [1] R.O. Schmidt, " Multiple emitter location and signal parameter estimation," *IEEE transactions On Antennas Propagation*, vol. 34, no. 3, pp. 276-280, March 1986

* [2] I. Debauchies, M. Defrise, and C.D. Mol, " An iterative soft thresholding algorithm for linear inverse problem with a sparsity constraint ", *Comm. Pure Appl. Math.*, vol. 57, pp. 1413-1457, November 2004

* [3] D. Cosandier-Rimelé, I. Merlet, J. Badier, P. Chauvel and F. Wendling, " The neuronal source of eeg: Modeling of simultaneous scalp and intracerebral recordings in epilepsy ", *NeuroImage*, vol. 42, no. 1, pp. 135-146, 2008

## METHODS

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**OBJECTIVE**
Improve the localization of highly correlated brain sources

**ASSUMPTION: SPARSITY OF THE EPILEPTIC SOURCES**
There are few sources involved in the epileptic network

* A **sparse vector** $v$ whose non-zero components select and weight the column vectors of $G$ corresponding to the epileptic dipoles.

### Our regularization problem

$$ \min_{v \in \mathbb{R}^P} \Upsilon(Gv) + \lambda ||v||_1 $$

* Minimization of $\Upsilon$ for searching the epileptic sources.

* L1 norm enforcing sparsity of the epileptic sources.

* Optimization solver: Iterative Soft Thresholding Algorithm [2].

## SIMULATION SETUP

### Data

* 257 channels EEG data generated from a realistic model [3].

* A three-shell realistic **head model**.

* A **source space** of 8000 dipoles corresponding to the triangles of the cortical surface mesh with orientations perpendicular to the cortical surface.

<figure><img src="image_11.png" /><figcaption>Fig 5. Frontal view of the brain mesh</figcaption></figure>

### Performance criteria

* **Euclidean distance** between the estimate source position ($\hat{\theta}_i$) and the true position ($\theta_i$):

$$ d_{\theta_i}^{\text{Euc}} = ||\hat{\theta}_i - \theta_i||_2 $$

### Scenarios

1. Influence of the Signal-to-Noise-Ratio (SNR) and deepness on source localization of one epileptic source: **1a.** a surface source and in the case of **1b.** a deep source.

2. Influence of the signal correlation on source localization of two synchronous epileptic sources with an SNR value of -5 dB.

## CONCLUSION

* An extension of the 2-MUSIC approach by regularizing the classical **2-MUSIC metric** and proposing an **automatic** minimization scheme.

* **Promising results** for dealing with **synchronous** brain current sources and **low SNR** ratios as demonstrated through realistic computer simulations.

## Discussion

* A higher SOS-MUSIC CPU times but a more accurate localization.

## Future works

* Validating on real clinical EEG data and on **distributed** sources.

* An extension of the SOS-MUSIC method to **fourth-order** data statistics.

<figure><img src="image_12.png" /><figcaption>Fig 6. Distributed sources</figcaption></figure>

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IRISA

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Université de Rennes

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Inserm

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EMPENN