## Single-Trial Analysis of Event-Related Brain Potentials

Event-related potentials (ERP) are used for studying brain activity associated with higher mental functions. They are caused by the electrical activity of the central nervous system as a response to a cognitive meaningful stimu-lation of the sensor system. The measured potentials are superpositions of all electrical activity in the head. This leads to the consideration of one of the fundamental problems in the analysis of evoked potentials: how can we extract information about the potentials from measurements that also include the ongoing background EEG?

The following method has been developed at the Department of Physics of the University of Kuopio and is based on so-called subspace regularisation.

- A linear model is used for the measurements. This means that the measured signal is modeled as superposition of the ERP and the background EEG. The backgroung EEG can be considered to represent the measurement error in the equation.
- In addition, ERPs are modeled as a combination of some generic basis functions e.g. Gaussian shaped humps. The solution will be in the subspace, R(H) in the terms used in the accompanying diagram, spanned by the basis vectors.
- The solution is then regularised in the direction
of the common properties of all the measured ERPs, i.e. to the subspace
R((H
_{S}H_{S}^{T})H) spanned by the eigenvectors corresponding the few largest eigenvalues of the data correlation matrix. The final solution is marked as Hθ in the diagram. The method can be expressed in the form of the following matrix equation: