## Evoked Potential analysis

Evoked potentials (EP) are originally defined to be potentials that are caused by the electrical activity in central nervous system after a stimulation of the sensoral system. The potentials are usually measured from outer layer, scalp, of the human head. The measured potential is then superposition of all electrical activity that is originated in head. Thus the fundamental problem in the analysis of evoked potentials is to extract information about the potential from measurements that contain also on-going background EEG.

### Analysis

Currently the goal in the analysis of the evoked potentials is to get information about single potentials e.g. obtain the best possible estimate for a single potential. The notion of single trial analysis can be used in this context.

Some of the various methods for the analysis of EPs are:

#### Averaging

The most widely used tool for the analysis of evoked potentials has been the averaging of the measurements over an ensemble of trials. In the mean square sense this is the optimal way to improve the signal-to-noise ratio (SNR) when the underlying model for the observations is that the evoked potential is a deterministic signal in independent additive background noise of zero mean.

However, for over three decades it has been evident that the nature of the evoked potentials is more or less stochastic. In particular, the latencies and the amplitudes of the peaks in the potentials may have stochastic variations between the repetitions of the stimuli. The information on these kinds of variations in evoked potentials is lost when the measurements are averaged. The resulting estimate for the evoked potential does not then possibly correspond to any physical or neuroanatomical situation.

#### Filtering

The most common way to do single trial analysis is to form a filter with
which the unwanted contribution of the on-going background activity
can be ltered out from the evoked potential. A major difficulty in this
task is often very low signal-to-noise ratio.

In any such filtering or estimation method the performance of the estimator depends on the properties of the underlying signals. In the most realistic methods some models for the measurements and the signals are assumed. The estimator that yields the minimum mean square criterion is then derived. The performance of the estimators then depends strongly on how realistic the assumptions are. The most common assumptions concern the second order statistics of the evoked potentials and the on-going background EEG.

In addition, we sometimes have prior information about the evoked potentials. The information can relate e.g. to the shape or the location of the peaks in potential. Generally, we should take this prior information into account to obtain best possible estimates. This can be done e.g. by using regularized least squares approach.

Click image to enlarge

Eight randomly selected single estimates calculated with transversal lter (thick) with noisy simulations (thin) and noiseless simulations (medium).

### Our methods

#### Single channel approach

One possibility to improve the estimates is to apply additional information
about the potentials to the estimation. The information can be concerned
with the assumed smoothness of the evoked potentials or be in the form
of limits for the possible locations of the peaks in the potentials.
A major problem in the implementation of prior information into the
estimation algorithm is how to express it in a feasible mathematical
form.

An effective way to add this kind of information to estimation is to use the so-called regularized least squares method. The desired solution for the estimates of the single potentials can then be written in the form

where H is
matrix which columns contain a simple model for the EP e.g. gaussian
shape humps, θ is a vector containing coefficient for each column
of H,
is
inverse of covariance matrix of background EEG, is regularization coefficient
(scalar), I
is eye matrix, H_{S}
is a matrix used for regularization and
z
is a matrix containing the measurements.
H_{S}
represents the covariance of the measurements.

Two measured Event Related Potentials (ERP) and corresponding estimates calculated with single channel approach.

A Graphical user interface is also under development.

#### Multi channel approach

In the case of multi channel measurements we form z by concatenating the measurements of different channels. Although the evoked potential and the background EEG are independent they both are highly correlated between different channels. In the case of multi channel measurements we can combine the spatial information between the channels by using the subspace regularization method.

As in single channel method we use generic
Gaussian basis for each channel. The channels are modeled separately,
that is, the matrix H
is a block diagonal matrix. This model does not contain any dependence
between the channels. We form next the eigendecomposition of the data
correlation matrix and use rst few eigenvectors to form the regularization
matrix H_{S}
. Because the correlation is calculated using the stacked measurements
z, the eigenvectors
model also the correlation between the separate channels. The solution
is again obtained with (2).

Visualization of equation (2) in case of three channels.