Dynamical estimation of Event Related Potentials

Tracking dynamic characteristics of Event Related Potentials

Event related potentials (ERPs) are voltage changes of brain electric activity due to stimulation. The measured responses are often considered as the combination of electric activity resulted by multiple brain generators, active in association with the eliciting event, and noises which are brain activity not related to the stimulus, together with interference from non neural sources, like eye blinks and other artifacts. Usually, ERPs are considered as transient-like smooth waveforms. Even though they are dominated by lower frequencies, compared to background electroencephalogram (EEG), due to poor signal to noise ratio conditions their form is difficult to be estimated in a trial to trial scheme. (see also : Section Event related potentials)

Of special interest is the case when some parameter of the ERP changes dynamically from stimulus to stimulus. Different levels of attention, sedation, fatigue or habituation may create dynamic changes form trial to trial. If we consider that the ERP is a vector valued random process with slow dynamic variability during the repetitions of the test, then past realizations contain information of relevance to future realizations and can be used in the estimation procedure. Dynamic changes can be modeled with a state-space model and the recursive mean square estimate for the states is given by Kalman filter.

Single-Trial Estimation of ERPs by Kalman filter

We use a linear time-varying observation model (1) for the measured ERP epochs \(z_{t}\) at every stimulus t. The state vector \(\theta_{t}\) is not observed directly but it is allowed to evolve according to a linear first order difference equation (2). Estimates for the ERPs \(\hat{s}_{t} \) can be obtained by equation (3) in terms of pre-selected observation matrices \(H_{t}\) and estimated parameters \(\hat{\theta}_{t}\).


\begin{displaymath}z_{t}=h_{t}\theta_{t}+\upsilon_{t}\end{displaymath} (1)

\begin{displaymath}\theta_{t+1}=f_{t}\theta_{t}+g_{t}\omega_{t}\end{displaymath} (2)

\begin{displaymath}\hat{s}_{t}=h_{t}\hat{\theta}_{t}\end{displaymath} (3)

The linear mean square estimator for the state, given present and past available observations, is equal to the conditional mean (4) and can be written in the form (5) which is a Bayesian MAP (Maximum aposteriori) estimate using the last available estimate as prior information.


\begin{displaymath}\hat{\theta}_{t}=e\{\theta_{t}\vert z_{1},\ldots,z_{t}\}\end{displaymath} (4)
\begin{displaymath}\hat{\theta}_{t} = ( h_{t}^{t} c_{\upsilon_{t}}^{-1} h_{t} +......{\tilde{\theta}_{t\vert t-1}}^{-1} \hat{\theta}_{t\vert t-1} )\end{displaymath} (5)

The sequential procedure to obtain this estimator is given by Kalman filter (see also: Section mathematics of dynamic spect). Some applications of dynamical single-trial estimation of ERPs are illustrated in the following figures.



Simulations
Figure 1 (Simulations): linear and random variations for the amplitudes, latencies and widths of the peaks. Simulated ERPs were constructed by superimposing upon to prestimulus real EEG epochs linear combinations of three Gaussian shaped functions. Dotted lines represent the simulated noisy ERPs, red lines the simulated ERPs and blue lines the estimates.


Real ERP measurements
Figure 2 (Real ERP measurements): dynamic variability of the N100 peak. ERPs were measured during different sedation levels. Dotted lines represent the measurements and solid lines the estimates.

Copyright © Biomedical Signal Analysis Group 2013