Stefanos D. Georgiadis Perttu O. Ranta-aho Mika P. Tarvainen Pasi A. Karjalainen. University of Kuopio Department of Applied Physics Kuopio, FINLAND

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1 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland Stefanos D. Georgiadis Perttu O. Ranta-aho Mika P. Tarvainen Pasi A. Karjalainen University of Kuopio Department of Applied Physics Kuopio, FINLAND Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 1

2 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland INTRODUCTION To understand human neurophysiology, we relay on several types of non-invasive neuroimaging techniques. These techniques include electroencephalography (EEG), magnetoencephalography (MEG), anatomical magnetic resonance imaging (MRI) and functional MRI (fmri). Neural activity in the cerebral cortex generates small electric currents which create potential differences on the surface of the scalp (detected by EEG). EEG recording is a useful tool for studying the functional states of the brain and for diagnosing certain neurophysiological states and disorders. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 2

3 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland NOISE One of the challenging tasks is how to reliably detect, enhance and estimate very week, non stationary brain signals corrupted by noise. E.g. De-noising or estimation of Event Related Potentials (ERPs). Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 3

4 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland EVENT RELATED POTENTIALS (ERPs) ERPs are voltage changes of brain electric activity due to stimulation, e.g external auditory. The simplest way to investigate them is to use ensemble averages of time-locked EEG data epochs. The investigation of the variability of ERP parameters can be used to reveal information related to changes of the cognitive state. Single-trial analysis methods are under concern. Different methods exist, e.g. digital filtering, wavelets, or multichannel methods, e.g Independent Component Analysis. Here we focus on single-trial single channel ERP estimation. We are interested in cases that the ERPs have dynamic changes form trial-to-trial, e.g. some trend at the amplitude or latency of some peak. Recursive estimation methods can be used to track such changes. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 4

5 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland SINGLE-TRIAL ERP EXTRACTION Preprocessing (e.g. bandpass 1-4Hz) Channel CZ Stim. no. 1 Stim. no. 2 Stim. no. 3 Stim. no. 4 Amplitude and latency of different peaks Ensemble Average 1 N1 N P3 Single-trial estimation Difficulties with low frequencies. Spectra of the unknown interesting activity and background noise overlap heavily. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 5

6 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland ERP ESTIMATION The sampled ERP measurement of length M at stimulus t, t = 1, 2,..., T can be denoted with the column vector: z t = (z t,1, z t,2,..., z t,m ) T, (measurement epoch) ERPs can be modeled as a linear combination of some preselected basis vectors z t = H t θ t + υ t, (additive noise model) Ht, observation model, basis vectors of length M it its columns. st = H t θ t, part of the activity that is related to the stimulation. υt, noise vector. θt parameter vector to be estimated, e.g. Least Squares. ŝt = H t ˆθt, the estimated response. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 6

7 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland STATE-SPACE FORMALISM Of special interest is the case that some characteristics of the ERPs change dynamically from stimulus to stimulus. This case can be naturally modeled with a state-space model θ t+1 = F t θ t + ω t, (state equation) The hidden states are not observed directly, but through the measurement model z t = H t θ t + υ t, (space equation) Ft, known matrices, for F t = I we have a random walk model. ωt is white noise vector process, independent of θ and υ t. The covariance matrices Cυt and C ωt are known. Estimators for the parameters optimal in the mean square sense are given by Kalman filter and smoother algorithms. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 7

8 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland RECURSIVE MEAN SQUARE ESTIMATION The mean square estimator for the state θt given the past and present observations z 1,..., z t is given by the conditional mean ˆθ t = E{θ t z 1,..., z t }. If the processes υt and ω t are Gaussian then this estimator is linear and maximizes the posterior density p(θ t z 1,..., z t ), Bayesian maximum a posteriori estimator (MAP). Recursive solution (linear mean square) for the problem is given by Kalman filter algorithm. If all the data set is available, the mean square estimator given all observations z 1,..., z T is given by the conditional mean ˆθ s t = E{θ t z 1,..., z T }. And solution is obtained by Kalman smoother algorithm. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 8

9 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland KALMAN FILTER AND SMOOTHER Kalman filter equations can be written as ˆθ t t 1 = F t 1 ˆθt 1 C θt t 1 = F t 1 C θt 1 F T t 1 + C wt 1 K t = C θt t 1 H T t (H t C θt t 1 H T t + C υt ) 1 C θt = (I K t H t )C θt t 1 ˆθ t = ˆθt t 1 + K t (z t H t ˆθt t 1 ), θ t is the state estimation error θ t = θ t ˆθ t, C denotes covariance matrices and K t is the Kalman gain matrix. The solution for the fixed-interval Kalman smoother is ˆθ t s = ˆθt + A t (ˆθ t+1 s ˆθ t+1 t ) A t = F T C θt t C 1, θ t+1 t error covariances and Kalman filter state estimates need to be stored. ˆθ s t are then obtained by running backward in time. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 9

10 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland MEASUREMENTS AND SIMULATIONS simulations noisy simulations 4 epochs and mean Simulations resembling P3 type responses. As linear combinations of 3 Gaussian shaped functions, sinusoidal+random variability for both amplitude, latency of the third peak. Noisy simulations, prestimulus EEG as additive noise real measurments P3 responses from an odd-ball paradigm with auditory stimuli, channel CZ, sampling rate 5Hz, bandpass 1-4Hz. Epochs form -1ms to 6ms relative to deviant stimuli time. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 1

11 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland RESULTS: IMAGE PLOTS Kalman filter Estimates for simulations Kalman smoother epochs and mean Estimates for real measurements Kalman filter epochs and mean Kalman smoother Columns Observation model H Observation model 3 time shifted Gaussian shaped functions, random walk model: H t = H, F t = I, C υt = I, C ωt =.1I, t. Same selections for Kalman filter and smoother and for simulations and real data. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 11

12 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland RESULTS: DYNAMIC VARIABILITY time interval data points Simulations Time interval for the computation of amplitudes and latencies of the 3 peak, simple max. Time-lag (noiseless) Amplitude trends (simulations) Amplitude trends (real data) Real data Latency trends (simulations) Latency trends (real data) MSE KF noisy KF KS noisy KS State noise variance parameter Average of the Mean Square Error C υt = I, C ωt = σ 2 ωi Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 12

13 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland RESULTS: SINGLE TRIALS Single-Trial Estimates simulations: stim real data: stim simulations: stim real data: stim Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 13

14 5 Finnish Signal Processing Symposium (Finsig 5) Kuopio, Finland CONCLUSIONS Recursive mean square estimations methods provide excellent single-trial ERP estimates in realistic noise conditions. The methods are suitable when some characteristics of the ERPs change dynamically from trial-to-trial, e.g. habituations to stimulation, fatigue, and other time varying effects. The benefit of the Kalman smoother approach is the avoidance of the time-lag in the estimates. Kalman filter can be used for on line estimation, e.g. measuring the depth of anesthesia. Data based observation models can also be used, e.g. eigenvectors of the data correlation matrix There are extensions of the method for multichannel simultaneous processing Improvement of the methods relates to the so called state-space identification procedures for the time evolution of the states. Stefanos D. Georgiadis, Department of Applied Physics, University of Kuopio Slide 14