Estimation of single event-related potentials utilizing the Prony method

Coding Prony's method in MATLAB and applying it to biomedical signal filtering

BACKGROUND

The response of many biomedical systems can be modelled using a linear combination of damped exponential functions. The approximation parameters, based on equally spaced samples, can be obtained using Prony's method and its variants (e.g. the matrix pencil method). This paper provides a tutorial on the main polynomial Prony and matrix pencil methods and their implementation in MATLAB and analyses how they perform with synthetic and multifocal visual-evoked potential (mfVEP) signals. This paper briefly describes the theoretical basis of four polynomial Prony approximation methods: classic, least squares (LS), total least squares (TLS) and matrix pencil method (MPM). In each of these cases, implementation uses general MATLAB functions. The features of the various options are tested by approximating a set of synthetic mathematical functions and evaluating filtering performance in the Prony domain when applied to mfVEP signals to improve diagnosis of patients with multiple sclerosis (MS).

RESULTS

The code implemented does not achieve 100%-correct signal approximation and, of the methods tested, LS and MPM perform best. When filtering mfVEP records in the Prony domain, the value of the area under the receiver-operating-characteristic (ROC) curve is 0.7055 compared with 0.6538 obtained with the usual filtering method used for this type of signal (discrete Fourier transform low-pass filter with a cut-off frequency of 35 Hz).

CONCLUSIONS

This paper reviews Prony's method in relation to signal filtering and approximation, provides the MATLAB code needed to implement the classic, LS, TLS and MPM methods, and tests their performance in biomedical signal filtering and function approximation. It emphasizes the importance of improving the computational methods used to implement the various methods described above.

Filtering multifocal VEP signals using Prony's method

BACKGROUND

This paper describes use of Prony's method as a filter applied to multifocal visual-evoked-potential (mfVEP) signals. Prony's method can be viewed as an extension of Fourier analysis that allows a signal to be decomposed into a linear combination of functions with different amplitudes, damping factors, frequencies and phase angles.

METHOD

By selecting Prony method parameters, a frequency filter has been developed which improves signal-to-noise ratio (SNR). Three different criteria were applied to data recorded from control subjects to produce three separate datasets: unfiltered raw data, data filtered using the traditional method (fast Fourier transform: FFT), and data filtered using Prony's method.

RESULTS

Filtering using Prony's method improved the signals' original SNR by 44.52%, while the FFT filter improved the SNR by 33.56%. The extent to which signal can be separated from noise was analysed using receiver-operating-characteristic (ROC) curves. The area under the curve (AUC) was greater in the signals filtered using Prony's method than in the original signals or in those filtered using the FFT.

CONCLUSION

filtering using Prony's method improves the quality of mfVEP signal pre-processing when compared with the original signals, or with those filtered using the FFT.

Comparing a template approach and complex bandpass filtering for single-trial analysis of auditory evoked M100

Two methods for single-trial analysis were compared, an established parametric template approach and a recently proposed non-parametric method based on complex bandpass filtering. The comparison was carried out by means of pseudo-real simulations based on magnetoencephalography measurements of cortical responses to auditory signals. The comparison focused on amplitude and latency estimation of the M100 response. The results show that both methods are well suited for single-trial analysis of the auditory evoked M100. While both methods performed similarly with respect to latency estimation, the non-parametric approach was observed to be more robust for amplitude estimation. The non-parametric approach can thus be recommended as an additional valuable tool for single-trial analysis.

Systematic latency variation of the auditory evoked M100: from average to single-trial data

Standard analyses of neurophysiologically evoked response data rely on signal averaging across many epochs associated with specific events. The amplitudes and latencies of these averaged events are subsequently interpreted in the context of the given perceptual, motor, or cognitive tasks. Can such critical timing properties of event-related responses be recovered from single-trial data? Here, we make use of the M100 latency paradigm used in previous magnetoencephalography (MEG) research to evaluate a novel single-trial analysis approach. Specifically, the latency of the auditory evoked M100 varies systematically with stimulus frequency over a well-defined time range (lower frequencies, e.g., 125 Hz, yield up to 25 ms longer latencies than higher frequencies, e.g., 1000 Hz). Here, we show that the complex filtering approach to single-trial analysis recovers this key characteristic of the M100 response, as well as some other important response properties relating to lateralization. The results illustrate (i) the utility of the complex filtering method and (ii) the potential of the M100 latency to be used for stimulus encoding, since the relevant variation can be observed in single trials.

Development and evaluation of the piecewise Prony method for evoked potential analysis

A new method is presented to decompose nonstationary signals into a summation of oscillatory components with time varying frequency, amplitude, and phase characteristics. This method, referred to as piecewise Prony method (PPM), is an improvement over the classical Prony method, which can only deal with signals containing components with fixed frequency, amplitude and phase, and monotonically increasing or decreasing rate of change. PPM allows the study of the temporal profile of post-stimulus signal changes in single-trial evoked potentials (EPs), which can lead to new insights in EP generation. We have evaluated this method on simulated data to test its limitations and capabilities, and also on single-trial EPs. The simulation experiments showed that the PPM can detect amplitude changes as small as 10%, rate changes as small as 10%, and 0.15 Hz of frequency changes. The capabilities of the PPM were demonstrated using single electroencephalogram/EP trials of flash visual EPs recorded from one normal subject. The trial-by-trial results confirmed that the stimulation drastically attenuates the alpha activity shortly after stimulus presentation, with the alpha activity returning about 0.5 s later. The PPM results also provided evidence that delta activity undergoes phase alignment following stimulus presentation.

Subspace averaging of steady-state visual evoked potentials

A new algorithm for doing signal averaging of steady-state visual evoked potentials (VEP's) is described. The subspace average is obtained by finding the orthogonal projection of the VEP measurement vector onto the signal subspace, which is based on a sinusoidal VEP signal model. The subspace average is seen to out-perform the conventional average using a new signal-to-noise-ratio-based performance measure on simulated and actual VEP data.

Estimation of evoked potentials using total least squares prony technique

The authors investigate the applicability of Prony modelling to the estimation of evoked potentials. Four types of total least squares (TLS) model are considered and their optimal parameters are defined based on ten visual averaged EPs. Simulations with various signal and noise characteristics show that the TLS-Prony estimation is superior to averaging for two of the models, namely the unconstrained and the stable models. Application of the TLS-Prony estimator as a post-processor to moderate averaging allows a reduction in the number of responses averaged, or equivalently of recording time, by a factor of two.