gms | German Medical Science

Introduction: Waveform analysis techniques are of increasing interest in many fields, especially in neuroscience and biomedical engineering. The waveform analysis process is often difficult and confusing. However, the key to successful signal analysis is to know when to use which method. We recently extended a novel method developed by Laleg-Kirati and colleagues which uses the time-independent Schrödinger equation for time-series analysis [1], [2], [3]. Here we give a digest of our method and report on its application to biomedical signals containing spike-like waves.

Materials and Methods: The time-independent Schrödinger equation H y(x) = E y(x) is an eigenvalue equation for the Hamilton or energy operator H = p²/2m + V with total energy E. Assume that the kinetic energy operator p²/2m = -ħ²/2m d²/dx² = -h² d²/dx² and the potential operator V, which represents the signal s of interest, i.e. V(x) = -s(x), are given. For signal analysis the variable x has no longer the meaning of position: instead of position it represents time. For discrete signals the time-independent Schrödinger equation is a common matrix eigenvalue problem. Solving this eigenvalue problem yields eigenvalues and associated eigenvectors for positive and negative signal components, provided that the energy operator matrix has an off-diagonal checkerboard pattern [3], [4]. Using some selected eigenvalues and associated eigenfunctions, the signal s can be reconstructed by a superposition of products of square roots of these eigenvalues µ(i) times squared associated eigenvectors v(x, i): s(x) ˜ -4 h ∑ [a(i) µ(i) v(x, i)], where the a(i) = ± 1 are related to positive and negative signal components [3], [4]. It is obvious that the parameter h of the kinetic energy operator -h² d²/dx² influences the properties of the Hamilton operator and thus the approximant or rather the reconstruction quality of the signal s. In addition, the impact of the kinetic energy operator to the approximation performance depends on signal's magnitude. Thus the parameter h should be selected with care. To investigate this more precisely we varied h systematically for test and biomedical signals and explored the resulting effects, especially with respect to approximation and filtering properties. Finally, we applied the method to analyse and to represent or extract action potentials present in extracellular neuronal micro-electrode recordings and QRS complexes present in ECG recordings.

Results: The quality of the approximation or reconstruction of discrete signals depends on the parameter h and on the number of eigenvalues and eigenfunctions used to compute an approximant. A sufficiently large h yields a smooth approximant. Reducing h initially produces a better overall approximation until a best optimum is achieved. A further reduction of h often yields minor approximation results. For appropriate h large spike-like waves as well as signal components with large amplitudes are better approximated than small spikes or signal components with small amplitudes. In other words: the "sharpness" of large spike-like waves is preserved, but small spike-like waves or signal components extenuated or filtered out. This filter effect is non-linear, i.e. magnitude dependend. Assume that signal and noise are linearly superimposed, then the smoothing is stronger for small signal components or noise but not for large signal components or noise. The contribution of eigenvalues and associated eigenvectors to approximation is as follows: the square root of an eigenvalue is related to the magnitude and the associated eigenfunction is related to the localization of a signal component. Large action potentials or QRS complexes are quite often well approximated by a few, say 2-5, superimposed products of square roots of eigenvalues and times squared associated eigenvectors. Thus this method can be used to represent or to extract spike-like waves present in biomedical signal recordings.

Discussion: We have launched a novel signal analysis method to analyze signals containing spike-like waves and applied it to test and biomedical signals. The method relies on a quantum mechanical approach to reconstruct a potential or rather a signal by using solutions of the time-independent Schrödinger equation. The quality of the reconstruction can be controlled by a parameter and the number of eigenvalues and eigenfunctions used for reconstruction. The method provides an interesting non-linear magnitude depending filtering and enables a startling smoothing or reduction of small signal components, e.g. noise, in combination with a localization, representation and moderate distortion of signal components. This might be advantageous for the detection, analysis, representation and processing of spike-like waves, or, more general, singular signals.