Adaptive Signal Complexity Analysis of Epileptic MEG

Transcription

1 Adaptive Signal Complexity Analysis of Epileptic MEG ADAM V. ADAMOPOULOS Medical Physics Laboratory, Department of Medicine Democritus University of Thrace GR-68 00, Alexandroupolis HELLAS Abstract: - The problem of complexity analysis of the Magneto-Encephalo-Gram (MEG) of epileptic patients is approached as a Non-Linear Time Series (NLTS) model identification problem. MEG can be recorded with the use of specific totally non-invasive Superconductive QUantum Interference Devices (SQUID). For the analysis and classification of the epileptic MEG signals a novel intelligent method was used. The method combines the Multi-Model Partitioning (MMP) theory and Extended Kalman Filtering (EKF) for the generation and evaluation of each candidate model. On the top, a Genetic Algorithm (GΑ) was used to evolve a population of candidate models for an optimal solution of the identification problem to be found. Simulations illustrated that the proposed method selected the correct model structure, identified the model parameters in a sufficiently small number of iterations and track successfully functional changes in the signal in real time. The neurologists can easily interpret the information provided detecting correlation with the clinical status of the epileptic patients. Key-Words: Magnetoencephalogram, Signal Identification, Multi-Model Analysis, Extended Kalman Filters, Epilepsy Introduction Complexity analysis and identification of a nonlinear system with unknown dynamics, has been a central issue in the field of signal processing for many years. System identification and model-based signal processing have been widely used in many application areas, and the related driving algorithms are part of many existing systems. The rapid growth of the fields of adaptive control, speech analysis and synthesis, neural networks, biomedical engineering, radar and sonar technology, fuzzy systems, and wavelets is offering a large variety of new methods for modeling static and dynamic nonlinear systems. Therefore, it is important to understand what are the opportunities, limitations, and pitfalls of the several approaches, in order to obtain reliable designs towards real-time applications. Thus, the subject of nonlinear time series modeling has attracted considerable attention in the past. So has been studied extensively from different points of view. Among others previous studies involved statistics and identification theory, approximation theory, signal processing, information theory, physics and optimization theory. In addition, a large number of numerous approximate nonlinear estimation algorithms, have been proposed for certain data models [, 2, 9, 24, 27]. In the present work the problem of signal complexity of the epileptic MEG timeseries was approached as a NonLinear Time Series (NLTS) model identification problem. This problem is closely related with the choice of the model structure and computation of the coefficients of the system (namely, brain structures) that generate epileptic behavior. In general the MEG signals are generated from the ionic micro-currents of the brain, originated at the cellular level [3, 6, 22]. The MEG analysis can provide information of vital importance for the monitoring of brain dynamics in both normal and pathological conditions of the Central Nervous System [6, 2, 25, 26]. The MEG signals are recorded with the use of specific Superconductive QUantum Interference Devices (SQUIDs) [4, 23]. SQUIDs are very sensitive superconductive magnetometers with the ability to detect and record extremely weak magnetic fields, of the order of ft (= 0-5 T). Therefore they can be used ideally for the recording of the MEG, since they do not emit any radiation and they are totally non- invasive. In addition to the feature of extremely high sensitivity, SQUID technology provides high spatial and temporal resolution and they can account as a

2 promising diagnostic technique for the investigation of neurological diseases and the exploration on normal brain function [7, 8, 2]. MEG recording and analysis can be used complementary to the Electro-Encephalo-Gram (EEG) method and other brain functioning techniques such as functional Magnetic Resonance Imaging (f-mri), and the novel method based on the combination of Positron Emmission Tomography and Computerized Tomography scans (PET/CT) that provides both functional/metabolic and anatomical information [6, 20]. For the purpose of complexity analysis, model identification, and classification of the MEG signals an intelligent method was used. The method combines intelligent Multi Model Partitioning (MMP) theory and the Extended Kalman Filtering (EKF) method. A bank of EKF was utilized for purpose of MEG modeling and identification, while the MMP theory was used to evaluate each particular candidate model. Following, a specifically designed Genetic Algorithms (GΑ) was used to evolve a population of models in order to provide the optimal solution on the MEG signal modeling problem. The developed methodology was designed to apply for general (not necessarily Gaussian) data probability distribution functions [, 8, 3, 5, 7]. The proposed algorithm was applied extensively on MEG signals of epileptic patients. The experimental results indicated that the method is selecting the correct model structure and complexity, and identifies the model parameters of the MEG signal. The task of MEG modeling and identification is accomplished in a sufficiently small number of iterations. In addition, the algorithm is capable to track successfully and in real time any change of dynamical complexity or any alteration of functional nature. By design, the algorithm can be parallel implemented. A VLSI implementation is also feasible in case that a real time application of the proposed method is considered. 2 Material and Methods The material considered for analysis in the present work is consisted of MEG signal of epileptic patients. Patients that had independently diagnosed by special neurologists to suffer from idiopathic epilepsy were referred to our Laboratory. All patients have had normal serum biochemical studies, as well as, normal CT or MRI scans. A 22-channel whole brain SQUID, model NEUROMAG-22, provided by 4-D Imaging Co., was used to record the MEG activity of these patients. During the recording procedure the patients were lying on a bed with eyes closed in order to avoid artifacts from eye flickering. For each patient, the MEG was recorded from 22 points of the skull. MEG recording have had 32 sec duration and were digitized and stored with a sampling frequency of 256 Hz for off-line analysis. A representative 4 seconds epoch (consisting of 024 samples) of the MEG activity from a point of the left temporal lobe of an epileptic patients suffering from temporal lobe epilepsy is shown in Fig. ( MEG measured in pt = 0-2 T). Fig.. A 4 sec MEG epoch of an epileptic patient. Considering the developed method we should state at first that in all signal-modeling problems, including nonlinear signal processing, the general problem is to find a good model structure and then estimate the parameters of some basis signals from the observations. Going from linear to nonlinear system identification also makes this problem much harder since the set of nonlinear models is much richer than the linear one. The topic of classifying the nonlinear time series models is based on many features in the literature such as the types of variables used in the model or the way the model parameters appear in the equations. The three basic groups of variables used in NLTS models include: The previous values of the dependent variable which lead to autoregressive (AR) terms. Sequences of independent random processes (white noise not necessarily Gaussian) which lead to moving average (MA) terms. Input variables, which are called external inputs and lead to exogenous (X) terms.

3 So a natural generalization of the linear autoregressive moving average with exogenous input model to the nonlinear case of NARMAX model can be written as: and, T ( h (, ( ) ) x( t+ ) = f ϑ t t + ( x( ) v( ) y( = gt + t x(,..., x( nx + ), T h ( = u( ),..., u( nu ), ),... n ) e ϑ = [,..., a, b,..., b, c c ] T a n x n u,..., ne () (2) (3) (4) where, x( is the signal produced by the pure dynamics of the system, is the ambient (intrinsic) dynamical noise of the system, y( is the observable values (in our case the MEG data), and v( is the external (observational) noise. In addition, f t and g t are known matrix-valued nonlinear functions of the states, and n=(n x n u, n e ) is the order of the NARMAX model. In the general case that we present here, and v( are uncorrelated, zero mean, white noise processes, not necessarily Gaussian, with variances R and V respectively; a i : i=,,n x, b j : j=,,n u, c k :k=,,n e are the predictor coefficients. The coefficients a i, b j and c k can be replaced by a i (, b j ( and c k ( to reflect the possibility that the coefficients are subject to random perturbations. This fact can be modeled by assuming that [7]: ϑ( t + ) =ϑ( + w( (5) where, w( is also a nx zero mean white noise process, not necessarily Gaussian, with variance W (we assume that processes, v( and w( are independen. The nonlinear time series model identification problem is now stated as follows: Given a set of observations y( and u(, where 0 t N, from an NLTS(n) process, we have to determine the unknown parameter vector: υ = [ x(,e (, ϑ(,n,r,v, W] (6) Clearly the problem is two-fold: based on the measurements y(, u(, one has first to implement a GA in order to estimate the unknown values of the noise sequences, variances, and the initial conditions, and then both to select the structure of the predictor, and to compute the signal x( and the predictor coefficients. The individuals of the population of the GA are EKFs, numbered from EKF() to EKF(max). Different individuals may have different order M and model coefficients. As a fitness function of each one of the candidate models for the GA is used the a-posteriori probability that a particular model is the correct one. The a-posteriori probability of each model is estimated according to Lainiotis MMP theory [5, 7]. An overview of the algorithm is pictorially presented in Fig. 2. Fig. 2. Pictorially representation of the algorithm. 3 Results The application of the algorithm described above on the MEG signals of epileptic patients shown that these signals can be modeled using formulas of the type: M x ( t + ) = i= 2 x ( t ) ( a i + bie ) x ( t i + ) + e ( t ) (7) y ( = x( + v( (8) which in the more general case, can be written as: M 2 x ( t k ) x( t ) a b e i + + = i + i x( i+ ) + i= EKF- Observables EKF-2 EKF-max Use a-posteriori probabilities to select (9) y ( = x( + v( (0) The order M, in Eqs. (7) and (9) was found in the range from 3 to 5. It is desirable for the algorithm to search of the terms in the exponential form in Eq. (9) and to estimate the correct number of k i values

4 that will help the algorithm to approximate the values of the signal as good as possible. The model easily calculated the corresponding coefficients. Results considering the procedure of tracking the correct order of the model are shown in Fig. 3. In that Figure, the a-posteriori probability of the candidate models with order M in the range to 5 is shown as a function of the number of signal samples that are processed. As it is indicated in Fig. 3 for the given MEG the algorithm was able to identify that the order M of the fittest model is 4. This is concluded from the fact that for the model with order M = 4 the a-posteriori probability settles to (solid line in Fig. 3) after a transient phase. This was done after processing about 90 samples of the MEG signal and did not change for the rest of the procedure. On the contrary, at the same processing time, the a-posteriori probabilities corresponding to the models of order different to 4 (dashed lines in Fig. 3) vanish to zero. Considering that the sampling frequency of MEG recording was 256 Hz, this shows the ability of the algorithm to indicate the correct model structure and complexity in a time less than 0.4 sec. Fig. 3. Evolution of a-posteriori probability for models with order M from to 5. 4 Conclusion From the performed experiments and the obtained results it appeared that the proposed algorithm is able not only to track and model the given MEG signals, but also to provide information on the features and characteristics of these signals. Specifically, the algorithm is capable to propose a possible non-linear model for the given timeseries and estimate the order, or in other terms, the order of complexity) of the given signal. Moreover, the algorithm is able to model the ambient as well as the extraneous noise that is incorporated in the pure dynamics of the system. These findings have to be related and compared to the ones obtained by non-linear analysis and chaotic methods for the analysis of the epileptic MEG [5], in order to obtain a better physical understanding of the underlying processes in epileptic brain dynamics. Specifically the low-order signal models that were obtained from our work are compared favorably to the results reported in [5] where low-dimensional non-linear dynamics were revealed to undergo the epileptic MEG. Thus, the hypothesis for highly synchronization neural dynamics in epileptic behavior seems to be justified also by the present work. From that point of view the algorithm proposed in the present work could account as a complexity measure of the underlying dynamics of the analyzed MEG recording. Therefore, it would be valuable to apply the proposed algorithm on MEG recordings generated under different normal or pathological brain conditions. Results of these investigations will be presented in the future. References: [] M.L. Andrade, L. Gimeno, & M.J. Mendes, On the optimal and suboptimal nonlinear filtering problem for discrete-time systems, IEEE Trans. Automat. Contr., Vol. 23, 978, pp [2] D. Andrisani, F.P. Kuhl & D. Gleason, A nonlinear tracker using attitude measurements, IEEE Trans. Aerosp. Electron. Syst., Vol. 22, 986, pp [3] P. Anninos, Electromagnetic fields generated from neuronal activity T.I.T Journal of Life Sciences, Vol. 3, 973, pp [4] P.A. Anninos, G. Anogianakis, K. Lenhertz, C. Pantev, & M. Hoke, Biomagnetic measurements using SQUID, International Journal of Neuroscience, Vol. 37, 987, pp [5] P.A. Anninos, A. Kotini, A. Adamopoulos, & Tsagas, The use of nonlinear analysis for differentiating brain biomagnetic activity in epileptic patients before and after magnetic stimulation, Hadronic Journal Supplement, Vol. 4, 999, pp [6] P.A. Anninos, & S. Raman, Derivation of a mathematical equation for the EEG and the general solution within the brain and space, International Journal of Theoretical Physics, Vol. 2, 973, pp. -9. [7] P.A. Anninos, N. Tsagas, R. Sandyk, & K. Derpapas, Magnetic stimulation in the treatment

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