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Background: Non-linear signal analysis has proven to be a technique that is capable of revealing qualitative and quan- titative differentiations between different dynamical systems (biological or otherwise). In the present work it has been demonstrated that this capability reveals quantitative differences in the Magnetoencephalograms (MEG) received from patients with Idiopathic Generalized Epilepsy (IGE) and from healthy volunteers. Method: We present MEG record- ings of 10 epileptic patients with IGE and the corresponding ones from 10 healthy volunteers. A 122-channel SQUID biomagnetometer in an electromagnetically shielded room was used to record the MEG signals and the Grassber- ger-Procaccia method for the estimation of the correlation dimension was applied in the phase space reconstruction of the recorded signal from each patient. Results: The aforementioned analysis demonstrates the existence of spatially diffused low dimensionality in the MEG signals of patients with IGE. Conclusion: The obtained results provide support for the hypothesis that low dimensionality in MEG signals is linked to functional brain pathogeny.

Non-linear analysis has been used in the past to assess magnetoencephalographic recordings (MEG) from patients with schizophrenia [

The recorded MEG activity is caused by ionic movements across the plasma membrane [^{−8} of the earth’s magnetic field which is equivalent to 50 μT), can be measured by means of a Superconducting Quantum Interference Device (SQUID) [

Chaos theory [

On the other hand despite the lack of understanding regarding the underlying pathogeny, IGE is a condition that is characterized by abnormal, rhythmic, electro-chemical brain activity without a single spatially determined epileptogenic focus [

According to the theory of nonlinear dynamical systems and chaos [10,11] the dynamics of any physical or biological system can be quantified and described by means of some new terms and concepts, such as the chaotic attractor and the correlation dimension of the reconstructed phase space. Of vital importance in the chaotic analysis of a dynamical system is the evidence for the existence of low dimension chaotic attractors and the estimation of the correlation dimension D of the attractor. The purpose of this study is to explore the diagnostic potential of this method [10,11] in the cases of IGE.

Magnetic recordings were obtained from 10 patients with IGE and 10 healthy volunteers. The 10 patients ,4 male (32 - 82 years old) and 6 female (20 - 55 years old) were classified as IGE according to the International League Against Epilepsy (ILAE) classifications. The selection of the IGE patients was random so the two gender groups (4 male and 6 female) were heterogeneous in their ages and clinical forms. This was our choice because in this preliminary study we wanted to explore the existence of correlation between low dimensionality chaos in MEG signals and the existence of IGE in the patients. In that context any demographic exclusion criteria would be a hindrance in sampling. The diagnoses were based on clinical manifestations, electroencephalography findings and brain MRI when it was necessary. All patients underwent an MEG examination. The 10 control subjects were volunteers with similar demogra phics with the patients. All the participants came from the county of Evros, Thrace, Greece. All controls underwent a standard neurological examination that, in all of them, resulted as normal. Informed consent was obtained from patients and volunteers prior to the procedure.

The method used for the recording of magnetic activity has been described in detail elsewhere [12,13]. In brief, we used a 122-channel SQUID gradiometer device and specifically the Neuromag-122 (Neuromag Ltd. Helsinki Finland). The 122 orthogonal thin-film planar gradiometers operate at low liquid helium temperatures (40 K) on the basis of the Josephson effect of superconductivity [

Nonlinear MEG signal analysis utilizes advanced mathematical methods to quantitatively correlate the characteristics of a given MEG signal with the underlying brain dynamics that produced this signal. The analysis method of this study was first proposed by Grassberger and Procaccia [10,11], based on the theorem of the reconstruction of the phase space introduced by Takens [

This method uses the observed MEG time series, in order to construct an appropriate geometrical representation of the dynamical properties of that signal. From this representation it is possible to derive geometrical parameters that are linked to the dynamical properties of the system that produced the given signal, in our case the human brain. One such geometrical parameter is the minimum saturation dimension (m_{minsat}). This parameter (m_{minsat}) is an index of the complexity of the dynamical system which produced the MEG signal. A more rigorous, although brief, overview of the mathematics of the method is presented in the Appendix.

Using the aforementioned method the correlation integrals were calculated according to Equation 2 (cf. Appendix) for all the MEG channels of both the patients and the volunteers. From these the slopes were derived and according to Equation 3 (cf. Appendix) the correlation dimensions were calculated for different embedding dimensions m.

The minimum saturation dimension (m_{minsat}) was the parameter that was estimated in our study for each MEG channel of each participant as a quantitative marker of the complexity of the MEG signal. It is worth noting that since the signal from each channel (sensor) was analyzed independently, no spatial sensor grouping was necessary.

An example of these correlation dimensions is plotted in

relation dimension thus providing these (varying by parameter variation) values. On the other hand, for the patient’s signal it is shown that a plateau of constant dimension reveals itself for m_{minsat} ³ 7. This signifies that an adequate embedding of the underlying system’s dynamic has been achieved in a vector space of 7 dimensions. It is worth noting that while an adequate embedding is achieved only at 7 dimensional vector space, the calculated correalation dimension is approx. D = 5.07. This deceptively low correlation dimension could be attributed to some, unavoidable in realistic time series, temporal correlation of the delay vectors. Even though the delay parameter τ was chosen as the first zero crossing of the autocorrelation function which excluded any linear correlation of the delay vectors, and (as discussed in the appendix) the parameter k of the closest neighbors was chosen so as to maximize the assessed correlation dimension, it appears unavoidable that realistic data would maintain some temporal correlation. For that reason we have chosen not the Correlation Dimension as the invariant of choice for our results but the minimum embedding dimension beyond which saturation occurs (m_{minsat}).

Using the aforementioned methods, MEG channel maps were derived for all the patients. These maps are presented in Figures 3 and 4 for the normal volunteers and the patients respectively. For each patient or volunteer a spatial layout presentation of the MEG channels was drawn with each channel color coded according to the m_{minsat} value. Details of the color scheme are presented in the legend of the figures.

_{minsat} = 24 since m = 23 was the maximum embedding dimension that our algorithm utilized. Additionally we have averaged the m_{minsat} for all channels of both patients and volunteers summarizing the results to