Non-linear analysis has been used in the past to assess magnetoencephalographic recordings (MEG) from patients with schizophrenia [1], Parkinson’s disease [2] and malignant tumours of the brain [3]. This paper provides evidence for a link between Idiopathic Generalized Epilepsy (IGE) and the characteristics of the strange attractor derived from the MEG recordings of the patients suffering from it.

The recorded MEG activity is caused by ionic movements across the plasma membrane [4]. This activity is weak as it is (≈10⁻⁸ of the earth’s magnetic field which is equivalent to 50 μT), can be measured by means of a Superconducting Quantum Interference Device (SQUID) [4]. The SQUID is one of the few devices which are capable of measuring the exceedingly weak magnetic fields emitted by living tissues. The method is non-invasive because the SQUID is a passive electromagnetic receiver and not a transmitter of any kind of radiation [5,6].

Chaos theory [7] provides us with measurable quantities of the complexity of dynamical systems. In the case of a time series obtained through a MEG measurement of the brain, these measurable quantities refer to the underlying dynamical system, namely the brain. In that context, chaos theory provides us with quantitative markers of the dynamical complexity of the brain. There are indications that the healthy brain, due to the statistical nature of its neuronal discharges, is a very highly, theoretically infinitely complex system [8].

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 [9].

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 [14] with a broadband gradient noise 5 fT/(cm) and max noise 10 fT/(cm) for the 95% of the channels for f > 10 Hz and a broadband gradient noise 15 fT/(cm) and max noise 20 fT/(cm)for 95% of the channels for 1 Hz < f < 10 Hz. The helmet-shaped sensor array contains 122 planar gradiometers that detect the MEG signals just above the local current sources. Figure 1 shows the channels in stereographic projection seen from above. The arrows indicate the direction of the gradient at which the sensor is sensitive. Each subject was comfortably seated on a non-magnetic chair in a magnetically shielded room. During the recordings the subject’s head was covered with the helmet-shaped dewar. All recordings were taken with the patients conscious and relaxed. There were no recordings taken from patients demonstrating either clinical or MEG signs of seizure. The aim is to avoid the known trivial MEG signal morphology that is associated with active seizure neuronal discharges. The MEG sampling frequency was 256 Hz and the associated Nyquist frequency was 128 Hz, which was well above constituent frequency components of interest in our MEG recordings thus avoiding aliasing artifacts. The MEG signal was filtered with cut-off frequencies between 0.3 to 40 Hz. The time taken for each recording was in the range of 1 min after all transients (due to patient placement and motion) have died off. The duration of the above records is the routinely chosen time interval that has proven long enough, in past studies [1,3], to cancel out, on average, all random events and to allow for only the persistent ones to remain.

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 [15].

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 Figure 2 both for a patient (solid line) and a healthy volunteer (dashed line). From Figure 2 it becomes clear that the assessed correlation dimension of the signal from the healthy volunteer rises according to the embedding dimension. Since the correlation dimension is an invariant of the underlying system’s dynamic, the fact that it scales according to the embedding dimension indicates that the underlying dynamic is not embedded in a sufficiently high vector space to reveal its true invariant cor-

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.

Figure 3 shows no low dimensionality signals in healthy volunteers. Figure 4 exhibits low dimensionality signals diffused throughout the whole of the brain. In order to quantify these results we assumed a worse case scenario that for all the channels not exhibiting saturation the m_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 Table 1. A paired samples Student’s t-test was performed resulting in t = −7.42 and p = 0.0002 thus proving that there is a statistically significant difference between the patient sample and the normal sample.