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In this paper, we present a rigorous, general definition of the nonlinear association index, known as [h.sup.2]. Proving equivalence between different definitions we show that the index measures the best dynamic range of any nonlinear map between signals. We present also a construction for removing the influence of one signal from another, providing, thus, the basis of an independent component analysis. Our definition applies to arbitrary multidimensional vector-valued signals and depends on an aperture function. In this way, the bin-related classic definition of [h.sup.2] can be generalized. We show that upon choosing suitable aperture functions the bin-related intuitive definition can be deduced. Special attention is dedicated to the direction of the association index that in general is taken in only one sense. We show that for linearly coupled signals high associations are always bidirectional. As a consequence, high asymmetric nonlinear associations are indicators of nonlinear relations, possibly critical, between the dynamic systems underlying the measured signals. We give a simple simulated example to illustrate this property. As a potential clinical application, we show that unidirectional associations between electroencephalogram (EEG) and electromyogram (EMG) recorded from patient with pharmacologically intractable epilepsy can be used to study the cortical involvement in the generation of motor seizures.

In this paper, we investigate the dynamical scenarios of transitions between normal and paroxysmal state in epilepsy. We assume that some epileptic neural network are bistable i.e, they feature two operational states, ictal and interictal that co-exist. The transitions between these two states may occur according to a Poisson process, a random walk process or as a result of deterministic time--dependent mechanisms. We analyze data from animal models of absence epilepsy, human epilepsies and in vitro models. The distributions of durations of ictal and interictal epochs are fitted with a gamma distribution. On the basis of qualitative features of the fits, we identify, the dynamical processes that may have generated the underlying data. The analysis showed that the following hold. 1) The dynamics of ictal epochs differ from those of interictal states. 2) Seizure initiation can be accounted for by a random walk process while seizure termination is often mediated by deterministic mechanisms. 3) In certain cases, the transitions between ictal and interictal states can be modeled by a Poisson process operating in a bistable network. These results imply that exact prediction of seizure occurrence is not possible but termination of an ictal state by appropriate counter stimulation might be feasible. Index Terms--Bistability, duration distribution, epilepsy.

In this overview, we consider epilepsies as dynamical diseases of brain systems since they are manifestations of the property of neuronal networks to display multistable dynamics. To illustrate this concept we may assume that at least two states of the epileptic brain are possible: the interictal state characterized by a normal, apparently random, steady-state electroencephalography (EEG) ongoing activity, and the ictal state, that is characterized by paroxysmal occurrence of synchronous oscillations and is generally called, in neurology, a seizure. The transition between these two states can either occur: 1) as a continuous sequence of phases, like in some cases of mesial temporal lobe epilepsy (MTLE); or 2) as a sudden leap, like in most cases of absence seizures. In the mathematical terminology of nonlinear systems, we can say that in the first case the system's attractor gradually deforms from an interictal to an ictal attractor. The causes for such a deformation can be either endogenous or external. In this type of ictal transition, the seizure possibly may be anticipated in its early, preclinical phases. In the second case, where a sharp critical transition takes place, we can assume that the system has at least two simultaneous interictal and ictal attractors all the time. To which attractor the trajectories converge, depends on the initial conditions and the system's parameters. An essential question in this scenario is how the transition between the normal ongoing and the seizure activity takes place. Such a transition can occur either due to the influence of external or endogenous factors or due to a random perturbation and, thus, it will be unpredictable. These dynamical changes may not be detectable from the analysis of the ongoing EEG, but they may be observable only by measuring the system's response to externally administered stimuli. In the special cases of reflex epilepsy, the leap between the normal ongoing attractor and the ictal attractor is caused by a well-defined external perturbation. Examples from these different scenarios are presented and discussed. Index Terms--Biomedical signal analysis, brain modeling, dynamics, electroencephalography, epilepsy, magnetoencephalography, nonlinear systems.

A new analytical method for quantifying brain activity from magnetoelectroencephalogram (MEG) and electroencephalogram (EEG) recordings during periodic light stimulation is proposed. It consists in estimating the phase clustering of harmonically related frequency components of a subject's MEG/EEG responses evoked by the light stimulation. The method was developed to test the hypothesis that changes in the dynamics of brain systems in the course of intermittent photic stimulation (IPS) may precede the transition to seizure activity in photosensitive patients. We assumed that such changes would be reflected in the phase of harmonic components of the evoked responses. Thus, we determined the phase clustering for different harmonic components of these MEG/EEG signals. We found that the patients who develop epileptiform discharges during IPS present an enhancement of the phase clustering index at the gamma frequency band, compared with that at the driving frequency. We introduce a quantity--relative phase clustering index (rPCI)--by means of which this enhancement can be quantified. We argue that this quantity reflects the degree of excitability of the underlying dynamical system and it can indicate presence of nonlinear dynamics, rPCI can be applied to detect transitions to epileptic seizure activity in patients with known sensitivity to IPS. Index Terms--EEG, epilepsy, IPS, MEG, nonlinear dynamics, phase clustering, seizure anticipation, synchrony, visual sensitivity.

We propose a new group-theoretical approach to the problem of alignment of time events in multichannel signal recordings. Such an alignment is an essential phase in the classification of transients in electroencephalogram/magnetoencephalogram (MEG) signals. A common reference frame is reconstructed applying a time translation transformation based on delayed mutual correlation functions of the individual events. The method is applied to MEG data sets recorded from epileptic patients showing paroxysmal interictal discharges. Index Terms--Alignment, clustering, epilepsy, MEG, paroxysmal discharges.

In this overview, we consider epilepsies as dynamical diseases of brain systems since they are manifestations of the property of neuronal networks to display multistable dynamics. To illustrate this concept we may assume that at least two states of the epileptic brain are possible: the interictal state characterized by a normal, apparently random, steady-state electroencephalography (EEG) ongoing activity, and the ictal state, that is characterized by paroxysmal occurrence of synchronous oscillations and is generally called, in neurology, a seizure. The transition between these two states can either occur: 1) as a continuous sequence of phases, like in some cases of mesial temporal lobe epilepsy (MTLE); or 2) as a sudden leap, like in most cases of absence seizures. In the mathematical terminology of nonlinear systems, we can say that in the first case the system's attractor gradually deforms from an interictal to an ictal attractor. The causes for such a deformation can be either endogenous or external. In this type of ictal transition, the seizure possibly may be anticipated in its early, preclinical phases. In the second case, where a sharp critical transition takes place, we can assume that the system has at least two simultaneous interictal and ictal attractors all the time. To which attractor the trajectories converge, depends on the initial conditions and the system's parameters. An essential question in this scenario is how the transition between the normal ongoing and the seizure activity takes place. Such a transition can occur either due to the influence of external or endogenous factors or due to a random perturbation and, thus, it will be unpredictable. These dynamical changes may not be detectable from the analysis of the ongoing EEG, but they may be observable only by measuring the system's response to externally administered stimuli. In the special cases of reflex epilepsy, the leap between the normal ongoing attractor and the ictal attractor is... [ABSTRACT FROM AUTHOR]