Parameter-dependent transitions and the optimal control of dynamical diseases

Seizure prediction

There is mounting evidence that seizures are preceded by characteristic changes in the EEG that are detectable minutes before seizure onset. Using novel signal analysis techniques, researchers are beginning to characterize the transition from the interictal to the ictal state in quantitative terms. This research has led to the development of automated seizure prediction algorithms. Active debate persists regarding the interpretation of research results, methods of signal analysis, as well as experimental and statistical methods for testing seizure prediction algorithms. Developments in this field have led to new theories on the mechanism of seizure development and resolution. The ability to predict seizures could lead the way to novel diagnostic and therapeutic methods for the treatment of patients with epilepsy.

An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent

Analysis of intracranial electroencephalographic (iEEG) recordings in patients with temporal lobe epilepsy (TLE) has revealed characteristic dynamical features that distinguish the interictal, ictal, and postictal states and inter-state transitions. Experimental investigations into the mechanisms underlying these observations require the use of an animal model. A rat TLE model was used to test for differences in iEEG dynamics between well-defined states and to test specific hypotheses: 1) the short-term maximum Lyapunov exponent (STL(max)), a measure of signal order, is lowest and closest in value among cortical sites during the ictal state, and highest and most divergent during the postictal state; 2) STL(max) values estimated from the stimulated hippocampus are the lowest among all cortical sites; and 3) the transition from the interictal to ictal state is associated with a convergence in STL(max) values among cortical sites. iEEGs were recorded from bilateral frontal cortices and hippocampi. STL(max) and T-index (a measure of convergence/divergence of STL(max) between recorded brain areas) were compared among the four different periods. Statistical tests (ANOVA and multiple comparisons) revealed that ictal STL(max) was lower (p<0.05) than other periods, STL(max) values corresponding to the stimulated hippocampus were lower than those estimated from other cortical regions, and T-index values were highest during the postictal period and lowest during the ictal period. Also, the T-index values corresponding to the preictal period were lower than those during the interictal period (p<0.05). These results indicate that a rat TLE model demonstrates several important dynamical signal characteristics similar to those found in human TLE and support future use of the model to study epileptic state transitions.

ABSENCE SEIZURES AS RESETTING MECHANISMS OF BRAIN DYNAMICS

To understand the increase in age-related incidence and frequency of absence seizures in the rat brain, we investigated the effect of these seizures on brain dynamics. This paper puts forward the hypothesis that age-related differences in the expression of absence seizures are associated with the ability of the seizures to reset brain dynamics.

Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model

It is a challenging problem to establish safe and simple therapeutic methods for various complicated diseases of the nervous system, particularly dynamical diseases such as epilepsy, Alzheimer's disease, and Parkinson's disease. From the viewpoint of nonlinear dynamical systems, a dynamical disease can be considered to be caused by a bifurcation induced by a change in the values of one or more regulating parameter. Therefore, the theory of bifurcation control may have potential applications in the diagnosis and therapy of dynamical diseases. In this study, we employ a washout filter-aided dynamic feedback controller to control the onset of Hopf bifurcation in the Hodgkin-Huxley (HH) model. Specifically, by the control scheme, we can move the Hopf bifurcation to a desired point irrespective of whether the corresponding steady state is stable or unstable. In other words, we are able to advance or delay the Hopf bifurcation, so as to prevent it from occurring in a certain range of the externally applied current. Moreover, we can control the criticality of the bifurcation and regulate the oscillation amplitude of the bifurcated limit cycle. In the controller, there are only two terms: the linear term and the nonlinear cubic term. We show that while the former determines the location of the Hopf bifurcation, the latter regulates the criticality of the Hopf bifurcation. According to the conditions of the occurrence of Hopf bifurcation and the bifurcation stability coefficient, we can analytically deduce the linear term and the nonlinear cubic term, respectively. In addition, we also show that mixed-mode oscillations (MMOs), featuring slow action potential generation, which are frequently observed in both experiments and models of chemical and biological systems, appear in the controlled HH model. It is well known that slow firing rates in single neuron models could be achieved only by type-I neurons. However, the controlled HH model is still classified as a type-II neuron, as is the original HH model. We explain that the occurrence of MMOs can be related to the presence of the canard explosion where a small oscillation grows through a sequence of canard cycles to a relaxation oscillation as the control parameter moves through an interval of exponentially small width.

Chaos or noise in EEG signals; dependence on state and brain site

EEG signals have been considered to result either from random processes or to be generated by non-linear dynamic systems exhibiting chaotic behaviour. In the latter case, the system may behave as a deterministic chaotic attractor. The complexity of the attractor can be characterized by the correlation dimension that can be computed from one signal generated by the system. A new procedure was developed and applied in order to test whether the correlation dimension, calculated from an EEG epoch, may correspond to a chaotic attractor or to a random process. This procedure was applied to EEG signals recorded from different sites of the limbic cortex of the rat during different states: wakeful rest, locomotion and in the course of an epileptic seizure induced by kindling. The signals recorded during the first two states had high dimensions and could not be distinguished from random noise. However, during an epileptic seizure the correlation dimension became low (between 2 and 4) indicating that in this state the networks behave as chaotic systems. A low correlation dimension appeared at different times and brain sites during an epileptic seizure. These results show that the computation of the correlation dimension may be useful in order to obtain insight into the dynamics of the propagation of an epileptic seizure in the brain.