Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep

Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calculation of dimensionality estimates the degrees of freedom of a signal. Nevertheless, it is difficult to decide from this kind of analysis whether a process is quasiperiodic or chaotic. Therefore, we performed a new analysis by calculating the first positive Lyapunov exponent L ₁ from sleep EEG data. Lyapunov exponents measure the mean exponential expansion or contraction of a flow in phase space. L ₁ is zero for periodic as well as quasiperiodic processes, but positive in the case of chaotic processes expressing the sensitive dependence on initial conditions. We calculated L ₁ for sleep EEG segments of 15 healthy men corresponding to the sleep stages I, II, III, IV, and REM (according to Rechtschaffen and Kales). Our investigations support the assumption that EEG signals are neither quasiperiodic waves nor a simple noise. Moreover, we found statistically significant differences between the values of L ₁ for different sleep stages. All together, this kind of analysis yields a useful extension of the characterization of EEG signals in terms of nonlinear dynamical system theory.     

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes—the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.     

Nonlinear dynamics of two-dimensional cardiac action potential duration mapping model with memory

The aim of this work is the analysis of the nonlinear dynamics of two-dimensional mapping model of cardiac action potential duration (2D-map APD) with memory derived from one dimensional map (1D-map). Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting cardiac arrhythmias. For a constant rate of stimulation the short action potential during alternans is followed by a longer DI and inversely. It has been suggested that these differences in DI are responsible for the occurrence and maintenance of APD alternans. We focus our attention on the observed bifurcations produced by a change in the stimulation period and a fixed value of a particular parameter in the model. This parameter provides new information about the dynamics of the APD with memory, such as the occurrence of bistabilities not previously described in the literature, as well as the fact that synchronization rhythms occur in different ways and in a new fashion as the stimulation frequency increases. Moreover, we show that this model is flexible enough as to accurately reflect the chaotic dynamics properties of the APD: we have highlighted the fractal structure of the strange attractor of the 2D-map APD, and we have characterized chaos by tools such as the calculation of the Lyapunov exponents, the fractal dimension and the Kolmogorov entropy, with the next objective of refining the study of the nonlinear dynamics of the duration of the action potential and to apply methods of controlling chaos.

     

Locally optimal extracellular stimulation for chaotic desynchronization of neural populations

We use optimal control theory to design a methodology to find locally optimal stimuli for desynchronization of a model of neurons with extracellular stimulation. This methodology yields stimuli which lead to positive Lyapunov exponents, and hence desynchronizes a neural population. We analyze this methodology in the presence of interneuron coupling to make predictions about the strength of stimulation required to overcome synchronizing effects of coupling. This methodology suggests a powerful alternative to pulsatile stimuli for deep brain stimulation as it uses less energy than pulsatile stimuli, and could eliminate the time consuming tuning process.     

Sound Asleep: Processing and Retention of Slow Oscillation Phase-Targeted Stimuli

The sleeping brain retains some residual information processing capacity. Although direct evidence is scarce, a substantial literature suggests the phase of slow oscillations during deep sleep to be an important determinant for stimulus processing. Here, we introduce an algorithm for predicting slow oscillations in real-time. Using this approach to present stimuli directed at both oscillatory up and down states, we show neural stimulus processing depends importantly on the slow oscillation phase. During ensuing wakefulness, however, we did not observe differential brain or behavioral responses to these stimulus categories, suggesting no enduring memories were formed. We speculate that while simpler forms of learning may occur during sleep, neocortically based memories are not readily established during deep sleep.     

Are Complexity Metrics Reliable in Assessing HRV Control in Obese Patients During Sleep?

Obesity is associated with cardiovascular mortality. Linear methods, including time domain and frequency domain analysis, are normally applied on the heart rate variability (HRV) signal to investigate autonomic cardiovascular control, whose imbalance might promote cardiovascular disease in these patients. However, given the cardiac activity non-linearities, non-linear methods might provide better insight. HRV complexity was hereby analyzed during wakefulness and different sleep stages in healthy and obese subjects. Given the short duration of each sleep stage, complexity measures, normally extracted from long-period signals, needed be calculated on short-term signals. Sample entropy, Lempel-Ziv complexity and detrended fluctuation analysis were evaluated and results showed no significant differences among the values calculated over ten-minute signals and longer durations, confirming the reliability of such analysis when performed on short-term signals. Complexity parameters were extracted from ten-minute signal portions selected during wakefulness and different sleep stages on HRV signals obtained from eighteen obese patients and twenty controls. The obese group presented significantly reduced complexity during light and deep sleep, suggesting a deficiency in the control mechanisms integration during these sleep stages. To our knowledge, this study reports for the first time on how the HRV complexity changes in obesity during wakefulness and sleep. Further investigation is needed to quantify altered HRV impact on cardiovascular mortality in obesity.     

Chaos theory: the transforming role of functional systems

The review of applications of chaos theory in physiology shows, that this theory makes possible to get quantitative measure of degree of the order for such processes as neuronal activity, heart rate, electroencephalogram etc. The basic concepts of chaos theory--fractal dimension. Liapunov exponents, entropy, some algorithms for obtaining of quantitative characteristics of degree of the order for different processes--have been considered in this paper. The main areas for application chaos theory in physiology and medicine have been revealed and some examples of practical and theoretical achievements in this new region of investigation are given, as well as existing limitations and problems with results interpretation. Entirely new possibilities for understanding of the order in physiological processes are shown when estimation is made from the point of view of the functional system theory.     

Ventilatory chaos is impaired in carotid atherosclerosis

Ventilatory chaos is strongly linked to the activity of central pattern generators, alone or influenced by respiratory or cardiovascular afferents. We hypothesized that carotid atherosclerosis should alter ventilatory chaos through baroreflex and autonomic nervous system dysfunctions. Chaotic dynamics of inspiratory flow was prospectively evaluated in 75 subjects undergoing carotid ultrasonography: 27 with severe carotid stenosis (>70%), 23 with moderate stenosis (<70%), and 25 controls. Chaos was characterized by the noise titration method, the correlation dimension and the largest Lyapunov exponent. Baroreflex sensitivity was estimated in the frequency domain. In the control group, 92% of the time series exhibit nonlinear deterministic chaos with positive noise limit, whereas only 68% had a positive noise limit value in the stenoses groups. Ventilatory chaos was impaired in the groups with carotid stenoses, with significant parallel decrease in the noise limit value, correlation dimension and largest Lyapunov exponent, as compared to controls. In multiple regression models, the percentage of carotid stenosis was the best in predicting the correlation dimension (p<0.001, adjusted R²: 0.35) and largest Lyapunov exponent (p<0.001, adjusted R²: 0.6). Baroreflex sensitivity also predicted the correlation dimension values (p = 0.05), and the LLE (p = 0.08). Plaque removal after carotid surgery reversed the loss of ventilatory complexity. To conclude, ventilatory chaos is impaired in carotid atherosclerosis. These findings depend on the severity of the stenosis, its localization, plaque surface and morphology features, and is independently associated with baroreflex sensitivity reduction. These findings should help to understand the determinants of ventilatory complexity and breathing control in pathological conditions.     

A thalamo-cortical neural mass model for the simulation of brain rhythms during sleep

Cortico-thalamic interactions are known to play a pivotal role in many brain phenomena, including sleep, attention, memory consolidation and rhythm generation. Hence, simple mathematical models that can simulate the dialogue between the cortex and the thalamus, at a mesoscopic level, have a great cognitive value. In the present work we describe a neural mass model of a cortico-thalamic module, based on neurophysiological mechanisms. The model includes two thalamic populations (a thalamo-cortical relay cell population, TCR, and its related thalamic reticular nucleus, TRN), and a cortical column consisting of four connected populations (pyramidal neurons, excitatory interneurons, inhibitory interneurons with slow and fast kinetics). Moreover, thalamic neurons exhibit two firing modes: bursting and tonic. Finally, cortical synapses among pyramidal neurons incorporate a disfacilitation mechanism following prolonged activity. Simulations show that the model is able to mimic the different patterns of rhythmic activity in cortical and thalamic neurons (beta and alpha waves, spindles, delta waves, K-complexes, slow sleep waves) and their progressive changes from wakefulness to deep sleep, by just acting on modulatory inputs. Moreover, simulations performed by providing short sensory inputs to the TCR show that brain rhythms during sleep preserve the cortex from external perturbations, still allowing a high cortical activity necessary to drive synaptic plasticity and memory consolidation. In perspective, the present model may be used within larger cortico-thalamic networks, to gain a deeper understanding of mechanisms beneath synaptic changes during sleep, to investigate the specific role of brain rhythms, and to explore cortical synchronization achieved via thalamic influences.     

Deterministic chaos in the dynamics of biomembrane ion channel current

Within the framework of the theory of deterministic chaos, a nonlinear first-order differential equation with delay and relaxation with periodic influence on channel current for the parameter of order (deviation of channel current from the equilibrium value) was obtained. The numerical solutions of the equation indicate a chaotic dynamics of the order parameter and conformation potential of the channel protein with positive Lyapunov indices. By integration in the time interval between the "jumps" of ions through energy barriers of the channel protein, a mapping was obtained that also results in chaotic solutions realized in experiments. Basic kinetic characteristics of ionic channels for the mapping were obtained: the probability for the channel to be in the open state, P0, and the mean duration of a pack of current pulses depending on controlling parameters. Algorithms for constructing bifurcation diagrams with the transition to chaos and for determining Lyapunov indices and Kholmogorov entropy, pulsation spectra, and other parameters of chaotic dymanics were developed.     

Research on the relation of EEG signal chaos characteristics with high-level intelligence activity of human brain

Using phase space reconstruct technique from one-dimensional and multi-dimensional time series and the quantitative criterion rule of system chaos, and combining the neural network; analyses, computations and sort are conducted on electroencephalogram (EEG) signals of five kinds of human consciousness activities (relaxation, mental arithmetic of multiplication, mental composition of a letter, visualizing a 3-dimensional object being revolved about an axis, and visualizing numbers being written or erased on a blackboard). Through comparative studies on the determinacy, the phase graph, the power spectra, the approximate entropy, the correlation dimension and the Lyapunov exponent of EEG signals of 5 kinds of consciousness activities, the following conclusions are shown: (1) The statistic results of the deterministic computation indicate that chaos characteristic may lie in human consciousness activities, and central tendency measure (CTM) is consistent with phase graph, so it can be used as a division way of EEG attractor. (2) The analyses of power spectra show that ideology of single subject is almost identical but the frequency channels of different consciousness activities have slight difference. (3) The approximate entropy between different subjects exist discrepancy. Under the same conditions, the larger the approximate entropy of subject is, the better the subject's innovation is. (4) The results of the correlation dimension and the Lyapunov exponent indicate that activities of human brain exist in attractors with fractional dimensions. (5) Nonlinear quantitative criterion rule, which unites the neural network, can classify different kinds of consciousness activities well. In this paper, the results of classification indicate that the consciousness activity of arithmetic has better differentiation degree than that of abstract.     

Neurophysiological Mechanisms of Attention

According to contemporary notions, the functional states of the brain are stages of a single continuum, the lower limit of which is a state of coma and sleep, and the upper limit, a state of hyperexcitation and alertness [2, 6, 24, 28, 32, and others].     

Chaotic dynamics of the surface potential of human skeletal muscles determined in electromyography

A new method for differential evaluation of electromyographic data on straited muscles of human lower extremities was developed. This method is based on nonlinear dynamics and thermodynamics and can be used for identification of pathologies. The distance between two trajectories of the potential of two symmetric muscles was the main measured characteristic of coordinated muscle work. These data were used to determine the Lyapunov exponent and the time of forgetting initial conditions, which reflect the generally chaotic dynamics of muscle activity. Application of the theory of deterministic chaos to analysis of electromyographic patterns can improve the diagnosis of peripheral nervous system diseases and the efficacy of treatment control. Quantitation of nonlinear dynamic parameters of muscle activity, clear data representation, high prognostic information content of the Lyapunov exponent and Kolmogorov entropy are among the advantages of the new method.     

Salubrinal, an inhibitor of protein synthesis, promotes deep slow wave sleep

Previous work showed that sleep is associated with increased brain protein synthesis and that arrest of protein synthesis facilitates sleep. Arrest of protein synthesis is induced during the endoplasmic reticulum (ER) stress response, through phosphorylation of eukaryotic initiation factor 2alpha (p-eIF2alpha). We tested a hypothesis that elevation of p-eIF2alpha would facilitate sleep. We studied the effects of intracerebroventricular infusion of salubrinal (Salub), which increases p-eIF2alpha by inhibiting its dephosphorylation. Salub increased deep slow wave sleep by 255%, while reducing active waking by 49%. Delta power within non-rapid eye movement (NREM) sleep was increased, while power in the sigma, beta, and gamma bands during NREM was reduced. We found that Salub increased expression of p-eIF2alpha in the basal forebrain (BF) area, a sleep-wake regulatory brain region. Therefore, we quantified the p-eIF2alpha-immunolabeled neurons in the BF area; Salub administration increased the number of p-eIF2alpha-expressing noncholinergic neurons in the caudal BF. In addition, Salub also increased the intensity of p-eIF2alpha expression in both cholinergic and noncholinergic neurons, but this was more widespread among the noncholinergic neurons. Our findings support a hypothesis that sleep is facilitated by signals associated with the ER stress response.     

Neural chaos and schizophrenia

Recent data indicate that random-like processes are related to the defects in the organization of semantic memory in schizophrenia which is more disorganized and less definable than those of controls with more semantic links and more bizarre and atypical associations. These aspects of schizophrenic cognition are similar to characteristics of chaotic nonlinear dynamical systems. In this context, the hypothesis tested in this study is that dynamic changes of electrodermal activity (EDA) as a measure of brain and autonomic activity may serve as a characteristic which can be used as an indicator of possible neural chaotic process in schizophrenia. In the present study, bilateral EDA in rest conditions were measured in 40 schizophrenic patients and 40 healthy subjects. Results of nonlinear and statistical analysis indicate left-side significant differences of positive largest Lyapunov exponents in schizophrenia patients compared to the control group. This might be interpreted that the neural activity during rest in schizophrenic patients is significantly more chaotic than in the control group. The relationship was confirmed by surrogate data testing. These data suggest that increased neural chaos in patients with schizophrenia may influence brain processes that can cause random-like disorganization of mental processes.     

Covert waking brain activity reveals instantaneous sleep depth

The neural correlates of the wake-sleep continuum remain incompletely understood, limiting the development of adaptive drug delivery systems for promoting sleep maintenance. The most useful measure for resolving early positions along this continuum is the alpha oscillation, an 8-13 Hz electroencephalographic rhythm prominent over posterior scalp locations. The brain activation signature of wakefulness, alpha expression discloses immediate levels of alertness and dissipates in concert with fading awareness as sleep begins. This brain activity pattern, however, is largely ignored once sleep begins. Here we show that the intensity of spectral power in the alpha band actually continues to disclose instantaneous responsiveness to noise--a measure of sleep depth--throughout a night of sleep. By systematically challenging sleep with realistic and varied acoustic disruption, we found that sleepers exhibited markedly greater sensitivity to sounds during moments of elevated alpha expression. This result demonstrates that alpha power is not a binary marker of the transition between sleep and wakefulness, but carries rich information about immediate sleep stability. Further, it shows that an empirical and ecologically relevant form of sleep depth is revealed in real-time by EEG spectral content in the alpha band, a measure that affords prediction on the order of minutes. This signal, which transcends the boundaries of classical sleep stages, could potentially be used for real-time feedback to novel, adaptive drug delivery systems for inducing sleep.     

Changes in seizure activity thresholds of the cat brain in various stages of sleep and wakefulness

Changes in seizure activity of the brain evoked by electrical stimulation of the dorsal hippocampus in various stages of sleep and wakefulness were studied in adult cats. During slow sleep, when the EEG is dominated by high-voltage slow waves, near-threshold epileptogenic hippocampal stimulation evokes well-marked paroxysmal discharges. During wakefulness or the paradoxical phase of sleep, when the EEG is desynchronized, this hippocampal stimulation is less effective: either no seizure discharges are produced or they are weak. Activation of the mesencephalic reticular formation before epileptogenic hippocampal stimulation hinders the appearance of seizure activity whereas activation after hippocampal stimulation does not inhibit paroxysmal discharges already in progress; on the contrary, in some cases they are actually strengthened a little. One of the main factors limiting the appearance and spread of seizure activity is considered to be the tonic inhibitory influence of the neocortex on other parts of the brain.     

Meteorological complexity in the Amazonian area of Ecuador: An approach based on dynamical system theory

Meteorological rhythms and trends are important components for ecosystem functioning. The complex time evolution of meteorology is often difficult to capture using linear methods. The objective of this work was to use basic concepts of dynamical system theory for assessing time evolution of daily records from six local meteorological variables collected at the Amazonian basin. We analysed rainfall, relative humidity, evaporation, minimum temperature, relative sunshine duration and evaporation/precipitation ratio. Data were collected from Puyo meteorological station, Pastaza Province, Ecuador. Data sets covered 4 years (from 1st January 2001 to 1st January 2005) (a total of 1460 data points). The TISEAN Software Package (public domain software available at http://www.mpipks-dresdren.mpg.de/～tisean) was used for deriving nonlinear parameters from each time series. We found interesting evidence of chaotic behaviour as maximal Lyapunov exponents were positive for all time series considered. These results were consistent with those computed from corresponding surrogate time series. Positive Lyapunov exponents allow an estimation of the lead time of correlation for making reliable predictions.     

Effect of decimation on the classification rate of non-linear analysis methods applied to uterine EMG signals

Recently, much attention has been paid to the use of non-linear analysis techniques for the characterization of biological signals. Several measures have been proposed to detect non-linear characteristics in time series. The effect of sampling frequency on the performance of these non-linear methods has rarely been evaluated. In this paper, we present a preliminary study of this effect for four methods that are widely used in non-linearity detection: Time reversibility, Sample Entropy, Lyapunov Exponents and Delay Vector Variance. These methods have been applied to real uterine EMG signals with the aim of distinguishing between pregnancy and labor contractions. The signals were used to classify contractions before and after decimating them by a factor 10. The results show that decimation improves the performance for sample entropy. It reduces it considerably for Delay Vector Variance and only slightly for Time reversibility and Lyapunov exponents. Time reversibility still gives the highest classification rate. The methods were much less computationally expensive after down sampling.