Non-linear analysis of the electroencephalogram in Creutzfeldt-Jakob disease

Creutzfeldt-Jakob disease is a rare, neurological, dementing disorder characterised by periodic sharp waves in the electroencephalogram (EEG). Non-linear analysis of these EEG changes may provide insight into the abnormal dynamics of cortical neural networks in this disorder. Babloyantz et al. have suggested that the periodic sharp waves reflect low-dimensional chaotic dynamics in the brain. In the present study this hypothesis was re-examined using newly developed techniques for non-linear time series analysis. We analysed the EEG of a patient with autopsy-proven Creutzfeldt-Jakob disease using the method of non-linear forecasting as introduced by Sugihara and May, and we tested for non-linearity with amplitude-adjusted, phase-randomised surrogate data. Two epochs with generalised periodic sharp waves showed clear evidence for non-linearity. These epochs could be predicted better and further ahead in time than most of the irregular background activity. Testing against cycle-randomised surrogate data and close inspection of the periodograms showed that the non-linearity of the periodic sharp waves may be better explained by quasi-periodicity than by low-dimensional chaos. The EEG further displayed at least one example of a sudden, large qualitative change in the dynamics, highly suggestive of a bifurcation. The presence of quasi-periodicity and bifurcations strongly argues for the use of a non-linear model to describe the EEG in Creutzfeldt-Jakob disease. Received: 28 October 1996 / Accepted in revised form: 8 July 1997     

Stimulus and network dynamics collide in a ratiometric model of the antennal lobe macroglomerular complex

Time is considered to be an important encoding dimension in olfaction, as neural populations generate odour-specific spatiotemporal responses to constant stimuli. However, during pheromone mediated anemotactic search insects must discriminate specific ratios of blend components from rapidly time varying input. The dynamics intrinsic to olfactory processing and those of naturalistic stimuli can therefore potentially collide, thereby confounding ratiometric information. In this paper we use a computational model of the macroglomerular complex of the insect antennal lobe to study the impact on ratiometric information of this potential collision between network and stimulus dynamics. We show that the model exhibits two different dynamical regimes depending upon the connectivity pattern between inhibitory interneurons (that we refer to as fixed point attractor and limit cycle attractor), which both generate ratio-specific trajectories in the projection neuron output population that are reminiscent of temporal patterning and periodic hyperpolarisation observed in olfactory antennal lobe neurons. We compare the performance of the two corresponding population codes for reporting ratiometric blend information to higher centres of the insect brain. Our key finding is that whilst the dynamically rich limit cycle attractor spatiotemporal code is faster and more efficient in transmitting blend information under certain conditions it is also more prone to interference between network and stimulus dynamics, thus degrading ratiometric information under naturalistic input conditions. Our results suggest that rich intrinsically generated network dynamics can provide a powerful means of encoding multidimensional stimuli with high accuracy and efficiency, but only when isolated from stimulus dynamics. This interference between temporal dynamics of the stimulus and temporal patterns of neural activity constitutes a real challenge that must be successfully solved by the nervous system when faced with naturalistic input.     

Nonlinear EEG analysis based on a neural mass model

The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model. The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter. On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al. (1978) is present in actual brain dynamics. Received: 11 August 1997 / Accepted in revised form: 20 April 1999     

Quasicycles revisited: apparent sensitivity to initial conditions

Environmental noise is known to sustain cycles by perturbing a deterministic approach to equilibrium that is itself oscillatory. Quasicycles produced in this way display a regular period but varied amplitude. They were proposed by Nisbet and Gurney (Nature 263 (1976) 319) as one possible explanation for population fluctuations in nature. Here, we revisit quasicyclic dynamics from the perspective of nonlinear time series analysis. Time series are generated with a predator-prey model whose prey's growth rate is driven by environmental noise. A method for the analysis of short and noisy data provides evidence for sensitivity to initial conditions, with a global Lyapunov exponent often close to zero characteristic of populations 'at the edge of chaos'. Results with methods restricted to long time series are consistent with a finite-dimensional attractor on which dynamics are sensitive to initial conditions. These results are compared with those previously obtained for quasicycles in an individual-based model with heterogeneous spatial distributions. Patterns of sensitivity to initial conditions are shown to differentiate phase-forgetting from phase-remembering quasicycles involving a periodic driver. The previously reported mode at zero of Lyapunov exponents in field and laboratory populations may reflect, in part, quasicyclic dynamics.     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Use of the fractal dimension for the analysis of electroencephalographic time series

 Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals are needed and consequently only long-term events can be analysed; also much calculation time is required. To analyse events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of fractal dimension behaviour, in this work we optimize and propose a new method for evaluating the fractal dimension. Moreover, we study the robustness of this evaluation in the presence of white or line noises and compare the results with those obtained with conventional spectral methods. The non-linear analysis carried out allows us to investigate relevant EEG events shorter than those detectable by means of other linear and non-linear techniques, thus achieving a better temporal resolution. An interesting link between the spectral distribution and the fractal dimension value is also pointed out. Received: 21 November 1996 / Accepted in revised form: 1 July 1997     

Hemispheric asymmetries in cortical electrical activity during sleep. I

The hypothesis of a predominance of the right hemisphere in stage REM as compared to NREM has been tested through a spectral analysis of the EEG recorded from left (T3) and right (T4) temporal sites in 5 young healthy right-handed male subjects. Variations in the asymmetry coefficient R - L/R + L in different sleep stages have been analyzed by one way ANOVAs and Sheffé's tests. The hypothesis of a progressive increase in left hemisphere activity throughout different REM cycles as one approaches final awakenings have been investigated by comparing variations in the asymmetry coefficient for epochs of REM and stage 2 NREM sampled in different phases of the REM cycle. EEG results do not support either the hypothesized stage dependent or cycle dependent variation in EEG activity during sleep. We question whether variations in EEG amplitude and synchronization can be used as indices of hemispheric asymmetries during sleep.     

An olfactory recognition model based on spatio-temporal encoding of odor quality in the olfactory bulb

In order to study the problem how the olfactory neural system processes the odorant molecular information for constructing the olfactory image of each object, we present a dynamic model of the olfactory bulb constructed on the basis of well-established experimental and theoretical results. The information relevant to a single odor, i.e. its constituent odorant molecules and their mixing ratios, are encoded into a spatio-temporal pattern of neural activity in the olfactory bulb, where the activity pattern corresponds to a limit cycle attractor in the mitral cell network. The spatio-temporal pattern consists of a temporal sequence of spatial firing patterns: each constituent molecule is encoded into a single spatial pattern, and the order of magnitude of the mixing ratio is encoded into the temporal sequence. The formation of a limit cycle attractor under the application of a novel odor is carried out based on the intensity-to-time-delay encoding scheme. The dynamic state of the olfactory bulb, which has learned many odors, becomes a randomly itinerant state in which the current firing state of the bulb itinerates randomly among limit cycle attractors corresponding to the learned odors. The recognition of an odor is generated by the dynamic transition in the network from the randomly itinerant state to a limit cycle attractor state relevant to the odor, where the transition is induced by the short-term synaptic changes made according to the Hebbian rule under the application of the odor stimulus. Received: 28 July 1997 / Accepted in revised form: 6 May 1998     

On the existence and the role of chaotic processes in the nervous system

Chaos theory is a rapidly growing field. As a technical term, “chaos” refers to deterministic but unpredictable processes being sensitively dependent upon initial conditions. Neurobiological models and experimental results are very complicated and some research groups have tried to pursue the “neuronal chaos”. Babloyantz's group has studied the fractal dimension (d) of electroencephalograms (EEG) in various physiological and pathological states. From deep sleep (d=4) to full awakening (d>8), a hierarchy of “strange” attractors paralles the hierarchy of states of consciousness. In epilepsy (petit mal), despite the turbulent aspect of a seizure, the attractor dimension was near to 2. In Creutzfeld-Jacob disease, the regular EEG activity corresponded to an attractor dimension less than the one measured in deep sleep. Is it healthy to be chaotic? An “active desynchronisation” could be favourable to a physiological system. Rapp's group reported variations of fractal dimension according to particular tasks. During a mental arithmetic task, this dimension increased. In another task, a P300 fractal index decreased when a target was identified. It is clear that the EEG is not representing noise. Its underlying dynamics depends on only a few degrees of freedom despite yet it is difficult to compute accurately the relevant parameters. What is the cognitive role of such a chaotic dynamics? Freeman has studied the olfactory bulb in rabbits and rats for 15 years. Multi-electrode recordings of a few mm² showed a chaotic hierarchy from deep anaesthesia to alert state. When an animal identified a previously learned odour, the fractal dimension of the dynamics dropped off (near limit cycles). The chaotic activity corresponding to an alert-and-waiting state seems to be a field of all possibilities and a focused activity corresponds to a reduction of the attractor in state space. For a couple of years, Freeman has developed a model of the olfactory bulb-cortex system. The behaviour of the simple model “without learning” was quite similar to the real behaviour and a model “with learning” is developed. Recently, more and more authors insisted on the importance of the dynamic aspect of nervous functioning in cognitive modelling. Most of the models in the neural-network field are designed to converge to a stable state (fixed point) because such behaviour is easy to understand and to control. However, some theoretical studies in physics try to understand how a chaotic behaviour can emerge from neural networks. Sompolinsky's group showed that a sharp transition from a stable state to a chaotic state occurred in totally interconnected networks depending on the value of one control parameter. Learning in such systems is an open field. In conclusion, chaos does exist in neurophysiological processes. It is neither a kind of noise nor a pathological sign. Its main role could be to provide diversity and flexibility to physiological processes. Could “strange” attractors in nervous system embody mental forms? This is a difficult but fascinating question.     

A Dynamic Gene Regulatory Network Model That Recovers the Cyclic Behavior of Arabidopsis thaliana Cell Cycle

Cell cycle control is fundamental in eukaryotic development. Several modeling efforts have been used to integrate the complex network of interacting molecular components involved in cell cycle dynamics. In this paper, we aimed at recovering the regulatory logic upstream of previously known components of cell cycle control, with the aim of understanding the mechanisms underlying the emergence of the cyclic behavior of such components. We focus on Arabidopsis thaliana, but given that many components of cell cycle regulation are conserved among eukaryotes, when experimental data for this system was not available, we considered experimental results from yeast and animal systems. We are proposing a Boolean gene regulatory network (GRN) that converges into only one robust limit cycle attractor that closely resembles the cyclic behavior of the key cell-cycle molecular components and other regulators considered here. We validate the model by comparing our in silico configurations with data from loss- and gain-of-function mutants, where the endocyclic behavior also was recovered. Additionally, we approximate a continuous model and recovered the temporal periodic expression profiles of the cell-cycle molecular components involved, thus suggesting that the single limit cycle attractor recovered with the Boolean model is not an artifact of its discrete and synchronous nature, but rather an emergent consequence of the inherent characteristics of the regulatory logic proposed here. This dynamical model, hence provides a novel theoretical framework to address cell cycle regulation in plants, and it can also be used to propose novel predictions regarding cell cycle regulation in other eukaryotes.     

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.     

Nonlinear analysis of EEG signals: Surrogate data analysis

ObjectivesThe electroencephalogram (EEG) signal contains information about the state and condition of the brain. The aim of the study is to conduct a nonlinear analysis of the EEG signals and to compare the differences in the nonlinear characteristics of the EEG during normal state and the epileptic state.DataThe EEG data used for this study – which consisted of epileptic EEG and normal EEG – were obtained from the EEG database available with the Bonn University, Germany.ResultsThe attractors seen in normal and epileptic human brain dynamics were studied and compared. Surrogate data analyses were conducted on two nonlinear measures, namely the largest Lyapunov exponent and the correlation dimension, to test the hypothesis whether EEG signals were in accordance with linear stochastic models.DiscussionsThe existence of deterministic chaos in brain activity is confirmed by the existence of a chaotic attractor; also, saturation of the correlation dimension towards a definite value is the manifestation of a deterministic dynamics. Also a reduction is observed between the dimensionalities of the brain attractors from normal state to the epileptic state. The evaluation of the largest Lyapunov exponent also confirms the lowering of complexity during an episode of seizure.ConclusionIn case of Lyapunov exponent of EEG data, the change due to surrogating is small suggesting that it is not representing the system complexity properly but there is a marked change in the case of correlation dimension value due to surrogating.     

Passive bipedal walking with phasic muscle contraction

The existence of self-organizing walking patterns is often considered the result of a mechanical system interacting with the environment and a (neural) oscillating unit. The pattern generators might be thought of as an indispensable component for the existence of limit cycle behavior. This paper shows that this is not a necessity for the existence of a self-organizing bipedal walking pattern. Stable walking cycles emerge from a simple passive bipedal structure, with an energy source inevitably present to sustain the oscillation. In this work the energy source is chosen to be phasic muscle contraction. A two-dimensional model is composed of two legs and a hip mass, symbolizing the trunk. The stance leg stiffness is generated by two muscles. The hip stiffness is generated by four muscles. Muscle activation is caused by two reflex-like trigger signals, without feedback control. Human equivalent model parameters such as geometry and mass distribution were assumed. With return map analysis, the model is analyzed on periodic behavior. Stable walking cycles were found and could be manipulated during walking by varying the muscle or reflex parameters, forcing the oscillation to converge to a new attractor. Received: 5 November 1998 / Accepted in revised form: 26 March 1999     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Methods of non-linear dynamics in estimation of electroencephalograms of healthy people and of patients with epilepsy

A possibility is discussed of use of methods of non-linear dynamics for analysis of spontaneous EEG and if the EEG caused by low acoustic stimuli in healthy people and in patients with epilepsy. A use of methods of non-linear dynamics--the fractal dimension of EEG--in clinical practice and in research is described.     

对脑电相关维数计算中有关参数的探讨

应用Ｇｒａｓｓｂｅｒｇｅｒ－Ｐｒｏｃａｃｉａ ａｌｇｏｒｉｔｈｍ算法,通过点列长度、时间延迟、嵌入维数等参数的变化,对人的８导脑电时间序列的相关维数值的影响进行了研究。结果表明不仅参数的选用是否合理对计算值影响很大,而且脑电信号不像是一个稳定的混沌吸引子,因此拟用相关维数作用一项客观指标来准确地表示大脑的不同功能状态,几乎是不可能的。     

Chaotic component of human high-frequency EEG in the state of quiet wakefulness

Chaotic component of human EEG oscillations in the high-frequency band (14.7-100 Hz) was investigated. EEG was recorded from four points in symmetrical frontal and occipital scalp areas. The results of the non-linear analysis of the high-frequency EEG indicate the existence of the deterministic chaotic component with a high attractor correlation dimension. It was significantly different from the respective values of the Gaussian noise filtered in the same frequency band. In the state of quiet wakefulness (eyes closed), the dimensions of chaotic components of the EEG in all derivations did not differ from each other. Analysis of correlation pairs between the ensembles of correlation dimensions of the high-frequency EEG revealed reliable patterns of significant connections between the neocortical areas with individual features in different subjects. When the functional state of the brain was changed by hyperventilation, both the values of the correlation dimensions and the structure of inter-area connection patterns changed. We believe that the nonlinear component of high-frequency EEG is a sensitive and local characteristic of the functional state of the human brain.     

Association between individual EEG characteristics and the level of intelligence

The aim of this study was to investigate the relationship between individual electroencephalogram (EEG) characteristics in the resting state and the level of nonverbal intelligence. The study involved 77 students of Demidov Yaroslavl State University. Analysis of the relationship between IQ and spectral parameters of EEG theta, alpha, and two subbands of beta oscillations revealed that the amplitude and power of alphaband EEG oscillations and low frequency beta-band EEG oscillations were positively correlated with the performance in the nonverbal intelligence test. The variety of brain periodic regimes was assessed using the correlation dimension (CD) of EEG. The correlation dimension can be used to quantify the degree of complexity of the nonlinear dynamical system. It was found that the value of the EEG correlation dimension was positively associated with the level of intelligence. The periodicity of the EEG signal was studied using autocorrelation analysis. It was shown that the autocorrelogram duration was negatively associated with IQ and the autocorrelogram amplitude was positively associated with IQ. A regression equation for predicting the level of nonverbal intelligence based on the power of theta- and beta-band oscillations, alpha-band oscillation indexes, and the amplitude and autocorrelation characteristics of the EEG signal was obtained.     

Dynamics from a time series: can we extract the phase resetting curve from a time series?

Recordings of the membrane potential from a bursting neuron were used to reconstruct the phase curve for that neuron for a limited set of perturbations. These perturbations were inhibitory synaptic conductance pulses able to shift the membrane potential below the most hyperpolarized level attained in the free running mode. The extraction of the phase resetting curve from such a one-dimensional time series requires reconstruction of the periodic activity in the form of a limit cycle attractor. Resetting was found to have two components. In the first component, if the pulse was applied during a burst, the burst was truncated, and the time until the next burst was shortened in a manner predicted by movement normal to the limit cycle. By movement normal to the limit cycle, we mean a switch between two well-defined solution branches of a relaxation-like oscillator in a hysteretic manner enabled by the existence of a singular dominant slow process (variable). In the second component, the onset of the burst was delayed until the end of the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied, resetting was independent of amplitude but increased linearly with pulse duration. The predicted and the experimental phase resetting curves for a pyloric dilator neuron show satisfactory agreement. The method was applied to only one pulse per cycle, but our results suggest it could easily be generalized to accommodate multiple inputs.