Is there chaos in the brain? II. Experimental evidence and related models

A computational study of the interdependencies between neuronal impulse pattern, noise effects and synchronization

Alterations of individual neurons dynamics and associated changes of the activity pattern, especially the transition from tonic firing (single-spikes) to bursts discharges (impulse groups), play an important role for neuronal information processing and synchronization in many physiological processes (sensory encoding, information binding, hormone release, sleep-wake cycles) as well as in disease (Parkinson, epilepsy). We have used Hodgkin-Huxley-type model neurons with subthreshold oscillations to examine the impact of noise on neuronal encoding and thereby have seen significant differences depending on noise implementation as well as on the neuron's dynamic state. The importance of the individual neurons' dynamics is further elucidated by simulation studies with electrotonically coupled model neurons which revealed mutual interdependencies between the alterations of the network's coupling strength and neurons' activity patterns with regard to synchronization. Remarkably, a pacemaker-like activity pattern which revealed to be much more noise sensitive than the bursting patterns also requires much higher coupling strengths for synchronization. This seemingly simple pattern is obviously governed by more complex dynamics than expected from a conventional pacemaker which may explain why neurons more easily synchronize in the bursting than in the tonic firing mode.

Self-organization in a parametrically coupled logistic map network: a model for information processing in the visual cortex

In this paper, a new model seeking to emulate the way the visual cortex processes information and interacts with subcortical areas to produce higher level brain functions is described. We developed a macroscopic approach that incorporates salient attributes of the cortex based on combining tools of nonlinear dynamics, information theory, and the known organizational and anatomical features of cortex. Justifications for this approach and demonstration of its effectiveness are presented. We also demonstrate certain capabilities of this model in producing efficient sparse representations and providing the cortical computational maps.

Dissociation, cognitive conflict and nonlinear patterns of heart rate dynamics in patients with unipolar depression

Recent findings in cognitive neuroscience indicate that activation of anterior cingulate cortex (ACC) is related to detecting cognitive conflict. Conflict related ACC activation elicits responses in central autonomic network which can be assessed by psychophysiological measures such as heart rate variability (i.e. beat to beat R-R intervals--RRI). Recent findings in neuroscience also suggest that cognitive conflict is related to specific nonlinear chaotic changes of the signal generated by the neural systems. The present study used Stroop word-colour test as an experimental approach to the study of cognitive conflict in connection with RRI measurement, psychometric measurement of dissociation (DES) and calculation of largest Lyapunov exponents in nonlinear data analysis of RRI time series in 40 patients with unipolar depression and 35 healthy controls. Significant correlation 0.58 (p<0.01) between largest Lyapunov exponents and DES found in depressive patients indicate that cognitive conflict related neural interference during conflicting Stroop task is closely related to dissociative processes. These results present first supportive evidence that degree of chaos could be related to dissociation.

Chaotic patterns of autonomic activity during hypnotic recall

Chaotic neural dynamics likely emerge in cognitive processes and may present time periods that are extremely sensitive to influences affecting the neural system. Recent findings suggest that this sensitivity may increase during retrieval of stressful emotional experiences reflecting underlying mechanism related to consolidation of traumatic memories. In this context, hypnotic recall of anxiety memories in 10 patients, simultaneously with ECG measurement was performed. The same measurement was performed during control cognitive task in 8 anxiety patients and 22 healthy controls. Nonlinear data analysis of ECG records indicates significant increase in the degree of chaos during retrieval of stressful memory in all the patients. The results suggest a role of chaotic neural dynamics during processing of anxiety-related stressful memories.

Chaos in schizophrenia associations, reality or metaphor?

There is evidence that schizophrenic associations display "chaotic", random-like behavior and decreased predictability. The evidence suggests a hypothesis that the "chaotic" mental disorganization could be explained within the concept of nonlinear dynamics and complexity in the brain that may cause chaotic neural organization. Testing of the hypothesis in the present study was performed using nonlinear analysis of bilateral electrodermal activity (EDA) during resting state and an association test in 56 schizophrenic patients and 44 healthy participants. EDA is a suitable measure of brain and autonomic activity reflecting neurobiological changes in schizophrenia that may indicate changes in nonlinear neural dynamics related to associative process. The results show that quantitative indices of chaotic dynamics (the largest Lyapunov exponents) calculated from EDA signals recorded during rest and the association test are significantly higher in schizophrenia patients than in the control group and increase during the test in comparison to the resting state. The difference was confirmed by statistical methods and using surrogate data testing that rejected an explanation within the linear statistical framework. The results provide supportive evidence that pseudo-randomness of schizophrenic associations and less predictability could be linked to increased complexity of nonlinear neural dynamics, although certain limitations in data interpretation must be taken into account.

Noise shaping in neural populations

Many neurons display intrinsic interspike interval correlations in their spike trains. However, the effects of such correlations on information transmission in neural populations are not well understood. We quantified signal processing using linear response theory supported by numerical simulations in networks composed of two different models: One model generates a renewal process where interspike intervals are not correlated while the other generates a nonrenewal process where subsequent interspike intervals are negatively correlated. Our results show that the fractional rate of increase in information rate as a function of network size and stimulus intensity is lower for the nonrenewal model than for the renewal one. We show that this is mostly due to the lower amount of effective noise in the nonrenewal model. We also show the surprising result that coupling has opposite effects in renewal and nonrenewal networks: Excitatory (inhibitory coupling) will decrease (increase) the information rate in renewal networks while inhibitory (excitatory coupling) will decrease (increase) the information rate in nonrenewal networks. We discuss these results and their applicability to other classes of excitable systems.

Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.

Deterministic Dynamics in Neuronal Discharge from Pallidotomy Targets

The non-linear dynamic specificity of the firing pattern discharged from neurons of the internal globus pallidus (GPi) was investigated by recording their spontaneous firing using a microelectrode during posteroventral pallidotomy in eight patients with Parkinson's disease. Raw data from the cells were processed to extract spiking events (discharges above a selected threshold) and the interspike interval was measured. Using the unstable periodic orbits extraction method, significant period-1, −2 and −3 orbits were identified in burst firing discharged from the GPi cells in all eight patients, suggesting that deterministic dynamics exist in the timing of the discharges. As well as providing a useful peri-operative technique for locating posteroventral pallidotomy targets in Parkinson's disease, this method also provides a promising basis for investigating characteristic neuronal discharges in other regions of the brain and for various other neurological disorders.

Instabilities in attractor networks with fast synaptic fluctuations and partial updating of the neurons activity

We present and study a probabilistic neural automaton in which the fraction of simultaneously-updated neurons is a parameter, rhoin(0,1). For small rho, there is relaxation towards one of the attractors and a great sensibility to external stimuli and, for rho > or = rho(c), itinerancy among attractors. Tuning rho in this regime, oscillations may abruptly change from regular to chaotic and vice versa, which allows one to control the efficiency of the searching process. We argue on the similarity of the model behavior with recent observations, and on the possible role of chaos in neurobiology.

Differential and chaotic calcium signatures in the symbiosis signaling pathway of legumes

Understanding how the cell uses a limited set of proteins to transduce very different signals into specific cellular responses is a central goal of cell biology and signal transduction disciplines. Although multifunctionality in signal transduction is widespread, the mechanisms that allow differential modes of signaling in multifunctional signaling pathways are not well defined. In legume plants, a common symbiosis signaling pathway composed of at least seven proteins mediates infection by both mycorrhizal fungi and rhizobial bacteria. Here we show that the symbiosis signaling pathway in legumes differentially transduces both bacterial and fungal signals (inputs) to generate alternative calcium responses (outputs). We show that these differential calcium responses are dependent on the same proteins, DMI1 and DMI2, for their activation, indicating an inherent flexibility in this signaling pathway. By using Lyapunov and other mathematical analyses, we discovered that both bacterial-induced and fungal-induced calcium responses are chaotic in nature. Chaotic systems require minimal energy to produce a wide spectrum of outputs in response to marginally different inputs. The flexibility provided by chaotic systems is consistent with the need to transduce two different signals, one from rhizobial bacteria and one from mycorrhizal fungi, by using common components of a single signaling pathway.

Special Section: The Skillful Body as a Concernful System of Possible Actions

For Merleau-Ponty, consciousness in skillful coping is a matter of prereflective `I can' and not explicit `I think that.' The body unifies many domain-specific capacities. There exists a direct link between the perceived possibilities for action in the situation (`affordances') and the organism's capacities. From Merleau-Ponty's descriptions it is clear that in a flow of skillful actions, the leading `I can' may change from moment to moment without explicit deliberation. How these transitions occur, however, is less clear. Given that Merleau-Ponty suggested that a better understanding of the self-organization of brain and behavior is important, I will re-read his descriptions of skillful coping in the light of recent ideas on neurodynamics. Affective processes play a crucial role in evaluating the motivational significance of objects and contribute to the individual's prereflective responsiveness to relevant affordances.

Progress in predicting seizure episodes with nonlinear methods

One of the most interesting and significant areas of epileptology has been the prediction of the onset of a seizure episode from preictal activity with nonlinear methods. Not only does this type of study have heuristic value for clinical neurophysiology, but it also has potential utilitarian value for the patient with seizures. In this review, 47 reports from 12 centers with multiple studies are presented in chronological order, as are single reports from 21 other centers. The chronological order was chosen to see if progress in the form of earlier prediction was made over time. Only 21% of these reports could provide specific times for the prediction of seizure onset. The range of values was several minutes to 4 hours, with an average (median) of 6-7 minutes. Some reports (16%) had negative or nonspecific findings that prediction times could not be provided. Thus, only limited progress has been made in predicting a seizure from preictal activity, but many other related phenomena have also been studied with nonlinear methods with some success.

Intraburst versus interburst locking in networks of driven nonidentical oscillators

We investigate the effect of common periodic drive applied to mean-field coupled oscillators and observe a specific realization of synchronization for particular ranges of drive frequency. This synchronization occurs when the phase difference variability between a pair of oscillators on a given cycle is larger than that between consecutive cycles. This synchrony may have implications for neural systems, in which case the apparent locking between neurons based on the magnitude of their interspike intervals may not be consistent with their dynamical locking.

Detecting determinism in EEG signals using principal component analysis and surrogate data testing

A novel method is proposed here to determine whether a time series is deterministic even in the presence of noise. The method is the extension of an existing method based on smoothness analysis of the signal in state space with surrogate data testing. While classical measures fail to detect determinism when the time series is corrupted by noise, the proposed method can clearly distinguish between pure stochastic and originally deterministic but noisy time series. A set of measures is defined here named partial smoothness indexes corresponding to principal components of the time series in state space. It is shown that when the time series is not pure stochastic, at least one of the indexes reflects determinism. The method is first successfully tested through simulation on a chaotic Lorenz time series contaminated with noise and then applied on EEG signals. Testing results on both our experimental recorded EEG signals and a benchmark EEG database verifies this hypothesis that EEG signals are deterministic in nature while contain some stochastic components as well.

Chaos breeds autonomy: connectionist design between bias and baby-sitting

In connectionism and its offshoots, models acquire functionality through externally controlled learning schedules. This undermines the claim of these models to autonomy. Providing these models with intrinsic biases is not a solution, as it makes their function dependent on design assumptions. Between these two alternatives, there is room for approaches based on spontaneous self-organization. Structural reorganization in adaptation to spontaneous activity is a well-known phenomenon in neural development. It is proposed here as a way to prepare connectionist models for learning and enhance the autonomy of these models.

Hypnotic abreaction releases chaotic patterns of electrodermal activity during dissociation

Chaotic transitions emerge in a wide variety of cognitive phenomena and may possibly be linked to specific changes during development of mental disorders. There are several hypotheses that link the dissociation to critical chaotic shifts with the resulting self-organization of behavioral patterns during critical periods. In 2 patients, hypnotic revivification of dissociated trauma along with measurement of bilateral electrodermal activity (EDA) for therapeutic and research purposes was performed. Nonlinear data analysis of EDA records shows a difference between degree of chaos in hypnotic relaxed state before revivification of the trauma and dissociated state after reliving the traumatic memory. Results suggest that the dissociated state after revivification of the trauma is significantly more chaotic than the state during the hypnotic relaxation before the event. Findings of this study suggest a possible role of neural chaos in the processing of the dissociated traumatic memory during hypnotic revivification.

Mountains and valleys: binocular rivalry and the flow of experience

Binocular rivalry provides a useful situation for studying the relation between the temporal flow of conscious experience and the temporal dynamics of neural activity. After proposing a phenomenological framework for understanding temporal aspects of consciousness, we review experimental research on multistable perception and binocular rivalry, singling out various methodological, theoretical, and empirical aspects of this research relevant to studying the flow of experience. We then review an experimental study from our group explicitly concerned with relating the temporal dynamics of rivalrous experience to the temporal dynamics of cortical activity. Drawing attention to the importance of dealing with ongoing activity and its inherent changing nature at both phenomenological and neurodynamical levels, we argue that the notions of recurrence and variability are pertinent to understanding rivalry in particular and the flow of experience in general.

The influence of low-frequency rTMS on EEG of rats

This study aimed to determine the effect of low-frequency repetitive transcranial magnetic stimulation (rTMS) on electroencephalograms (EEGs) of rats. Fifteen Sprague-Dawley rats were subject to 100 pulses of 0.5 Hz rTMS, or sham stimulation. EEGs were recorded before stimulation and within 1 min after rTMS or sham stimulation. Estimates of the EEG correlation dimension (D(2)) and power spectra were calculated. Results show that the D(2) reduced significantly after low-frequency rTMS, but not after sham stimulation. Mean absolute power (MAP) of the gamma band and relative power (RP) of the beta and gamma bands reduce markedly after low-frequency rTMS, but there are no changes with sham stimulation. These results indicate that low-frequency rTMS could affect cortical activities significantly, but effects were markedly different from those of high-frequency rTMS.

Chaotic hopping between attractors in neural networks

We present a neurobiologically-inspired stochastic cellular automaton whose state jumps with time between the attractors corresponding to a series of stored patterns. The jumping varies from regular to chaotic as the model parameters are modified. The resulting irregular behavior, which mimics the state of attention in which a system shows a great adaptability to changing stimulus, is a consequence in the model of short-time presynaptic noise which induces synaptic depression. We discuss results from both a mean-field analysis and Monte Carlo simulations.

Chaos via Shilnikov's saddle-node bifurcation in a theory of the electroencephalogram

We study the bifurcation diagram of a mesoscopic model of the human cortex. This model is known to exhibit robust chaotic behavior in the space of parameters that model exterior forcing. We show that the bifurcation diagram has an unusual degree of organization. In particular, we show that the chaos is spawned by a codimension-one homoclinic bifurcation that was analyzed by Shilnikov in 1969 but has never before been found in a physical application.

Nonlinear analysis of discharge patterns in monkey basal ganglia

Spontaneous discharge of basal ganglia neurons is often analyzed with time- or frequency-domain methods. However, it has been shown that sequences of inter-spike interval series are not fully described by such linear procedures. We therefore carried out a characterization of the nonlinear features of spontaneous discharge of neurons in the primate basal ganglia. We studied the spontaneous activity of neurons in the subthalamic nucleus (22 cells), as well as neurons in the external and internal pallidal segments (53 and 39 cells, respectively), recorded with standard extracellular recording methods in two awake Rhesus monkeys. As a measure of the statistical irregularity of neuronal discharge, we compared the approximate entropy of inter-spike interval sequences with that of shuffled representations of the same data. In all three basal ganglia structures, approximately 95% of the original data showed lower approximate entropy values than the shuffled data, suggesting a temporal organization in the original sequence. Fano factor analysis confirmed the presence of a temporal organization of inter-spike interval sequences, and indicated the presence of self-similarity in the great majority of them. In addition, Hurst exponent analysis showed that the inter-spike interval series are persistent. Hurst exponents often differ between short and long scaling ranges. Subsequent principal component analyses allowed us to identify three distinct patterns of the temporal evolution of inter-spike interval sequences in the phase space. These types were found in varying distributions in all three nuclei. Our analyses demonstrate that the discharge of most neurons in the basal ganglia of awake monkeys has nonlinear features that may be important for information coding in the basal ganglia.

QUANTUM MECHANICS AND BRAIN UNCERTAINTY

This paper argues that molecular governing structures (such as receptors, gating molecules, or ionic channels) which operate pervasively in the brain, often with small number particle systems (as, for example, at the surfaces of membranes, synaptic clefts, or macromolecules), may plausibly be vehicles for the transmutation of quantum mechanical fluctuations to normal-level neural signaling.

EEG-like signals generated by a simple chaotic model based on the logistic equation

We introduce a simple model to produce EEG-like signals. The model is based on the assumption that the number of active nerve cells that generate an electric field, at a given time, is essentially chaotic. In accordance, we use the logistic equation together with a spike-like function to simulate the neuronal activity processes. With this model, we are able to generate EEG-like patterns, with quite a short time of calculation. Real pre-recorded neuronal and simulated signals, as well as their power spectra, are compared in terms of the main conventional EEG frequency peaks.

ON INDETERMINISM, CHAOS, AND SMALL NUMBER PARTICLE SYSTEMS IN THE BRAIN

This paper presents rational, theoretical, and empirical grounds for doubting the principle of determinism in nature and in the brain, and discusses implications of this for free will and the chaos model of the brain. Small number particle systems are practically indeterministic and may be intrinsically indeterministic. Determinism in nature has often been taken to preclude free will. Strict determinism is a concept frequently applied to systems theory, establishing, e.g., the uniqueness of state-space trajectories. In order to consider determinism as a law of nature, however, one must be able to subject it to empirical tests. Presently, one is not able to and whether this can be shown to enable free will or not is not clear. It does remove, at least for the present, determinism itself as a rationale for precluding free will. The work partially supports the chaos model, but weakens the computational computer metaphor of brain function.

Experimental study of electrical FitzHugh–Nagumo neurons with modified excitability

We present an electronical circuit modelling a FitzHugh-Nagumo neuron with a modified excitability. To characterize this basic cell, the bifurcation curves between stability with excitation threshold, bistability and oscillations are investigated. An electrical circuit is then proposed to realize a unidirectional coupling between two cells, mimicking an inter-neuron synaptic coupling. In such a master-slave configuration, we show experimentally how the coupling strength controls the dynamics of the slave neuron, leading to frequency locking, chaotic behavior and synchronization. These phenomena are then studied by phase map analysis. The architecture of a possible neural network is described introducing different kinds of coupling between neurons.

Adaptive targeting of chaotic response in periodically stimulated neural systems

We demonstrate a technique for the enhancement of chaos in a computational model of a periodically stimulated excitable neuron. "Anticontrol" of chaos is achieved through intermittent adaptive intervention, which is based on finite-time Lyapunov exponents measured from the time series. Our results suggest that an adaptive strategy for chaos anticontrol is viable for increasing the complexity in physiological systems that are typically both noisy and nonstationary.

fMRI MODELS OF DENDRITIC AND ASTROCYTIC NETWORKS

In order to elucidate the relationships between hierarchical structures within the neocortical neuropil and the information carried by an ensemble of neurons encompassing a single voxel, it is essential to predict through volume conductor modeling LFPs representing average extracellular potentials, which are expressed in terms of interstitial potentials of individual cells in networks of gap-junctionally connected astrocytes and synaptically connected neurons. These relationships have been provided and can then be used to investigate how the underlying neuronal population activity can be inferred from the measurement of the BOLD signal through electrovascular coupling mechanisms across the blood-brain barrier. The importance of both synaptic and extrasynaptic transmission as the basis of electrophysiological indices triggering vascular responses between dendritic and astrocytic networks, and sequential configurations of firing patterns in composite neural networks is emphasized. The purpose of this review is to show how fMRI data may be used to draw conclusions about the information transmitted by individual neurons in populations generating the BOLD signal.

A distributed computing system for multivariate time series analyses of multichannel neurophysiological data

We present a client-server application for the distributed multivariate analysis of time series using standard PCs. We here concentrate on analyses of multichannel EEG/MEG data, but our method can easily be adapted to other time series. Due to the rapid development of new analysis techniques, the focus in the design of our application was not only on computational performance, but also on high flexibility and expandability of both the client and the server programs. For this purpose, the communication between the server and the clients as well as the building of the computational tasks has been realized via the Extensible Markup Language (XML). Running our newly developed method in an asynchronous distributed environment with random availability of remote and heterogeneous resources, we tested the system's performance for a number of different univariate and bivariate analysis techniques. Results indicate that for most of the currently available analysis techniques, calculations can be performed in real time, which, in principle, allows on-line analyses at relatively low cost.

Synchronization and propagation of bursts in networks of coupled map neurons

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Transient information flow in a network of excitatory and inhibitory model neurons: role of noise and signal autocorrelation

We investigate the performance of sparsely-connected networks of integrate-and-fire neurons for ultra-short term information processing. We exploit the fact that the population activity of networks with balanced excitation and inhibition can switch from an oscillatory firing regime to a state of asynchronous irregular firing or quiescence depending on the rate of external background spikes. We find that in terms of information buffering the network performs best for a moderate, non-zero, amount of noise. Analogous to the phenomenon of stochastic resonance the performance decreases for higher and lower noise levels. The optimal amount of noise corresponds to the transition zone between a quiescent state and a regime of stochastic dynamics. This provides a potential explanation of the role of non-oscillatory population activity in a simplified model of cortical micro-circuits.

Nonlinear dynamical analysis of EEG and MEG: review of an emerging field

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.

Chaotic channel

This work combines the theory of chaotic synchronization with the theory of information in order to introduce the chaotic channel, an active medium formed by connected chaotic systems. This subset of a large chaotic net represents the path along which information flows. We show that the possible amount of information exchange between the transmitter, where information enters the net, and the receiver, the destination of the information, is proportional to the level of synchronization between these two special subsystems.

Nonlinear multivariate analysis of neurophysiological signals

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependence between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions

We model the dynamical behavior of the neuropil, the densely interconnected neural tissue in the cortex, using neuropercolation approach. Neuropercolation generalizes phase transitions modeled by percolation theory of random graphs, motivated by properties of neurons and neural populations. The generalization includes (1) a noisy component in the percolation rule, (2) a novel depression function in addition to the usual arousal function, (3) non-local interactions among nodes arranged on a multi-dimensional lattice. This paper investigates the role of non-local (axonal) connections in generating and modulating phase transitions of collective activity in the neuropil. We derived a relationship between critical values of the noise level and non-locality parameter to control the onset of phase transitions. Finally, we propose a potential interpretation of ontogenetic development of the neuropil maintaining a dynamical state at the edge of criticality.

Electroreceptor neuron dynamics shape information transmission

The gymnotiform weakly electric fish Apteronotus leptorhynchus can capture prey using electrosensory cues that are dominated by low temporal frequencies. However, conventional tuning curves predict poor electroreceptor afferent responses to low-frequency stimuli. We compared conventional tuning curves with information tuning curves and found that the latter predicted substantially improved responses to these behaviorally relevant stimuli. Analysis of receptor afferent baseline activity showed that negative correlations reduced low-frequency noise levels, thereby increasing information transmission. Multiunit recordings from receptor afferents showed that this increased information transmission could persist at the population level. Finally, we verified that this increased low-frequency information is preserved in the spike trains of central neurons that receive receptor afferent input. Our results demonstrate that conventional tuning curves can be misleading when certain noise reduction strategies are used by the nervous system.

Anticipating epileptic seizures: from mathematics to clinical applications

The study of dynamical changes in the neural activity preceding an epileptic seizure allows the characterization of a preictal state several minutes prior to seizure onset. This opens new perspectives for studying the mechanisms of ictogenesis as well as for possible therapeutic interventions that represent a major breakthrough. In this review we present and discuss the results from our group in this domain using nonlinear analysis of brain signals, as well as its limitation and open questions.

"Dynamic" connectivity in neural systems: theoretical and empirical considerations

The study of functional interdependences between brain regions is a rapidly growing focus of neuroscience research. This endeavor has been greatly facilitated by the appearance of a number of innovative methodologies for the examination of neurophysiological and neuroimaging data. The aim of this article is to present an overview of dynamical measures of interdependence and contrast these with statistical measures that have been more widely employed. We first review the motivation, conceptual basis, and experimental approach of dynamical measures of interdependence and their application to the study of neural systems. A consideration of boot-strap "surrogate data" techniques, which facilitate hypothesis testing of dynamical measures, is then used to clarify the difference between dynamical and statistical measures of interdependence. An overview of some of the most active research areas such as the study of the "synchronization manifold," dynamical interdependence in neurophysiology data and the putative role of nonlinear desynchronization is then given. We conclude by suggesting that techniques based on dynamical interdependence--or "dynamical connectivity"--show significant potential for extracting meaningful information from functional neuroimaging data.