Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble

Dimension of interaction dynamics

A method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated. Some facts concerning generalized Rényi dimensions that form the basis of our method are proved. We show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts. We use our method to distinguish interacting from noninteracting systems on the examples of logistic and Hénon maps. A classification of all possible interaction schemes is given.

Regularization of synchronized chaotic bursts

The onset of regular bursts in a group of irregularly bursting neurons with different individual properties is one of the most interesting dynamical properties found in neurobiological systems. In this paper we show how synchronization among chaotically bursting cells can lead to the onset of regular bursting. In order to clearly present the mechanism behind such regularization we model the individual dynamics of each cell with a simple two-dimensional map that produces chaotic bursting behavior similar to biological neurons.

Information transmission and recovery in neural communications channels

Biological neural communications channels transport environmental information from sensors through chains of active dynamical neurons to neural centers for decisions and actions to achieve required functions. These kinds of communications channels are able to create information and to transfer information from one time scale to the other because of the intrinsic nonlinear dynamics of the component neurons. We discuss a very simple neural information channel composed of sensory input in the form of a spike train that arrives at a model neuron, then moves through a realistic synapse to a second neuron where the information in the initial sensory signal is read. Our model neurons are four-dimensional generalizations of the Hindmarsh-Rose neuron, and we use a model of chemical synapse derived from first-order kinetics. The four-dimensional model neuron has a rich variety of dynamical behaviors, including periodic bursting, chaotic bursting, continuous spiking, and multistability. We show that, for many of these regimes, the parameters of the chemical synapse can be tuned so that information about the stimulus that is unreadable at the first neuron in the channel can be recovered by the dynamical activity of the synapse and the second neuron. Information creation by nonlinear dynamical systems that allow chaotic oscillations is familiar in their autonomous oscillations. It is associated with the instabilities that lead to positive Lyapunov exponents in their dynamical behavior. Our results indicate how nonlinear neurons acting as input/output systems along a communications channel can recover information apparently "lost" in earlier junctions on the channel. Our measure of information transmission is the average mutual information between elements, and because the channel is active and nonlinear, the average mutual information between the sensory source and the final neuron may be greater than the average mutual information at an earlier neuron in the channel. This behavior is strikingly different than the passive role communications channels usually play, and the "data processing theorem" of conventional communications theory is violated by these neural channels. Our calculations indicate that neurons can reinforce reliable transmission along a chain even when the synapses and the neurons are not completely reliable components. This phenomenon is generic in parameter space, robust in the presence of noise, and independent of the discretization process. Our results suggest a framework in which one might understand the apparent design complexity of neural information transduction networks. If networks with many dynamical neurons can recover information not apparent at various waystations in the communications channel, such networks may be more robust to noisy signals, may be more capable of communicating many types of encoded sensory neural information, and may be the appropriate design for components, neurons and synapses, which can be individually imprecise, inaccurate "devices."

Asymptotically stable phase synchronization revealed by autoregressive circle maps

A specially designed of nonlinear time series analysis is introduced based on phases, which are defined as polar angles in spaces spanned by a finite number of delayed coordinates. A canonical choice of the polar axis and a related implicit estimation scheme for the potentially underlying autoregressive circle map (next phase map) guarantee the invertibility of reconstructed phase space trajectories to the original coordinates. The resulting Fourier approximated, invertibility enforcing phase space map allows us to detect conditional asymptotic stability of coupled phases. This comparatively general synchronization criterion unites two existing generalizations of the old concept and can successfully be applied, e.g., to phases obtained from electrocardiogram and airflow recordings characterizing cardiorespiratory interaction.

Neural modeling and functional brain imaging: an overview

This article gives an overview of the different functional brain imaging methods, the kinds of questions these methods try to address and some of the questions associated with functional neuroimaging data for which neural modeling must be employed to provide reasonable answers.

Learning driver-response relationships from synchronization patterns

We test recent claims that causal (driver-response) relationships can be deduced from interdependencies between simultaneously measured time series. We apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors. The systems that we study are asymmetrically coupled simple models (Lorenz, Roessler, and Hénon models), the couplings being such that they lead to generalized synchronization. If the data were perfect (noise-free, infinitely long), we should be able to detect, at least in some cases, which of the coupled systems is the driver and which the response. This might no longer be true if the time series has finite length. Instead, estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them, and causal relationships are more difficult to detect. We also show that slightly different variants of the interdependence measure can have quite different sensitivities.

Robust synchronization of chaotic systems

The question of robustness of synchronization with respect to small arbitrary perturbations of the underlying dynamical systems is addressed. We present examples of chaos synchronization demonstrating that normal hyperbolicity is a necessary and sufficient condition for the synchronization manifold to be smooth and persistent under small perturbations. The same examples, however, show that in real applications normal hyperbolicity is not sufficient to give quantitative bounds for deformations of the synchronization manifold, i.e., even in the case of normal hyperbolicity two almost identical systems may cause large synchronization errors.

The labile brain. II. Transients, complexity and selection

The successive expression of neuronal transients is related to dynamic correlations and, as shown in this paper, to dynamic instability. Dynamic instability is a form of complexity, typical of neuronal systems, which may be crucial for adaptive brain function from two perspectives. The first is from the point of view of neuronal selection and self-organizing systems: if selective mechanisms underpin the emergence of adaptive neuronal responses then dynamic instability is, itself, necessarily adaptive. This is because dynamic instability is the source of diversity on which selection acts and is therefore subject to selective pressure. In short, the emergence of order, through selection, depends almost paradoxically on the instabilities that characterize the diversity of brain dynamics. The second perspective is provided by information theory.

The labile brain. I. Neuronal transients and nonlinear coupling

In this, the first of three papers, the nature of, and motivation for, neuronal transients is described in relation to characterizing brain dynamics. This paper deals with some basic aspects of neuronal dynamics, interactions, coupling and implicit neuronal codes. The second paper develops neuronal transients and nonlinear coupling in the context of dynamic instability and complexity, and suggests that instability or lability is necessary for adaptive self-organization. The final paper addresses the role of neuronal transients through information theory and the emergence of spatio-temporal receptive fields and functional specialization. By considering the brain as an ensemble of connected dynamic systems one can show that a sufficient description of neuronal dynamics comprises neuronal activity at a particular time and its recent history This history constitutes a neuronal transient. As such, transients represent a fundamental metric of neuronal interactions and, implicitly, a code employed in the functional integration of brain systems. The nature of transients, expressed conjointly in distinct neuronal populations, reflects the underlying coupling among populations. This coupling may be synchronous (and possibly oscillatory) or asynchronous. A critical distinction between synchronous and asynchronous coupling is that the former is essentially linear and the latter is nonlinear. The nonlinear nature of asynchronous coupling enables the rich, context-sensitive interactions that characterize real brain dynamics, suggesting that it plays a role in functional integration that may be as important as synchronous interactions. The distinction between linear and nonlinear coupling has fundamental implications for the analysis and characterization of neuronal interactions, most of which are predicated on linear (synchronous) coupling (e.g. cross-correlograms and coherence). Using neuromagnetic data it is shown that nonlinear (asynchronous) coupling is, in fact, more abundant and can be more significant than synchronous coupling.

Synchronization in the human cardiorespiratory system

We investigate synchronization between cardiovascular and respiratory systems in healthy humans under free-running conditions. For this aim we analyze nonstationary irregular bivariate data, namely, electrocardiograms and measurements of respiratory flow. We briefly discuss a statistical approach to synchronization in noisy and chaotic systems and illustrate it with numerical examples; effects of phase and frequency locking are considered. Next, we present and discuss methods suitable for the detection of hidden synchronous epochs from such data. The analysis of the experimental records reveals synchronous regimes of different orders n:m and transitions between them; the physiological significance of this finding is discussed.

Low doses of ethanol reduce evidence for nonlinear structure in brain activity

Recent theories of the effects of ethanol on the brain have focused on its direct actions on neuronal membrane proteins. However, neuromolecular mechanisms whereby ethanol produces its CNS effects in low doses typically used by social drinkers (e.g., 2-3 drinks, 10-25 mM, 0.05-0.125 gm/dl) remain less well understood. We propose the hypothesis that ethanol may act by introducing a level of randomness or "noise" in brain electrical activity. We investigated the hypothesis by applying a battery of tests originally developed for nonlinear time series analysis and chaos theory to EEG data collected from 32 men who had participated in an ethanol/placebo challenge protocol. Because nonlinearity is a prerequisite for chaos and because we can detect nonlinearity more reliably than chaos, we concentrated on a series of measures that quantitated different aspects of nonlinearity. For each of these measures the method of surrogate data was used to assess the significance of evidence for nonlinear structure. Significant nonlinear structure was found in the EEG as evidenced by the measures of time asymmetry, determinism, and redundancy. In addition, the evidence for nonlinear structure in the placebo condition was found to be significantly greater than that for ethanol. Nonlinear measures, but not spectral measures, were found to correlate with a subject's overall feeling of intoxication. These findings are consistent with the notion that ethanol may act by introducing a level of randomness in neuronal processing as assessed by EEG nonlinear structure.

Periodic orbits: a new language for neuronal dynamics

A new nonlinear dynamical analysis is applied to complex behavior from neuronal systems. The conceptual foundation of this analysis is the abstraction of observed neuronal activities into a dynamical landscape characterized by a hierarchy of "unstable periodic orbits" (UPOs). UPOs are rigorously identified in data sets representative of three different levels of organization in mammalian brain. An analysis based on UPOs affords a novel alternative method of decoding, predicting, and controlling these neuronal systems.

Nonlinear interdependencies of EEG signals in human intracranially recorded temporal lobe seizures

The degree of interdependence between intracranial EEG channels was investigated in four epileptic patients with complex partial seizures of mesial temporal lobe origin. With a new method to characterize nonlinear dynamical interdependence-the mutual nonlinear prediction-we demonstrated here a possibility to quantify, during epileptic seizures, the relationships between EEG signals of electrode contacts in the epileptogenic area. During the interictal period, the degree of nonlinear interdependences were very low or absent. In contrast, it was found that transient patterns of nonlinear interdependences emerge at the initial spread of the seizure, during essential parts of its development, and at seizure end, but the maintenance of these interactions are not observed throughout the seizure activity. These results suggest that the nonlinear associations plays an important role in epileptogenesis, and that the process of neuronal entrainment during seizure onset involves a transient interaction between a distributed network of neuronal aggregates, but the maintenance of this interaction is not required for sustained seizure activity. Furthermore, this technique can describe properly the spatio-temporal organisation of the seizures of medio-temporal lobe origin and could become a very useful tool to aid the localization of the epileptogenic regions at the origin of epileptic seizures and their pathways of propagation.

Synchronizing spatiotemporal chaos

We show analytically and numerically that a pair of uni-directionally coupled spatially extended systems can synchronize. For the case of partial differential equations the synchronization can be achieved by applying the scalar driving signals only at finite number of space points. Our approach is very general and can be useful for practical applications since the synchronization is achieved via feeding in the response system only the information from certain (discrete) spatial locations of the drive system. We also stress some open problems in the field of synchronization of spatiotemporal chaos. (c) 1997 American Institute of Physics.

Fundamentals of synchronization in chaotic systems, concepts, and applications

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.