Synchronized action of synaptically coupled chaotic model neurons

Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity

Although neuron models have been well studied for their rich dynamics and biological properties, limited research has been done on the complex geometries that emerge from the basins of attraction and basin boundaries of multistable neuron systems. In this paper, we investigate the geometrical properties of the strange attractors, four-dimensional basins, and fractal basin boundaries of an asymmetrically electrically coupled system of two identical nonchaotic Rulkov neurons. We discover a quasimultistability in the system emerging from the existence of a chaotic spiking-bursting pseudo-attractor, and we classify and quantify the system's basins of attraction, which are found to have complex fractal geometries. Using the method of uncertainty exponents, we also find that the system exhibits extreme final state sensitivity, which results in a dynamical uncertainty that could have important applications in neurobiology.

Transient chaos and periodic structures in a model of neuronal early afterdepolarization

The presence of chaos is ubiquitous in mathematical models of neuroscience. In experimental neural systems, chaos was convincingly demonstrated in membranes, neurons, and small networks. However, its effects on the brain have long been debated. In this work, we use a three-dimensional map-based membrane potential model, the logistic KTz, to study chaos in single and coupled neurons. We first obtain an alternative phase diagram for the model using the interspike interval (ISI), evidencing a region of slow spikes (SS), missing from the original diagram of the KTz model. A large chaotic region is found inside the SS phase. Embedded in chaos are several self-similar periodic structures, such as shrimp-shaped domains and other structures. Sampling the behavior of neurons in this diagram, we detect a novel type of action potential, the neuronal early afterdepolarization (nEAD). EADs are pathological oscillations during the action potential, commonly found in cardiac cells and believed to be chaotic and responsible for generating arrhythmias in the heart. nEAD was found experimentally in neurons in a type of epilepsy. We study two chemically coupled neurons with this behavior. We identify and characterize transient chaos in their interaction. A phase diagram for this system presents a novel type of self-similar periodic structures, where the structures appear "chopped" in pieces.

Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, n-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

Variations of the spontaneous electrical activities of the neuronal networks imposed by the exposure of electromagnetic radiations using computational map-based modeling

The interaction between neurons in a neuronal network develops spontaneous electrical activities. But the effects of electromagnetic radiation on these activities have not yet been well explored. In this study, a ring of three coupled 1-dimensional Rulkov neurons and the generated electromagnetic field (EMF) are considered to investigate how the spontaneous activities might change regarding the EMF exposure. By employing the bifurcation analysis and time series, a comprehensive view of neuronal behavioral changes due to electromagnetic inductions is provided. The main findings of this study are as follows: 1) When a neuronal network is showing a spontaneous chaotic firing manner (without any external stimuli), a generated magnetic field inhibits this type of behavior. In fact, EMF completely eliminated the chaotic intrinsic behaviors of the neuronal loop. 2) When the network is exhibiting regular period-3 spiking patterns, the generated magnetic field changes its firing pattern to chaotic spiking, which is similar to epileptic seizures. 3) With weak synaptic connections, electromagnetic radiation inhibits and suppresses neuronal activities. 4) If the external magnetic flux has a high amplitude, it can change the shape of the induction current according to its shape 5) when there are weak synaptic connections in the network, a high-frequency external magnetic flux engenders high-frequency fluctuations in the membrane voltages. On the whole, electromagnetic radiation changes the pattern of the spontaneous activities of neuronal networks in the brain according to synaptic strengths and initial states of the neurons.

A tale of two rhythms: Locked clocks and chaos in biology

The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.

Robust dynamical invariants in sequential neural activity

By studying different sources of temporal variability in central pattern generator (CPG) circuits, we unveil fundamental aspects of the instantaneous balance between flexibility and robustness in sequential dynamics -a property that characterizes many systems that display neural rhythms. Our analysis of the triphasic rhythm of the pyloric CPG (Carcinus maenas) shows strong robustness of transient dynamics in keeping not only the activation sequences but also specific cycle-by-cycle temporal relationships in the form of strong linear correlations between pivotal time intervals, i.e. dynamical invariants. The level of variability and coordination was characterized using intrinsic time references and intervals in long recordings of both regular and irregular rhythms. Out of the many possible combinations of time intervals studied, only two cycle-by-cycle dynamical invariants were identified, existing even outside steady states. While executing a neural sequence, dynamical invariants reflect constraints to optimize functionality by shaping the actual intervals in which activity emerges to build the sequence. Our results indicate that such boundaries to the adaptability arise from the interaction between the rich dynamics of neurons and connections. We suggest that invariant temporal sequence relationships could be present in other networks, including those shaping sequences of functional brain rhythms, and underlie rhythm programming and functionality.

Asymmetry in electrical coupling between neurons alters multistable firing behavior

The role of asymmetry in electrical synaptic connection between two neuronal oscillators is studied in the Hindmarsh-Rose model. We demonstrate that the asymmetry induces multistability in spiking dynamics of the coupled neuronal oscillators. The coexistence of at least three attractors, one chaotic and two periodic orbits, for certain coupling strengths is demonstrated with time series, phase portraits, bifurcation diagrams, basins of attraction of the coexisting states, Lyapunov exponents, and standard deviations of peak amplitudes and interspike intervals. The experimental results with analog electronic circuits are in good agreement with the results of numerical simulations.

Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (I_h) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the I_h is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

Asymmetry Factors Shaping Regular and Irregular Bursting Rhythms in Central Pattern Generators

Central Pattern Generator (CPG) circuits are neural networks that generate rhythmic motor patterns. These circuits are typically built of half-center oscillator subcircuits with reciprocally inhibitory connections. Another common property in many CPGs is the remarkable rich spiking-bursting dynamics of their constituent cells, which balance robustness and flexibility to generate their joint coordinated rhythms. In this paper, we use conductance-based models and realistic connection topologies inspired by the crustacean pyloric CPG to address the study of asymmetry factors shaping CPG bursting rhythms. In particular, we assess the role of asymmetric maximal synaptic conductances, time constants and gap-junction connectivity to establish the regularity of half-center oscillator based CPGs. We map and characterize the synaptic parameter space that lead to regular and irregular bursting activity in these networks. The analysis indicates that asymmetric configurations display robust regular rhythms and that large regions of both regular and irregular but coordinated rhythms exist as a function of the asymmetry in the circuit. Our results show that asymmetry both in the maximal conductances and in the temporal dynamics of mutually inhibitory neurons can synergistically contribute to shape wide regimes of regular spiking-bursting activity in CPGs. Finally, we discuss how a closed-loop protocol driven by a regularity goal can be used to find and characterize regular regimes when there is not time to perform an exhaustive search, as in most experimental studies.

Application of Higuchi's fractal dimension from basic to clinical neurophysiology: A review

BACKGROUND AND OBJECTIVE

For more than 20 years, Higuchi's fractal dimension (HFD), as a nonlinear method, has occupied an important place in the analysis of biological signals. The use of HFD has evolved from EEG and single neuron activity analysis to the most recent application in automated assessments of different clinical conditions. Our objective is to provide an updated review of the HFD method applied in basic and clinical neurophysiological research.

METHODS

This article summarizes and critically reviews a broad literature and major findings concerning the applications of HFD for measuring the complexity of neuronal activity during different neurophysiological conditions. The source of information used in this review comes from the PubMed, Scopus, Google Scholar and IEEE Xplore Digital Library databases.

RESULTS

The review process substantiated the significance, advantages and shortcomings of HFD application within all key areas of basic and clinical neurophysiology. Therefore, the paper discusses HFD application alone, combined with other linear or nonlinear measures, or as a part of automated methods for analyzing neurophysiological signals.

CONCLUSIONS

The speed, accuracy and cost of applying the HFD method for research and medical diagnosis make it stand out from the widely used linear methods. However, only a combination of HFD with other nonlinear methods ensures reliable and accurate analysis of a wide range of neurophysiological signals.

Synchronization properties of coupled chaotic neurons: The role of random shared input

Spike-time correlations of neighbouring neurons depend on their intrinsic firing properties as well as on the inputs they share. Studies have shown that periodically firing neurons, when subjected to random shared input, exhibit asynchronicity. Here, we study the effect of random shared input on the synchronization of weakly coupled chaotic neurons. The cases of so-called electrical and chemical coupling are both considered, and we observe a wide range of synchronization behaviour. When subjected to identical shared random input, there is a decrease in the threshold coupling strength needed for chaotic neurons to synchronize in-phase. The system also supports lag-synchronous states, and for these, we find that shared input can cause desynchronization. We carry out a master stability function analysis for a network of such neurons and show agreement with the numerical simulations. The contrasting role of shared random input for complete and lag synchronized neurons is useful in understanding spike-time correlations observed in many areas of the brain.

Systems biology, emergence and antireductionism

This study explores the conceptual history of systems biology and its impact on philosophical and scientific conceptions of reductionism, antireductionism and emergence. Development of systems biology at the beginning of 21st century transformed biological science. Systems biology is a new holistic approach or strategy how to research biological organisms, developed through three phases. The first phase was completed when molecular biology transformed into systems molecular biology. Prior to the second phase, convergence between applied general systems theory and nonlinear dynamics took place, hence allowing the formation of systems mathematical biology. The second phase happened when systems molecular biology and systems mathematical biology, together, were applied for analysis of biological data. Finally, after successful application in science, medicine and biotechnology, the process of the formation of modern systems biology was completed.

Systems and molecular reductionist views on organisms were completely opposed to each other. Implications of systems and molecular biology on reductionist–antireductionist debate were quite different. The analysis of reductionism, antireductionism and emergence issues, in the era of systems biology, revealed the hierarchy between methodological, epistemological and ontological antireductionism. Primarily, methodological antireductionism followed from the systems biology. Only after, epistemological and ontological antireductionism could be supported.

Cluster synchronization of spiking induced by noise and interaction delays in homogenous neuronal ensembles

Properties of spontaneously formed clusters of synchronous dynamics in a structureless network of noisy excitable neurons connected via delayed diffusive couplings are studied in detail. Several tools have been applied to characterize the synchronization clusters and to study their dependence on the neuronal and the synaptic parameters. Qualitative explanation of the cluster formation is discussed. The interplay between the noise, the interaction time-delay and the excitable character of the neuronal dynamics is shown to be necessary and sufficient for the occurrence of the synchronization clusters. We have found the two-cluster partitions where neurons are firmly bound to their subsets, as well as the three-cluster ones, which are dynamical by nature. The former turn out to be stable under small disparity of the intrinsic neuronal parameters and the heterogeneity in the synaptic connectivity patterns.

Stability, bifurcations, and dynamics of global variables of a system of bursting neurons

An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.

Drug effect on EEG connectivity assessed by linear and nonlinear couplings

Quantitative analysis of human electroencephalogram (EEG) is a valuable method for evaluating psychopharmacological agents. Although the effects of different drug classes on EEG spectra are already known, interactions between brain locations remain unclear. In this work, cross mutual information function and appropriate surrogate data were applied to assess linear and nonlinear couplings between EEG signals. The main goal was to evaluate the pharmacological effects of alprazolam on brain connectivity during wakefulness in healthy volunteers using a cross-over, placebo-controlled design. Eighty-five pairs of EEG leads were selected for the analysis, and connectivity was evaluated inside anterior, central, and posterior zones of the scalp. Connectivity between these zones and interhemispheric connectivity were also measured. Results showed that alprazolam induced significant changes in EEG connectivity in terms of information transfer in comparison with placebo. Trends were opposite depending on the statistical characteristics: decreases in linear connectivity and increases in nonlinear couplings. These effects were generally spread over the entire scalp. Linear changes were negatively correlated, and nonlinear changes were positively correlated with drug plasma concentrations; the latter showed higher correlation coefficients. The use of both linear and nonlinear approaches revealed the importance of assessing changes in EEG connectivity as this can provide interesting information about psychopharmacological effects.

Robust microcircuit synchronization by inhibitory connections

Microcircuits in different brain areas share similar architectural and biophysical properties with compact motor networks known as central pattern generators (CPGs). Consequently, CPGs have been suggested as valuable biological models for understanding of microcircuit dynamics and particularly, their synchronization. We use a well known compact motor network, the lobster pyloric CPG to study principles of intercircuit synchronization. We couple separate pyloric circuits obtained from two animals via artificial synapses and observe how their synchronization depends on the topology and kinetic parameters of the computer-generated synapses. Stable in-phase synchronization appears when electrically coupling the pacemaker groups of the two networks, but reciprocal inhibitory connections produce more robust and regular cooperative activity. Contralateral inhibitory connections offer effective synchronization and flexible setting of the burst phases of the interacting networks. We also show that a conductance-based mathematical model of the coupled circuits correctly reproduces the observed dynamics illustrating the generality of the phenomena.

Synchronization of bursting neurons with delayed chemical synapses

The synchronization of bursting Hindmarsh-Rose neurons coupled by a time-delayed fast threshold modulation synapse was studied. It is shown that there is a domain of the coupling parameter and nonzero time-lag values such that the bursting neurons are exactly synchronized. Furthermore, and contrary to the case of electrical synapses, such synchronous bursting is stochastically stable.

Neuronal synchrony: peculiarity and generality

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their "dynamical repertoire" includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale).

Probing the dynamics of identified neurons with a data-driven modeling approach

In controlling animal behavior the nervous system has to perform within the operational limits set by the requirements of each specific behavior. The implications for the corresponding range of suitable network, single neuron, and ion channel properties have remained elusive. In this article we approach the question of how well-constrained properties of neuronal systems may be on the neuronal level. We used large data sets of the activity of isolated invertebrate identified cells and built an accurate conductance-based model for this cell type using customized automated parameter estimation techniques. By direct inspection of the data we found that the variability of the neurons is larger when they are isolated from the circuit than when in the intact system. Furthermore, the responses of the neurons to perturbations appear to be more consistent than their autonomous behavior under stationary conditions. In the developed model, the constraints on different parameters that enforce appropriate model dynamics vary widely from some very tightly controlled parameters to others that are almost arbitrary. The model also allows predictions for the effect of blocking selected ionic currents and to prove that the origin of irregular dynamics in the neuron model is proper chaoticity and that this chaoticity is typical in an appropriate sense. Our results indicate that data driven models are useful tools for the in-depth analysis of neuronal dynamics. The better consistency of responses to perturbations, in the real neurons as well as in the model, suggests a paradigm shift away from measuring autonomous dynamics alone towards protocols of controlled perturbations. Our predictions for the impact of channel blockers on the neuronal dynamics and the proof of chaoticity underscore the wide scope of our approach.

Detailed numerical investigation of the dissipative stochastic mechanics based neuron model

Recently, a physical approach for the description of neuronal dynamics under the influence of ion channel noise was proposed in the realm of dissipative stochastic mechanics (Güler, Phys Rev E 76:041918, 2007). Led by the presence of a multiple number of gates in an ion channel, the approach establishes a viewpoint that ion channels are exposed to two kinds of noise: the intrinsic noise, associated with the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, and the topological noise, associated with the uncertainty in accessing the permissible topological states of open gates. Renormalizations of the membrane capacitance and of a membrane voltage dependent potential function were found to arise from the mutual interaction of the two noisy systems. The formalism therein was scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. In this paper, the resultant computational neuron model of the above approach is investigated in detail numerically for its dynamics using time-independent input currents. The following are the major findings obtained. The intrinsic noise gives rise to two significant coexisting effects: it initiates spiking activity even in some range of input currents for which the corresponding deterministic model is quiet and causes bursting in some other range of input currents for which the deterministic model fires tonically. The renormalization corrections are found to augment the above behavioral transitions from quiescence to spiking and from tonic firing to bursting, and, therefore, the bursting activity is found to take place in a wider range of input currents for larger values of the correction coefficients. Some findings concerning the diffusive behavior in the voltage space are also reported.

Connection topology selection in central pattern generators by maximizing the gain of information

A study of a general central pattern generator (CPG) is carried out by means of a measure of the gain of information between the number of available topology configurations and the output rhythmic activity. The neurons of the CPG are chaotic Hindmarsh-Rose models that cooperate dynamically to generate either chaotic or regular spatiotemporal patterns. These model neurons are implemented by computer simulations and electronic circuits. Out of a random pool of input configurations, a small subset of them maximizes the gain of information. Two important characteristics of this subset are emphasized: (1) the most regular output activities are chosen, and (2) none of the selected input configurations are networks with open topology. These two principles are observed in living CPGs as well as in model CPGs that are the most efficient in controlling mechanical tasks, and they are evidence that the information-theoretical analysis can be an invaluable tool in searching for general properties of CPGs.

Revealing direction of coupling between neuronal oscillators from time series: phase dynamics modeling versus partial directed coherence

The problem of determining directional coupling between neuronal oscillators from their time series is addressed. We compare performance of the two well-established approaches: partial directed coherence and phase dynamics modeling. They represent linear and nonlinear time series analysis techniques, respectively. In numerical experiments, we found each of them to be applicable and superior under appropriate conditions: The latter technique is superior if the observed behavior is "closer" to limit-cycle dynamics, the former is better in cases that are closer to linear stochastic processes.

Synchronization of spontaneous bursting in a CO2 laser

We present experimental and numerical evidence of synchronization of burst events in two different modulated CO2 lasers. Bursts appear randomly in each laser as trains of large amplitude spikes intercalated by a small amplitude chaotic regime. Experimental data and model show the frequency locking of bursts in a suitable interval of coupling strength. We explain the mechanism of this phenomenon and demonstrate the inhibitory properties of the implemented coupling.

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.

Energy aspects of the synchronization of model neurons

We have deduced an energy function for a Hindmarsh-Rose model neuron and we have used it to evaluate the energy consumption of the neuron during its signaling activity. We investigate the balance of energy in the synchronization of two bidirectional linearly coupled neurons at different values of the coupling strength. We show that when two neurons are coupled there is a specific cost associated to the cooperative behavior. We find that the energy consumption of the neurons is incoherent until very near the threshold of identical synchronization, which suggests that cooperative behaviors without complete synchrony could be energetically more advantageous than those with complete synchrony.

Invertebrate central pattern generation moves along

Central pattern generators (CPGs) are circuits that generate organized and repetitive motor patterns, such as those underlying feeding, locomotion and respiration. We summarize recent work on invertebrate CPGs which has provided new insights into how rhythmic motor patterns are produced and how they are controlled by higher-order command and modulatory interneurons.

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.

Electrochemical bursting oscillations on a high-dimensional slow subsystem

Experiments are carried out with a chemical burster, the electrodissolution of iron in sulfuric acid solution. The system exhibits bursting oscillations in which fast periodic spiking is superimposed on chaotic, slow oscillations. Regularization of the slow dynamics, i.e., transition from chaotic to periodic bursting oscillations, is investigated through changes in the experimental parameters (circuit potential, external resistance, and electrode diameter). These transitions are accompanied by changes in the fast dynamics; a 'Hopf-Hopf' spiking is transformed to 'homoclinic-Hopf' spiking. The periodic bursting is destroyed through a period lengthening process in which the fast spiking region extends to a large fraction of the slow oscillatory cycle until there is no clear distinction between the fast and slow oscillations. Finally, it is shown that the time-scales of the fast spiking and, to a lesser extent, of the slow oscillations (or the occurrence of fast spiking) can be controlled with periodic perturbation of an experimental parameter, the circuit potential.

A Network of Electronic Neural Oscillators Reproduces the Dynamics of the Periodically Forced Pyloric Pacemaker Group

Low-dimensional oscillators are a valuable model for the neuronal activity of isolated neurons. When coupled, the self-sustained oscillations of individual free oscillators are replaced by a collective network dynamics. Here, dynamical features of such a network, consisting of three electronic implementations of the Hindmarsh-Rose mathematical model of bursting neurons, are compared to those of a biological neural motor system, specifically the pyloric CPG of the crustacean stomatogastric nervous system. We demonstrate that the network of electronic neurons exhibits realistic synchronized bursting behavior comparable to the biological system. Dynamical properties were analyzed by injecting sinusoidal currents into one of the oscillators. The temporal bursting structure of the electronic neurons in response to periodic stimulation is shown to bear a remarkable resemblance to that observed in the corresponding biological network. These findings provide strong evidence that coupled nonlinear oscillators realistically reproduce the network dynamics experimentally observed in assemblies of several neurons.

A neural infrastructure for rhythmic motor patterns

It is possible to work out the neural circuity of many invertebrate central pattern generators (CPGs) thereby providing a basis for linking cellular processes to actual behaviors. This review summarizes the infrastructure of the two CPGs in the lobster stomatogastric ganglion in terms of circuitry, ionic conductances and chemical modulation by amines and peptides. Analysis of the circuit using modeling techniques including the use of electronic neurons closes the chapter.

Phase synchronization in ensembles of bursting oscillators

We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles.

A model for emergent complex order in small neural networks

A new neural network model is introduced in this paper. The aim of the proposed Sierpinski neural networks is to provide a simple and biologically plausible neural network architecture that produces emergent complex spatio-temporal patterns through the activity of the output neurons of the network and is able to perform computational tasks. Such networks may play an important role in the analysis and understanding of complex dynamic activity observed at various levels of biological neural systems. The proposed Sierpinski neural networks are described in detail and their functioning is analyzed. We discuss about emerging neural activity patterns and their interpretations, neuro-computation with such emerging activity patterns, and also possible implications for computational neuroscience.

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

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773-793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system. We first provide a short mathematical description of the models of excitable cells and of the different modes of firing of bursting neurons (Section 1). The deterministic behavior reported in giant axons (principally squid), in pacemaker cells, in isolated or in paired neurons of Invertebrates acting as coupled oscillators is then described (Section 2). We also consider chaotic processes exhibited by coupled Vertebrate neurons and of several components of Central Pattern Generators (Section 3). It is then shown that as indicated by studies of synaptic noise, deterministic patterns of firing in presynaptic interneurons are reliably transmitted, to their postsynaptic targets, via probabilistic synapses (Section 4). This raises the more general issue of chaos as a possible neuronal code and of the emerging concept of stochastic resonance Considerations on cortical dynamics and of EEGs are divided in two parts. The first concerns the early attempts by several pioneer authors to demonstrate chaos in experimental material such as the olfactory system or in human recordings during various forms of epilepsies, and the belief in 'dynamical diseases' (Section 5). The second part explores the more recent period during which surrogate-testing, definition of unstable periodic orbits and period-doubling bifurcations have been used to establish more firmly the nonlinear features of retinal and cortical activities and to define predictors of epileptic seizures (Section 6). Finally studies of multidimensional systems have founded radical hypothesis on the role of neuronal attractors in information processing, perception and memory and two elaborate models of the internal states of the brain (i.e. 'winnerless competition' and 'chaotic itinerancy'). Their modifications during cognitive functions are given special attention due to their functional and adaptive capabilities (Section 7) and despite the difficulties that still exist in the practical use of topological profiles in a state space to identify the physical underlying correlates. The reality of 'neurochaos' and its relations with information theory are discussed in the conclusion (Section 8) where are also emphasized the similarities between the theory of chaos and that of dynamical systems. Both theories strongly challenge computationalism and suggest that new models are needed to describe how the external world is represented in the brain.

Inhibitory synchronization of bursting in biological neurons: dependence on synaptic time constant

Using the dynamic clamp technique, we investigated the effects of varying the time constant of mutual synaptic inhibition on the synchronization of bursting biological neurons. For this purpose, we constructed artificial half-center circuits by inserting simulated reciprocal inhibitory synapses between identified neurons of the pyloric circuit in the lobster stomatogastric ganglion. With natural synaptic interactions blocked (but modulatory inputs retained), these neurons generated independent, repetitive bursts of spikes with cycle period durations of approximately 1 s. After coupling the neurons with simulated reciprocal inhibition, we selectively varied the time constant governing the rate of synaptic activation and deactivation. At time constants <or=100 ms, bursting was coordinated in an alternating (anti-phase) rhythm. At longer time constants (>400 ms), bursts became phase-locked in a fully overlapping pattern with little or no phase lag and a shorter period. During the in-phase bursting, the higher-frequency spiking activity was not synchronized. If the circuit lacked a robust periodic burster, increasing the time constant evoked a sharp transition from out-of-phase oscillations to in-phase oscillations with associated intermittent phase-jumping. When a coupled periodic burster neuron was present (on one side of the half-center circuit), the transition was more gradual. We conclude that the magnitude and stability of phase differences between mutually inhibitory neurons varies with the ratio of burst cycle period duration to synaptic time constant and that cellular bursting (whether periodic or irregular) can adopt in-phase coordination when inhibitory synaptic currents are sufficiently slow.

Modeling of spiking-bursting neural behavior using two-dimensional map

A simple model that replicates the dynamics of spiking and spiking-bursting activity of real biological neurons is proposed. The model is a two-dimensional map that contains one fast and one slow variable. The mechanisms behind generation of spikes, bursts of spikes, and restructuring of the map behavior are explained using phase portrait analysis. The dynamics of two coupled maps that model the behavior of two electrically coupled neurons is discussed. Synchronization regimes for spiking and bursting activities of these maps are studied as a function of coupling strength. It is demonstrated that the results of this model are in agreement with the synchronization of chaotic spiking-bursting behavior experimentally found in real biological neurons.

Synaptic heterogeneity and stimulus-induced modulation of depression in central synapses

Short-term plasticity is a pervasive feature of synapses. Synapses exhibit many forms of plasticity operating over a range of time scales. We develop an optimization method that allows rapid characterization of synapses with multiple time scales of facilitation and depression. Investigation of paired neurons that are postsynaptic to the same identified interneuron in the buccal ganglion of Aplysia reveals that the responses of the two neurons differ in the magnitude of synaptic depression. Also, for single neurons, prolonged stimulation of the presynaptic neuron causes stimulus-induced increases in the early phase of synaptic depression. These observations can be described by a model that incorporates two availability factors, e.g., depletable vesicle pools or desensitizing receptor populations, with different time courses of recovery, and a single facilitation component. This model accurately predicts the responses to novel stimuli. The source of synaptic heterogeneity is identified with variations in the relative sizes of the two availability factors, and the stimulus-induced decrement in the early synaptic response is explained by a slowing of the recovery rate of one of the availability factors. The synaptic heterogeneity and stimulus-induced modifications in synaptic depression observed here emphasize that synaptic efficacy depends on both the individual properties of synapses and their past history.

Regularization mechanisms of spiking-bursting neurons

An essential question raised after the observation of highly variable bursting activity in individual neurons of Central Pattern Generators (CPGs) is how an assembly of such cells can cooperatively act to produce regular signals to motor systems. It is well known that some neurons in the lobster stomatogastric ganglion have a highly irregular spiking-bursting behavior when they are synaptically isolated from any connection in the CPG. Experimental recordings show that periodic stimuli on a single neuron can regulate its firing activity. Other evidence demonstrates that specific chemical and/or electrical synapses among neurons also induce the regularization of the rhythms. In this paper we present a modeling study in which a slow subcellular dynamics, the exchange of calcium between an intracellular store and the cytoplasm, is responsible for the origin and control of the irregular spiking-bursting activity. We show this in simulations of single cells under periodic driving and in minimal networks where the cooperative activity can induce regularization. While often neglected in the description of realistic neuron models, subcellular processes with slow dynamics may play an important role in information processing and short-term memory of spiking-bursting neurons.

Synaptic efficacy and the transmission of complex firing patterns between neurons

In central neurons, the summation of inputs from presynaptic cells combined with the unreliability of synaptic transmission produces incessant variations of the membrane potential termed synaptic noise (SN). These fluctuations, which depend on both the unpredictable timing of afferent activities and quantal variations of postsynaptic potentials, have defied conventional analysis. We show here that, when applied to SN recorded from the Mauthner (M) cell of teleosts, a simple method of nonlinear analysis reveals previously undetected features of this signal including hidden periodic components. The phase relationship between these components is compatible with the notion that the temporal organization of events comprising this noise is deterministic rather than random and that it is generated by presynaptic interneurons behaving as coupled periodic oscillators. Furthermore a model of the presynaptic network shows how SN is shaped both by activities in incoming inputs and by the distribution of their synaptic weights expressed as mean quantal contents of the activated synapses. In confirmation we found experimentally that long-term tetanic potentiation (LTP), which selectively increases some of these synaptic weights, permits oscillating temporal patterns to be transmitted more effectively to the postsynaptic cell. Thus the probabilistic nature of transmitter release, which governs the strength of synapses, may be critical for the transfer of complex timing information within neuronal assemblies.

Diffusion of nitric oxide can facilitate cerebellar learning: A simulation study

The gaseous second messenger nitric oxide (NO), which readily diffuses in brain tissue, has been implicated in cerebellar long-term depression (LTD), a form of synaptic plasticity thought to be involved in cerebellar learning. Can NO diffusion facilitate cerebellar learning? The inferior olive (IO) cells, which provide the error signals necessary for modifying the granule cell-Purkinje cell (PC) synapses by LTD, fire at ultra-low firing rates in vivo, rarely more than 2-4 spikes within a second. In this paper, we show that NO diffusion can improve the transmission of sporadic IO error signals to PCs within cerebellar cortical functional units, or microzones. To relate NO diffusion to adaptive behavior, we add NO diffusion and a "volumic" LTD learning rule, i.e., a learning rule that depends both on the synaptic activity and on the NO concentration at the synapse, to a cerebellar model for arm movement control. Our results show that biologically plausible diffusion leads to an increase in information transfer of the error signals to the PCs when the IO firing rate is ultra-low. This, in turn, enhances cerebellar learning as shown by improved performance in an arm-reaching task.

Reliable circuits from irregular neurons: a dynamical approach to understanding central pattern generators

Central pattern generating neurons from the lobster stomatogastric ganglion were analyzed using new nonlinear methods. The LP neuron was found to have only four or five degrees of freedom in the isolated condition and displayed chaotic behavior. We show that this chaotic behavior could be regularized by periodic pulses of negative current injected into the neuron or by coupling it to another neuron via inhibitory connections. We used both a modified Hindmarsh-Rose model to simulate the neurons behavior phenomenologically and a more realistic conductance-based model so that the modeling could be linked to the experimental observations. Both models were able to capture the dynamics of the neuron behavior better than previous models. We used the Hindmarsh-Rose model as the basis for building electronic neurons which could then be integrated into the biological circuitry. Such neurons were able to rescue patterns which had been disabled by removing key biological neurons from the circuit.

Synchronous behavior of two coupled electronic neurons

We report on experimental studies of synchronization phenomena in a pair of analog electronic neurons (ENs). The ENs were designed to reproduce the observed membrane voltage oscillations of isolated biological neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus. The ENs are simple analog circuits which integrate four-dimensional differential equations representing fast and slow subcellular mechanisms that produce the characteristic regular/chaotic spiking-bursting behavior of these cells. In this paper we study their dynamical behavior as we couple them in the same configurations as we have done for their counterpart biological neurons. The interconnections we use for these neural oscillators are both direct electrical connections and excitatory and inhibitory chemical connections: each realized by analog circuitry and suggested by biological examples. We provide here quantitative evidence that the ENs and the biological neurons behave similarly when coupled in the same manner. They each display well defined bifurcations in their mutual synchronization and regularization. We report briefly on an experiment on coupled biological neurons and four-dimensional ENs, which provides further ground for testing the validity of our numerical and electronic models of individual neural behavior. Our experiments as a whole present interesting new examples of regularization and synchronization in coupled nonlinear oscillators.

Electrophysiological properties of inferior olive neurons: A compartmental model

As a step in exploring the functions of the inferior olive, we constructed a biophysical model of the olivary neurons to examine their unique electrophysiological properties. The model consists of two compartments to represent the known distribution of ionic currents across the cell membrane, as well as the dendritic location of the gap junctions and synaptic inputs. The somatic compartment includes a low-threshold calcium current (I(Ca_l)), an anomalous inward rectifier current (I(h)), a sodium current (I(Na)), and a delayed rectifier potassium current (I(K_dr)). The dendritic compartment contains a high-threshold calcium current (I(Ca_h)), a calcium-dependent potassium current (I(K_Ca)), and a current flowing into other cells through electrical coupling (I(c)). First, kinetic parameters for these currents were set according to previously reported experimental data. Next, the remaining free parameters were determined to account for both static and spiking properties of single olivary neurons in vitro. We then performed a series of simulated pharmacological experiments using bifurcation analysis and extensive two-parameter searches. Consistent with previous studies, we quantitatively demonstrated the major role of I(Ca_l) in spiking excitability. In addition, I(h) had an important modulatory role in the spike generation and period of oscillations, as previously suggested by Bal and McCormick. Finally, we investigated the role of electrical coupling in two coupled spiking cells. Depending on the coupling strength, the hyperpolarization level, and the I(Ca_l) and I(h) modulation, the coupled cells had four different synchronization modes: the cells could be in-phase, phase-shifted, or anti-phase or could exhibit a complex desynchronized spiking mode. Hence these simulation results support the counterintuitive hypothesis that electrical coupling can desynchronize coupled inferior olive cells.

Dynamic control of irregular bursting in an identified neuron of an oscillatory circuit

In the oscillatory circuits known as central pattern generators (CPGs), most synaptic connections are inhibitory. We have assessed the effects of inhibitory synaptic input on the dynamic behavior of a component neuron of the pyloric CPG in the lobster stomatogastric ganglion. Experimental perturbations were applied to the single, lateral pyloric neuron (LP), and the resulting voltage time series were analyzed using an entropy measure obtained from power spectra. When isolated from phasic inhibitory input, LP generates irregular spiking-bursting activity. Each burst begins in a relatively stereotyped manner but then evolves with exponentially increasing variability. Periodic, depolarizing current pulses are poor regulators of this activity, whereas hyperpolarizing pulses exert a strong, frequency-dependent regularizing action. Rhythmic inhibitory inputs from presynaptic pacemaker neurons also regularize the bursting. These inputs 1) reset LP to a similar state at each cycle, 2) extend and further stabilize the initial, quasi-stable phase of its bursts, and 3) at sufficiently high frequencies terminate ongoing bursts before they become unstable. The dynamic time frame for stabilization overlaps the normal frequency range of oscillations of the pyloric CPG. Thus, in this oscillatory circuit, the interaction of rhythmic inhibitory input with intrinsic burst properties affects not only the phasing, but also the dynamic stability of neural activity.

Experimentally determined chaotic phase synchronization in a neuronal system

Mathematical analysis of the subthreshold oscillatory properties of inferior olivary neurons in vitro indicates that the oscillation is nonlinear and supports low dimensional chaotic dynamics. This property leads to the generation of complex functional states that can be attained rapidly via phase coherence that conform to the category of "generalized synchronization." Functionally, this translates into neuronal ensemble properties that can support maximum functional permissiveness and that rapidly can transform into robustly determined multicellular coherence.