International Journal of Bifurcation and Chaos

We present some recent investigations resulting from the modeling of neural networks as dynamical systems, and deal with the following questions, adressed in the context of specific models.

(i) Characterizing the collective dynamics;

(ii) Statistical analysis of spike trains;

(iii) Interplay between dynamics and network structure;

(iv) Effects of synaptic plasticity.

This tutorial reviews a new important class of mathematical phenomenological models of neural activity generated by iterative dynamical systems: the so-called map-based systems. We focus on 1-D and 2-D maps for the replication of many features of the neural activity of a single neuron. It was shown that such systems can reproduce the basic activity modes such as spiking, bursting, chaotic spiking-bursting, subthreshold oscillations, tonic and phasic spiking, normal excitability, etc. of the real biological neurons. We emphasize on the representation of chaotic spiking-bursting oscillations by chaotic attractors of 2-D models. We also explain the dynamical mechanism of formation of such attractors and transition from one mode to another. We briefly present some synchronization mehanisms of chaotic spiking-bursting activity for two coupled neurons described by 1-D maps.

Understanding and predicting the behavior of complex multiagent systems like brain or ecological food net requires new approaches and paradigms. Traditional analyses based on just asymptotic results of behavior as time goes to infinity, or on straightforward mathematical images that can accommodate only fixed points or limit cycles do not tell much about these systems. To obtain sensible dynamical models of natural phenomena, such as the reproducible order observed in ecological, cognitive or behavioral experiments, one cannot afford to neglect the transient dynamics of the underlying complex network. In disclosing such dynamical mechanisms, the focus of interest must be on reproducible or, even, structurally stable transients. In this tutorial, we formulate the Winnerless Competition (WLC) principle that induces robust transient dynamics in open complex networks. The main point of WLC principle is the transformation of the acquired information into ensemble (spatio)-temporal output via intrinsic transient dynamics of the network. Such encoding provides a reproducible transient response, whose geometrical image in phase space is a stable heteroclinic sequence. We compile a diverse list of natural phenomena which can be rigorously modeled by the WLC. Together with the experimental and numerical results of the networks with different levels of complexity, we evaluate the robustness and reproducibility of the WLC dynamics and discuss the advantages of future possible application of the discussed approach.

Today, the human brain can be studied as a whole. Electroencephalography, magnetoencephalography, or functional magnetic resonance imaging (fMRI) techniques provide functional connectivity patterns between different brain areas, and during different pathological and cognitive neuro-dynamical states. In this tutorial, we review novel complex networks approaches to unveil how brain networks can efficiently manage local processing and global integration for the transfer of information, while being at the same time capable of adapting to satisfy changing neural demands.

Nonlinear time series analyses have suggested that the human electroencephalogram (EEG) may share statistical and dynamical properties with chaotic systems. During slow-wave sleep or pathological states like epilepsy, correlation dimension measurements display low values, while in awake and attentive subjects, there is no such low dimensionality, and the EEG is more similar to a stochastic variable. We briefly review these results and contrast them with recordings in cat cerebral cortex, as well as with theoretical models. In awake or sleeping cats, recordings with microelectrodes inserted in cortex show that global variables such as local field potentials (local EEG) are similar to the human EEG. However, neuronal discharges are highly irregular and exponentially distributed, similar to Poisson stochastic processes. To reconcile these results, we investigate models of randomly-connected networks of integrate-and-fire neurons, and also contrast global (averaged) variables, with neuronal activity. The network displays different states, such as "synchronous regular" (SR) or "asynchronous irregular" (AI) states. In SR states, the global variables display coherent behavior with low dimensionality, while in AI states, the global activity is high-dimensionally chaotic with exponentially distributed neuronal discharges, similar to awake cats. Scale-dependent Lyapunov exponents and ∊-entropies show that the seemingly stochastic nature at small scales (neurons) can coexist with more coherent behavior at larger scales (averages). Thus, we suggest that brain activity obeys a similar scheme, with seemingly stochastic dynamics at small scales (neurons), while large scales (EEG) display more coherent behavior or high-dimensional chaos.

We classified performance-related mental states from EEG-derived measurements. We investigated the usefulness of massively distributed source reconstruction, comparing scalp and cortical scales. This approach provides a more detailed picture of the functional brain networks underlying the changes related to the mental state of interest. Local and distant synchrony measurements (coherence, phase locking value) were used for both scalp measurements and cortical current density sources, and were fed into a SVM-based classifier. We designed two simulations where classification scores increased when our 60-electrode scalp measurements were reconstructed on 60 sources and on a 500-source cortex. Source reconstruction appeared to be most useful in these simulations, in particular, when distant synchronies were involved and local synchronies did not prevail. Despite the simplicity of the model used, certain flaws in accuracy were observed in the localization of informative activities, due to the relationship between amplitude and phase for mixed signals. Our results with real EEG data suggested that the phenomenon of interest was characterized merely by modulations in local amplitudes, but also in strength of distant couplings. After source reconstruction, classification rates also increased for real EEG data when seeking distant phase-related couplings. When reconstructing a large number of sources, the regularization coefficient should be carefully selected on a subject-by-subject basis. We showed that training classifiers using such high-dimension data is useful for localizing discriminating patterns of activity.

We calculate a measure of statistical complexity from the global dynamics of electroencephalographic (EEG) signals from healthy subjects and epileptic patients, and are able to establish a criterion to characterize the collective behavior in both groups of individuals. It is found that the collective dynamics of EEG signals possess relatively higher values of complexity for healthy subjects in comparison to that for epileptic patients. To interpret these results, we propose a model of a network of coupled chaotic maps where we calculate the complexity as a function of a parameter and relate this measure with the emergence of nontrivial collective behavior in the system. Our results show that the presence of nontrivial collective behavior is associated to high values of complexity; thus suggesting that similar dynamical collective process may take place in the human brain. Our findings also suggest that epilepsy is a degenerative illness related to the loss of complexity in the brain.

This paper introduces an extension of recurrence analysis to symbolic sequences. Heuristic arguments based on Shannon–McMillan–Breiman theorem suggest several relations between the statistical features of "symbolic" recurrence plots and the entropy per unit time of the dynamics; their practical efficiency for experimental sequences of finite length is checked numerically on two paradigmatic models, namely discretized logistic map trajectories and Markov chains, and also on experimental behavioral sequences. Specific advantages of RP analysis are presented, among which is the detection of nonstationary features.

Synchronization of coupled oscillators is by now a very well studied subject. A large number of analytical and computational tools are available for the treatment of experimental results. This article focuses on a method recently proposed to construct a phase model from experimental data. The advantage of this method is that it extracts a phase model in a noninvasive manner without any prior knowledge of the experimental system. The aim of the present research is to apply this methodology to a network of electronic genetic oscillators. These electronic circuits mimic the dynamics of a synthetic genetic oscillator, called "repressilator", which is capable of synthesizing autonomous biological rhythms. The obtained phase model is shown to be capable of recovering the route leading to synchronization of genetic oscillators. The predicted onset point of synchronization agrees quite well with the experiment. This encourages further application of the present method to synthetic biological systems.

This paper investigates hydrodynamic processes of suspended planktonic population within a thermal stratification to represent a thermocline, in using a model of coupled equations: Navier–Stokes' equation for the fluid dynamic motion, Fisher's equation for the planktonic population with logistic growth, energy conservation equation for thermal effects and the Boussinesq approximation for the dependence on mass density of plankton. The planktonic population is assumed to be slightly denser than the fluid and to follow logistic growth. A numerical investigation is conducted to study patterns of plankton distribution under thermal stratification, and we adopt the finite volume method for solving the system of governing equations in simulations. It is found that thermal stratification constrains the feedback between hydrodynamic and plankton dynamics. More precisely, when temperature is lower in the upper than in the lower layer, the temperature gradient is destabilizing and both convection mechanisms — plankton motility (i.e. taxis behavior that can induce bioconvection) and temperature gradient — are additive. In the case of higher temperature in the upper than the lower layer, which applies to most natural systems, thermal stratification can dampen bioconvection and induce oscillatory dynamics if plankton diffuses faster than temperature. In all cases, if hydrodynamic and demographic time scales overlap, plankton growth rate has a strong effect on hydrodynamic processes involving both thermal convection and convection created by gravitaxis mechanism. Our study thus reveals the potential feedback between hydrodynamic and local demographic processes in plankton populations.

Noise-induced synchronization is an unexpected phenomenon which we show here to have biological relevance for multitrophic ecological systems. We focus on two uncoupled Hastings–Powell systems having oscillatory dynamics. Despite the fact the systems are uncoupled, it is shown that they can nevertheless synchronize by applying common Gaussian noise forcing to a suitably chosen variable in both systems. Complete synchronization is studied in the case of identical oscillators, and intermittent synchronization is analyzed when the oscillators are nonidentical. We discuss how this unusual synchronization phenomenon has features akin to Ecology's well-known "Moran effect."

The continuous Bonhoeffer–van der Pol (BVP for short) oscillator is transformed into a map-based BVP model by using the forward Euler scheme. At first, the bifurcations and chaos of the map-based BVP model are investigated when the step size varies as a bifurcation parameter. By using the fast-slow decomposition technique, a two-parameter bifurcation diagram is obtained to give insight into the effect of the step size on bifurcations and chaos of the map-based BVP model. The investigation shows that the period-doubling bifurcation is dependent on the step size, while the saddle-node bifurcation is independent of the step size. Second, when the fast–slow decomposition technique cannot be used, we rigorously prove that in the map-based BVP model there exists chaos in the sense of Marotto when the discrete step size varies as a bifurcation parameter. These results show that the discrete step sizes play a vital role between the continuous-time dynamical system and the corresponding discrete dynamical system. Much attention should be paid on the step size when a map-based neuron model is used as an alternative to a continuous neuron model.

The influence of coupling time delay on synchronization processes is studied within the framework of a model describing the dynamics of two inferior olive cells coupled electronically through gap-junctions surrounded by synaptic inhibitory terminals that block these couplings. It was found that different types of synchronous behavior exist in the system, depending on parameters, and that even small changes of coupling time delay may change a dynamic regime to another one. This may be the transition between the regimes of different synchronization types, spike time binding, nonsynchronous behavior or the regime of activity suppression of an inferior olive cell.

The phenomenon of self-referential phase reset is investigated in ensembles of interacting FitzHugh–Nagumo neurons with different topologies of couplings. It is shown that the reset phase of neurons oscillation is independent of the initial phase and is determined by parameters of the stimulus. This process does not require direct influence on all elements of the ensemble and takes place when the stimulus is applied to one of the interacting neurons only. The influence of inter-neuron couplings and stimulus parameters on ensemble dynamics and phase reset phenomenon is studied.

The paper describes the appearance of a novel, high-dimensional chaotic regime, called phase chaos, in a time-discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It arises from the nonlinear interaction among the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional time-discrete Kuramoto model, we outline the region of phase chaos in the parameter plane and determine the regions where phase chaos coexists with different periodic attractors. We also study the subcritical frequency-splitting bifurcation at the onset of desynchronization and demonstrate that the transition to phase chaos takes place via a torus destruction process.

We determine the boundary of the synchronization domain of a large number of one-dimensional continuous stochastic elements with time delayed nonhomogeneous mean-field coupling. The exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker–Planck equation. Here the synchronization threshold is found by solving this BVP using the continuation technique (AUTO). Approximate analytics is obtained using expansion into eigenfunctions of the stationary Fokker–Planck operator. Multistability and hysteresis are demonstrated for the case of bistable elements with a polynomial potential.

The problem of phase synchronization of coupled chaotic bursting oscillatory systems is studied using the probability distribution of return time to the quiscent state. It is shown that, in some range of the coupling strength, the continuous type distribution becomes discrete and is supported by few values around a preferred one. This shows that the shape and the amplitudes of the spikes are not important and the main aspect of coupling is the change of distribution of the units' time return difference.

We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first-return times" in various settings: when phase space is made of sequences of symbols from a finite alphabet (with application, for instance, to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two-dimensional torus. We derive relations linking these statistics with Rényi entropies and Lyapunov exponents.