Heteroclinic synchronization: ultrasubharmonic locking

Ghost Channels and Ghost Cycles Guiding Long Transients in Dynamical Systems

Dynamical descriptions and modeling of natural systems have generally focused on fixed points, with saddles and saddle-based phase-space objects such as heteroclinic channels or cycles being central concepts behind the emergence of quasistable long transients. Reliable and robust transient dynamics observed for real, inherently noisy systems is, however, not met by saddle-based dynamics, as demonstrated here. Generalizing the notion of ghost states, we provide a complementary framework that does not rely on the precise knowledge or existence of (un)stable fixed points, but rather on slow directed flows organized by ghost sets in ghost channels and ghost cycles. Moreover, we show that the appearance of these novel objects is an emergent property of a broad class of models typically used for description of natural systems.

Non-Markovian processes on heteroclinic networks

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values.

Perturbed Ion Temperature and Toroidal Flow Profile Measurements in Rotating Neoclassical Tearing Mode Magnetic Islands

The perturbed ion temperature and toroidal flow were measured in rotating neoclassical tearing modes (NTM) in a tokamak for the first time. These toroidally and radially resolved profiles were obtained by impurity ion spectroscopy in a 2,1 NTM in DIII-D. In agreement with drift-kinetic simulations, the electron temperature profile is flat, while the ion temperature gradient is restored across the magnetic island O point in the presence of fast ions; the perturbed flow has minima in the O points and maxima at the X points. These measurements provide the first confirmation of the theoretically expected ion temperature and flow response to a magnetic island needed to predict the NTM onset threshold scaling for ITER and other future tokamaks.

Heteroclinic networks for brain dynamics

Heteroclinic networks are a mathematical concept in dynamic systems theory that is suited to describe metastable states and switching events in brain dynamics. The framework is sensitive to external input and, at the same time, reproducible and robust against perturbations. Solutions of the corresponding differential equations are spatiotemporal patterns that are supposed to encode information both in space and time coordinates. We focus on the concept of winnerless competition as realized in generalized Lotka-Volterra equations and report on results for binding and chunking dynamics, synchronization on spatial grids, and entrainment to heteroclinic motion. We summarize proposals of how to design heteroclinic networks as desired in view of reproducing experimental observations from neuronal networks and discuss the subtle role of noise. The review is on a phenomenological level with possible applications to brain dynamics, while we refer to the literature for a rigorous mathematical treatment. We conclude with promising perspectives for future research.

The Sync-Fire/deSync model: Modelling the reactivation of dynamic memories from cortical alpha oscillations

We propose a neural network model to explore how humans can learn and accurately retrieve temporal sequences, such as melodies, movies, or other dynamic content. We identify target memories by their neural oscillatory signatures, as shown in recent human episodic memory paradigms. Our model comprises three plausible components for the binding of temporal content, where each component imposes unique limitations on the encoding and representation of that content. A cortical component actively represents sequences through the disruption of an intrinsically generated alpha rhythm, where a desynchronisation marks information-rich operations as the literature predicts. A binding component converts each event into a discrete index, enabling repetitions through a sparse encoding of events. A timing component - consisting of an oscillatory "ticking clock" made up of hierarchical synfire chains - discretely indexes a moment in time. By encoding the absolute timing between discretised events, we show how one can use cortical desynchronisations to dynamically detect unique temporal signatures as they are reactivated in the brain. We validate this model by simulating a series of events where sequences are uniquely identifiable by analysing phasic information, as several recent EEG/MEG studies have shown. As such, we show how one can encode and retrieve complete episodic memories where the quality of such memories is modulated by the following: alpha gate keepers to content representation; binding limitations that induce a blink in temporal perception; and nested oscillations that provide preferential learning phases in order to temporally sequence events.

Experimental Observation of Magnetic Island Heteroclinic Bifurcation in Tokamaks

We report empirical observations of magnetic island heteroclinic bifurcation for the first time. This behavior is observed in interacting coupled 2/1 tearing modes in the core of a DIII-D tokamak plasma. Poincaré maps constrained by measured magnetic amplitudes and phasing show bifurcation from heteroclinic to homoclinic topology in the 2/1 island as the 4/2 relative amplitude (R_{4/2}) decreases. Initially, the local electron temperature peak in the 2/1 island splits, consistent with two O points. As R_{4/2} decreases a single peak forms, consistent with one O point. These call for developing tearing stability theory and control solutions for heteroclinic islands in tokamaks.

Coupled heteroclinic networks in disguise

We consider diffusively coupled heteroclinic networks, ranging from two coupled heteroclinic cycles to small numbers of heteroclinic networks, each composed of two connected heteroclinic cycles. In these systems, we analyze patterns of synchronization as a function of the coupling strength. We find synchronized limit cycles, slowing-down states, as well as quasiperiodic motion of rotating tori solutions, transient chaos, and chaos, in general along with multistable behavior. This means that coupled heteroclinic networks easily come in disguise even when they constitute the main building blocks of the dynamics. The generated spatial patterns are rotating waves with on-site limit cycles and perturbed traveling waves from on-site quasiperiodic behavior. The bifurcation diagrams of these simple systems are in general quite intricate.

Discrete Sequential Information Coding: Heteroclinic Cognitive Dynamics

Discrete sequential information coding is a key mechanism that transforms complex cognitive brain activity into a low-dimensional dynamical process based on the sequential switching among finite numbers of patterns. The storage size of the corresponding process is large because of the permutation capacity as a function of control signals in ensembles of these patterns. Extracting low-dimensional functional dynamics from multiple large-scale neural populations is a central problem both in neuro- and cognitive- sciences. Experimental results in the last decade represent a solid base for the creation of low-dimensional models of different cognitive functions and allow moving toward a dynamical theory of consciousness. We discuss here a methodology to build simple kinetic equations that can be the mathematical skeleton of this theory. Models of the corresponding discrete information processing can be designed using the following dynamical principles: (i) clusterization of the neural activity in space and time and formation of information patterns; (ii) robustness of the sequential dynamics based on heteroclinic chains of metastable clusters; and (iii) sensitivity of such sequential dynamics to intrinsic and external informational signals. We analyze sequential discrete coding based on winnerless competition low-frequency dynamics. Under such dynamics, entrainment, and heteroclinic coordination leads to a large variety of coding regimes that are invariant in time.

Mind-to-mind heteroclinic coordination: Model of sequential episodic memory initiation

Retrieval of episodic memory is a dynamical process in the large scale brain networks. In social groups, the neural patterns, associated with specific events directly experienced by single members, are encoded, recalled, and shared by all participants. Here, we construct and study the dynamical model for the formation and maintaining of episodic memory in small ensembles of interacting minds. We prove that the unconventional dynamical attractor of this process-the nonsmooth heteroclinic torus-is structurally stable within the Lotka-Volterra-like sets of equations. Dynamics on this torus combines the absence of chaos with asymptotic instability of every separate trajectory; its adequate quantitative characteristics are length-related Lyapunov exponents. Variation of the coupling strength between the participants results in different types of sequential switching between metastable states; we interpret them as stages in formation and modification of the episodic memory.

Synchronization of oscillators through time-shifted common inputs

Shared upstream dynamical processes are frequently the source of common inputs in various physical and biological systems. However, due to finite signal transmission speeds and differences in the distance to the source, time shifts between otherwise common inputs are unavoidable. Since common inputs can be a source of correlation between the elements of multi-unit dynamical systems, regardless of whether these elements are directly connected with one another or not, it is of importance to understand their impact on synchronization. As a canonical model that is representative for a variety of different dynamical systems, we study limit-cycle oscillators that are driven by stochastic time-shifted common inputs. We show that if the oscillators are coupled, time shifts in stochastic common inputs do not simply shift the distribution of the phase differences, but rather the distribution actually changes as a result. The best synchronization is therefore achieved at a precise intermediate value of the time shift, which is due to a resonance-like effect with the most probable phase difference that is determined by the deterministic dynamics.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Data-driven inference of network connectivity for modeling the dynamics of neural codes in the insect antennal lobe

The antennal lobe (AL), olfactory processing center in insects, is able to process stimuli into distinct neural activity patterns, called olfactory neural codes. To model their dynamics we perform multichannel recordings from the projection neurons in the AL driven by different odorants. We then derive a dynamic neuronal network from the electrophysiological data. The network consists of lateral-inhibitory neurons and excitatory neurons (modeled as firing-rate units), and is capable of producing unique olfactory neural codes for the tested odorants. To construct the network, we (1) design a projection, an odor space, for the neural recording from the AL, which discriminates between distinct odorants trajectories (2) characterize scent recognition, i.e., decision-making based on olfactory signals and (3) infer the wiring of the neural circuit, the connectome of the AL. We show that the constructed model is consistent with biological observations, such as contrast enhancement and robustness to noise. The study suggests a data-driven approach to answer a key biological question in identifying how lateral inhibitory neurons can be wired to excitatory neurons to permit robust activity patterns.

Transient dynamics and rhythm coordination of inferior olive spatio-temporal patterns

The inferior olive (IO) is a neural network belonging to the olivo-cerebellar system whose neurons are coupled with electrical synapses and display subthreshold oscillations and spiking activity. The IO is frequently proposed as the generator of timing signals to the cerebellum. Electrophysiological and imaging recordings show that the IO network generates complex spatio-temporal patterns. The generation and modulation of coherent spiking activity in the IO is one key issue in cerebellar research. In this work, we build a large scale IO network model of electrically coupled conductance-based neurons to study the emerging spatio-temporal patterns of its transient neuronal activity. Our modeling reproduces and helps to understand important phenomena observed in IO in vitro and in vivo experiments, and draws new predictions regarding the computational properties of this network and the associated cerebellar circuits. The main factors studied governing the collective dynamics of the IO network were: the degree of electrical coupling, the extent of the electrotonic connections, the presence of stimuli or regions with different excitability levels and the modulatory effect of an inhibitory loop (IL). The spatio-temporal patterns were analyzed using a discrete wavelet transform to provide a quantitative characterization. Our results show that the electrotonic coupling produces quasi-synchronized subthreshold oscillations over a wide dynamical range. The synchronized oscillatory activity plays the role of a timer for a coordinated representation of spiking rhythms with different frequencies. The encoding and coexistence of several coordinated rhythms is related to the different clusterization and coherence of transient spatio-temporal patterns in the network, where the spiking activity is commensurate with the quasi-synchronized subthreshold oscillations. In the presence of stimuli, different rhythms are encoded in the spiking activity of the IO neurons that nevertheless remains constrained to a commensurate value of the subthreshold frequency. The stimuli induced spatio-temporal patterns can reverberate for long periods, which contributes to the computational properties of the IO. We also show that the presence of regions with different excitability levels creates sinks and sources of coordinated activity which shape the propagation of spike wave fronts. These results can be generalized beyond IO studies, as the control of wave pattern propagation is a highly relevant problem in the context of normal and pathological states in neural systems (e.g., related to tremor, migraine, epilepsy) where the study of the modulation of activity sinks and sources can have a potential large impact.

Synaptic inhibition controls transient oscillatory synchronization in a model of the insect olfactory system

In a variety of neuronal systems it has been hypothesized that inhibitory interneurons corral principal neurons into synchronously firing groups that encode sensory information and sub-serve behavior (Buzsáki and Chrobak, 1995; Buzsáki, 2008). This mechanism is particularly relevant to the olfactory system where spatiotemporal patterns of projection neuron (PN) activity act as robust markers of odor attributes (Laurent et al., 1996; Wehr and Laurent, 1996). In the insect antennal lobe (AL), a network of local inhibitory interneurons arborizes extensively throughout the AL (Leitch and Laurent, 1996) providing inhibitory input to the cholinergic PNs. Our theoretical work has attempted to elaborate the exact role of inhibition in the generation of odor specific PN responses (Bazhenov et al., 2001a,b; Assisi et al., 2011). In large-scale AL network models we characterized the inhibitory sub-network by its coloring (Assisi et al., 2011) and showed that it can entrain excitatory PNs to the odor specific patterns of transient synchronization. In this focused review, we further examine the dynamics of entrainment in more detail by simulating simple model networks in various parameter regimes. Our simulations in conjunction with earlier studies point to the key role played by lateral (between inhibitory interneurons) and feedback (from inhibitory interneurons to principal cells) inhibition in the generation of experimentally observed patterns of transient synchrony.

Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.

Organization of Anti-Phase Synchronization Pattern in Neural Networks: What are the Key Factors?

Anti-phase oscillation has been widely observed in cortical neural network. Elucidating the mechanism underlying the organization of anti-phase pattern is of significance for better understanding more complicated pattern formations in brain networks. In dynamical systems theory, the organization of anti-phase oscillation pattern has usually been considered to relate to time delay in coupling. This is consistent to conduction delays in real neural networks in the brain due to finite propagation velocity of action potentials. However, other structural factors in cortical neural network, such as modular organization (connection density) and the coupling types (excitatory or inhibitory), could also play an important role. In this work, we investigate the anti-phase oscillation pattern organized on a two-module network of either neuronal cell model or neural mass model, and analyze the impact of the conduction delay times, the connection densities, and coupling types. Our results show that delay times and coupling types can play key roles in this organization. The connection densities may have an influence on the stability if an anti-phase pattern exists due to the other factors. Furthermore, we show that anti-phase synchronization of slow oscillations can be achieved with small delay times if there is interaction between slow and fast oscillations. These results are significant for further understanding more realistic spatiotemporal dynamics of cortico-cortical communications.

Transient dynamics between displaced fixed points: an alternate nonlinear dynamical framework for olfaction

Significant insights into the dynamics of neuronal populations have been gained in the olfactory system where rich spatio-temporal dynamics is observed during, and following, exposure to odours. It is now widely accepted that odour identity is represented in terms of stimulus-specific rate patterning observed in the cells of the antennal lobe (AL). Here we describe a nonlinear dynamical framework inspired by recent experimental findings which provides a compelling account of both the origin and the function of these dynamics. We start by analytically reducing a biologically plausible conductance based model of the AL to a quantitatively equivalent rate model and construct conditions such that the rate dynamics are well described by a single globally stable fixed point (FP). We then describe the AL's response to an odour stimulus as rich transient trajectories between this stable baseline state (the single FP in absence of odour stimulation) and the odour-specific position of the single FP during odour stimulation. We show how this framework can account for three phenomena that are observed experimentally. First, for an inhibitory period often observed immediately after an odour stimulus is removed. Second, for the qualitative differences between the dynamics in the presence and the absence of odour. Lastly, we show how it can account for the invariance of a representation of odour identity to both the duration and intensity of an odour stimulus. We compare and contrast this framework with the currently prevalent nonlinear dynamical framework of 'winnerless competition' which describes AL dynamics in terms of heteroclinic orbits. This article is part of a Special Issue entitled "Neural Coding".

Robust transient dynamics and brain functions

In the last few decades several concepts of dynamical systems theory (DST) have guided psychologists, cognitive scientists, and neuroscientists to rethink about sensory motor behavior and embodied cognition. A critical step in the progress of DST application to the brain (supported by modern methods of brain imaging and multi-electrode recording techniques) has been the transfer of its initial success in motor behavior to mental function, i.e., perception, emotion, and cognition. Open questions from research in genetics, ecology, brain sciences, etc., have changed DST itself and lead to the discovery of a new dynamical phenomenon, i.e., reproducible and robust transients that are at the same time sensitive to informational signals. The goal of this review is to describe a new mathematical framework - heteroclinic sequential dynamics - to understand self-organized activity in the brain that can explain certain aspects of robust itinerant behavior. Specifically, we discuss a hierarchy of coarse-grain models of mental dynamics in the form of kinetic equations of modes. These modes compete for resources at three levels: (i) within the same modality, (ii) among different modalities from the same family (like perception), and (iii) among modalities from different families (like emotion and cognition). The analysis of the conditions for robustness, i.e., the structural stability of transient (sequential) dynamics, give us the possibility to explain phenomena like the finite capacity of our sequential working memory - a vital cognitive function -, and to find specific dynamical signatures - different kinds of instabilities - of several brain functions and mental diseases.

Small-world properties of nonlinear brain activity in schizophrenia

A disturbance in the interactions between distributed cortical regions may underlie the cognitive and perceptual dysfunction associated with schizophrenia. In this article, nonlinear measures of cortical interactions and graph-theoretical metrics of network topography are combined to investigate this schizophrenia "disconnection hypothesis." This is achieved by analyzing the spatiotemporal structure of resting state scalp EEG data previously acquired from 40 young subjects with a recent first episode of schizophrenia and 40 healthy matched controls. In each subject, a method of mapping the topography of nonlinear interactions between cortical regions was applied to a widely distributed array of these data. The resulting nonlinear correlation matrices were converted to weighted graphs. The path length (a measure of large-scale network integration), clustering coefficient (a measure of "cliquishness"), and hub structure of these graphs were used as metrics of the underlying brain network activity. The graphs of both groups exhibited high levels of local clustering combined with comparatively short path lengths--features consistent with a "small-world" topology--as well as the presence of strong, central hubs. The graphs in the schizophrenia group displayed lower clustering and shorter path lengths in comparison to the healthy group. Whilst still "small-world," these effects are consistent with a subtle randomization in the underlying network architecture--likely associated with a greater number of links connecting disparate clusters. This randomization may underlie the cognitive disturbances characteristic of schizophrenia.

Winnerless competition principle and prediction of the transient dynamics in a Lotka-Volterra model

Predicting the evolution of multispecies ecological systems is an intriguing problem. A sufficiently complex model with the necessary predicting power requires solutions that are structurally stable. Small variations of the system parameters should not qualitatively perturb its solutions. When one is interested in just asymptotic results of evolution (as time goes to infinity), then the problem has a straightforward mathematical image involving simple attractors (fixed points or limit cycles) of a dynamical system. However, for an accurate prediction of evolution, the analysis of transient solutions is critical. In this paper, in the framework of the traditional Lotka-Volterra model (generalized in some sense), we show that the transient solution representing multispecies sequential competition can be reproducible and predictable with high probability.

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).

Transient cognitive dynamics, metastability, and decision making

The idea that cognitive activity can be understood using nonlinear dynamics has been intensively discussed at length for the last 15 years. One of the popular points of view is that metastable states play a key role in the execution of cognitive functions. Experimental and modeling studies suggest that most of these functions are the result of transient activity of large-scale brain networks in the presence of noise. Such transients may consist of a sequential switching between different metastable cognitive states. The main problem faced when using dynamical theory to describe transient cognitive processes is the fundamental contradiction between reproducibility and flexibility of transient behavior. In this paper, we propose a theoretical description of transient cognitive dynamics based on the interaction of functionally dependent metastable cognitive states. The mathematical image of such transient activity is a stable heteroclinic channel, i.e., a set of trajectories in the vicinity of a heteroclinic skeleton that consists of saddles and unstable separatrices that connect their surroundings. We suggest a basic mathematical model, a strongly dissipative dynamical system, and formulate the conditions for the robustness and reproducibility of cognitive transients that satisfy the competing requirements for stability and flexibility. Based on this approach, we describe here an effective solution for the problem of sequential decision making, represented as a fixed time game: a player takes sequential actions in a changing noisy environment so as to maximize a cumulative reward. As we predict and verify in computer simulations, noise plays an important role in optimizing the gain.

Generation and reshaping of sequences in neural systems

The generation of informational sequences and their reorganization or reshaping is one of the most intriguing subjects for both neuroscience and the theory of autonomous intelligent systems. In spite of the diversity of sequential activities of sensory, motor, and cognitive neural systems, they have many similarities from the dynamical point of view. In this review we discus the ideas, models, and mathematical image of sequence generation and reshaping on different levels of the neural hierarchy, i.e., the role of a sensory network dynamics in the generation of a motor program (hunting swimming of marine mollusk Clione), olfactory dynamical coding, and sequential learning and decision making. Analysis of these phenomena is based on the winnerless competition principle. The considered models can be a basis for the design of biologically inspired autonomous intelligent systems.

Frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle

Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invariant hyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanical instabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce the dynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability, intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes strongly stable, the dynamics remains chaotic and no frequency locking is observed.