We examine the emergence of collective dynamical structures and complexity in a network of interacting populations of neuronal oscillators. Each population consists of a heterogeneous collection of globally-coupled theta neurons, which are a canonical representation of Type-1 neurons. For simplicity, the populations are arranged in a fully autonomous driver-response configuration and our analysis allows us to have a full description of the asymptotic macroscopic dynamics of the network. We find that in this arrangement, the collective macroscopic behavior of the response population can exhibit equilibrium and limit cycle states, multistability, quasiperiodicity, and chaos. Furthermore, we obtain detailed bifurcation diagrams illustrating transitions among these macrostates and show that despite the many complications that are observed, it is possible to understand the emergence of such complexity based on an understanding of the collective behavior of a single population of theta neurons and the inter-population coupling. This work is a first step in the construction of a mathematically-tractable network-of-networks representation of neuronal network dynamics.
Using recently developed analytical techniques, we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons' excitability parameter varies in time. We demonstrate that such periodic variation can lead to the emergence of macroscopic chaos, multistability, and final-state unertainty in the collective behavior of the network. Finite-size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated.
We design and analyze the dynamics of a large network of theta neurons, which are idealized type I neurons. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global, via pulselike synapses of adjustable sharpness. Using recently developed analytical methods, we identify all possible asymptotic states that can be exhibited by a mean field variable that captures the network’s macroscopic state. These consist of two equilibrium states that reflect partial synchronization in the network and a limit cycle state in which the degree of network synchronization oscillates in time. Our approach also permits a complete bifurcation analysis, which we carry out with respect to parameters that capture the degree of excitability of the neurons, the heterogeneity in the population, and the coupling strength (which can be excitatory or inhibitory).We find that the network typically tends toward the two macroscopic equilibrium states when the neuron’s intrinsic dynamics and the network interactions reinforce one another. In contrast, the limit cycle state, bifurcations, and multistability tend to occur when there is competition among these network features. Finally,we show that our results are exhibited by finite network realizations of reasonable size.
We consider an infinite network of globally-coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.
We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.
In many networks of interest including technological, biological, and social networks, the connectivity between the interacting elements is not static, but changes in time. Furthermore, the elements themselves are often not identical, but rather display a variety of behaviors, and may come in different classes. Here, we investigate the dynamics of such systems. Specifically, we study a network of two large interacting heterogeneous populations of limit-cycle oscillators whose connectivity switches between two fixed arrangements at a particular frequency. We show that for sufficiently high switching frequency, this system behaves as if the connectivity were static and equal to the time average of the switching connectivity. We also examine the mechanisms by which this fast-switching limit is approached in several nonintuitive cases. The results illuminate novel mechanisms by which synchronization can arise or be thwarted in large populations of coupled oscillators with nonstatic coupling.
The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.
We examine the effects of applied electric fields on neuronal synchronization. Two-compartment model neurons were synaptically coupled and embedded within a resistive array, thus allowing the neurons to interact both chemically and electrically. In addition, an external electric field was imposed on the array. The effects of this field were found to be nontrivial, giving rise to domains of synchrony and asynchrony as a function of the heterogeneity among the neurons. A simple phase oscillator reduction was successfulin qualitatively reproducing these domains. The findings form several readily testable experimental predictions, and the model can be extended to a larger scale in which the effects of electric fields on seizure activity may be simulated.
Spiral waves are a basic feature of excitable systems. Although such waves have been observed in a variety of biological systems, they have not been observed in the mammalian cortex during neuronal activity. Here, we report stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging. Tissue from the occipital cortex (visual) was sectioned parallel to cortical lamina to preserve horizontal connections in layers III-V (500 mm thick, ~ 4 x 6 mm^{2}). In such tangential slices, excitation waves propagated in two dimensions during cholinergic oscillations. Spiral waves occurred spontaneously and alternated with plane, ring, and irregular waves. The rotation rate of the spirals was ~ 10 turns per second, and the rotation was linked to the oscillations in a one-cycle-one-rotation manner. A small (<128 mm) phase singularity occurred at the center of the spirals, about which were observed oscillations of widely distributed phases. The phase singularity drifted slowly across the tissue (~ 1 mm/10 turns). We introduced a computational model of a cortical layer that predicted and replicated many of the features of our experimental findings. We speculate that rotating spiral waves may provide a spatial framework to organize cortical oscillations.
A rigorous analytical approach is developed to test for the existence of a continuous nonlinear functional realtionship between systems. We compare the application of this nonlinear local technique to the existing analytical linear global approach in the setting of increasing additive noise. For natural systems with unknown levels of noise and nonlinearity, we propose a general framework for detecting coupling. Lastly, we demonstrate the applicability of this method to detect coupling between simultaneous, experimentally measured, intracellular voltages between neurons within a mammalian neuronal network.
Motived by the observation that applied fields modulate hippocampal seizures, and that seizures may be asynchronous, we modeled synaptically-coupled 2-compartment hippocampal pyramidal neurons embedded within an electrically resistive lattice in order to examine network sychronization properties under the influence of externally applied electric fields. Excitatory electric fields were shown to synchronize or desynchronize the network dependin on the natural frequency mismatch between the neurons. Such frequency mismatch was found to decrease as a function of increasing electric field amplitude. These findings provide testable hypotheses for future seizure control experiments.
Chaos synchronization in coupled systems is often characterized by a map $\phi$ between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set -- by which we mean graph($\phi$) -- can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. We also discuss how these features can be quantified, and how they interfere with standard methods for detecting synchronization in measured data.
The onset of collective synchronous behavior in globally coupled ensembles of oscillators is discussed. We present a formalism that is applicable to general ensembles of heterogeneous, continuous time dynamical units that, when uncoupled, are chaotic, periodic, or a mixture of both. A discussion of convergence issues, important for the proper implementation of our method, is included. Our work leads to a quantitative prediction for the critical coupling value at the onset of collective synchrony and for the growth rate of the resulting coherent state.
Synchronization between CA1 pyramidal neurons was studied using dual-cell patch-clamp technique simultaneous with an extracellular measurement of network activity. We explored various linear and nonlinear methods to detect weak synchronization in this network, using cross-correlation, mutual information in one and two dimensions, and phase correlation in both broad and narrow band. The linear and nonlinear methods demonstrated different patterns of sensitivity to detect synchrony in this network, depending on the dynamical state of the network. Bursts in 4-amino-pyridine (4AP) were highly synchronous events. Unexpectedly, seizure-like events in 4AP were desynchronous events, both in comparsion with interictal periods preceding the seizure without bursts (cut Schaffer collateral tract) and in comparsion with bursts preceding the seizures (intact Schaffer collateral tract). The finding that seizure-like events are associated with desynchronization in such networks is consistent with recent theoretical work, suggesting that asynchrony is necessary to maintain a high level of activity in neuronal networks for sustained periods of time and that synchrony may disrupt such activity.
A general stability analysis is presented for the determination of the transition from incoherent to coherent behavior in an ensemble of globally coupled, heterogeneous, continuous-time dynamical systems. The formalism allows for the simultaneous presence of ensemble members exhibiting chaotic and periodic behavior, and, in a special case, yields the Kuramoto model for globally coupled periodic oscillators described by a phase. Numerical experiments using different types of ensembles of Lorenz equations with a distribution of parameters are presented.
Chaos synchronization is often characterized by the existence of a continuous function between the states of the components. However, in coupled systems without inherent symmetries, the synchronization set might be extremely complicated. For coupled invertible systems, the synchronization set can be nondifferentiable; in the more severe case of coupled noninvertible systems, the synchronization set will in general be a multivalued relation. We will discuss how existing methods for detecting synchronization will be hampered by these features.
The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persists as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of non-identical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of "subsystems" which are derived from the unstable periodic orbit structure. Our results are illustrated with data from physical experiments.
We present a powerful method of analyzing coupled chaotic systems in terms of a ``subsystem" decomposition based on unstable periodic orbits. We focus on two important phenomena that have received a great deal of attention: synchronization and shadowability. This formalism forms the foundation for discussing these subjects in systems of non-identical coupled elements without special symmetries or invariant manifolds.
A chaotic attractor containing unstable periodic orbits with different numbers of unstable directions is said to exhibit unstable dimension variability (UDV). We present a general mechanism for the progressive development of UDV in uni- and bidirectionally coupled systems of chaotic elements. Our results are applicable to systems of dissimilar elements without invariant manifolds. We also quantify the severity of UDV to identify coupling ranges where the shadowability and modelability of such systems are significantly compromised.
Transitions that occur as systems of coupled maps are gradually decoupled hae been described in the literature in terms of the dynamics on a synchronization manifold. In this Latter we consider the evolution of the unstable periodic orbit structure of such systems and describe a new transition, which we call the decoherence transition. This is mediated by a set that develops off the synchronization manifold, the emergent set. The proposed framework is applicable to coupled systems of non-identical elements without obvious symmetries. For such systems, we also describe a spreading-out of the unstable periodic orbits in the synchronization set that occurs before the creation of the emergent set.
Nonlinear Mathematical Techniques can be used to Analyze the EEG Patterns of Epilepsy Patients enabling the Prediction of Seizures prior to the Onset of Symptoms
We propose an iterative scheme for calculating the box-counting (capacity) dimension of a chaotic attractor in terms of its average expansion rates. Similar to the Kaplan-Yorke conjecture for the information dimension, this scheme provides a connection between a geometric property of a strange set and its underlying dynamical properties. Our conjecture is demonstrated analytically with an exactly solvable two dimensional hyperbolic map and numerically with a more complicated higher dimensional non-hyperbolic map.
A method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction. Systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail. Mutual nonlinear prediction also provides information on the directionality of the coupling between systems. Furthermore, the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony. Numerical examples studied include three classes of unidirectionally coupled systems: systems with identical parameters, nonidentical parameters, and stochastic driving of a nonlinear system. This technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool. The interrelationships examined include single neuron unit firing, the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex, and intracellular measurements of integrated excitatory postsynaptic potentials (EPSPs). Dynamical interdependence, perhaps generalized synchrony, was identified in this neuronal network between simultaneous single unit firings, between units and the population, and between units and intracellular EPSPs.
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