Clustering in globally coupled phase oscillators

Two distinct mechanisms shape the reliability of neural responses

Despite intrinsic noise sources, neurons can generate action potentials with remarkable reliability. This reliability is influenced by the characteristics of sensory or synaptic inputs, such as stimulus frequency. Here we use conductance-based models to study the frequency dependence of reliability in terms of the underlying single-cell properties. We are led to distinguish a mean-driven firing regime, where the stimulus mean is sufficient to elicit continuous firing, and a fluctuation-driven firing regime, where spikes are generated by transient stimulus fluctuations. In the mean-driven regime, the stimulus frequency that induces maximum reliability coincides with the firing rate of the cell, whereas in the fluctuation-driven regime, it is determined by the resonance properties of the subthreshold membrane potential. When the stimulus frequency does not match the optimal frequency, the two firing regimes exhibit different "symptoms" of decreased reliability: reduced spike-time precision and reduced spike probability, respectively. As a signature of stochastic resonance, reliable spike generation in the fluctuation-driven regime can benefit from intermediate amounts of noise that boost spike probability without significantly impairing spike-time precision. Our analysis supports the view that neurons are endowed with selection mechanisms that allow only certain stimulus frequencies to induce reliable spiking. By modulating the intrinsic cell properties, the nervous system can thus tune individual neurons to pick out specific input frequency bands with enhanced spike precision or spike probability.

Exact results for the Kuramoto model with a bimodal frequency distribution

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.

Clustering in a network of non-identical and mutually interacting agents

Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behaviour. It is observed in fields ranging from the exact sciences to social and life sciences; consider, for example, swarm behaviour of animals or social insects, the dynamics of opinion formation or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modelling brain or heart cells. We consider a clustering model with a general network structure and saturating interaction functions. We derive both necessary and sufficient conditions for clustering behaviour of the model and we investigate the cluster structure for varying coupling strength. Generically, each cluster asymptotically reaches a (relative) equilibrium state. We discuss the relationship of the model to swarming, and we explain how the model equations naturally arise in a system of interconnected water basins. We also indicate how the model applies to opinion formation dynamics.

Design principles for phase-splitting behaviour of coupled cellular oscillators: clues from hamsters with 'split' circadian rhythms

Nonlinear interactions among coupled cellular oscillators are likely to underlie a variety of complex rhythmic behaviours. Here we consider the case of one such behaviour, a doubling of rhythm frequency caused by the spontaneous splitting of a population of synchronized oscillators into two subgroups each oscillating in anti-phase (phase-splitting). An example of biological phase-splitting is the frequency doubling of the circadian locomotor rhythm in hamsters housed in constant light, in which the pacemaker in the suprachiasmatic nucleus (SCN) is reconfigured with its left and right halves oscillating in anti-phase. We apply the theory of coupled phase oscillators to show that stable phase-splitting requires the presence of negative coupling terms, through delayed and/or inhibitory interactions. We also find that the inclusion of real biological constraints (that the SCN contains a finite number of non-identical noisy oscillators) implies the existence of an underlying non-uniform network architecture, in which the population of oscillators must interact through at least two types of connections. We propose that a key design principle for the frequency doubling of a population of biological oscillators is inhomogeneity of oscillator coupling.

Routes to synchrony between asymmetrically interacting oscillator ensembles

We report that asymmetrically interacting ensembles of oscillators follow novel routes to synchrony. These routes seem to be a characteristic feature of coupling asymmetry. We show that they are unaffected by white noise except that the entrainment frequencies are shifted. The probability of occurrence of the routes is determined by phase asymmetry. The identification of these phenomena offers new insight into synchrony between oscillator ensembles and suggest new ways in which it may be controlled.

Resonances and entrainment breakup in Kuramoto models with multimodal frequency densities

We characterize some intriguing aspects of the entrainment behavior of coupled oscillators by means of a perturbation analysis of the partially synchronized solution of the classical Kuramoto-Sakaguchi model. The analysis reveals that partial entrainment may disappear with increasing coupling strength. It also predicts the occurrence of resonances: partial entrainment is induced in oscillators with natural frequencies in specific intervals not corresponding to high oscillator densities. The results are illustrated by simulations.

Synchronization engineering: theoretical framework and application to dynamical clustering

A method for engineering the global behavior of populations of rhythmic elements is presented. The framework, which is based on phase models, allows a nonlinear time-delayed global feedback signal to be constructed which produces an interaction function corresponding to the desired behavior of the system. It is shown theoretically and confirmed in numerical simulations that a polynomial, delayed feedback is a versatile tool to tune synchronization patterns. Dynamical states consisting of one to four clusters were engineered to demonstrate the application of synchronization engineering in an experimental electrochemical system.

Clusters and switchers in globally coupled photochemical oscillators

We experimentally investigate the transition to synchronization in a population of photochemical oscillators with weak global coupling. Above a critical coupling strength the oscillators join a one-phase group or two-phase clusters. The number of oscillators in each cluster depends on the initial phase distribution, and irregular switching of oscillators between clusters is observed. The fully synchronized state emerges above a second critical coupling strength. In agreement with earlier theory, the experiments demonstrate the importance of population heterogeneity in cluster multistability.

Interplay between a phase response curve and spike-timing-dependent plasticity leading to wireless clustering

A phase response curve (PRC) characterizes the signal transduction between oscillators such as neurons on a fixed network in a minimal manner, while spike-timing-dependent plasiticity (STDP) characterizes the way of rewiring networks in an activity-dependent manner. This paper demonstrates that these two key properties both related to the interaction times of oscillators work synergetically to carve functionally useful circuits. STDP working on neurons that prefer asynchrony converts the initial asynchronous firing to clustered firing with synchrony within a cluster. They get synchronized within a cluster despite their preference to asynchrony because STDP selectively disrupts intracluster connections, which we call wireless clustering. Our PRC analysis reveals a triad mechanism: the network structure affects how the PRC is read out to determine the synchrony tendency, the synchrony tendency affects how the STDP works, and STDP affects the network structure, closing the loop.

Resonance clustering in globally coupled electrochemical oscillators with external forcing

Experiments are carried out with a globally coupled, externally forced population of limit-cycle electrochemical oscillators with an approximately unimodal distribution of heterogeneities. Global coupling induces mutually entrained (at frequency omega1) states; periodic forcing produces forced-entrained (omegaF) states. As a result of the interaction of mutual and forced entrainment, resonant cluster states occur with equal spacing of frequencies that have discretized frequencies following a resonance rule omegan congruent with nomega1-(n-1)omegaF. Resonance clustering requires an optimal, intermediate global coupling strength; at weak coupling the clusters have smaller sizes and do not strictly follow the resonance rule, while at strong coupling the population behaves similar to a single, giant oscillator.

Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication

We show that phase-repulsive coupling eliminates oscillations in a population of synthetic genetic clocks. For this, we propose an experimentally feasible synthetic genetic network that contains phase repulsively coupled repressilators with broken temporal symmetry. As the coupling strength increases, silencing of oscillations is found to occur via the appearance of an inhomogeneous limit cycle, followed by oscillation death. Two types of oscillation death are observed: For lower couplings, the cells cluster in one of two stationary states of protein expression; for larger couplings, all cells end up in a single (stationary) cellular state. Several multistable regimes are observed along this route to oscillation death.

Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators

We study synchronization properties of general uncoupled limit-cycle oscillators driven by common and independent Gaussian white noises. Using phase reduction and averaging methods, we analytically derive the stationary distribution of the phase difference between oscillators for weak noise intensity. We demonstrate that in addition to synchronization, clustering, or more generally coherence, always results from arbitrary initial conditions, irrespective of the details of the oscillators.

Aging and clustering in globally coupled oscillators

A population of coupled nonlinear oscillators may age in such a way that the fraction of non-self-oscillatory elements increases. Following our previous paper [Phys. Rev. Lett. 93, 104101 (2004)], we study the effect of aging in this sense mainly for globally coupled Stuart-Landau oscillators with the emphasis on the structure of the (K,p) phase diagram, where K is the coupling strength and p is the ratio of inactive oscillators. In addition to the aging transition reported previously, such a diagram is shown to be characterized by a hornlike region, which we call a "desynchronization horn," where active oscillators desynchronize to form a number of clusters, provided that uncoupled active oscillators are sufficiently nonisochronous. We also show that desynchronization in such a region can be captured as a type of diffusion-induced inhomogeneity based on a "swing-by mechanism." Our results suggest that the desynchronization horn with some curious properties may be a fairly common feature in aging systems of globally and diffusively coupled periodic oscillators.

Inherent multistability in arrays of autoinducer coupled genetic oscillators

Rhythm generation mechanisms are very important for genetic network functions as well as for the design of synthetic genetic circuits. A significant attention to date has been focused on the synchronization of communicating genetic units, which results in the production of an unified rhythm. In contrast to this we address the question: what mechanisms of intercell communication can be responsible for multirhythmicity in globally coupled genetic units? Here, we show that an autoinducer intercell communication system that provides coupling between synthetic genetic oscillators will inherently lead to multirhythmicity and the appearance of several coexisting dynamical regimes, if the time evolution of the genetic network can be split in two well-separated time scales. We investigate in detail a variety of dynamical regimes in a genetic population and show the possibility for multiple element distributions between clusters, as well as the possibility of generating complex oscillations with different return times in one limit cycle.

Noise accelerates synchronization of coupled nonlinear oscillators

For a chain of homogeneous nonlinear oscillators starting from different initial phases, a certain amount of time is required for the system to evolve to complete phase synchronization. The effect of independent noise in such a system was investigated, and an optimal noise intensity was found that minimized the average synchronization time. Both threshold noise and connection noise show similar effects. The features of the phenomenon and the underlying mechanism are discussed through the analysis of a two-unit system and the numerical studies of chains up to 30 units in length.

Localized structures in a nonlinear wave equation stabilized by negative global feedback: one-dimensional and quasi-two-dimensional kinks

We study the evolution of fronts in a nonlinear wave equation with global feedback. This equation generalizes the Klein-Gordon and sine-Gordon equations. Extending previous work, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the classical case (with no global feedback), leading in most cases to a localized solution; i.e., the stabilization of one phase inside the other. The nature of the localized solution depends on the strength of the global feedback as well as on other parameters of the model.

How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons

GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.

Electrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons

Information processing in higher brain structures is thought to rely on the synchronization of spiking neurons. Increasing evidence indicates that, within these structures, inhibitory neurons are linked by both chemical and electrical synapses. However, how synchronized states may emerge from such circuits is not fully understood. Using snail neurons interconnected through a dynamic-clamp system, we show that networks of spiking neurons linked by both reciprocal inhibition and electrical coupling can express two coexisting coordination patterns of different rhythms. One of these patterns consists of antiphase firing of the network partners whereas, in the other, neurons fire synchronously. Switching between patterns may be evoked immediately by transient stimuli, demonstrating bistability of the network. Thus electrical coupling can provide a potent way for instantaneous reconfiguration of activity patterns in inhibitory spiking networks without alteration of intrinsic network properties by modulatory processes.

Synchrony and clustering in heterogeneous networks with global coupling and parameter dispersion

Networks with nonidentical nodes and global coupling may display a large variety of dynamic behaviors, such as phase clustered solutions, synchrony, and oscillator death. The network dynamics is a function of the parameter dispersion and may be captured by conventional mean field approaches if it is close to the completely synchronous state. In this Letter we introduce a novel method based on a mode decomposition in the parameter space, which provides a low-dimensional network description for more complex dynamic behaviors and captures the mean field approach as a special case. The example of globally coupled Fitzhugh-Nagumo neurons is discussed.

A hierarchy of global coupling induced cluster patterns during the oscillatory H2-electrooxidation reaction on a Pt ring-electrode

We report experimental results on spatiotemporal pattern formation during the oscillatory hydrogen electrooxidation reaction on a Pt ring-electrode under negative (desynchronizing) global coupling (GC). Spatially one-dimensional profiles of the interfacial potential drop along the angular direction of the ring electrode are recorded by means of a potential probe. The dynamics is investigated as a function of two control parameters, the applied voltage U and the strength of the global coupling. The latter is adjusted either by varying the distance between the working electrode (WE) and the reference electrode (RE) or by inserting a negative impedance device in series with the WE. In the absence of global coupling, uniform oscillations were destabilized by migration coupling, and electrochemical turbulence developed at large values of U (H. Varela, C. Beta, A. Bonnefont and K. Krischer, Phys. Rev. Lett., 2005, 94, 174104; ). Already low global coupling strengths sufficed to suppress turbulence. Instead, regular two-phase clusters formed. At higher coupling strength, a second type of two-phase cluster was observed as well as two types of irregular cluster patterns, which were connected with an irregular motion of the cluster boundaries and the emergence and disappearance of clusters through splitting and merging of the boundaries, respectively. Upon increasing the coupling strength even further, five-phase clusters were stabilized and at the highest coupling strength applied the cluster patterns transformed into strongly modulated pulses. The two types of two-phase clusters and the five-phase clusters are analyzed employing several signal processing techniques.

Cooperative action of coherent groups in broadly heterogeneous populations of interacting chemical oscillators

We present laboratory experiments on the effects of global coupling in a population of electrochemical oscillators with a multimodal frequency distribution. The experiments show that complex collective signals are generated by this system through spontaneous emergence and joint operation of coherently acting groups representing hierarchically organized resonant clusters. Numerical simulations support these experimental findings. Our results suggest that some forms of internal self-organization, characteristic for complex multiagent systems, are already possible in simple chemical systems.

Clustering through postinhibitory rebound in synaptically coupled neurons

Postinhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyperpolarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases (i) where the rebound mechanism is due to anode break excitation and (ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean-type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, while slow T-type calcium currents predominantly give rise to clusters of just two (antisynchronous) populations.

Synchronization of active mechanical oscillators by an inertial load

Motivated by the operation of myogenic (self-oscillatory) insect flight muscle, we study a model consisting of a large number of identical oscillatory contractile elements joined in a chain, whose end is attached to a damped mass-spring oscillator. When the inertial load is small, the serial coupling favors an antisynchronous state in which the extension of one oscillator is compensated by the contraction of another, in order to preserve the total length. However, a sufficiently massive load can synchronize the oscillators and can even induce oscillation in situations where isolated elements would be stable. The system has a complex phase diagram displaying quiescent, synchronous and antisynchronous phases, as well as an unusual asynchronous phase in which the total length of the chain oscillates at a different frequency from the individual active elements.

Synchrony of fast-spiking interneurons interconnected by GABAergic and electrical synapses

Fast-spiking (FS) interneurons have specific types (Kv3.1/3.2 type) of the delayed potassium channel, which differ from the conventional Hodgkin-Huxley (HH) type potassium channel (Kv1.3 type) in several aspects. In this study, we show dramatic effects of the Kv3.1/3.2 potassium channel on the synchronization of the FS interneurons. We show analytically that two identical electrically coupled FS interneurons modeled with Kv3.1/3.2 channel fire synchronously at arbitrary firing frequencies, unlike similarly coupled FS neurons modeled with Kv1.3 channel that show frequency-dependent synchronous and antisynchronous firing states. Introducing GABA(A) receptor-mediated synaptic connections into an FS neuron pair tends to induce an antisynchronous firing state, even if the chemical synapses are bidirectional. Accordingly, an FS neuron pair connected simultaneously by electrical and chemical synapses achieves both synchronous firing state and antisynchronous firing state in a physiologically plausible range of the conductance ratio between electrical and chemical synapses. Moreover, we find that a large-scale network of FS interneurons connected by gap junctions and bidirectional GABAergic synapses shows similar bistability in the range of gamma frequencies (30-70 Hz).

Clustering in small networks of excitatory neurons with heterogeneous coupling strengths

Excitatory coupling with a slow rise time destabilizes synchrony between coupled neurons. Thus, the fully synchronous state is usually unstable in networks of excitatory neurons. Phase-clustered states, in which neurons are divided into multiple synchronized clusters, have also been found unstable in numerical studies of excitatory networks in the presence of noise. The question arises as to whether synchrony is possible in networks of neurons coupled through slow, excitatory synapses. In this paper, we show that robust, synchronous clustered states can occur in such networks. The effects of non-uniform distributions of coupling strengths are explored. Conditions for the existence and stability of clustered states are derived analytically. The analysis shows that a multi-cluster state can be stable in excitatory networks if the overall interactions between neurons in different clusters are stabilizing and strong enough to counter-act the destabilizing interactions between neurons within each cluster. When heterogeneity in the coupling strengths strengthens the stabilizing inter-cluster interactions and/or weakens the destabilizing in-cluster interactions, robust clustered states can occur in excitatory networks of all known model neurons. Numerical simulations were carried out to support the analytical results.

Spiking neuron models with excitatory or inhibitory synaptic couplings and synchronization phenomena

We investigate synchronization phenomena in a system of two piecewise-linear-type model neurons with excitatory or inhibitory synaptic couplings. Employing the phase plane analysis and a singular perturbation approach to split the dynamics into slow and fast ones, we construct analytically the Poincaré map of the solution to the piecewise-linear equations. We investigate conditions for the occurrence of synchronized oscillations of in phase as well as of antiphase in terms of parameters representing the strength of the synaptic coupling and the decaying relaxation rate of the synaptic dynamics. We present the results of numerical simulations that agree with our theoretical ones.

Object selection by an oscillatory neural network

We describe a new solution to the problem of consecutive selection of objects in a visual scene by an oscillatory neural network with the global interaction realised through a central executive element (central oscillator). The frequency coding is used to represent greyscale images in the network. The functioning of the network is based on three main principles: (1) the synchronisation of oscillators via phase-locking, (2) adaptation of the natural frequency of the central oscillator, and (3) resonant increase of the amplitudes of the oscillators which work in-phase with the central oscillator. Examples of network simulations are presented to show the reliability of the results of consecutive selection of objects under conditions of constant and varying brightness of the objects.

Effective desynchronization with bipolar double-pulse stimulation

This paper is devoted to the desynchronizing effects of bipolar stimuli on a synchronized cluster of globally coupled phase oscillators. The bipolar pulses considered here are symmetrical and consist of a positive and a negative monopolar pulse. A bipolar single pulse with the right intensity and duration desynchronizes a synchronized cluster provided the stimulus is administered at a vulnerable initial phase of the cluster's order parameter. A considerably more effective desynchronization is achieved with a bipolar double pulse consisting of two qualitatively different bipolar pulses. The first bipolar pulse is stronger and resets the cluster, so that the second bipolar pulse, which follows after a constant delay, hits the cluster in a vulnerable state and desynchronizes it. A bipolar double pulse desynchronizes the cluster independently of the cluster's dynamical state at the beginning of the stimulation. The dynamics of the order parameter during a bipolar single pulse or a bipolar double pulse is different from the dynamics during a monopolar single pulse or a monopolar double pulse. Nevertheless, concerning their desynchronizing effects the monopolar and the bipolar stimuli are comparable, respectively. This is significant for applications where bipolar stimulation is required. For example, in medicine and physiology charge-balanced stimulation is typically necessary in order to avoid tissue damage. Based on the results presented here, demand-controlled bipolar double-pulse stimulation is suggested as a milder and more efficient therapy compared to the standard permanent high-frequency deep brain stimulation in neurological patients.

Bifurcations of synchronized responses in synaptically coupled Bonhöffer-van der Pol neurons

The Bonhöffer-van der Pol (BvdP) equation is considered as an important model for studying dynamics in a single neuron. In this paper, we investigate bifurcations of periodic solutions in model equations of four and five BvdP neurons coupled through the characteristics of synaptic transmissions with a time delay. The model can be considered as a dynamical system whose solution includes jumps depending on a condition related to the behavior of the trajectory. Although the solution is discontinuous, we can define the Poincaré map as a synthesis of successive submaps, and give its derivatives for obtaining periodic points and their bifurcations. Using our proposed numerical method, we clarify mechanisms of bifurcations among synchronized oscillations with phase-locking patterns by analyzing periodic solutions observed in the coupling system and its subsystems. Moreover, we show that a global behavior of chaotic itinerancy or a phenomenon of chaotic transitions among several quasiattracting states can be observed in higher-dimensional systems of the synaptically four and five coupled neurons.

Distributed synchrony in a cell assembly of spiking neurons

We investigate the formation of a Hebbian cell assembly of spiking neurons, using a temporal synaptic learning curve that is based on recent experimental findings. It includes potentiation for short time delays between pre- and post-synaptic neuronal spiking, and depression for spiking events occurring in the reverse order. The coupling between the dynamics of synaptic learning and that of neuronal activation leads to interesting results. One possible mode of activity is distributed synchrony, implying spontaneous division of the Hebbian cell assembly into groups, or subassemblies, of cells that fire in a cyclic manner. The behavior of distributed synchrony is investigated both by simulations and by analytic calculations of the resulting synaptic distributions.

Firing patterns and correlations of spontaneous discharge of pallidal neurons in the normal and the tremulous 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine vervet model of parkinsonism

To investigate the role of the basal ganglia in parkinsonian tremor, we recorded hand tremor and simultaneous activity of several neurons in the external and internal segments of the globus pallidus (GPe and GPi) in two vervet monkeys, before and after systemic treatment with 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP) and development of parkinsonism with tremor of 5 and 11 Hz. In healthy monkeys, only 11% (20/174) of the GPe cells and 3% (1/29) of the GPi cells displayed significant 3-19 Hz oscillations. After MPTP treatment, 39% (107/271) of the GPe cells and 43% (26/61) of the GPi cells developed significant oscillations. Oscillation frequencies of single cells after MPTP treatment were bimodally distributed around 7 and 13 Hz. For 10% of the oscillatory cells that were recorded during tremor periods, there was a significant tendency for the tremor and neuronal oscillations to appear simultaneously. Cross-correlation analysis revealed a very low level of correlated activity between pallidal neurons in the normal state; 95.6% (477/499) of the pairs were not correlated, and oscillatory cross-correlograms were found in only 1% (5/499) of the pairs. After MPTP treatment, the correlations increased dramatically, and 40% (432/1080) of the cross-correlograms had significant oscillations, centered around 13-14 Hz. Phase shifts of the cross-correlograms of GPe pairs, but not of GPi, were clustered around 0 degrees. The results illustrate that MPTP treatment changes the pattern of activity and synchronization in the GPe and GPi. These changes are related to the symptoms of Parkinson's disease and especially to the parkinsonian tremor.

Oscillatory clusters in a model of the photosensitive belousov-zhabotinsky reaction system with global feedback

Oscillatory cluster patterns are studied numerically in a reaction-diffusion model of the photosensitive Belousov-Zhabotinsky reaction supplemented with a global negative feedback. In one- and two-dimensional systems, families of cluster patterns arise for intermediate values of the feedback strength. These patterns consist of spatial domains of phase-shifted oscillations. The phase of the oscillations is nearly constant for all points within a domain. Two-phase clusters display antiphase oscillations; three-phase clusters contain three sets of domains with a phase shift equal to one-third of the period of the local oscillation. Border (nodal) lines between domains for two-phase clusters become stationary after a transient period, while borders drift in the case of three-phase clusters. We study the evolving border movement of the clusters, which, in most cases, leads to phase balance, i.e., equal areas of the different phase domains. Border curling of three-phase clusters results in formation of spiral clusters-a combination of a fast oscillating cluster with a slow spiraling movement of the domain border. At higher feedback coefficient, irregular cluster patterns arise, consisting of domains that change their shape and position in an irregular manner. Localized irregular and regular clusters arise for parameters close to the boundary between the oscillatory region and the reduced steady state region of the phase space.

Oscillatory cluster patterns in a homogeneous chemical system with global feedback

Oscillatory clusters are sets of domains in which nearly all elements in a given domain oscillate with the same amplitude and phase. They play an important role in understanding coupled neuron systems. In the simplest case, a system consists of two clusters that oscillate in antiphase and can each occupy multiple fixed spatial domains. Examples of cluster behaviour in extended chemical systems are rare, but have been shown to resemble standing waves, except that they lack a characteristic wavelength. Here we report the observation of so-called 'localized clusters'--periodic antiphase oscillations in one part of the medium, while the remainder appears uniform--in the Belousov-Zhabotinsky reaction-diffusion system with photochemical global feedback. We also observe standing clusters with fixed spatial domains that oscillate periodically in time and occupy the entire medium, and irregular clusters with no periodicity in either space or time, with standing clusters transforming into irregular clusters and then into localized clusters as the strength of the global negative feedback is gradually increased. By incorporating the effects of global feedback into a model of the reaction, we are able to simulate successfully the experimental data.

Synchrony in heterogeneous networks of spiking neurons

The emergence of synchrony in the activity of large, heterogeneous networks of spiking neurons is investigated. We define the robustness of synchrony by the critical disorder at which the asynchronous state becomes linearly unstable. We show that at low firing rates, synchrony is more robust in excitatory networks than in inhibitory networks, but excitatory networks cannot display any synchrony when the average firing rate becomes too high. We introduce a new regime where all inputs, external and internal, are strong and have opposite effects that cancel each other when averaged. In this regime, the robustness of synchrony is strongly enhanced, and robust synchrony can be achieved at a high firing rate in inhibitory networks. On the other hand, in excitatory networks, synchrony remains limited in frequency due to the intrinsic instability of strong recurrent excitation.

The number of synaptic inputs and the synchrony of large, sparse neuronal networks

The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses, M, that a cell receives is larger than a critical value, Mc. Below Mc, the system is in an asynchronous state. In the limit of weak coupling, assuming identical neurons, we reduce the model to a system of phase oscillators that are coupled via an effective interaction, gamma. In this framework, we develop an approximate theory for sparse networks of identical neurons to estimate Mc analytically from the Fourier coefficients of gamma. Our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it. We apply this theory to compute Mc for a model of inhibitory networks of integrate-and-fire (I&F) neurons as a function of the intrinsic neuronal properties (e.g., the refractory period Tr), the synaptic time constants, and the strength of the external stimulus, Iext. The number Mc is found to be nonmonotonous with the strength of Iext. For Tr = 0, we estimate the minimum value of Mc over all the parameters of the model to be 363.8. Above Mc, the neurons tend to fire in smeared one-cluster states at high firing rates and smeared two-or-more-cluster states at low firing rates. Refractoriness decreases Mc at intermediate and high firing rates. These results are compared to numerical simulations. We show numerically that systems with different sizes, N, behave in the same way provided the connectivity, M, is such that 1/Meff = 1/M - 1/N remains constant when N varies. This allows extrapolating the large N behavior of a network from numerical simulations of networks of relatively small sizes (N = 800 in our case). We find that our theory predicts with remarkable accuracy the value of Mc and the patterns of synchrony above Mc, provided the synaptic coupling is not too large. We also study the strong coupling regime of inhibitory sparse networks. All of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity. Above a critical coupling strength, the network activity is asynchronous. We point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone, if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account.

Method for determining antiphase dynamics in a multimode laser

We measure the cross spectrum of the intensity fluctuations of pairs of modes for a multilongitudinal mode neodymium-doped yttrium aluminum garnet laser operating in the steady state regime. From the data we build up a picture of how the longitudinal mode fluctuations interfere and directly show the antiphase dynamics of the intensity fluctuations.

Population dynamics of spiking neurons: fast transients, asynchronous states, and locking

An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized"locked"solutions is analyzed.

Synchronized family dynamics in globally coupled maps

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many stable solutions, except for sufficiently large coupling strengths, where the lattice elements always synchronize, and for small map parameter, where only simple fixed points are found. For smaller varepsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of synchronized families. The dynamics and stability of two-family states (solutions partitioned into two families) are explored in detail. (c) 1999 American Institute of Physics.

Response of coupled noisy excitable systems to weak stimulation

It is known that coupling can enhance the response of noisy bistable devices to weak periodic modulation. This work examines whether a similar phenomenon occurs in the active rotator model for excitable systems. We study the dynamics of assemblies of weakly periodically modulated active rotators. The addition of noise to these brings about a number of behaviors that have no counterpart in networks of bistable systems. The analysis of the dynamics of the solution of the Fokker-Planck equation of active rotator networks shows that these new behaviors are similar to generic responses of periodically forced autonomous oscillators. This is because noise alone, in the absence of other inputs, can regularize the dynamics of single active rotators through coherence resonance, and lead to regular synchronous activity at the level of networks. We argue that similar phenomena take place in a broad class of excitable systems.

Physiology of MPTP tremor

Rhesus and vervet monkeys respond differently to treatment with 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine hydrochloride neurotoxin (MPTP). Both species develop akinesia, rigidity, and severe postural instability. However, rhesus monkeys only develop infrequent, short episodes of high-frequency tremor, whereas vervet monkeys have many prolonged episodes of low-frequency tremor. After MPTP treatment, the spiking activity of many pallidal neurons became oscillatory and highly correlated. Oscillatory autocorrelation functions were dominated by lower frequencies, cross-correlograms by higher frequencies. The phase shift distribution of the oscillatory cross-correlograms of pallidal cells in MPTP-treated vervet monkey were clustered around 0 phase shift, unlike the oscillatory correlograms in the MPTP-treated rhesus monkey, which were widely distributed between 0 degrees and 180 degrees. Analysis of the instantaneous phase differences between tremors of two limbs in the MPTP monkeys and human parkinsonian patients showed short periods of tremor synchronization. We thus concluded that the rhesus and the vervet models of MPTP-induced parkinsonism may represent the tremulous and nontremulous variants of human parkinsonism. We suggest that the tremor phenomena of Parkinson's disease (PD) are related to the emergence of synchronous neuronal oscillations in the basal ganglia. Finally, the oscillating neuronal assemblies in the pallidum of tremulous parkinsonian primates are more stable (in time and in space) than those of parkinsonian primates without overt tremor.

The role of axonal delay in the synchronization of networks of coupled cortical oscillators

Coupled oscillator models use a single phase variable to approximate the voltage oscillation of each neuron during repetitive firing where the behavior of the model depends on the connectivity and the interaction function chosen to describe the coupling. We introduce a network model consisting of a continuum of these oscillators that includes the effects of spatially decaying coupling and axonal delay. We derive equations for determining the stability of solutions and analyze the network behavior for two different interaction functions. The first is a sine function, and the second is derived from a compartmental model of a pyramidal cell. In both cases, the system of coupled neural oscillators can undergo a bifurcation from synchronous oscillations to waves. The change in qualitative behavior is due to the axonal delay, which causes distant connections to encourage a phase shift between cells. We suggest that this mechanism could contribute to the behavior observed in several neurobiological systems.

What matters in neuronal locking?

Exploiting local stability, we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessary and, in the limit of a large number of interacting neighbors, also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem, we present a simple geometric method to verify the existence and local stability of a coherent oscillation.