Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli

The response of neurons to external stimuli greatly depends on the intrinsic dynamics of the network. Here, the intrinsic dynamics are modeled as coupling and the external input is modeled as shared and unshared noise. We assume the neurons are repetitively firing action potentials (i.e., neural oscillators), are weakly and identically coupled, and the external noise is weak. Shared noise can induce bistability between the synchronous and anti-phase states even though the anti-phase state is the only stable state in the absence of noise. We study the Fokker-Planck equation of the system and perform an asymptotic reduction ρ ₀. The ρ ₀ solution is more computationally efficient than both the Monte Carlo simulations and the 2D Fokker-Planck solver, and agrees remarkably well with the full system with weak noise and weak coupling. With moderate noise and coupling, ρ ₀ is still qualitatively correct despite the small noise and coupling assumption in the asymptotic reduction. Our phase model accurately predicts the behavior of a realistic synaptically coupled Morris-Lecar system.

Transitory behaviors in diffusively coupled nonlinear oscillators

We study collective behaviors of diffusively coupled oscillators which exhibit out-of-phase synchrony for the case of weakly interacting two oscillators. In large populations of such oscillators interacting via one-dimensionally nearest neighbor couplings, there appear various collective behaviors depending on the coupling strength, regardless of the number of oscillators. Among others, we focus on an intermittent behavior consisting of the all-synchronized state, a weakly chaotic state and some sorts of metachronal waves. Here, a metachronal wave means a wave with orderly phase shifts of oscillations. Such phase shifts are produced by the dephasing interaction which produces the out-of-phase synchronized states in two coupled oscillators. We also show that the abovementioned intermittent behavior can be interpreted as in-out intermittency where two saddles on an invariant subspace, the all-synchronized state and one of the metachronal waves play an important role.     

Entrainment boundaries of relaxation oscillators coupled by low pass filters

Abstract

A simple electronic network employing unijunction transistors has been used to obtain qualitative information regarding the entrainment behavior of relaxation oscillators coupled by low pass filters. Using arbitrary criteria, entrainment boundaries have been determined over the frequency range from 0.14 to 0.5 Hz for filters having time constants of 0.55,1.1 and 1.7 s. It is shown that the efficacy of entrainment is related to filter time constant and the Fourier structure of the bilaterally coupled signals     

Analysis of clustered firing patterns in synaptically coupled networks of oscillators

Oscillators in networks may display a variety of activity patterns. This paper presents a geometric singular perturbation analysis of clustering, or alternate firing of synchronized subgroups, among synaptically coupled oscillators. We consider oscillators in two types of networks: mutually coupled, with all-to-all inhibitory connections, and globally inhibitory, with one excitatory and one inhibitory population of oscillators, each of arbitrary size. Our analysis yields existence and stability conditions for clustered states, along with formulas for the periods of such firing patterns. By using two different approaches, we derive complementary conditions, the first set stated in terms of time lengths determined by intrinsic and synaptic properties of the oscillators and their coupling and the second set stated in terms of model parameters and phase space structures directly linked to parameters. These results suggest how biological components may interact to produce the spindle sleep rhythm in thalamocortical networks. Received: 9 September 1999 / Revised version: 7 July 2000 / Published online: 24 November 2000     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Phase resetting and coupling of noisy neural oscillators

A number of experimental groups have recently computed Phase Response Curves (PRCs) for neurons. There is a great deal of noise in the data. We apply methods from stochastic nonlinear dynamics to coupled noisy phase-resetting maps and obtain the invariant density of phase distributions. By exploiting the special structure of PRCs, we obtain some approximations for the invariant distributions. Comparisons to Monte-Carlo simulations are made. We show how phase-dependence of the noise can move the peak of the invariant density away from the peak expected from the analysis of the deterministic system and thus lead to noise-induced bifurcations. B. Ermentrout supported in part by NIMH and NSF. Action Editor: Wulfram Gerstner     

Power-rate synchronization of coupled genetic oscillators with unbounded time-varying delay

In this paper, a new synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay is investigated. The dynamical system under consideration consists of an array of linearly coupled identical genetic oscillators with each oscillators having unbounded time-delays. A new concept called power-rate synchronization, which is different from both the asymptotical synchronization and the exponential synchronization, is put forward to facilitate handling the unbounded time-varying delays. By using a combination of the Lyapunov functional method, matrix inequality techniques and properties of Kronecker product, we derive several sufficient conditions that ensure the coupled genetic oscillators to be power-rate synchronized. The criteria obtained in this paper are in the form of matrix inequalities. Illustrative example is presented to show the effectiveness of the obtained results.     

Synchronization of non-linear biochemical oscillators coupled by diffusion

The behaviour of similar coupled non-linear oscillators of the type \(\dot x\) =f(x, y, µ \(\dot y\) =g(x, y, µ is to be investigated. The oscillators are assumed to be coupled by diffusion gradients. If some conditions on the magnitude of the diffusion coefficients are satisfied, it is proved that: 1) if the oscillators have the same period (identical value of the parameter μ) and different phases before coupling, after coupling they tend to synchronize the phases; 2) if the periods of the oscillators are not too different (in terms of the values of the parameter μ) before coupling, after coupling they tend to oscillate with the same period. It is suggested the possible role of diffusion as a synchronizing mechanism in some biological phenomena.     

Intercellular synchronization of diffusively coupled Ca2+ oscillators

We examine the synchrony in the dynamics of localized [Ca^2 +]_i oscillations among a group of cells exhibiting such complex Ca^2 + oscillations, connected in the form of long chain, via diffusing coupling where cytosolic Ca^2 + and inositol 1,4,5-triphosphate are coupling molecules. Based on our numerical results, we could able to identify three regimes, namely desynchronized, transition and synchronized regimes in the (T − k_e) (time period-coupling constant) and (A − k_e) (amplitude-coupling constant) spaces which are supported by phase plots (Δϕ verses time) and recurrence plots, respectively. We further show the increase of synchronization among the cells as the number of coupling molecules increases in the (T − k_e) and (A − k_e) spaces.     

Systems and synthetic biology approaches in understanding biological oscillators

Background: Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators. Results: Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations. Conclusion: With the recent development of biological and computational tools, both approaches have made significant achievements.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Synchronization in a pool of mutually coupled oscillators with random frequencies

An exact solution to a model of mutually interacting sinusoidal oscillators is found. Limits on the variation of the native frequencies are determined in order for synchronization to occur. These limits are computed for different distributions of native frequencies.This research was supported by NSF Award No. MCS8300885 and the Alfred Sloan Foundation.     

Molecular dynamics simulations of carbon nanotube oscillators deformed by encapsulated copper nanowires

Pure carbon nanotube (CNT) oscillators are compared to the corresponding CNT oscillators encapsulating copper nanowires (Cu@CNTs) by molecular dynamics simulations. The classical oscillation theory provides a fairly good estimate of the mass dependence of the operating frequency when the CNT surface is not deformed by the Cu nanowire. The structural deformations of the CNT induced by the encapsulated copper nanowire have a greater effect on the oscillation frequency than the mass of the copper nanowire. The excess forces of the Cu@CNT oscillator are slightly higher than those of the CNT oscillator and the excess van der Waals forces induced by the inter-wall interactions are 17 times higher than the excess forces induced by the Cu nanowire–CNT interactions.     

Multiple oscillators in the suprachiasmatic nucleus

The suprachiasmatic nucleus (SCN) of the hypothalamus is the site of the pacemaker that controls circadian rhythms of a variety of physiological functions. Data strongly indicate the majority of the SCN neurons express self-sustaining oscillations that can be detected as rhythms in the spontaneous firing of individual neurons. The period of single SCN neurons in a dissociated cell culture is dispersed in a wide range (from 20h to 28h in rats), but that of the locomotor rhythm is close to 24h, suggesting individual oscillators are coupled to generate an averaged circadian period in the nucleus. Electrical coupling via gap junctions, glial regulation, calcium spikes, ephaptic interactions, extracellular ion flux, and diffusible substances have been discussed as possible mechanisms that mediate the interneuronal rhythm synchrony. Recently, GABA (γ-aminobutyric acid), a major neurotransmitter in the SCN, was reported to regulate cellular communication and to synchronize rhythms through GABA_A receptors. At present, subsequent intracellular processes that are able to reset the genetic loop of oscillations are unknown. There may be diverse mechanisms for integrating the multiple circadian oscillators in the SCN. This article reviews the knowledge about the various circadian oscillations intrinsic to the SCN, with particular focus on the intercellular signaling of coupled oscillators. (Chronobiology International, 18(3), 371-387, 2001)     

Dynamics of coupled stomatal oscillators

Stomata are microscopic openings in the leaves of green plants which permit gas exchange. Stomata exhibit oscillatory opening and closing behavior under certain environmental conditions in addition to a daily (diurnal) cycle. In order to explore the effects of coupling between neighboring stomata we present a mathematical model of the dynamics of a system of N coupled stomatal oscillators. An individual stomate is modeled to either remain closed, oscillate periodically, or remain open, depending on the local water potential. Coupling between neighboring stomata is accomplished in the model by taking into account the flow of water in the leaf as well as by oscillator phase coupling.Analysis of the model shows that under certain conditions it exhibits a stable spatially uniform synchronized behavior, referred to here as the in-phase mode. It is also shown that under non-uniform illumination the system may behave in a more complicated fashion.     

The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division

Intercellular communication of molecules between normal cells by tight junctions, and lack of this in some cancer cells (Loewenstein), can explain contact inhibition of cellular division in tissues. A general theory has been based on assuming the continual rise and fall (intrinsic oscillation) of a key substance x in each cell, with the period of the cell cycle. Periods are asynchronous in different cells, and x is exchanged between cells in contact by diffusion. A reduction in the resultant amplitude of fluctuation of x results, so that it does not reach the threshold x_t required for division to ensue; hence contact inhibition.The mathematical model is defined in its simplest form, and the sets of differential equations for arrays of cells are solved, from the isolated cell to the cell in an infinite sheet. The relative probability of division, P, is computed by numerical analysis from the area of resultant curves of x that lies above the threshold x_t. P depends on four dimensionless parameters, the order of coupling n (the number of cells directly communicating with a given cell), the total number of cells N in the aggregate, the communication constant K, and x_t, as a fraction of the amplitude of the intrinsic oscillation. The degree of synchrony, measured by the coefficient of variation σ of the periods, is important. If σ < ± 4%, contact inhibition is much reduced. The theory predicts that a paradoxical “contact-facilitation” is possible for very small aggregates of cells. For a cell in an infinite sheet, the amplitude of oscillation of x is reduced approximately by the factor $\text{1}\text{nK}$. For normal cells K is probably > 1, for cancer cells that lack communication, K is probably «< 1. However, two other basic causes for lack of regulation of tissue growth (cancer) could be excessive intrinsic oscillation of x, cf. x_t, and partial or complete synchronization of groups of cells by some unknown mechanism.     

Coupled oscillators coordinate collective germline growth

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Quantum-like model of behavioral response computation using neural oscillators

In this paper we propose the use of neural interference as the origin of quantum-like effects in the brain. We do so by using a neural oscillator model consistent with neurophysiological data. The model used was shown elsewhere to reproduce well the predictions of behavioral stimulus-response theory. The quantum-like effects are brought about by the spreading activation of incompatible oscillators, leading to an interference-like effect mediated by inhibitory and excitatory synapses.