Emergent global oscillations in heterogeneous excitable media: the example of pancreatic beta cells

Biological Rhythms Generated by a Single Activator-Repressor Loop with Inhomogeneity and Diffusion

Common models of circadian rhythms are typically constructed as compartmental reactions of well-mixed biochemicals, incorporating a negative-feedback loop consisting of several intermediate reaction steps essentially required to produce oscillations. Spatial transport of each reactant is often represented as an extra compartmental reaction step. Contrary to this traditional understanding, in this Letter we demonstrate that a single activation-repression biochemical reaction pair is sufficient to generate sustained oscillations if the sites of both reactions are spatially separated and molecular transport is mediated by diffusion. Our proposed scenario represents the simplest configuration in terms of the participating chemical reactions and offers a conceptual basis for understanding biological oscillations and inspiring in vitro assays aimed at constructing minimal clocks.

Diversity-enhanced stability

We give compelling evidence that diversity, represented by a quenched disorder, can produce a resonant collective transition between two unsteady states in a network of coupled oscillators. The stability of a metastable state is optimized and the mean first-passage time maximized at an intermediate value of diversity. This finding shows that a system can benefit from inherent heterogeneity by allowing it to maximize the transition time from one state to another at the appropriate degree of heterogeneity.

Determinants of collective failure in excitable networks

We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.

Diversity-induced decoherence

We analyze the effect of small-amplitude noise and heterogeneity in a network of coupled excitable oscillators with strong timescale separation. Using mean-field analysis, we uncover the mechanism of a nontrivial effect-diversity-induced decoherence (DIDC)-in which heterogeneity modulates the mechanism of self-induced stochastic resonance to inhibit the coherence of oscillations. We argue that DIDC may offer one possible mechanism via which, in excitable neural systems, generic heterogeneity and background noise can synergistically prevent unwanted resonances that may be related to hyperkinetic movement disorders.

Spontaneous generation of persistent activity in diffusively coupled cellular assemblies

The spontaneous generation of electrical activity underpins a number of essential physiological processes, and is observed even in tissues where specialized pacemaker cells have not been identified. The emergence of periodic oscillations in diffusively coupled assemblies of excitable and electrically passive cells (which are individually incapable of sustaining autonomous activity) has been suggested as a possible mechanism underlying such phenomena. In this paper we investigate the dynamics of such assemblies in more detail by considering simple motifs of coupled electrically active and passive cells. The resulting behavior encompasses a wide range of dynamical phenomena, including chaos. However, embedding such assemblies in a lattice yields spatiotemporal patterns that either correspond to a quiescent state or to partial or globally synchronized oscillations. The resulting reduction in dynamical complexity suggests an emergent simplicity in the collective dynamics of such large, spatially extended systems. Furthermore, we show that such patterns can be reproduced by a reduced model comprising only excitatory and oscillatory elements. Our results suggest a generalization of the mechanism by which periodic activity can emerge in a heterogeneous system comprising nonoscillatory elements by coupling them diffusively, provided their steady states in isolation are sufficiently dissimilar.

Hubs, diversity, and synchronization in FitzHugh-Nagumo oscillator networks: Resonance effects and biophysical implications

Using the FitzHugh-Nagumo equations to represent the oscillatory electrical behavior of β-cells, we develop a coupled oscillator network model with cubic lattice topology, showing that the emergence of pacemakers or hubs in the system can be viewed as a natural consequence of oscillator population diversity. The optimal hub to nonhub ratio is determined by the position of the diversity-induced resonance maximum for a given set of FitzHugh-Nagumo equation parameters and is predicted by the model to be in a range that is fully consistent with experimental observations. The model also suggests that hubs in a β-cell network should have the ability to "switch on" and "off" their pacemaker function. As a consequence, their relative amount in the population can vary in order to ensure an optimal oscillatory performance of the network in response to environmental changes, such as variations of an external stimulus.

Cooperative maintenance of cellular identity in systems with intercellular communication defects

The cooperative dynamics of cellular populations emerging from the underlying interactions determines cellular functions and thereby their identity in tissues. Global deviations from this dynamics, on the other hand, reflect pathological conditions. However, how these conditions are stabilized from dysregulation on the level of the single entities is still unclear. Here, we tackle this question using the generic Hodgkin-Huxley type of models that describe physiological bursting dynamics of pancreatic β-cells and introduce channel dysfunction to mimic pathological silent dynamics. The probability for pathological behavior in β-cell populations is ∼100% when all cells have these defects, despite the negligible size of the silent state basin of attraction for single cells. In stark contrast, in a more realistic scenario for a mixed population, stabilization of the pathological state depends on the size of the subpopulation which acquired the defects. However, the probability to exhibit stable pathological dynamics in this case is less than 10%. These results, therefore, suggest that the physiological bursting dynamics of a population of β-cells is cooperatively maintained, even under intercellular communication defects induced by dysfunctional channels of single cells.

Complex oscillation modes in the Belousov-Zhabotinsky reaction by weak diffusive coupling

We study the diffusive coupling of oscillating or excitable Belousov-Zhabotinsky reaction units arranged in a square lattice array and show that for certain sizes of the units and for certain distances between the units, complex oscillation modes of individual spots occur, which manifest themselves in multi-periodic, amplitude-modulated, and multi-mode oscillations. This experimental finding can be reproduced in simulations of the FitzHugh-Nagumo model mimicking the experimental setup, suggesting that it is a generic phenomenon in systems of coupled excitable units such as excitable cell tissues or coupled oscillators such as neurons. Further analysis let us conclude that the complex oscillation modes occur close to the transition from quiescent to coupling-induced oscillations states if this transition is taking place at weak coupling strength.

Autonomous cycling between excitatory and inhibitory coupling in photosensitive chemical oscillators

Photochemically coupled Belousov-Zhabotinsky micro-oscillators are studied in experiments and simulations. The photosensitive oscillators exhibit excitatory or inhibitory coupling depending on the surrounding reaction mixture composition, which can be systematically varied. In-phase or out-of-phase synchronization is observed with predominantly excitatory or inhibitory coupling, respectively, and complex frequency cycling between excitatory and inhibitory coupling is found between these extremes. The dynamical behavior is characterized in terms of the corresponding phase response curves, and a map representation of the dynamics is presented.

Coexistence of critical sensitivity and subcritical specificity can yield optimal population coding

The vicinity of phase transitions selectively amplifies weak stimuli, yielding optimal sensitivity to distinguish external input. Along with this enhanced sensitivity, enhanced levels of fluctuations at criticality reduce the specificity of the response. Given that the specificity of the response is largely compromised when the sensitivity is maximal, the overall benefit of criticality for signal processing remains questionable. Here, it is shown that this impasse can be solved by heterogeneous systems incorporating functional diversity, in which critical and subcritical components coexist. The subnetwork of critical elements has optimal sensitivity, and the subnetwork of subcritical elements has enhanced specificity. Combining segregated features extracted from the different subgroups, the resulting collective response can maximize the trade-off between sensitivity and specificity measured by the dynamic-range-to-noise ratio. Although numerous benefits can be observed when the entire system is critical, our results highlight that optimal performance is obtained when only a small subset of the system is at criticality.

Extreme events in excitable systems and mechanisms of their generation

We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems' dynamics that may qualify as precursors to extreme events. We investigate these features and elucidate mechanisms that may be responsible for the generation of the extreme events.

Crystal growth as an excitable medium

Crystal growth has been widely studied for many years, and, since the pioneering work of Burton, Cabrera and Frank, spirals and target patterns on the crystal surface have been understood as forms of tangential crystal growth mediated by defects and by two-dimensional nucleation. Similar spirals and target patterns are ubiquitous in physical systems describable as excitable media. Here, we demonstrate that this is not merely a superficial resemblance, that the physics of crystal growth can be set within the framework of an excitable medium, and that appreciating this correspondence may prove useful to both fields. Apart from solid crystals, we discuss how our model applies to the biomaterial nacre, formed by layer growth of a biological liquid crystal.

Mathematical models for insulin secretion in pancreatic β-cells

Insulin secretion is one of the most characteristic features of β-cell physiology. As it plays a central role in glucose regulation, a number of experimental and theoretical studies have been performed since the discovery of the pancreatic β-cell. This review article aims to give an overview of the mathematical approaches to insulin secretion. Beginning with the bursting electrical activity in pancreatic β-cells, we describe effects of the gap-junction coupling between β-cells on the dynamics of insulin secretion. Then, implications of paracrine interactions among such islet cells as α-, β-, and δ-cells are discussed. Finally, we present mathematical models which incorporate effects of glycolysis and mitochondrial glucose metabolism on the control of insulin secretion.

Self-sustaining oscillations in complex networks of excitable elements

Random networks of symmetrically coupled, excitable elements can self-organize into coherently oscillating states if the networks contain loops (indeed loops are abundant in random networks) and if the initial conditions are sufficiently random. In the oscillating state, signals propagate in a single direction and one or a few network loops are selected as driving loops in which the excitation periodically circulates. We analyze the mechanism, describe the oscillating states, identify the pacemaker loops, and explain key features of their distribution.

Synchronization and entrainment of coupled circadian oscillators

Circadian rhythms in mammals are controlled by the neurons located in the suprachiasmatic nucleus of the hypothalamus. In physiological conditions, the system of neurons is very efficiently entrained by the 24 h light-dark cycle. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light-dark cycle in neuronal synchronization. Nevertheless, heterogeneity as a natural and permanent ingredient of these cellular interactions seemingly plays a major role in these biochemical processes. In this paper, we use a model that considers the neurons of the suprachiasmatic nucleus as chemically coupled modified Goodwin oscillators, and introduce non-negligible heterogeneity in the periods of all neurons in the form of quenched noise. The system response to the light-dark cycle periodicity is studied as a function of the interneuronal coupling strength, external forcing amplitude and neuronal heterogeneity. Our results indicate that the right amount of heterogeneity helps the extended system to respond globally in a more coherent way to the external forcing. Our proposed mechanism for neuronal synchronization under external periodic forcing is based on heterogeneity-induced oscillator death, damped oscillators being more entrainable by the external forcing than the self-oscillating neurons with different periods.

Distant synchronization through a passive medium

This paper deals with the phenomenon of synchronization of oscillatory ensembles interacting distantly through the passive medium. Main characteristics of such a kind of synchronization are studied. The results of this work can be applied to describe the synchronization of cardiac oscillatory cells separated by the passive fibroblasts. In this work the phenomenological models (Bonhoeffer-Van der Pol) of cardiac cells as well as biologically relevant (Luo-Rudy, Sachse) models are used. We also propose equivalent model of distant synchronization and derive on its basis an analytical scaling of the frequency of synchronous oscillations.

Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing

Spontaneous spatiotemporal wave activity occurs in groups of excitable particles for groups larger than a critical size. Experiments are carried out with particles loaded with the catalyst of the Belousov-Zhabotinsky reaction that are immersed in catalyst-free reaction mixture. The particles diffusively exchange activator and inhibitor species with the surrounding solution. All particles are nonoscillatory when separated from the other particles; however, target and spiral waves are exhibited in sufficiently large groups. A cellular particle model of the system also exhibits transitions from excitable steady state behavior to spatiotemporal wave activity with increasing group size.

Self-sustained collective oscillation generated in an array of nonoscillatory cells

Oscillations are ubiquitous phenomena in biological systems. Conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops. Some specific proteins function as transcription factors, which in turn negatively regulate the expression of the genes that encode these "clock proteins." These loops may lead to rhythmic changes in gene expression in a cell. In the case of multicellular tissue, collective oscillation is often due to the synchronization of these cells, which manifest themselves as autonomous oscillators. In contrast, we propose here a different scenario for the occurrence of collective oscillation in a group of nonoscillatory cells. Neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in the present system. By adopting a spatially inhomogeneous active factor, we observe and analyze a coupling-induced oscillation, inherent to the phenomenon of wave propagation due to intracellular communication.

Influence of passive elements on the dynamics of oscillatory ensembles of cardiac cells

In this paper we focus on the influence of passive elements on the collective dynamics of oscillatory ensembles. Two major effects considered are (i) the influence of passive elements on the synchronization properties of ensembles of coupled nonidentical oscillators and (ii) the influence of passive elements on the wave dynamics of such systems. For the first effect, it is demonstrated that the introduction of passive elements may lead to both an increase or decrease in the global synchronization threshold. For the second effect, it is also demonstrated that the steady state of the passive element is a key parameter which defines how this passive element affects the wave dynamics of the oscillatory ensemble. It was shown that for different values of this parameter, one can observe increase or decrease in wave propagation velocity and increase or decrease in synchronization frequency in oscillatory ensembles with the growth of influence of passive elements. The results are obtained for the models of cardiac cells dynamics as well as for the Bonhoeffer-Van der Pol model and are compared with data of real biological experiments.

Robust synchrony and rhythmogenesis in endocrine neurons via autocrine regulations in vitro and in vivo

Episodic pulses of gonadotropin-releasing hormone (GnRH) are essential for maintaining reproductive functions in mammals. An explanation for the origin of this rhythm remains an ultimate goal for researchers in this field. Some plausible mechanisms have been proposed among which the autocrine-regulation mechanism has been implicated by numerous experiments. GnRH binding to its receptors in cultured GnRH neurons activates three types of G-proteins that selectively promote or inhibit GnRH secretion (Krsmanovic et al. in Proc. Natl. Acad. Sci. 100:2969-2974, 2003). This mechanism appears to be consistent with most data collected so far from both in vitro and in vivo experiments. Based on this mechanism, a mathematical model has been developed (Khadra and Li in Biophys. J. 91:74-83, 2006) in which GnRH in the extracellular space plays the roles of a feedback regulator and a synchronizing agent. In the present study, we show that synchrony between different neurons through sharing a common pool of GnRH is extremely robust. In a diversely heterogeneous population of neurons, the pulsatile rhythm is often maintained when only a small fraction of the neurons are active oscillators (AOs). These AOs are capable of recruiting nonoscillatory neurons into a group of recruited oscillators while forcing the nonrecruitable neurons to oscillate along. By pointing out the existence of the key elements of this model in vivo, we predict that the same mechanism revealed by experiments in vitro may also operate in vivo. This model provides one plausible explanation for the apparently controversial conclusions based on experiments on the effects of the ultra-short feedback loop of GnRH on its own release in vivo.

Mathematical modelling in neuroendocrinology

In neuroendocrinology, mathematical modelling is about formalising our understanding of the behaviour of the complex biological systems with which we deal. Formulating our explanations mathematically ensures their logical consistency, and makes them open to structured analysis; it is a stringent test of their intellectual coherence. In addition, however, modellers are seeking to extend our understanding in new ways, by seeking novel, simple explanations for complex behaviour. Here we discuss some styles of modelling as they have been applied to neuroendocrine systems, and discuss some of their strengths and limitations.

Synchronization and firing death in the dynamics of two interacting excitable units with heterogeneous signals

We study the response of two coupled FitzHugh-Nagumo systems to heterogeneous external inputs. The latter, modeled by periodic parametric stimuli, force the uncoupled excitable systems into a regime of chaotic firing. Due to parameter dispersion involved in randomly distributed amplitudes and/or phases of the external forces the units are nonidentical and their firing events will be asynchronous. Interest is focused on mutually synchronized spikings arising through the coupling. It is demonstrated that the phase difference of the two external forces crucially affects the onset of spike synchronization as well as the resulting degree of synchrony. For large phase differences the degree of spike synchrony is constricted to a maximal possible value and cannot be enhanced upon increasing the coupling strength. We even found that overcritically strong couplings lead to suppression of firing so that the units perform synchronous subthreshold oscillations. This effect, which we call "firing death," is due to a coupling-induced modification of the excitation threshold impeding spiking of the units. In clear contrast, when only the amplitudes of the forces are distributed perfect spike synchrony is achieved for sufficiently strong coupling.

Theory of collective firing induced by noise or diversity in excitable media

Large variety of physical, chemical, and biological systems show excitable behavior, characterized by a nonlinear response under external perturbations: only perturbations exceeding a threshold induce a full system response (firing). It has been reported that in coupled excitable identical systems noise may induce the simultaneous firing of a macroscopic fraction of units. However, a comprehensive understanding of the role of noise and that of natural diversity present in realistic systems is still lacking. Here we develop a theory for the emergence of collective firings in nonidentical excitable systems subject to noise. Three different dynamical regimes arise: subthreshold motion, where all elements remain confined near the fixed point; coherent pulsations, where a macroscopic fraction fire simultaneously; and incoherent pulsations, where units fire in a disordered fashion. We also show that the mechanism for collective firing is generic: it arises from degradation of entrainment originated either by noise or by diversity.

Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study nonclustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small worlds that includes declustered networks and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.

Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model

By employing a semianalytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by Langevin equations with delays. In an AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM (AMMm), which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N=1 is nicely compared to the exact result. In the nonlinear Langevin ensemble, the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semianalytical augmented moment method (AMM) which has been proposed in a preceding paper [H. Hasegawa, Phys. Rev. E 70, 021911 (2004)]. For N-unit FN neuron ensembles, AMM transforms original 2N-dimensional stochastic delay differential equations (SDDEs) to infinite-dimensional deterministic DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level m in the level-m AMM (AMMm), yielding 8(m+1)-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the fluctuation-induced enhancement at the transition between nonoscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.

Analysis of the noise-induced bursting-spiking transition in a pancreatic beta-cell model

A stochastic model of the electrophysiological behavior of the pancreatic beta cell is studied, as a paradigmatic example of a bursting biological cell embedded in a noisy environment. The analysis is focused on the distortion that a growing noise causes to the basic properties of the membrane potential signals, such as their periodic or chaotic nature, and their bursting or spiking behavior. We present effective computational tools to obtain as much information as possible from these signals, and we suggest that the methods could be applied to real time series. Finally, a universal dependence of the main characteristics of the membrane potential on the size of the considered cell cluster is presented.

Oscillations by symmetry breaking in homogeneous networks with electrical coupling

In many biological systems, the electrical coupling of nonoscillating cells generates synchronized membrane potential oscillations. This work describes a dynamical mechanism in which the electrical coupling of identical nonoscillating cells destabilizes the homogeneous fixed point and leads to network oscillations via a Hopf bifurcation. Each cell is described by a passive membrane potential and additional internal variables. The dynamics of the internal variables, in isolation, is oscillatory, but their interaction with the membrane potential damps the oscillations and therefore constructs nonoscillatory cells. The electrical coupling reveals the oscillatory nature of the internal variables and generates network oscillations. This mechanism is analyzed near the bifurcation point, where the spatial structure of the membrane potential oscillations is determined by the network architecture and in the limit of strong coupling, where the membrane potentials of all cells oscillate in-phase and multiple cluster states dominate the dynamics. In particular, we have derived an asymptotic behavior for the spatial fluctuations in the limit of strong coupling in fully connected networks and in a one-dimensional lattice architecture.

Triggering synchronized oscillations through arbitrarily weak diversity in close-to-threshold excitable media

It is shown that an arbitrarily weak (frozen) heterogeneity can induce global synchronized oscillations in excitable media close to threshold. The work is carried out on networks of coupled van der Pol-FitzHugh-Nagumo oscillators. The result is shown to be robust against the presence of internal dynamical noise.

Bursting as an emergent phenomenon in coupled chaotic maps

A two-dimensional map exhibiting chaotic bursting behavior similar to the bursting electrical activity observed in biological neurons and endocrine cells is examined. Model parameters are changed so that the bursting behavior is destroyed. We show that bursting can be recovered in a population of such nonbursting cells when they are coupled via the mean field. The phenomenon is explained with a geometric bifurcation analysis. The analysis reveals that emergent bursting in the network is due to coupling alone and is very robust to changes in the coupling strength, and that heterogeneity in the model parameters does not play a role.

The generation of oscillations in networks of electrically coupled cells

In several biological systems, the electrical coupling of nonoscillating cells generates synchronized membrane potential oscillations. Because the isolated cell is nonoscillating and electrical coupling tends to equalize the membrane potentials of the coupled cells, the mechanism underlying these oscillations is unclear. Here we present a dynamic mechanism by which the electrical coupling of identical nonoscillating cells can generate synchronous membrane potential oscillations. We demonstrate this mechanism by constructing a biologically feasible model of electrically coupled cells, characterized by an excitable membrane and calcium dynamics. We show that strong electrical coupling in this network generates multiple oscillatory states with different spatio-temporal patterns and discuss their possible role in the cooperative computations performed by the system.