Multifrequency forcing of a Hopf oscillator model of the inner ear

In response to a sound stimulus, the inner ear emits sounds called otoacoustic emissions. While the exact mechanism for the production of otoacoustic emissions is not known, active motion of individual hair cells is thought to play a role. Two possible sources for otoacoustic emissions, both localized within individual hair cells, include somatic motility and hair bundle motility. Because physiological models of each of these systems are thought to be poised near a Hopf bifurcation, the dynamics of each can be described by the normal form for a system near a Hopf bifurcation. Here we demonstrate that experimental results from three-frequency suppression experiments can be predicted based on the response of an array of noninteracting Hopf oscillators tuned at different frequencies. This supports the idea that active motion of individual hair cells contributes to active processing of sounds in the ear. Interestingly, the model suggests an explanation for differing results recorded in mammals and nonmammals.     

Two coupled neural oscillators as a model of the circadian pacemaker

Pittendrigh first found that the circadian rhythm of locomotor activity in nocturnal rodents split into two components. Hoffman then reported that the splitting phenomenon was even more reproducible in the small diurnal primate Tupaia. These “splitting” experiments and many other experiments suggest that two coupled oscillators may constitute the circadian pacemaker system. Pittendrigh proposed a phenomenological two-oscillator model. Daan and Berde developed a quantitative model assuming that the interaction between the two constituent oscillators is by instantaneous resets. Their model system can simulate several qualitative features in the experimental data. As the assumption of instantaneous resets seems to be unnatural, we study two limit cycle oscillators, which are coupled continuously to each other, as a model of the circadian pacemaker. We assume the following points, (i) One oscillator in a resting state does not affect another oscillator, (ii) Two oscillators are identical, (iii) The coupling is symmetrical. By the theory of Hopf bifurcation it is found that the general two-oscillator system has two stable periodic solutions. One is the in-phase solution where the two constituent oscillators oscillate in phase synchrony. Another is the anti-phase solution where the two oscillators oscillate 180 ° out of phase. The former corresponds to a single pattern of locomotor activity and the latter corresponds to a splitting pattern. Furthermore, we study specific two-neural oscillators, which are linearly coupled to each other. By the method of secondary bifurcation we find that the model shows simultaneous stability of the two alternative phase relationships and the hysteresis phenomena found in Tupaia. A natural period of the uncoupled constituent oscillator is longer than that of the in-phase solution but it is shorter than that of the anti-phase solution. This is in agreement with the data of Tupaia.     

Mechanical Amplification Exhibited by Quiescent Saccular Hair Bundles

Spontaneous oscillations exhibited by free-standing hair bundles from the Bullfrog sacculus suggest the existence of an active process that might underlie the exquisite sensitivity of the sacculus to mechanical stimulation. However, this spontaneous activity is suppressed by coupling to an overlying membrane, which applies a large mechanical load on the bundle. How a quiescent hair bundle utilizes its active process is still unknown. We studied the dynamics of motion of individual hair bundles under different offsets in the bundle position, and observed the occurrence of spikes in hair-bundle motion, associated with the generation of active work. These mechanical spikes can be evoked by a sinusoidal stimulus, leading to an amplified movement of the bundle with respect to the passive response. Amplitude gain reached as high as 100-fold at small stimulus amplitudes. Amplification of motion decreased with increasing amplitude of stimulation, ceasing at ∼6–12 pN stimuli. Results from numerical simulations suggest that the adaptation process, mediated by myosin 1c, is not required for the production of mechanical spikes.     

A computational fluid dynamics model of viscous coupling of hairs

Arrays of arthropod filiform hairs form highly sensitive mechanoreceptor systems capable of detecting minute air disturbances, and it is unclear to what extent individual hairs interact with one another within sensor arrays. We present a computational fluid dynamics model for one or more hairs, coupled to a rigid-body dynamics model, for simulating both biological (e.g., a cricket cercal hair) and artificial MEMS-based systems. The model is used to investigate hair–hair interaction between pairs of hairs and quantify the extent of so-called viscous coupling. The results show that the extent to which hairs are coupled depends on the mounting properties of the hairs and the frequency at which they are driven. In particular, it is shown that for equal length hairs, viscous coupling is suppressed when they are driven near the natural frequency of the undamped system and the damping coefficient at the base is small. Further, for certain configurations, the motion of a hair can be enhanced by the presence of nearby hairs. The usefulness of the model in designing artificial systems is discussed.     

Dendritic and Synaptic Effects in Systems of Coupled Cortical Oscillators

We explore the influence of synaptic location and form on the behavior of networks of coupled cortical oscillators. First, we develop a model of two coupled somatic oscillators that includes passive dendritic cables. Using a phase model approach, we show that the synchronous solution can change from a stable solution to an unstable one as the cable lengthens and the synaptic position moves further from the soma. We confirm this prediction using a system of coupled compartmental models. We also demonstrate that when the synchronous solution becomes unstable, a bifurcation occurs and a pair of asynchronous stable solutions appear, causing a phase lag between the cells in the system. Then using a variety of coupling functions and different synaptic positions, we show that distal connections and broad synaptic time courses encourage phase lags that can be reduced, eliminated, or enhanced by the presence of active currents in the dendrite. This mechanism may appear in neural systems where proximal connections could be used to encourage synchrony, and distal connections and broad synaptic time courses could be used to produce phase lags that can be modulated by active currents.     

Coupled Van der Pol oscillators — A model of excitatory and inhibitory neural interactions

A system of mutually coupled Van der Pol equations is derived from an extended version of the Wilson and Cowan model for the dynamics of a number of excitatory and inhibitory neural subsets. In the lowest order of approximation, interactions between excitatory and inhibitory subsets appear as linear elastic coupling, while those within and between excitatory and excitatory subsets appear as nonlinear frictional coupling. The case of two coupled oscillators is investigated by the method of averaging and the stability conditions for two mode oscillations are obtained. Internal resonance is also discussed briefly in the case of identical oscillators.     

Synchronization of Spontaneous Active Motility of Hair Cell Bundles

Hair cells of the inner ear exhibit an active process, believed to be crucial for achieving the sensitivity of auditory and vestibular detection. One of the manifestations of the active process is the occurrence of spontaneous hair bundle oscillations in vitro. Hair bundles are coupled by overlying membranes in vivo; hence, explaining the potential role of innate bundle motility in the generation of otoacoustic emissions requires an understanding of the effects of coupling on the active bundle dynamics. We used microbeads to connect small groups of hair cell bundles, using in vitro preparations that maintain their innate oscillations. Our experiments demonstrate robust synchronization of spontaneous oscillations, with either 1:1 or multi-mode phase-locking. The frequency of synchronized oscillation was found to be near the mean of the innate frequencies of individual bundles. Coupling also led to an improved regularity of entrained oscillations, demonstrated by an increase in the quality factor.     

Neural networks for computing eigenvalues and eigenvectors

We consider two electrically coupled oscillators described by modified Fitzhugh-Nagumo equations. We study the relative influence of the individual cellular characteristics and the electrical coupling on the behavior of the coupled system. We show that, for similar oscillators, the load effect of the slow oscillator increases with the coupling strength. We prove that an asymmetry between the uncoupled bursters can accelerate the system with respect to the free cells, this effect depending on the characteristics of the coupling.On leave from Centre de Physique Théoruique (UPR A0014 CNRS), Palaiseau, France     

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 repetitivefiring where the behavior of the model depends on the connectivityand the interaction function chosen to describe the coupling. Weintroduce a network model consisting of a continuum of theseoscillators 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.     

Coupling and elastic loading affect the active response by the inner ear hair cell bundles

Active hair bundle motility has been proposed to underlie the amplification mechanism in the auditory endorgans of non-mammals and in the vestibular systems of all vertebrates, and to constitute a crucial component of cochlear amplification in mammals. We used semi-intact in vitro preparations of the bullfrog sacculus to study the effects of elastic mechanical loading on both natively coupled and freely oscillating hair bundles. For the latter, we attached glass fibers of different stiffness to the stereocilia and observed the induced changes in the spontaneous bundle movement. When driven with sinusoidal deflections, hair bundles displayed phase-locked response indicative of an Arnold Tongue, with the frequency selectivity highest at low amplitudes and decreasing under stronger stimulation. A striking broadening of the mode-locked response was seen with increasing stiffness of the load, until approximate impedance matching, where the phase-locked response remained flat over the physiological range of frequencies. When the otolithic membrane was left intact atop the preparation, the natural loading of the bundles likewise decreased their frequency selectivity with respect to that observed in freely oscillating bundles. To probe for signatures of the active process under natural loading and coupling conditions, we applied transient mechanical stimuli to the otolithic membrane. Following the pulses, the underlying bundles displayed active movement in the opposite direction, analogous to the twitches observed in individual cells. Tracking features in the otolithic membrane indicated that it moved in phase with the bundles. Hence, synchronous active motility evoked in the system of coupled hair bundles by external input is sufficient to displace large overlying structures.     

Noise-Induced Coherence in Multicellular Circadian Clocks

In higher organisms, circadian rhythms are generated by a multicellular genetic clock that is entrained very efficiently to the 24-h light-dark cycle. Most studies done so far of these circadian oscillators have considered a perfectly periodic driving by light, in the form of either a square wave or a sinusoidal modulation. However, in natural conditions, organisms are subject to nonnegligible fluctuations in the light level all through the daily cycle. In this article, we investigate how the interplay between light fluctuations and intercellular coupling affects the dynamics of the collective rhythm in a large ensemble of nonidentical, globally coupled cellular clocks modeled as Goodwin oscillators. On the basis of experimental considerations, we assume an inverse dependence of the cell-cell coupling strength on the light intensity, in such a way that the larger the light intensity, the weaker the coupling. Our results show a noise-induced rhythm generation for constant light intensities at which the clock is arrhythmic in the noise-free case. Importantly, the rhythm shows a resonancelike phenomenon as a function of the noise intensity. Such improved coherence can be only observed at the level of the overt rhythm and not at the level of the individual oscillators, thus suggesting a cooperative effect of noise, coupling, and the emerging synchronization between the oscillators.     

Modeling the active process of the cochlea: phase relations, amplification, and spontaneous oscillation.

The high sensitivity and sharp frequency selectivity of acoustical signal transduction in the cochlea suggest that an active process pumps energy into the basilar membrane's oscillations. This function is generally attributed to outer hair cells, but its exact mechanism remains uncertain. Several classical models of amplification represent the load upon the basilar membrane as a single mass. Such models encounter a fundamental difficulty, however: the phase difference between basilar-membrane movement and the force generated by outer hair cells inhibits, rather than amplifies, the modeled basilar-membrane oscillations. For this reason, modelers must introduce artificially either negative impedance or an appropriate phase shift, neither of which is justified by physical analysis of the system. We consider here a physical model based upon the recent demonstration that the basilar membrane and reticular lamina can move independently, albeit with elastic coupling through outer hair cells. The mechanical model comprises two resonant masses, representing the basilar membrane and the reticular lamina, coupled through an intermediate spring, the outer hair cells. The spring's set point changes in response to displacement of the reticular lamina, which causes deflection of the hair bundles, variation of outer hair cell length and, hence, force production. Depending upon the frequency of the acoustical input, the basilar membrane and reticular lamina can oscillate either in phase or in counterphase. In the latter instance, the force produced by hair cells leads basilar-membrane oscillation, energy is pumped into basilar-membrane movement, and an external input can be strongly amplified. The model is also capable of producing spontaneous oscillation. In agreement with experimental observations, the model describes mechanical relaxation of the basilar membrane after electrical stimulation causes outer hair cells to change their length.     

Diffusively coupled bursters: Effects of cell heterogeneity

The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity. An erratum to this article is available at .     

Nonlinear dynamics and bifurcations of a coupled oscillator model for calling behavior of Japanese tree frogs (Hyla japonica)

Synchronization has been observed in various systems, including living beings. In a previous study, we reported a new phenomenon with antisynchronization in calling behavior of two interacting Japanese tree frogs. In this paper, we theoretically analyse nonlinear dynamics in a system of three coupled oscillators, which models three interacting frogs, where the oscillators of each pair have the property of antisynchronization; in particular, we perform bifurcation analysis and Lyapunov function analysis.     

Nonlinear dynamics and bifurcations of a coupled oscillator model for calling behavior of Japanese tree frogs (Hyla japonica)

Synchronization has been observed in various systems, including living beings. In a previous study, we reported a new phenomenon with antisynchronization in calling behavior of two interacting Japanese tree frogs. In this paper, we theoretically analyse nonlinear dynamics in a system of three coupled oscillators, which models three interacting frogs, where the oscillators of each pair have the property of antisynchronization; in particular, we perform bifurcation analysis and Lyapunov function analysis.     

Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junctions in the nervous system, a model system consisting of two nearly identical neurons electrotonically coupled is studied. We assume that each neuron discharges a train of impulses or bursts either spontaneously or under constant stimulus via chemical synapses. It is known that not only an electric current but also chemical substances whose molecular weight is about 1000 can pass through the junction of an electrical synapse (gap junction). So, our model system is regarded as a set of non-linear oscillators coupled by diffusion, and it may be described by a system of ordinary differential equations. Neurons are excited constantly when they are stimulated by an electric current above the threshold level. Therefore, we expect Hopf bifurcation to occur at the critical magnitude of a stimulating electric current in the system of differential equations which describes the dynamics of a single neuron. Studying our model system according to the theory of Hopf bifurcation, we found regions of diffusion constants of the electrical junction which give two kinds of periodic solutions. One is the solution where two neurons oscillate in phase synchrony. The other is where two neurons oscillate 180° out of phase. In the case where one neuron is described by the BVP model, the following was found by computer simulation. When the initial difference between the phase of two neurons is small, the two neurons come to oscillate synchronously. If the initial difference is large, however, the two come to be excited alternately. The physiological implications of these results are discussed.Department of Behaviorology, Faculty of Human Sciences     

Gβ Regulates Coupling between Actin Oscillators for Cell Polarity and Directional Migration

For directional movement, eukaryotic cells depend on the proper organization of their actin cytoskeleton. This engine of motility is made up of highly dynamic nonequilibrium actin structures such as flashes, oscillations, and traveling waves. In Dictyostelium, oscillatory actin foci interact with signals such as Ras and phosphatidylinositol 3,4,5-trisphosphate (PIP₃) to form protrusions. However, how signaling cues tame actin dynamics to produce a pseudopod and guide cellular motility is a critical open question in eukaryotic chemotaxis. Here, we demonstrate that the strength of coupling between individual actin oscillators controls cell polarization and directional movement. We implement an inducible sequestration system to inactivate the heterotrimeric G protein subunit Gβ and find that this acute perturbation triggers persistent, high-amplitude cortical oscillations of F-actin. Actin oscillators that are normally weakly coupled to one another in wild-type cells become strongly synchronized following acute inactivation of Gβ. This global coupling impairs sensing of internal cues during spontaneous polarization and sensing of external cues during directional motility. A simple mathematical model of coupled actin oscillators reveals the importance of appropriate coupling strength for chemotaxis: moderate coupling can increase sensitivity to noisy inputs. Taken together, our data suggest that Gβ regulates the strength of coupling between actin oscillators for efficient polarity and directional migration. As these observations are only possible following acute inhibition of Gβ and are masked by slow compensation in genetic knockouts, our work also shows that acute loss-of-function approaches can complement and extend the reach of classical genetics in Dictyostelium and likely other systems as well.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Smoking epidemic eradication in a eco-epidemiological dynamical model

Smoking is perceived as a major epidemic with regard to mortality. Modelling is a major tool used to obtain insight in the dynamics and possible solutions to decrease or even eradicate this epidemic. Most models on smoking consider the epidemiological context explicitly, in which smoking is regarded as an ‘infectious disease’, in which individuals ‘infect’ each other. However, the population dynamics are often ignored, while these occur at roughly the same timescale as smoking, and hence should explicitly be considered in the modelling of smoking. We present a simple but dynamical eco-epidemiological model. The model formulation consists of a resource-population dynamic part coupled to an epidemiological part resembling a SIR type model for the three compartments: non-smokers, smokers and ex-smokers. The coupling is via birth of non-smokers and death of the three classes with different death rates. The final four-dimensional system of ordinary differential equations are studied using brute force simulations for the short term dynamics and bifurcation analysis for the long-term dynamics. Due to a feed-back mechanism of the two coupling terms there is a codim-two tangent-transcritical bifurcation. This leads to bi-stability of one smoker endemic interior equilibrium and a smoker free boundary equilibrium. Changing parameters beyond the emerging tangent bifurcation leads on the short term to eradicating smoking. We consider the Netherlands in this paper for parametrization, but the modelling approach may be generally applicable.