Synchronization of active mechanical oscillators by an inertial load

Something in Our Ears Is Oscillating, but What? A Modeller's View of Efforts to Model Spontaneous Emissions

When David Kemp discovered "spontaneous ear noise" in 1978, it opened up a whole new perspective on how the cochlea works. The continuous tonal sound emerging from most healthy human ears, now called spontaneous otoacoustic emissions or SOAEs, was an unmistakable sign that our hearing organ must be considered an active detector, not just a passive microphone, just as Thomas Gold had speculated some 30 years earlier. Clearly, something is oscillating as a byproduct of that sensitive inbuilt detector, but what exactly is it? Here, we give a chronological account of efforts to model SOAEs as some form of oscillator, and at intervals, we illustrate key concepts with numerical simulations. We find that after many decades there is still no consensus, and the debate extends to whether the oscillator is local, confined to discrete local sources on the basilar membrane, or global, in which an assembly of micro-mechanical elements and basilar membrane sections, coupled by inner ear fluid, interact over a wide region. It is also undecided whether the cochlear oscillator is best described in terms of the well-known Van der Pol oscillator or the less familiar Duffing or Hopf oscillators. We find that irregularities play a key role in generating the emissions. This paper is not a systematic review of SOAEs and their properties but more a historical survey of the way in which various oscillator configurations have been applied to modelling human ears. The conclusion is that the difference between the local and global approaches is not clear-cut, and they are probably not mutually exclusive concepts. Nevertheless, when one sees how closely human SOAEs can be matched to certain arrangements of oscillators, Gold would no doubt say we are on the right track.

The self-oscillation paradox in the flight motor of Drosophila melanogaster

Tiny flying insects, such as Drosophila melanogaster, fly by flapping their wings at frequencies faster than their brains are able to process. To do so, they rely on self-oscillation: dynamic instability, leading to emergent oscillation, arising from muscle stretch-activation. Many questions concerning this vital natural instability remain open. Does flight motor self-oscillation necessarily lead to resonance-a state optimal in efficiency and/or performance? If so, what state? And is self-oscillation even guaranteed in a motor driven by stretch-activated muscle, or are there limiting conditions? In this work, we use data-driven models of wingbeat and muscle behaviour to answer these questions. Developing and leveraging novel analysis techniques, including symbolic computation, we establish a fundamental condition for motor self-oscillation common to a wide range of motor models. Remarkably, D. melanogaster flight apparently defies this condition: a paradox of motor operation. We explore potential resolutions to this paradox, and, within its confines, establish that the D. melanogaster flight motor is probably not resonant with respect to exoskeletal elasticity: instead, the muscular elasticity plays a dominant role. Contrary to common supposition, the stiffness of stretch-activated muscle is an obstacle to, rather than an enabler of, the operation of the D. melanogaster flight motor.

Sunlight-powered self-excited oscillators for sustainable autonomous soft robotics

As the field of soft robotics advances, full autonomy becomes highly sought after, especially if robot motion can be powered by environmental energy. This would present a self-sustained approach in terms of both energy supply and motion control. Now, autonomous movement can be realized by leveraging out-of-equilibrium oscillatory motion of stimuli-responsive polymers under a constant light source. It would be more advantageous if environmental energy could be scavenged to power robots. However, generating oscillation becomes challenging under the limited power density of available environmental energy sources. Here, we developed fully autonomous soft robots with self-sustainability based on self-excited oscillation. Aided by modeling, we have successfully reduced the required input power density to around one-Sun level through a liquid crystal elastomer (LCE)-based bilayer structure. The autonomous motion of the low-intensity LCE/elastomer bilayer oscillator "LiLBot" under low energy supply was achieved by high photothermal conversion, low modulus, and high material responsiveness simultaneously. The LiLBot features tunable peak-to-peak amplitudes from 4 to 72 degrees and frequencies from 0.3 to 11 hertz. The oscillation approach offers a strategy for designing autonomous, untethered, and sustainable small-scale soft robots, such as a sailboat, walker, roller, and synchronized flapping wings.

Model simulation of the SPOC wave in a bundle of striated myofibrils

SPOC (spontaneous oscillatory contraction) is a phenomenon observed in striated muscle under intermediate activation conditions. Recently, we constructed a theoretical model of SPOC for a sarcomere, a unit sarcomere model, which explains the behavior of SPOC at each sarcomere level. We also constructed a single myofibril model, which visco-elastically connects the unit model in series, and explains the behaviors of SPOC at the myofibril level. In the present study, to understand the SPOC properties in a bundle of myofibrils, we extended the single myofibril model to a two-dimensional (2D) model and a three-dimensional (3D) model, in which myofibrils were elastically connected side-by-side through cross-linkers between the Z-lines and M-lines. These 2D and 3D myofibril models could reproduce various patterns of SPOC waves experimentally observed in a 2D sheet and a 3D bundle of myofibrils only by choosing different values of elastic constants of the cross-linkers and the external spring. The results of these 2D and 3D myofibril models provide insight into the SPOC properties of the higher-ordered assembly of myofibrils.

Elastic interactions synchronize beating in cardiomyocytes

Motivated by recent experimental results, we study theoretically the synchronization of the beating phase and frequency of two nearby cardiomyocyte cells. Each cell is represented as an oscillating force dipole in an infinite, viscoelastic medium and the propagation of the elastic signal within the medium is predicted. We examine the steady-state beating of two nearby cells, and show that elastic interactions result in forces that synchronize the phase and frequency of beating in a manner that depends on their mutual orientation. The theory predicts both in-phase and anti-phase steady-state beating depending on the relative cell orientations, as well as how synchronized beating varies with substrate elasticity and the inter-cell distance. These results suggest how mechanics plays a role in cardiac efficiency, and may be relevant for the design of cardiomyocyte based micro devices and other biomedical applications.

From Quasiperiodic Partial Synchronization to Collective Chaos in Populations of Inhibitory Neurons with Delay

Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model's dynamics with fast neuronal oscillations is discussed.

Spontaneous oscillatory contraction (SPOC) in cardiomyocytes

SPOC (spontaneous oscillatory contraction) is a characteristic state of the contractile system of striated (skeletal and cardiac) muscle that exists between the states of relaxation and contraction. For example, Ca-SPOCs occur at physiological Ca2+ levels (pCa ∼6.0), whereas ADP-SPOC occurs in the virtual absence of Ca2+ (pCa ≥ 8; relaxing conditions in the presence of MgATP), but in the presence of inorganic phosphate (Pi) and a high concentration of MgADP. The concentration of Mg-ADP necessary for SPOC is nearly equal to or greater than the MgATP concentration for cardiac muscle and is several times higher for skeletal muscle. Thus, the cellular conditions for SPOC are broader in cardiac muscle than in skeletal muscle. During these SPOCs, each sarcomere in a myofibril undergoes length oscillation that has a saw-tooth waveform consisting of a rapid lengthening and a slow shortening phase. The lengthening phase of one half of a sarcomere is transmitted to the adjacent half of the sarcomere successively, forming a propagating wave (termed a SPOC wave). The SPOC waves are synchronized across the cardiomyocytes resulting in a visible wave of successive contractions and relaxations termed the SPOC wave. Experimentally, the SPOC period (and therefore the velocity of SPOC wave) is observed in demembranated cardiomyocytes and can be prepared from a wide range of animal hearts. These periods correlate well with the resting heartbeats of a wide range of mammals (rat, rabbit, dog, pig and cow). Preliminary experiments showed that the SPOC properties of human cardiomyocytes are similar to the heartbeat of a large dog or a pig. This correlation suggests that SPOCs may play a fundamental role in the heart. Here, we briefly summarize a range of SPOC parameters obtained experimentally, and relate them to a theoretical model to explain those characteristics. Finally, we discuss the possible significance of these SPOC properties in each and every heartbeat.

Spontaneous oscillations of elastic contractile materials with turnover

Single and collective cellular oscillations driven by the actomyosin cytoskeleton have been observed in numerous biological systems. Here, we propose that these oscillations can be accounted for by a generic oscillator model of a material turning over and contracting against an elastic element. As an example, we show that during dorsal closure of the Drosophila embryo, experimentally observed changes in actomyosin concentration and oscillatory cell shape changes can, indeed, be captured by the dynamic equations studied here. We also investigate the collective dynamics of an ensemble of such contractile elements and show that the relative contribution of viscous and friction losses yields different regimes of collective oscillations. Taking into account the diffusion of force-producing molecules between contractile elements, our theoretical framework predicts the appearance of traveling waves, resembling the propagation of actomyosin waves observed during morphogenesis.

Microscopic mechanism for self-organized quasiperiodicity in random networks of nonlinear oscillators

Self-organized quasiperiodicity is one of the most puzzling dynamical phases observed in systems of nonlinear coupled oscillators. The single dynamical units are not locked to the periodic mean field they produce, but they still feature a coherent behavior, through an unexplained complex form of correlation. We consider a class of leaky integrate-and-fire oscillators on random sparse and massive networks with dynamical synapses, featuring self-organized quasiperiodicity, and we show how complex collective oscillations arise from constructive interference of microscopic dynamics. In particular, we find a simple quantitative relationship between two relevant microscopic dynamical time scales and the macroscopic time scale of the global signal. We show that the proposed relation is a general property of collective oscillations, common to all the partially synchronous dynamical phases analyzed. We argue that an analogous mechanism could be at the origin of similar network dynamics.

Power-stroke-driven actomyosin contractility

In ratchet-based models describing actomyosin contraction the activity is usually associated with actin binding potential while the power-stroke mechanism, residing inside myosin heads, is viewed as passive. To show that contraction can be propelled directly through a conformational change, we propose an alternative model where the power stroke is the only active mechanism. The asymmetry, ensuring directional motion, resides in steric interaction between the externally driven power-stroke element and the passive nonpolar actin filament. The proposed model can reproduce all four discrete states of the minimal actomyosin catalytic cycle even though it is formulated in terms of continuous Langevin dynamics. We build a conceptual bridge between processive and nonprocessive molecular motors by demonstrating that not only the former but also the latter can use structural transformation as the main driving force.

Locally and globally coupled oscillators in muscle

At an intermediate activation level, striated muscle exhibits autonomous oscillations called SPOC, in which the basic contractile units, sarcomeres, oscillate in length, and various oscillatory patterns such as traveling waves and their disrupted forms appear in a myofibril. Here we show that these patterns are reproduced by mechanically connecting in series the unit model that explains characteristics of SPOC at the single-sarcomere level. We further reduce the connected model to phase equations, revealing that the combination of local and global couplings is crucial to the emergence of these patterns.

Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: an experimental study

We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not.

Experiments on oscillator ensembles with global nonlinear coupling

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.

SPontaneous Oscillatory Contraction (SPOC): auto-oscillations observed in striated muscle at partial activation

Striated muscle is well known to exist in either of two states-contraction or relaxation-under the regulation of Ca2+ concentration. Described here is a less well-known third, intermediate state induced under conditions of partial activation, known as SPOC (SPontaneous Oscillatory Contraction). This state is characterised by auto-oscillation between rapid-lengthening and slow-shortening phases. Notably, SPOC occurs in skinned muscle fibres and is therefore not the result of fluctuating Ca2+ levels, but is rather an intrinsic and fundamental phenomenon of the actomyosin motor. Summarised in this review are the experimental data on SPOC and its fundamental mechanism. SPOC presents a novel technique for studying independent communication and coordination between sarcomeres. In cardiac muscle, this auto-oscillatory property may work in concert with electro-chemical signalling to coordinate the heartbeat. Further, SPOC may represent a new way of demonstrating functional defects of sarcomeres in human heart failure.

Effect of periodic parametric excitation on an ensemble of force-coupled self-oscillators

We report the synchronization behavior in an ensemble of identical limit cycle oscillators coupled to a mass-spring load via a force relation. We consider the effect of periodic parametric modulation on the final states of the system. Two types of external parametric excitations are investigated numerically: periodic modulation of the stiffness of the inertial oscillator and periodic excitation of the frequency of the self-oscillatory element. We show that the synchronization scenarios are not only ruled by the choice of parameters of the excitation force but also depend on the initial collective state in the ensemble. We give detailed analysis of entrainment behavior for initially synchronous or antiphase synchronous states as well as for disorganized states.

Coordination and collective properties of molecular motors: theory

Many cellular processes require molecular motors to produce motion and forces. Single molecule experiments have led to a precise description of how a motor works. Under most physiological conditions, however, molecular motors operate in groups. Interactions between motors yield collective behaviors that cannot be explained only from single molecule properties. The aim of this paper is to review the various theoretical descriptions that explain the emergence of collective effects in molecular motor assemblies. These include bidirectional motion, hysteretic behavior, spontaneous oscillations, and self-organization into dynamical structures. We discuss motors acting on the cytoskeleton both in a prescribed geometry such as in muscles or flagella and in the cytoplasm.

Invertebrate muscles: thin and thick filament structure; molecular basis of contraction and its regulation, catch and asynchronous muscle

This is the second in a series of canonical reviews on invertebrate muscle. We cover here thin and thick filament structure, the molecular basis of force generation and its regulation, and two special properties of some invertebrate muscle, catch and asynchronous muscle. Invertebrate thin filaments resemble vertebrate thin filaments, although helix structure and tropomyosin arrangement show small differences. Invertebrate thick filaments, alternatively, are very different from vertebrate striated thick filaments and show great variation within invertebrates. Part of this diversity stems from variation in paramyosin content, which is greatly increased in very large diameter invertebrate thick filaments. Other of it arises from relatively small changes in filament backbone structure, which results in filaments with grossly similar myosin head placements (rotating crowns of heads every 14.5 nm) but large changes in detail (distances between heads in azimuthal registration varying from three to thousands of crowns). The lever arm basis of force generation is common to both vertebrates and invertebrates, and in some invertebrates this process is understood on the near atomic level. Invertebrate actomyosin is both thin (tropomyosin:troponin) and thick (primarily via direct Ca(++) binding to myosin) filament regulated, and most invertebrate muscles are dually regulated. These mechanisms are well understood on the molecular level, but the behavioral utility of dual regulation is less so. The phosphorylation state of the thick filament associated giant protein, twitchin, has been recently shown to be the molecular basis of catch. The molecular basis of the stretch activation underlying asynchronous muscle activity, however, remains unresolved.

Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling

We describe a transition from fully synchronous periodic oscillations to partially synchronous quasiperiodic dynamics in ensembles of identical oscillators with all-to-all coupling that nonlinearly depends on the generalized order parameters. We present an analytically solvable model that predicts a regime where the mean field does not entrain individual oscillators, but has a frequency incommensurate to theirs. The self-organized onset of quasiperiodicity is illustrated with Landau-Stuart oscillators and a Josephson junction array with a nonlinear coupling.

Oscillations in molecular motor assemblies

Autonomous oscillations in biological systems may have a biochemical origin or result from an interplay between force-generating and visco-elastic elements. In molecular motor assemblies the force-generating elements are molecular engines and the visco-elastic elements are stiff cytoskeletal polymers. The physical mechanism leading to oscillations depends on the particular architecture of the assembly. Existing models can be grouped into two distinct categories: systems with a delayed force activation and anomalous force-velocity relations. We discuss these systems within phase plane analysis known from the theory of dynamic systems and by adopting methods from control theory, the Nyquist stability criterion.

Local nanomechanical motion of the cell wall of Saccharomyces cerevisiae

We demonstrate that the cell wall of living Saccharomyces cerevisiae (baker's yeast) exhibits local temperature-dependent nanomechanical motion at characteristic frequencies. The periodic motions in the range of 0.8 to 1.6 kHz with amplitudes of approximately 3 nm were measured using the cantilever of an atomic force microscope (AFM). Exposure of the cells to a metabolic inhibitor causes the periodic motion to cease. From the strong frequency dependence on temperature, we derive an activation energy of 58 kJ/mol, which is consistent with the cell's metabolism involving molecular motors such as kinesin, dynein, and myosin. The magnitude of the forces observed ( approximately 10 nN) suggests concerted nanomechanical activity is operative in the cell.