Low rattling: A predictive principle for self-organization in active collectives.

Self-organization is frequently observed in active collectives as varied as ant rafts and molecular motor assemblies. General principles describing self-organization away from equilibrium have been challenging to identify. We offer a unifying framework that models the behavior of complex systems as largely random while capturing their configuration-dependent response to external forcing. This allows derivation of a Boltzmann-like principle for understanding and manipulating driven self-organization. We validate our predictions experimentally, with the use of shape-changing robotic active matter, and outline a methodology for controlling collective behavior. Our findings highlight how emergent order depends sensitively on the matching between external patterns of forcing and internal dynamical response properties, pointing toward future approaches for the design and control of active particle mixtures and metamaterials.

Emergent, linked traits of fluctuation feedback systems.

A variety of nonequilibrium systems display intermittent switching between semistable macroscopic behaviors. We identify a certain type of indeterminacy, with episodes of patterned behavior irregularly punctuated by transitions. It appears that the long-lived patterns are, not coincidentally, also low-fluctuation states. We describe these linked traits with a small set of examples.

Dynamics of tinnitus and coordinated reset therapy.

A clinical study of tinnitus patients found promising results using a noninvasive therapy. We introduce a dynamical model to explore both the onset of tinnitus and the effects of coordinated reset therapy. Our model extends an existing theory of individual outer hair cell dynamics to include their mutual interaction, and considers how sustained activity can inhibit the natural recovery exhibited by normal (healthy) individuals. The model is investigated through numerical simulations and shows behavior broadly similar to that reported in the clinical study.

Dynamics of scattering in undulatory active collisions.

Natural and artificial self-propelled systems must manage environmental interactions during movement. In complex environments, these interactions include active collisions, in which propulsive forces create persistent contacts with heterogeneities. Due to the driven and dissipative nature of these systems, such collisions are fundamentally different from those typically studied in classical physics. Here we experimentally and numerically study the effects of active collisions on a laterally undulating sensory-deprived robophysical model, whose dynamics are relevant to self-propelled systems across length scales and environments. Interactions with a single rigid post scatter the robot, and this deflection is dominated by head-post contact. These results motivate a model which reduces the snake to a circular particle with two key features: The collision dynamics are set by internal driving subject to the geometric constraints of the post, and the particle has an effective length equal to the wavelength of the snake. Interactions with a single row of evenly spaced posts (with interpost spacing d) produce distributions reminiscent of far-field diffraction patterns: As d decreases, distinct secondary peaks emerge as large deflections become more likely. Surprisingly, we find that the presence of multiple posts does not change the nature of individual collisions; instead, multimodal scattering patterns arise from multiple posts altering the likelihood of individual collisions to occur. As d decreases, collisions near the leading edges of the posts become more probable, and we find that these interactions are associated with larger deflections. Our results, which highlight the surprising dynamics that can occur during active collisions of self-propelled systems, can inform control principles for locomotors in complex terrain and facilitate design of task-capable active matter.

A robot made of robots: Emergent transport and control of a smarticle ensemble.

Robot locomotion is typically generated by coordinated integration of single-purpose components, like actuators, sensors, body segments, and limbs. We posit that certain future robots could self-propel using systems in which a delineation of components and their interactions is not so clear, becoming robust and flexible entities composed of functional components that are redundant and generic and can interact stochastically. Control of such a collective becomes a challenge because synthesis techniques typically assume known input-output relationships. To discover principles by which such future robots can be built and controlled, we study a model robophysical system: planar ensembles of periodically deforming smart, active particles-smarticles. When enclosed, these individually immotile robots could collectively diffuse via stochastic mechanical interactions. We show experimentally and theoretically that directed drift of such a supersmarticle could be achieved via inactivation of individual smarticles and used this phenomenon to generate endogenous phototaxis. By numerically modeling the relationship between smarticle activity and transport, we elucidated the role of smarticle deactivation on supersmarticle dynamics from little data-a single experimental trial. From this mapping, we demonstrate that the supersmarticle could be exogenously steered anywhere in the plane, expanding supersmarticle capabilities while simultaneously enabling decentralized closed-loop control. We suggest that the smarticle model system may aid discovery of principles by which a class of future "stochastic" robots can rely on collective internal mechanical interactions to perform tasks.

Emergence of the advancing neuromechanical phase in a resistive force dominated medium.

Undulatory locomotion, a gait in which thrust is produced in the opposite direction of a traveling wave of body bending, is a common mode of propulsion used by animals in fluids, on land, and even within sand. As such, it has been an excellent system for discovery of neuromechanical principles of movement. In nearly all animals studied, the wave of muscle activation progresses faster than the wave of body bending, leading to an advancing phase of activation relative to the curvature toward the tail. This is referred to as "neuromechanical phase lags" (NPL). Several multiparameter neuromechanical models have reproduced this phenomenon, but due to model complexity, the origin of the NPL has proved difficult to identify. Here, we use perhaps the simplest model of undulatory swimming to predict the NPL accurately during sand-swimming by the sandfish lizard, with no fitting parameters. The sinusoidal wave used in sandfish locomotion, the friction-dominated and noninertial granular resistive force environment, and the simplicity of the model allow detailed analysis, and reveal the fundamental mechanism responsible for the phenomenon: the combination of synchronized torques from distant points on the body and local traveling torques. This general mechanism should help explain the NPL in organisms in other environments; we therefore propose that sand-swimming could be an excellent system with which to generate and test other neuromechanical models of movement quantitatively. Such a system can also provide guidance for the design and control of robotic undulatory locomotors in complex environments.

Critical cell wall hole size for lysis in Gram-positive bacteria.

Gram-positive bacteria can transport molecules necessary for their survival through holes in their cell wall. The holes in cell walls need to be large enough to let critical nutrients pass through. However, the cell wall must also function to prevent the bacteria's membrane from protruding through a large hole into the environment and lysing the cell. As such, we hypothesize that there exists a range of cell wall hole sizes that allow for molecule transport but prevent membrane protrusion. Here, we develop and analyse a biophysical theory of the response of a Gram-positive cell's membrane to the formation of a hole in the cell wall. We predict a critical hole size in the range of 15-24 nm beyond which lysis occurs. To test our theory, we measured hole sizes in Streptococcus pyogenes cells undergoing enzymatic lysis via transmission electron microscopy. The measured hole sizes are in strong agreement with our theoretical prediction. Together, the theory and experiments provide a means to quantify the mechanisms of death of Gram-positive cells via enzymatically mediated lysis and provides insights into the range of cell wall hole sizes compatible with bacterial homeostasis.

Phase noise of oscillators with unsaturated amplifiers.

We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operating point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback.

Lift-off dynamics in a simple jumping robot.

We study vertical jumping in a simple robot comprising an actuated mass-spring arrangement. The actuator frequency and phase are systematically varied to find optimal performance. Optimal jumps occur above and below (but not at) the robot's resonant frequency f(0). Two distinct jumping modes emerge: a simple jump, which is optimal above ff(0), is achievable with a squat maneuver, and a peculiar stutter jump, which is optimal below f(0), is generated with a countermovement. A simple dynamical model reveals how optimal lift-off results from nonresonant transient dynamics.

Effects of heterogeneity in synaptic conductance between weakly coupled identical neurons.

A significant degree of heterogeneity in synaptic conductance is present in neuron to neuron connections. We study the dynamics of weakly coupled pairs of neurons with heterogeneities in synaptic conductance using Wang-Buzsaki and Hodgkin-Huxley model neurons which have Types I and II excitability, respectively. This type of heterogeneity breaks a symmetry in the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant when compared to the identical case. For weakly coupled neurons coupled with identical values of synaptic conductance a phase locked solution exists for all values of the synaptic rate constant, α. In particular, in-phase and anti-phase solutions are guaranteed to exist for all α. Heterogeneity in synaptic conductance results in regions where no phase locked solution exists and the general loss of the ubiquitous in-phase and anti-phase solutions of the identically coupled case. We explain these results through examination of interaction functions using the weak coupling approximation and an in-depth analysis of the underlying multiple cusp bifurcation structure of the systems of coupled neurons.

Huygens (and others) revisited.

We develop a generic iterative map model of coupled oscillators based on simple physical processes common to many such systems. The model allows us to understand, from a unified perspective, the range of different outcomes reported for experiments by Huygens and modern realizations of his two coupled clocks.

Coherence between two coupled lasers from a dynamics perspective.

We compare a simple dynamical model of fiber laser arrays with independent experiments on two coupled lasers. The degree of agreement with experimental observations is excellent. Collectively the evidence presented supports this dynamical approach as an alternative to the traditional static eigenmode analysis of the coupled laser cavities.

Refined fiber laser model.

With modification, a recently proposed laser array model is found to agree quantitatively with fiber laser experiments. Comparisons of transient behavior, stable dynamical states, and transitions are made using both previously published and new experiments. While the original model agrees well for fibers with relatively low losses, achieving quantitative agreement over a wide range of operating conditions requires more physically appropriate descriptions of gain dynamics. The refined model is derived, and its predictions are found to be in excellent agreement with experiments.

Robust synchronization in fiber laser arrays.

Synchronization of coupled fiber lasers has been reported in recent experiments [Bruesselbach, Opt. Lett. 30, 1339 (2005); Minden, Proc. SPIE 5335, 89 (2004)]. While these results may lead to dramatic advances in laser technology, the mechanism by which these lasers synchronize is not understood. We analyze a recently proposed [Rogers, IEEE J. Quantum Electron. 41, 767 (2005)] iterated map model of fiber laser arrays to explore this phenomenon. In particular, we look at synchronous solutions of the maps when the gain fields are constant. Determining the stability of these solutions is analytically tractable for a number of different coupling schemes. We find that in the most symmetric physical configurations the most symmetric solution is either unstable or stable over insufficient parameter range to be practical. In contrast, a lower symmetry configuration yields surprisingly robust coherence. This coherence persists beyond the pumping threshold for which the gain fields become time dependent.

Weak-link synchronization.

We describe a mutual synchronization mechanism observed in a model of fiber laser arrays. Though suboptimal in terms of total coherent power, the weak-link-synchronized state is far more robust than the in-phase state of a uniformly pumped array, with respect to parameter mismatch among the individual elements. We find similar dynamical behavior in a more general system of coupled oscillators where the amplitude dynamics is crucial.

Potential energy landscape and finite-state models of array-enhanced stochastic resonance.

Noise and coupling can optimize the response of arrays of nonlinear elements to periodic signals. We analyze such array-enhanced stochastic resonance (AESR) using finite-state transition rate models. We simply derive the transition rate matrices from the underlying potential energy function of the corresponding Langevin problem. Our implementation exploits Floquet theory and provides useful theoretical and numerical tools. Our framework both facilitates analysis and elucidates the mechanism of AESR. In particular, we show how sublinear coupling diminishes AESR, but superlinear coupling enhances it.

Stochastic resonance in the mechanoelectrical transduction of hair cells.

In transducing mechanical stimuli into electrical signals, at least some hair cells in vertebrate auditory and vestibular systems respond optimally to weak periodic signals at natural, nonzero noise intensities. We understand this stochastic resonance by constructing a faithful mechanical model reflecting the hair cell geometry and described by a nonlinear stochastic differential equation. This Langevin description elucidates the mechanism of hair cell stochastic resonance while supporting the hypothesis that noise plays a functional role in hearing.

Synchronization law for a van der Pol array.

We explore the transition to in-phase synchronization in globally coupled oscillator arrays, and compare results for van der Pol arrays with Josephson junction arrays. Our approach yields in each case an analytically tractable iterative map; the resulting stability formulas are simple because the expansion procedure identifies natural parameter groups. A third example, an array of Duffing-van der Pol oscillators, is found to be of the same fundamental type as the van der Pol arrays, but the Josephson arrays are fundamentally different owing to the absence of self-resonant interactions.

Neural correlates of the complexity of rhythmic finger tapping.

Using functional magnetic resonance imaging (fMRI), we studied the neural correlates of the complexity of rhythmic finger tapping. Our experiments measured the brain activity of 13 subjects performing rhythmic tapping on a response box with multistable rhythms of 1 to 5 different interresponse intervals. From the button press response times, we constructed phase portraits where we identified the number of clusters of periodic points in a rhythm that corresponded to the number of different beats of the rhythm performed. We then constructed a statistical model for correlation analysis involving the following behavioral parameters: rate of tapping and number of beats in a rhythm. The tapping rate correlated with the brain activity in the ipsilateral pre/postcentral gyrus, and the number of beats (complexity) was correlated with activations in the primary motor cortex, supplementary motor area, basal ganglia, thalamus, and cerebellum. A region of interest (ROI) average analysis showed that the complexity of a rhythm had a differential correlation with the activity in these regions. The cerebellum and the thalamus showed increasing activity, and the basal ganglia showed decreasing activity with complexity of a rhythm. These results identify the areas involved in a rhythm generation and the modulation of brain activity with the complexity.

Measurements of brain activity complexity for varying mental loads.

Using functional magnetic resonance imaging, we investigate the variation in dynamical complexity of human brain activity for different mental loads. Our experiments measured the activity of ten subjects under three experimental conditions: a rest condition, a periodic task of finger opposition, and a task of finger opposition alternated with mathematical serial calculation. We used the correlation dimension to gauge the spatiotemporal complexity of brain activity. The experiments show a direct relationship between this complexity and the difficulty of the task. A natural interpretation is that higher levels of mental load recruit a larger number of independent neural processes that contribute to complex brain dynamics. These results suggest the possibility that the relative change in correlation dimension can be a useful global measure of brain dynamics, e.g., in determining the levels of mental activity, even if little is known about the underlying neurological processes.