Noise and critical behavior of the pupil light reflex at oscillation onset

Dynamics of a colloidal particle driven by continuous time-delayed feedback

We perform feedback experiments and simulations in which a colloidal dumbbell particle, acting as a particle on a ring, is followed by a repulsive optical trap controlled by a continuous-time-delayed feedback protocol. The dynamics are described by a persistent random walk similarly to that of an active Brownian particle, with a transition from predominantly diffusive to driven behavior at a critical delay time. We model the dynamics in the short and long delay regimes using stochastic delay differential equations and derive a condition for stable driven motion. We study the stochastic thermodynamic properties of the system, finding that the maximum work done by the trap coincides with a local minimum in the mutual information between the trap and the particle position at the onset of stable driven dynamics.

Complexity Collapse, Fluctuating Synchrony, and Transient Chaos in Neural Networks With Delay Clusters

Neural circuits operate with delays over a range of time scales, from a few milliseconds in recurrent local circuitry to tens of milliseconds or more for communication between populations. Modeling usually incorporates single fixed delays, meant to represent the mean conduction delay between neurons making up the circuit. We explore conditions under which the inclusion of more delays in a high-dimensional chaotic neural network leads to a reduction in dynamical complexity, a phenomenon recently described as multi-delay complexity collapse (CC) in delay-differential equations with one to three variables. We consider a recurrent local network of 80% excitatory and 20% inhibitory rate model neurons with 10% connection probability. An increase in the width of the distribution of local delays, even to unrealistically large values, does not cause CC, nor does adding more local delays. Interestingly, multiple small local delays can cause CC provided there is a moderate global delayed inhibitory feedback and random initial conditions. CC then occurs through the settling of transient chaos onto a limit cycle. In this regime, there is a form of noise-induced order in which the mean activity variance decreases as the noise increases and disrupts the synchrony. Another novel form of CC is seen where global delayed feedback causes “dropouts,” i.e., epochs of low firing rate network synchrony. Their alternation with epochs of higher firing rate asynchrony closely follows Poisson statistics. Such dropouts are promoted by larger global feedback strength and delay. Finally, periodic driving of the chaotic regime with global feedback can cause CC; the extinction of chaos can outlast the forcing, sometimes permanently. Our results suggest a wealth of phenomena that remain to be discovered in networks with clusters of delays.

Brain rhythm bursts are enhanced by multiplicative noise

Many healthy and pathological brain rhythms, including beta and gamma rhythms and essential tremor, are suspected to be induced by noise. This yields randomly occurring, brief epochs of higher amplitude oscillatory activity known as "bursts," the statistics of which are important for proper neural function. Here, we consider a more realistic model with both multiplicative and additive noise instead of only additive noise, to understand how state-dependent fluctuations further affect rhythm induction. For illustrative purposes, we calibrate the model at the lower end of the beta band that relates to movement; parameter tuning can extend the relevance of our analysis to the higher frequency gamma band or to lower frequency essential tremors. A stochastic Wilson-Cowan model for reciprocally as well as self-coupled excitatory (E) and inhibitory (I) populations is analyzed in the parameter regime where the noise-free dynamics spiral in to a fixed point. Noisy oscillations known as quasi-cycles are then generated by stochastic synaptic inputs. The corresponding dynamics of E and I local field potentials can be studied using linear stochastic differential equations subject to both additive and multiplicative noises. As the prevalence of bursts is proportional to the slow envelope of the E and I firing activities, we perform an envelope-phase decomposition using the stochastic averaging method. The resulting envelope dynamics are uni-directionally coupled to the phase dynamics as in the case of additive noise alone but both dynamics involve new noise-dependent terms. We derive the stationary probability and compute power spectral densities of envelope fluctuations. We find that multiplicative noise can enhance network synchronization by reducing the magnitude of the negative real part of the complex conjugate eigenvalues. Higher noise can lead to a "virtual limit cycle," where the deterministically stable eigenvalues around the fixed point acquire a positive real part, making the system act more like a noisy limit cycle rather than a quasi-cycle. Multiplicative noise can thus exacerbate synchronization and possibly contribute to the onset of symptoms in certain motor diseases.

Control of dynamics via identical time-lagged stochastic inputs

We investigate the impact of a stochastic forcing, comprised of a sum of time-lagged copies of a single source of noise, on the system dynamics. This type of stochastic forcing could be made artificially, or it could be the result of shared upstream inputs to a system through different channel lengths. By means of a rigorous mathematical framework, we show that such a system is, in fact, equivalent to the classical case of a stochastically-driven dynamical system with time-delayed intrinsic dynamics but without a time lag in the input noise. We also observe a resonancelike effect between the intrinsic period of the oscillation and the time lag of the stochastic forcing, which may be used to determine the intrinsic period of oscillations or the inherent time delay in dynamical systems with oscillatory behavior or delays. As another useful application of imposing time-lagged stochastic forcing, we show that the dynamics of a system can be controlled by changing the time lag of this stochastic forcing, in a fashion similar to the classical case of Pyragas control via delayed feedback. To confirm these results experimentally, we set up a laser diode system with such stochastic inputs, which effectively behaves as a Langevin system. As in the theory, a peak emerged in the autocorrelation function of the output signal that could be tuned by the lag of the stochastic input. Our findings, thus, indicate a new approach for controlling useful instabilities in dynamical systems.

Virtual stick balancing: sensorimotor uncertainties related to angular displacement and velocity

Sensory uncertainties and imperfections in motor control play important roles in neural control and Bayesian approaches to neural encoding. However, it is difficult to estimate these uncertainties experimentally. Here, we show that magnitude of the uncertainties during the generation of motor control force can be measured for a virtual stick balancing task by varying the feedback delay, τ. It is shown that the shortest stick length that human subjects are able to balance is proportional to τ  2. The proportionality constant can be related to a combined effect of the sensory uncertainties and the error in the realization of the control force, based on a delayed proportional-derivative (PD) feedback model of the balancing task. The neural reaction delay of the human subjects was measured by standard reaction time tests and by visual blank-out tests. Experimental observations provide an estimate for the upper boundary of the average sensorimotor uncertainty associated either with angular position or with angular velocity. Comparison of balancing trials with 27 human subjects to the delayed PD model suggests that the average uncertainty in the control force associated purely with the angular position is at most 14% while that associated purely with the angular velocity is at most 40%. In the general case when both uncertainties are present, the calculations suggest that the allowed uncertainty in angular velocity will always be greater than that in angular position.

Heat flow due to time-delayed feedback

Many stochastic systems in biology, physics and technology involve discrete time delays in the underlying equations of motion, stemming, e. g., from finite signal transmission times, or a time lag between signal detection and adaption of an apparatus. From a mathematical perspective, delayed systems represent a special class of non-Markovian processes with delta-peaked memory kernels. It is well established that delays can induce intriguing behaviour, such as spontaneous oscillations, or resonance phenomena resulting from the interplay between delay and noise. However, the thermodynamics of delayed stochastic systems is still widely unexplored. This is especially true for continuous systems governed by nonlinear forces, which are omnipresent in realistic situations. We here present an analytical approach for the net steady-state heat rate in classical overdamped systems subject to time-delayed feedback. We show that the feedback inevitably leads to a finite heat flow even for vanishingly small delay times, and detect the nontrivial interplay of noise and delay as the underlying reason. To illustrate this point, and to provide an understanding of the heat flow at small delay times below the velocity-relaxation timescale, we compare with the case of underdamped motion where the phenomenon of "entropy pumping" has already been established. Application to an exemplary (overdamped) bistable system reveals that the feedback induces heating as well as cooling regimes and leads to a maximum of the medium entropy production at coherence resonance conditions. These observations are, in principle, measurable in experiments involving colloidal suspensions.

The primacy of rhythm: how discrete actions merge into a stable rhythmic pattern

This study examined how humans spontaneously merge a sequence of discrete actions into a rhythmic pattern, even when periodicity is not required. Two experiments used a virtual throwing task, in which subjects performed a long sequence of discrete throwing movements, aiming to hit a virtual target. In experiment 1, subjects performed the task for 11 sessions. Although there was no instruction to perform rhythmically, the variability of the interthrow intervals decreased to a level comparable to that of synchronizing with a metronome; furthermore, dwell times shortened or even disappeared with practice. Floquet multipliers and decreasing variability of the arm trajectories estimated in state space indicated an increasing degree of dynamic stability. Subjects who achieved a higher level of periodicity and stability also displayed higher accuracy in the throwing task. To directly test whether rhythmicity affected performance, experiment 2 disrupted the evolving continuity and periodicity by enforcing a pause between successive throws. This discrete group performed significantly worse and with higher variability in their arm trajectories than the self-paced group. These findings are discussed in the context of previous neuroimaging results showing that rhythmic movements involve significantly fewer cortical and subcortical activations than discrete movements and therefore may pose a computationally more parsimonious solution. Such emerging stable rhythms in neuromotor subsystems may serve as building blocks or dynamic primitives for complex actions. The tendency for humans to spontaneously fall into a rhythm in voluntary movements is consistent with the ubiquity of rhythms at all levels of the physiological system. NEW & NOTEWORTHY When performing a series of throws to hit a target, humans spontaneously merged successive actions into a continuous approximately periodic pattern. The degree of rhythmicity and stability correlated with hitting accuracy. Enforcing irregular pauses between throws to disrupt the rhythm deteriorated performance. Stable rhythmic patterns may simplify control of movement and serve as dynamic primitives for more complex actions. This observation reveals that biological systems tend to exhibit rhythmic behavior consistent with a plethora of physiological processes.

Quantization improves stabilization of dynamical systems with delayed feedback

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possible when the fixed point in the absence of feedback is an unstable node, and in the presence of feedback, it is an unstable focus (spiral). The results are illustrated with numerical simulation of the unstable Hayes equation. The solutions of the quantized Hayes equation take the form of oscillations in which the amplitude is a function of the size of the quantization step. If the quantization step is sufficiently small, the amplitude of the oscillations can be small enough to practically approximate the dynamics around a stable fixed point.

A self-avoiding walk with neural delays as a model of fixational eye movements

Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.

Force-linearization closure for non-Markovian Langevin systems with time delay

This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various n-time joint distribution functions. Here, we present an approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Considering two generic nonlinear systems, a colloidal particle in a sinusoidal or bistable potential supplemented by a linear delay force, we demonstrate that our approach yields a very accurate representation of the density as compared to quasiexact numerical results from direct solution of the Langevin equation. Moreover, the results are significantly improved against those from a small-delay approximation and a perturbation-theoretical approach. We also discuss the possibility of accessing transport-related quantities, such as escape times, based on an additional Kramers approximation. Our approach applies to a wide class of models with nonlinear deterministic forces.

Identification of Bifurcations from Observations of Noisy Biological Oscillators

Hair bundles are biological oscillators that actively transduce mechanical stimuli into electrical signals in the auditory, vestibular, and lateral-line systems of vertebrates. A bundle’s function can be explained in part by its operation near a particular type of bifurcation, a qualitative change in behavior. By operating near different varieties of bifurcation, the bundle responds best to disparate classes of stimuli. We show how to determine the identity of and proximity to distinct bifurcations despite the presence of substantial environmental noise. Using an improved mechanical-load clamp to coerce a hair bundle to traverse different bifurcations, we find that a bundle operates within at least two functional regimes. When coupled to a high-stiffness load, a bundle functions near a supercritical Hopf bifurcation, in which case it responds best to sinusoidal stimuli such as those detected by an auditory organ. When the load stiffness is low, a bundle instead resides close to a subcritical Hopf bifurcation and achieves a graded frequency response—a continuous change in the rate, but not the amplitude, of spiking in response to changes in the offset force—a behavior that is useful in a vestibular organ. The mechanical load in vivo might therefore control a hair bundle’s responsiveness for effective operation in a particular receptor organ. Our results provide direct experimental evidence for the existence of distinct bifurcations associated with a noisy biological oscillator, and demonstrate a general strategy for bifurcation analysis based on observations of any noisy system.

Using pupil size and heart rate to infer affective states during behavioral neurophysiology and neuropsychology experiments

BACKGROUND

Nonhuman primates (NHPs) are a valuable research model because of their behavioral, physiological and neuroanatomical similarities to humans. In the absence of language, autonomic activity can provide crucial information about cognitive and affective states during single-unit recording, inactivation and lesion studies. Methods standardized for use in humans are not easily adapted to NHPs and detailed guidance has been lacking.

NEW METHOD

We provide guidance for monitoring heart rate and pupil size in the behavioral neurophysiology setting by addressing the methodological issues, pitfalls and solutions for NHP studies. The methods are based on comparative physiology to establish a rationale for each solution. We include examples from both electrophysiological and lesion studies.

RESULTS

Single-unit recording, pupil responses and heart rate changes represent a range of decreasing temporal resolution, a characteristic that impacts experimental design and analysis. We demonstrate the unexpected result that autonomic measures acquired before and after amygdala lesions are comparable despite disruption of normal autonomic function.

COMPARISON WITH EXISTING METHODS

Species and study design differences can render standard techniques used in human studies inappropriate for NHP studies. We show how to manage data from small groups typical of NHP studies, data from the short behavioral trials typical of neurophysiological studies, issues associated with longitudinal studies, and differences in anatomy and physiology.

CONCLUSIONS

Autonomic measurement to infer cognitive and affective states in NHP is neither off-the-shelf nor onerous. Familiarity with the issues and solutions will broaden the use of autonomic signals in NHP single unit and lesion studies.

Incomplete phase-space method to reveal time delay from scalar time series

A computationally quick and conceptually simple method to recover time delay of the chaotic system from scalar time series is developed in this paper. We show that the orbits in the incomplete two-dimensional reconstructed phase-space will show local clustering phenomenon after the component reordering procedure proposed in this work. We find that information captured by the incomplete two-dimensional reconstructed phase-space is related to the time delay τ_{0} present in the system, and will be transferred to the reordered component by the procedure of component reordering. We then propose the segmented mean variance (SMV) from the reordered component to identify the time delay τ_{0} of the system. The proposed SMV shows clear maximum when the embedding delay τ of the incomplete reconstruction matches the time delay τ_{0} of the chaotic system. Numerical data generated by a time-delay system based on the Mackey-Glass equation operating in the chaotic regime are used to illustrate the effectiveness of the proposed SMV. Experimental results show that the proposed SMV is robust to additive observational noise and is able to recover the time delay of the chaotic system even though the amount of data is relatively small and the feedback strength is weak. Moreover, the time complexity of the proposed method is quite low.

Perturbations of linear delay differential equations at the verge of instability

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

Nonlinear analysis of pupillary dynamics

Pupil size reflects autonomic response to different environmental and behavioral stimuli, and its dynamics have been linked to other autonomic correlates such as cardiac and respiratory rhythms. The aim of this study is to assess the nonlinear characteristics of pupil size of 25 normal subjects who participated in a psychophysiological experimental protocol with four experimental conditions, namely “baseline”, “anger”, “joy”, and “sadness”. Nonlinear measures, such as sample entropy, correlation dimension, and largest Lyapunov exponent, were computed on reconstructed signals of spontaneous fluctuations of pupil dilation. Nonparametric statistical tests were performed on surrogate data to verify that the nonlinear measures are an intrinsic characteristic of the signals. We then developed and applied a piecewise linear regression model to detrended fluctuation analysis (DFA). Two joinpoints and three scaling intervals were identified: slope α0, at slow time scales, represents a persistent nonstationary long-range correlation, whereas α1 and α2, at middle and fast time scales, respectively, represent long-range power-law correlations, similarly to DFA applied to heart rate variability signals. Of the computed complexity measures, α0 showed statistically significant differences among experimental conditions (p<0.001). Our results suggest that (a) pupil size at constant light condition is characterized by nonlinear dynamics, (b) three well-defined and distinct long-memory processes exist at different time scales, and (c) autonomic stimulation is partially reflected in nonlinear dynamics.

Exact dynamics of stochastic linear delayed systems: application to spatiotemporal coordination of comoving agents

Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.

Linear noise approximation for oscillations in a stochastic inhibitory network with delay

Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm.

Stochastic description of delayed systems

We study general stochastic birth and death processes including delay. We develop several approaches for the analytical treatment of these non-Markovian systems, valid, not only for constant delays, but also for stochastic delays with arbitrary probability distributions. The interplay between stochasticity and delay and, in particular, the effects of delay in the fluctuations and time correlations are discussed.

Time delay implies cost on task switching: a model to investigate the efficiency of task partitioning

Task allocation, and task switching have an important effect on the efficiency of distributed, locally controlled systems such as social insect colonies. Both efficiency and workload distribution are global features of the system which are not directly accessible to workers and can only be sampled locally by an individual in a distributed system. To investigate how the cost of task switching affects global performance we use social wasp societies as a metaphor to construct a simple model system with four interconnected tasks. Our goal is not the accurate description of the behavior of a given species, but to seek general conclusions on the effect of noise and time delay on a behavior that is partitioned into subtasks. In our model a nest structure needs to be constructed by the cooperation of individuals that carry out different tasks: builders, pulp and water foragers, and individuals storing water. We report a simulation study based on a model using delay-differential equations to analyze the trade-off between task switching costs and keeping a high degree of adaptivity in a dynamic, noisy environment. Combining the methods of time-delayed equations and stochastic processes we are able to represent the influence of swarm size and task switching sensitivity. We find that the system is stable for reasonable choices of parameters but shows oscillations for extreme choices of parameters and we find that the system is resilient to perturbations. We identify a trade-off between reaching equilibria of high performance and having short transients.

Reduced dynamics for delayed systems with harmonic or stochastic forcing

The analysis of nonlinear delay-differential equations (DDEs) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimension. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDEs. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes.

ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

TIME SERIES ANALYSIS OF MULTIVARIATE TIME-DELAYED SYSTEMS WITH NOISE: APPLICATIONS TO LASER PHYSICS AND HUMAN MOVEMENT

A data analysis method is proposed to reconstruct evolution equations of stochastic systems with time-delayed feedback from experimental time series. Multivariate systems involving noise sources with arbitrary correlation times are considered.

Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations

We consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler–Maruyama scheme.

Balancing with positive feedback: the case for discontinuous control

Experimental observations indicate that positive feedback plays an important role for maintaining human balance in the upright position. This observation is used to motivate an investigation of a simple switch-like controller for postural sway in which corrective movements are made only when the vertical displacement angle exceeds a certain threshold. This mechanism is shown to be consistent with the experimentally observed variations in the two-point correlation for human postural sway. Analysis of first-passage times for this model suggests that this control strategy may slow escape by taking advantage of two intrinsic properties of a stochastic unstable first-order delay differential equation: (i) time delay and (ii) the possibility that the dynamics can be 'temporarily confined' near the origin.

How do spontaneous pupillary oscillations in light relate to light intensity?

Characteristics of light-induced pupillary oscillations at constant light intensities have been investigated sparsely compared to sleepiness-related pupillary oscillations in darkness. This study presents the first controlled analysis of light-induced pupillary oscillations and their relationship to illumination. Pupillary oscillations of alert subjects were recorded by infrared video pupillography in different background lighting. Although showing obvious relationships of mean frequency and amplitude to light intensity, there were considerable inter- and intra-individual differences in the appearance of light-induced oscillations. As they looked rather similar to sleepiness waves, the question remains to identify light-induced oscillations in day light and to differentiate them from sleepiness-related oscillations.

Practical criterion for delay estimation using random perturbations

Many systems contain an internal time delay, which significantly influences their dynamical properties. Methods to estimate this delay from times series in the presence of dynamical noise are not systematically studied. Addressing this problem, we demonstrate that it is sufficient to analyze the system's response to short-correlated external disturbances or internal noise. Following this idea, it is shown for linear and nonlinear systems, as well as for periodic dynamics, that the delay can be estimated by analyzing the correlation function. This method covers the case of strong noise and multiple delays.

Modeling the control of fixational eye movements with neurophysiological delays

We propose a model for the control of fixational eye movements using time-delayed random walks. Fixational eye movements produce random displacements of the retinal image to prevent perceptual fading. First, we demonstrate that a transition from persistent to antipersistent correlations occurs in data recorded from a visual fixation task. Second, we propose and investigate a delayed random-walk model and get, by comparison of the transition points, an estimate of the neurophysiological delay. Differences between horizontal and vertical components of eye movements are found which can be explained neurophysiologically. Finally, we compare our numerical results with analytic approximations.

Stochastic order parameter equation of isometric force production revealed by drift-diffusion estimates

We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production.

Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.

Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations

We study time-delayed stochastic systems that can be described by means of so-called delay Fokker-Planck equations. Using Novikov's theorem, we first show that the theory of delay Fokker-Planck equations is on an equal footing with the theory of ordinary Fokker-Planck equations. Subsequently, we derive stationary distributions in the case of small time delays. In the case of additive noise systems, these distributions can be cast into the form of Boltzmann distributions involving effective potential functions.

Delay Fokker-Planck equations, perturbation theory, and data analysis for nonlinear stochastic systems with time delays

We study nonlinear stochastic systems with time-delayed feedback using the concept of delay Fokker-Planck equations introduced by Guillouzic, L'Heureux, and Longtin. We derive an analytical expression for stationary distributions using first-order perturbation theory. We demonstrate how to determine drift functions and noise amplitudes of this kind of systems from experimental data. In addition, we show that the Fokker-Planck perspective for stochastic systems with time delays is consistent with the so-called extended phase-space approach to time-delayed systems.

Functional characterization of linear delay Langevin equations

We present an exact functional characterization of linear delay Langevin equations driven by any noise structure defined through its characteristic functional. This method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise. General properties of the transient dissipative dynamics are analyzed. The corresponding interplay with a color Gaussian noise is presented. As a full application of our functional method we study a model for population growth with non-Gaussian fluctuations: the Gompertz model driven by multiplicative white shot noise.

Human stick balancing: tuning Lèvy flights to improve balance control

State-dependent, or parametric, noise is an essential component of the neural control mechanism for stick balancing at the fingertip. High-speed motion analysis in three dimensions demonstrates that the controlling movements made by the fingertip during stick balancing can be described by a Lèvy flight. The Lèvy index, alpha, is approximately 0.9; a value close to optimal for a random search. With increased skill, the index alpha does not change. However, the tails of the Lèvy distribution become broader. These observations suggest a Lèvy flight that is truncated by the properties of the nervous and musculoskeletal system; the truncation decreasing as skill level increases. Measurements of the cross-correlation between the position of the tip of the stick and the fingertip demonstrate that the role of closed-loop feedback changes with increased skill. Moreover, estimation of the neural latencies for stick balancing show that for a given stick length, the latency increases with skill level. It is suggested that the neural control for stick balancing involves a mechanism in which brief intervals of consciously generated, corrective movements alternate with longer intervals of prediction-free control. With learning the truncation of the Lèvy flight becomes better optimized for balance control and hence the time between successive conscious corrections increases. These observations provide the first evidence that changes in a Lèvy flight may have functional significance for the nervous system. This work has implications for the control of balancing problems ranging from falling in the elderly to the design of two-legged robots and earthquake proof buildings.

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.

Analytical results for fundamental time-delayed feedback systems subjected to multiplicative noise

We study the stochastic behavior of fundamental time-delayed feedback systems subjected to multiplicative noise. We derive exact results for the first and second moments and the autocorrelation function. For a particular class of systems we show how the variance depends on the amplitude of the multiplicative noise. Furthermore, we identify parameter regions of stationary solutions with finite and infinite variances. Finally, we suggest that delay-induced Lévy flights may occur in time-delayed feedback systems involving multiplicative noise.

On the Road to Automatic: Dynamic Aspects in the Development of Expertise

One of the important steps on the road to becoming expert in a motor skill occurs when the individual can perform the movements in a seemingly effortless and automatic fashion. The authors review two lines of investigations, namely, fMRI and mathematically guided studies of the dynamics of skill acquisition, that suggest that this road to automatic involves two steps: (1) an increasing reliance on the self-regulatory aspects of the motor task, and (2) a minimization of the role of mechanisms based on intentionally directed corrective movements. The interplay between these two mechanisms implies that, at a given skill level, performance decreases whenever intention intervenes. The observation that psychological factors may be as important as mechanical repetition for the development of expertise has important implications for the design of neurorehabilitative strategies.

Fokker-Planck perspective on stochastic delay systems: exact solutions and data analysis of biological systems

Stochastic delay systems with additive noise are examined from the perspective of Fokker-Planck equations. For a linear system, the exact stationary probability density is derived by means of a delay Fokker-Planck equation. We show how to determine the delay equation of the linear system from experimental data, and corroborate a fundamental result previously obtained by Küchler and Mensch. We also propose a method to derive delay equations of nonlinear systems from experimental data. To this end, the theory of multivariate Fokker-Planck equations is used. The applicability of this method is demonstrated for stochastic models describing tracking and pointing movements of humans.

Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity

We reduce the Lang-Kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay-differential equation in the long delay-time limit. The reduced equation has a time-delayed linear term and a cubic instantaneous nonlinearity. There are only two parameters, the real linewidth enhancement factor and the complex feedback strength. The equation displays a very rich dynamics and can sustain steady, periodic, quasiperiodic, and chaotic regimes. We study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method. This leads to a bifurcation diagram of the steady and periodic solutions, stable and unstable. We illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur.

On-off intermittency in a human balancing task

Motion analysis in three dimensions demonstrate that the fluctuations in the vertical displacement angle of a stick balanced at the fingertip obey a scaling law characteristic of on-off intermittency and that >98% of the corrective movements occur fast compared to the measured time delay. These experimental observations are reproduced by a model for an inverted pendulum with time-delayed feedback in which parametric noise forces a control parameter across a particular stability boundary. Our observations suggest that parametric noise is an essential, but up until now underemphasized, component of the neural control of balance.

Coherence and anticoherence resonance tuned by noise

We present numerical evidence and a theoretical analysis of the appearance of anticoherence resonance induced by noise, not predicted in former analysis of coherence resonance. We have found that this phenomenon occurs for very small values of the intensity of the noise acting on an excitable system, and we claim that this is a universal signature of a nonmonotonous relaxational behavior near its oscillatory regime. Moreover, we demonstrate that this new phenomenon is totally compatible with the standard situation of coherence resonance appearing at intermediate values of noise intensity.

Multivariate Markov processes for stochastic systems with delays: application to the stochastic Gompertz model with delay

Using the method of steps, we describe stochastic processes with delays in terms of Markov diffusion processes. Thus, multivariate Langevin equations and Fokker-Planck equations are derived for stochastic delay differential equations. Natural, periodic, and reflective boundary conditions are discussed. Both Ito and Stratonovich calculus are used. In particular, our Fokker-Planck approach recovers the generalized delay Fokker-Planck equation proposed by Guillouzic et al. The results obtained are applied to a model for population growth: the Gompertz model with delay and multiplicative white noise.

Noise-induced dynamics in bistable systems with delay

Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. Analytical solutions for the autocorrelation function and the power spectrum have been found. The power spectrum has a peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.

Dynamics of Parkinsonian tremor during deep brain stimulation

The mechanism by which chronic, high frequency, electrical deep brain stimulation (HF-DBS) suppresses tremor in Parkinson's disease is unknown. Rest tremor in subjects with Parkinson's disease receiving HF-DBS was recorded continuously throughout switching the deep brain stimulator on (at an effective frequency) and off. These data suggest that the stimulation induces a qualitative change in the dynamics, called a Hopf bifurcation, so that the stable oscillations are destabilized. We hypothesize that the periodic stimulation modifies a parameter affecting the oscillation in a time dependent way and thereby induces a Hopf bifurcation. We explore this hypothesis using a schematic network model of an oscillator interacting with periodic stimulation. The mechanism of time-dependent change of a control parameter in the model captures two aspects of the dynamics observed in the data: (1) a gradual increase in tremor amplitude when the stimulation is switched off and a gradual decrease in tremor amplitude when the stimulation is switched on and (2) a time delay in the onset and offset of the oscillations. This mechanism is consistent with these rest tremor transition data and with the idea that HF-DBS acts via the gradual change of a network property. (c) 2001 American Institute of Physics.