Delayed stochastic differential model for quiet standing

Noise-induced quasiperiod and period switching

We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.

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.

An active balance board system with real-time control of stiffness and time-delay to assess mechanisms of postural stability

Increased time-delay in the neuromuscular system caused by neurological disorders, concussions, or advancing age is an important factor contributing to balance loss (Chagdes et al., 2013, 2016a,b). We present the design and fabrication of an active balance board system that allows for a systematic study of stiffness and time-delay induced instabilities in standing posture. Although current commercial balance boards allow for variable stiffness, they do not allow for manipulation of time-delay. Having two controllable parameters can more accurately determine the cause of balance deficiencies, and allows us to induce instabilities even in healthy populations. An inverted pendulum model of human posture on such an active balance board predicts that reduced board rotational stiffness destabilizes upright posture through board tipping, and limit cycle oscillations about the upright position emerge as feedback time-delay is increased. We validate these two mechanisms of instability on the designed balance board, showing that rotational stiffness and board time-delay induced the predicted postural instabilities in healthy, young adults. Although current commercial balance boards utilize control of rotational stiffness, real-time control of both stiffness and time-delay on an active balance board is a novel and innovative manipulation to reveal balance deficiencies and potentially improve individualized balance training by targeting multiple dimensions contributing to standing balance.

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.

The relationship between intermittent limit cycles and postural instability associated with Parkinson's disease

BACKGROUND

Many disease-specific factors such as muscular weakness, increased muscle stiffness, varying postural strategies, and changes in postural reflexes have been shown to lead to postural instability and fall risk in people with Parkinson's disease (PD). Recently, analytical techniques, inspired by the dynamical systems perspective on movement control and coordination, have been used to examine the mechanisms underlying the dynamics of postural declines and the emergence of postural instabilities in people with PD.

METHODS

A wavelet-based technique was used to identify limit cycle oscillations (LCOs) in the anterior-posterior (AP) postural sway of people with mild PD (n = 10) compared to age-matched controls (n = 10). Participants stood on a foam and on a rigid surface while completing a dual task (speaking).

RESULTS

There was no significant difference in the root mean square of center of pressure between groups. Three out of 10 participants with PD demonstrated LCOs on the foam surface, while none in the control group demonstrated LCOs. An inverted pendulum model of bipedal stance was used to demonstrate that LCOs occur due to disease-specific changes associated with PD: time-delay and neuromuscular feedback gain.

CONCLUSION

Overall, the LCO analysis and mathematical model appear to capture the subtle postural instabilities associated with mild PD. In addition, these findings provide insights into the mechanisms that lead to the emergence of unstable posture in patients with PD.

Failure of Arm Movement Control in Stroke Patients, Characterized by Loss of Complexity

We study the mechanism of human arm-posture control by means of nonlinear dynamics and quantitative time series analysis methods. Utilizing linear and nonlinear measures in combination, we find that pathological tremors emerge in patient dynamics and serve as a main feature discriminating between normal and patient groups. The deterministic structure accompanied with loss of complexity inherent in the tremor dynamics is also revealed. To probe the underlying mechanism of the arm-posture dynamics, we further analyze the coupling patterns between joints and components, and discuss their roles in breaking of the organization structure. As a result, we elucidate the mechanisms in the arm-posture dynamics of normal subjects responding to the gravitational force and for the reduction of the dynamic degrees of freedom in the patient dynamics. This study provides an integrated framework for the origin of the loss of complexity in the dynamics of patients as well as the coupling structure in the arm-posture dynamics.

Delay-induced stochastic bifurcations in a bistable system under white noise

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

Causality, criticality, and reading words: distinct sources of fractal scaling in behavioral sequences

The finding of fractal scaling (FS) in behavioral sequences has raised a debate on whether FS is a pervasive property of the cognitive system or is the result of specific processes. Inferences about the origins of properties in time sequences are causal. That is, as opposed to correlational inferences reflecting instantaneous symmetrical relations, causal inferences concern asymmetric relations lagged in time. Here, I integrate Granger-causality with inferences about FS. Four simulations illustrate that causal analyses can isolate distinct FS sources, whereas correlational techniques cannot. I then analyze three simultaneous sequences of responses from a database of word-naming trials. I find that two, or perhaps three, distinct sources account for the presence of FS in these sequences, but FS is not a general property of the system. This suggests that FS arises due to the properties of a limited number of identifiable psychological and/or neural processes. Finally, I reanalyze a previously published dataset of acoustic frequency spectra using the new tools. The causality/criticality combination introduced here offers a new important perspective in the study of cognition.

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.

Delay effects in the human sensory system during balancing

Mechanical models of human self-balancing often use the Newtonian equations of inverted pendula. While these mathematical models are precise enough on the mechanical side, the ways humans balance themselves are still quite unexplored on the control side. Time delays in the sensory and motoric neural pathways give essential limitations to the stabilization of the human body as a multiple inverted pendulum. The sensory systems supporting each other provide the necessary signals for these control tasks; but the more complicated the system is, the larger delay is introduced. Human ageing as well as our actual physical and mental state affects the time delays in the neural system, and the mechanical structure of the human body also changes in a large range during our lives. The human balancing organ, the labyrinth, and the vision system essentially adapted to these relatively large time delays and parameter regions occurring during balancing. The analytical study of the simplified large-scale time-delayed models of balancing provides a Newtonian insight into the functioning of these organs that may also serve as a basis to support theories and hypotheses on balancing and vision.

Modeling the dynamics of a free boundary on turbulent thermal convection

Based on our previous experimental study, we present a one-dimensional phenomenological model of a thermal blanket floating on the upper surface of a thermally convecting fluid. The model captures the most important interactions between the floating solid and the fluid underneath. By the thermal blanketing effect, the presence of the solid plate modifies the flow structure below; in turn, the flow exerts a viscous drag that causes the floating boundary to move. An oscillatory state and a trapped state are found in this model, which is in excellent agreement with experimental observations. The model also offers details on the transition between the states, and gives useful insights on this coupled system without the need for full-scale simulations.

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.

Predicting the dynamic postural control response from quiet-stance behavior in elderly adults

Human postural sway, as measured by fluctuations of the center of pressure (COP) under the feet of a quietly standing individual, can be characterized as a stochastic process. The fluctuation-dissipation theorem (FDT) provides a linear relationship between the fluctuations of a quasi-static, stochastic system to the same system's relaxation to equilibrium following a perturbation. We applied a similar linear relationship, based on the FDT, to the human postural control system to explore whether anterior-posterior (AP) fluctuations of the COP during quiet stance can be used to predict the AP response of the postural control system to a weak posteriorly directed mechanical perturbation (tug or pull at the waist). We tested 10 healthy elderly (mean age of 69yr) and 10 healthy young (mean age of 25yr) adult subjects. We found that this linear relationship was applicable to the postural control system of all 10 young and eight of the 10 elderly adult subjects. These results suggest that it is possible to predict an individual's dynamic response to a mild perturbation using quiet-stance data, regardless of age. The existence of this FDT-based linear relationship with respect to the human postural control system suggests that, for a given individual, the postural control system may use the same control mechanisms during quiet stance and mild-perturbation conditions, regardless of age.