Delay effects in the human sensory system during balancing

Human performance in virtual stabilization of a fractional-order system with reaction delay

Virtual balancing tasks facilitate the study of human motion control: human reaction to the change of artificially introduced parameters can be studied in a computer environment. In this article, the dynamics of human stick balancing are generalized using fractional-order derivatives. Reaction delay sets a strong limitation on the length of the shortest stick that human subjects can balance. Human processing of visual input also exhibits a memory effect, which can be modelled by fractional-order derivatives. Therefore, we hypothesize a delayed fractional-order PD control of the unstable fractional-order process. The resulting equation of motion is investigated in a dimensionless framework, and stabilizability limits are determined as a function of the dynamics's order. These theoretical limits are then compared with the results of a systematic series of virtual balancing tests performed by 18 subjects. The comparison shows that the theoretical stabilizability limits for controllers with fixed fractional order correspond to the measured data points. The best fit is obtained if the fractional order of the underlying control law is 0.475.

Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects

Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers.

Practice walking on a treadmill-mounted balance beam modifies beam walking sacral movement and alters performance in other balance tasks

The goals of this study were to determine if a single 30-minute session of practice walking on a treadmill mounted balance beam: 1) altered sacral marker movement kinematics during beam walking, and 2) affected measures of balance during treadmill walking and standing balance. Two groups of young, healthy human subjects practiced walking on a treadmill mounted balance beam for thirty minutes. One group trained with intermittent visual occlusions and the other group trained with unperturbed vision. We hypothesized that the subjects would show changes in sacrum movement kinematics after training and that there would be group differences due to larger improvements in beam walking performance by the visual occlusions group. We also investigated if there was any balance transfer from training on the beam to treadmill walking (margin of stability) and to standing static balance (center of pressure excursion). We found significant differences in sacral marker maximal velocity after training for both groups, but no significant differences between the two groups from training. There was limited evidence of balance transfer from beam-walking practice to gait margin of stability for treadmill walking and for single leg standing balance, but not for tandem stance balance. The number of step-offs while walking on a narrow beam had the largest change with training (partial η2 = 0.7), in accord with task specificity. Other balance metrics indicative of transfer had lower effect sizes (partial η2<0.5). Given the limited transfer across balance training tasks, future work should examine how intermittent visual occlusions during multi-task training improve real world functional outcomes.

Controlling stick balancing on a linear track: Delayed state feedback or delay-compensating predictor feedback?

A planar stick balancing task was investigated using stabilometry parameters (SP); a concept initially developed to assess the stability of human postural sway. Two subject groups were investigated: 6 subjects (MD) with many days of balancing a 90 cm stick on a linear track and 25 subjects (OD) with only one day of balancing experience. The underlying mechanical model is a pendulum-cart system. Two control force models were investigated by means of numerical simulations: (1) delayed state feedback (DSF); and (2) delay-compensating predictor feedback (PF). Both models require an internal model and are subject to certainty thresholds with delayed switching. Measured and simulated time histories were compared quantitatively using a cost function in terms of some essential SPs for all subjects. Minimization of the cost function showed that the control strategy of both OD and MD subjects can better be described by DSF. The control mechanism for the MD subjects was superior in two aspects: (1) they devoted less energy to controlling the cart's position; and (2) their perception threshold for the stick's angular velocity was found to be smaller. Findings support the concept that when sufficient sensory information is readily available, a delay-compensating PF strategy is not necessary.

Synchronization of non-identical systems by non-invasive mutual time-delayed feedback

Inspired by time-delayed feedback control, it is shown that synchronization of non-identical systems can be achieved by mutual time-delayed feedback with an asymptotically vanishing interaction. An analytic perturbation scheme is developed, which uncovers the merits as well as the constraints of such an approach. As an application, the use of the concept for a secure communication channel is considered.

Learn one size to infer all: Exploiting translational symmetries in delay-dynamical and spatiotemporal systems using scalable neural networks

We design scalable neural networks adapted to translational symmetries in dynamical systems, capable of inferring untrained high-dimensional dynamics for different system sizes. We train these networks to predict the dynamics of delay-dynamical and spatiotemporal systems for a single size. Then, we drive the networks by their own predictions. We demonstrate that by scaling the size of the trained network, we can predict the complex dynamics for larger or smaller system sizes. Thus, the network learns from a single example and by exploiting symmetry properties infers entire bifurcation diagrams.

Parameter identification of a delayed infinite-dimensional heat-exchanger process based on relay feedback and root loci analysis

The focus of this contribution is twofold. The first part aims at the rigorous and complete analysis of pole loci of a simple delayed model, the characteristic function of which is represented by a quasi-polynomial with a non-delay and a delay parameter. The derived spectrum constitutes an infinite set, making it a suitable and simple-enough representative of even high-order process dynamics. The second part intends to apply the simple infinite-dimensional model for relay-based parameter identification of a more complex model of a heating–cooling process with heat exchangers. Processes of this type and construction are widely used in industry. The identification procedure has two substantial steps. The first one adopts the simple model with a low computational effort using the saturated relay that provides a more accurate estimation than the standard on/off test. Then, this result is transformed to the estimation of the initial characteristic equation parameters of the complex infinite-dimensional heat-exchanger model using the exact dominant-pole-loci assignment. The benefit of this technique is that multiple model parameters can be estimated under a single relay test. The second step attempts to estimate the remaining model parameters by various numerical optimization techniques and also to enhance all model parameters via the Autotune Variation Plus relay experiment for comparison. Although the obtained unordinary time and frequency domain responses may yield satisfactory results for control tasks, the identified model parameters may not reflect the actual values of process physical quantities.

Predictor feedback models for stick balancing with delay mismatch and sensory dead zones

Human stick balancing is investigated in terms of reaction time delay and sensory dead zones for position and velocity perception using a special combination of delayed state feedback and mismatched predictor feedback as a control model. The corresponding mathematical model is a delay-differential equation with event-driven switching in the control action. Due to the sensory dead zones, initial conditions of the actual state cannot always be provided for an internal-model-based prediction, which indicates that (1) perfect prediction is not possible and (2) the delay in the switching condition cannot be compensated. The imperfection of the predictor is described by the delay mismatch, which is treated as a lumped parameter that creates a transition between perfect predictor feedback (zero delay mismatch) and delayed state feedback (mismatch equal to switching delay). The maximum admissible switching delay (critical delay) is determined numerically based on a practical stabilizability concept. This critical delay is compared to a realistic reference value of 230 ms in order to assess the possible regions of the threshold values for position and velocity perception. The ratio of the angular position and angular velocity for 44 successful balancing trials by 8 human subjects was used to validate the numerical results. Comparison of actual human stick balancing data and numerical simulations based on the mismatched predictor feedback model provided a plausible range of parameters: position detection threshold 1°, velocity detection threshold between 4.24 and 9.35°/s, and delay mismatch around 100-150 ms.

Virtual stick balancing: skill development in Newtonian and Aristotelian dynamics

Human reaction delay significantly limits manual control of unstable systems. It is more difficult to balance a short stick on a fingertip than a long one, because a shorter stick falls faster and therefore requires faster reactions. In this study, a virtual stick balancing environment was developed where the reaction delay can be artificially modulated and the law of motion can be changed between second-order (Newtonian) and first-order (Aristotelian) dynamics. Twenty-four subjects were separated into two groups and asked to perform virtual stick balancing programmed according to either Newtonian or Aristotelian dynamics. The shortest stick length (critical length, L_c) was determined for different added delays in six sessions of balancing trials performed on different days. The observed relation between L_c and the overall reaction delay τ reflected the feature of the underlying mathematical models: (i) for the Newtonian dynamics L_c is proportional to τ²; (ii) for the Aristotelian dynamics L_c is proportional to τ. Deviation of the measured L_c(τ) function from the theoretical one was larger for the Newtonian dynamics for all sessions, which suggests that, at least in virtually controlled tasks, it is more difficult to adopt second-order dynamics than first-order dynamics.

Slacklining: A narrative review on the origins, neuromechanical models and therapeutic use

Slacklining, the neuromechanical action of balance retention on a tightened band, is achieved through self-learned strategies combining dynamic stability with optimal energy expenditure. Published slacklining literature is recent and limited, including for neuromechanical control strategy models. This paper explores slacklining's definitions and origins to provide background that facilitates understanding its evolution and progressive incorporation into both prehabilitation and rehabilitation. Existing explanatory slacklining models are considered, their application to balance and stability, and knowledge-gaps highlighted. Current slacklining models predominantly derive from human quiet-standing and frontal plane movement on stable surfaces. These provide a multi-tiered context of the unique and complex neuro-motoric requirements for slacklining's multiple applications, but are not sufficiently comprehensive. This consequently leaves an incomplete understanding of how slacklining is achieved, in relation to multi-directional instability and complex multi-dimensional human movement and behavior. This paper highlights the knowledge-gaps and sets a foundation for the required explanatory control mechanisms that evolve and expand a more detailed model of multi-dimensional slacklining and human functional movement. Such a model facilitates a more complete understanding of existing performance and rehabilitation applications that opens the potential for future applications into broader areas of movement in diverse fields including prostheses, automation and machine-learning related to movement phenotypes.

Response to perturbation during quiet standing resembles delayed state feedback optimized for performance and robustness

Postural sway is a result of a complex action–reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the response to large enough perturbations during quiet standing such that the resulting responses can clearly be distinguished from the local postural sway. Measurements show that typical responses very closely resemble those of a critically damped oscillator. The recovery dynamics are modelled by an inverted pendulum subject to delayed state feedback and is described in the space of the control parameters. We hypothesize that the control gains are tuned such that (H1) the response is at the border of oscillatory and nonoscillatory motion similarly to the critically damped oscillator; (H2) the response is the fastest possible; (H3) the response is a result of a combined optimization of fast response and robustness to sensory perturbations. Parameter fitting shows that H1 and H3 are accepted while H2 is rejected. Thus, the responses of human postural balance to “large” perturbations matches a delayed feedback mechanism that is optimized for a combination of performance and robustness.

Slacklining as therapy to address non-specific low back pain in the presence of multifidus arthrogenic muscle inhibition

Low back pain (LBP) represents the most prevalent, problematic and painful of musculoskeletal conditions that affects both the individual and society with health and economic concerns. LBP is a heterogeneous condition with multiple diagnoses and causes. In the absence of consensus definitions, partly because of terminology inconsistency, it is further referred to as non-specific LBP (NSLBP). In NSLBP patients, the lumbar multifidus (MF), a key stabilizing muscle, has a depleted role due to recognized myocellular lipid infiltration and wasting, with the potential primary cause hypothesized as arthrogenic muscle inhibition (AMI). This link between AMI and NSLBP continues to gain increasing recognition. To date there is no 'gold standard' or consensus treatment to alleviate symptoms and disability due to NSLBP, though the advocated interventions are numerous, with marked variations in costs and levels of supportive evidence. However, there is consensus that NSLBP management be cost-effective, self-administered, educational, exercise-based, and use multi-modal and multi-disciplinary approaches. An adjuvant therapy fulfilling these consensus criteria is 'slacklining', within an overall rehabilitation program. Slacklining, the neuromechanical action of balance retention on a tightened band, induces strategic indirect-involuntary therapeutic muscle activation exercise incorporating spinal motor control. Though several models have been proposed, understanding slacklining's neuro-motor mechanism of action remains incomplete. Slacklining has demonstrated clinical effects to overcome AMI in peripheral joints, particularly the knee, and is reported in clinical case-studies as showing promising results in reducing NSLBP related to MF deficiency induced through AMI (MF-AMI). Therefore, this paper aims to: rationalize why and how adjuvant, slacklining therapeutic exercise may positively affect patients with NSLBP, due to MF-AMI induced depletion of spinal stabilization; considers current understandings and interventions for NSLBP, including the contributing role of MF-AMI; and details the reasons why slacklining could be considered as a potential adjuvant intervention for NSLBP through its indirect-involuntary action. This action is hypothesized to occur through an over-ride or inhibition of central down-regulatory induced muscle insufficiency, present due to AMI. This subsequently allows neuroplasticity, normal neuro-motor sequencing and muscle re-activation, which facilitates innate advantageous spinal stabilization. This in-turn addresses and reduces NSLBP, its concurrent symptoms and functional disability. This process is hypothesized to occur through four neuro-physiological processing pathways: finite neural delay; movement-control phenotypes; inhibition of action and the innate primordial imperative; and accentuated corticospinal drive. Further research is recommended to investigate these hypotheses and the effect of slacklining as an adjuvant therapy in cohort and control studies of NSLBP populations.

Rolling balance board of adjustable geometry as a tool to assess balancing skill and to estimate reaction time delay

The relation between balancing performance and reaction time is investigated for human subjects balancing on rolling balance board of adjustable physical parameters: adjustable rolling radius R and adjustable board elevation h. A well-defined measure of balancing performance is whether a subject can or cannot balance on balance board with a given geometry (R, h). The balancing ability is linked to the stabilizability of the underlying two-degree-of-freedom mechanical model subject to a delayed proportional-derivative feedback control. Although different sensory perceptions involve different reaction times at different hierarchical feedback loops, their effect is modelled as a single lumped reaction time delay. Stabilizability is investigated in terms of the time delay in the mechanical model: if the delay is larger than a critical value (critical delay), then no stabilizing feedback control exists. Series of balancing trials by 15 human subjects show that it is more difficult to balance on balance board configuration associated with smaller critical delay, than on balance boards associated with larger critical delay. Experiments verify the feature of the mechanical model that a change in the rolling radius R results in larger change in the difficulty of the task than the same change in the board elevation h does. The rolling balance board characterized by the two well-defined parameters R and h can therefore be a useful device to assess human balancing skill and to estimate the corresponding lumped reaction time delay.

Slacklining: An explanatory multi-dimensional model considering classical mechanics, biopsychosocial health and time

This paper aims to overcome slacklining's limited formulated explanatory models. Slacklining is an activity with increasing recreational use, but also has progressive adoption into prehabilitation and rehabilitation. Slacklining is achieved through self-learned strategies that optimize energy expenditure without conceding dynamic stability, during the neuromechanical action of balance retention on a tightened band. Evolved from rope-walking or 'Funambulus', slacklining has an extensive history, yet limited and only recent published research, particularly for clinical interventions and in-depth hypothesized multi-dimensional models describing the neuromechanical control strategies. These 'knowledge-gaps' can be overcome by providing an, explanatory model, that evolves and progresses existing standards, and explains the broader circumstances of slacklining's use. This model details the individual's capacity to employ control strategies that achieve stability, functional movement and progressive technical ability. The model considers contributing entities derived from: Self-learned control of movement patterns; subjected to classical mechanical forces governed by Newton's physical laws; influenced by biopsychosocial health factors; and within time's multi-faceted perspectives, including as a quantified unit and as a spatial and cortical experience. Consequently, specific patient and situational uses may be initiated within the framework of evidence based medicine that ensures a multi-tiered context of slacklining applications in movement, balance and stability. Further research is required to investigate and mathematically define this proposed model and potentially enable an improved understanding of human functional movement. This will include its application in other diverse constructed and mechanical applications in varied environments, automation levels, robotics, mechatronics and artificial-intelligence factors, including machine learning related to movement phenotypes and applications.

The effects of sensory quantization and control torque saturation on human balance control

The effect of reaction delay, temporal sampling, sensory quantization, and control torque saturation is investigated numerically for a single-degree-of-freedom model of postural sway with respect to stability, stabilizability, and control effort. It is known that reaction delay has a destabilizing effect on the balancing process: the later one reacts to a perturbation, the larger the possibility of falling. If the delay is larger than a critical value, then stabilization is not even possible. In contrast, numerical analysis showed that quantization and control torque saturation have a stabilizing effect: the region of stabilizing control gains is greater than that of the linear model. Control torque saturation allows the application of larger control gains without overcontrol while sensory quantization plays a role of a kind of filter when sensory noise is present. These beneficial effects are reflected in the energy demand of the control process. On the other hand, neither control torque saturation nor sensory quantization improves stabilizability properties. In particular, the critical delay cannot be increased by adding saturation and/or sensory quantization.

Establishing metrics and control laws for the learning process: ball and beam balancing

Understanding how dexterity improves with practice is a fundamental challenge of motor control and neurorehabilitation. Here we investigate a ball and beam implementation of a dexterity puzzle in which subjects stabilize a ball at the mid-point of a beam by manipulating the angular position of the beam. Stabilizability analysis of different biomechanical models for the ball and beam task with time-delayed proportional-derivative feedback identified the angular position of the beam as the manipulated variable. Consequently, we monitored the changes in the dynamics with learning by measuring changes in the control parameters. Two types of stable motion are possible: node type (nonoscillatory) and spiral type (oscillatory). Both types of motion are observed experimentally and correspond to well-defined regions in the parameter space of the control gains. With practice the control gains for each subject move close to or on the portion of the boundary which separates the node-type and spiral-type solutions and which is associated with the rightmost characteristic exponent of smallest real part. These observations suggest that with learning the control gains for ball and beam balancing change in such a way that minimizes overshoot and the settling time. This study provides an example of how mathematical analysis together with careful experimental observations can shed light onto the early stages of skill acquisition. Since the difficulty of this task depends on the length of the beam, ball and beam balancing tasks may be useful for the rehabilitation of children with dyspraxia and those recovering from a stroke.

Anticipatory coadaptation of ankle stiffness and sensorimotor gain for standing balance

External perturbation forces may compromise standing balance. The nervous system can intervene only after a delay greater than 100 ms, during which the body falls freely. With ageing, sensorimotor delays are prolonged, posing a critical threat to balance. We study a generic model of stabilisation with neural delays to understand how the organism should adapt to challenging balance conditions. The model suggests that ankle stiffness should be increased in anticipation of perturbations, for example by muscle co-contraction, so as to slow down body fall during the neural response delay. Increased ankle muscle co-contraction is indeed observed in young adults when standing in challenging balance conditions, and in older relative to young adults during normal stance. In parallel, the analysis of the model shows that increases in either stiffness or neural delay must be coordinated with decreases in spinal sensorimotor gains, otherwise the feedback itself becomes destabilizing. Accordingly, a decrease in spinal feedback is observed in challenging conditions, and with age-related increases in neural delay. These observations have been previously interpreted as indicating an increased reliance on cortical rather than spinal control of balance, despite the fact that cortical responses have a longer latency. Our analysis challenges this interpretation by showing that these observations are consistent with a functional coadaptation of spinal feedback gains to functional changes in stiffness and neural delay.

Being able to stand still can be difficult when faced with an unexpected push. It takes the nervous system more than a tenth of a second to respond to such a perturbation, and during this delay the body falls under the influence of its own weight. By co-contracting their ankle muscles in anticipation of a perturbation, subjects can increase their ankle stiffness, which slows down their fall during the neural delay. Young subjects indeed adopt this strategy when they need to remain particularly still (for example when they stand in front of a cliff). Older subjects adopt this strategy even during normal standing. We present a model of standing balance that shows that this postural strategy provides partial compensation for the increase in neural delays with ageing. According to our model, increasing ankle stiffness only improves balance if it is accompanied by a decrease in sensorimotor gain. This provides a novel and functional interpretation for the decrease in spinal feedback observed during ageing, and observed in young subjects when they stand in challenging balance conditions.

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.

Acting together, destabilizing influences can stabilize human balance

The causes of falling in the elderly are multi-factorial. Three factors that influence balance stability are the time delay, a sensory dead zone and the maximum ankle torque that can be generated by muscular contraction. Here, the effects of these contributions are evaluated in the context of a model of an inverted pendulum stabilized by time-delayed proportional-derivative (PD) feedback. The effect of the sensory dead zone is to produce a hybrid type of control in which the PD feedback is switched ON or OFF depending on whether or not the controlled variable is larger or smaller than the detection threshold, Π. It is shown that, as Π increases, the region in the plane of control parameters where the balance time (BT) is greater than 60 s is increased slightly. However, when maximum ankle torque is also limited, there is a dramatic increase in the parameter region associated with BTs greater than 60 s. This increase is due to the effects of a torque limitation on over-control associated with bang-bang type switching controllers. These observations show that acting together influences, which are typically thought to destabilize balance, can actually stabilize balance. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Feedback Stabilization of First Order Neutral Delay Systems Using the Lambert W Function

This paper presents a new approach to stabilize the first order neutral delay differential systems with two time delays. First, we provided a few oscillation and non-oscillation criteria for the neutral delay differential equations using spectrum analysis and the Lambert W function. These conditions were explicit and the real roots were analytically expressed in terms of the Lambert W function in the case of non-oscillation. Second, we designed a stabilizing state feedback controller for the neutral delay differential systems with two time delays, wherein the proportional and derivative gains were analytically determined using the results of the non-oscillation criteria. A few examples are given to illustrate the main results.

Saturation limits the contribution of acceleration feedback to balancing against reaction delay

A nonlinear model for human balancing subjected to a saturated delayed proportional-derivative-acceleration (PDA) feedback is analysed. Compared to the proportional-derivative (PD) controller, it is confirmed that the PDA controller improves local stability even for large feedback delays. However, it is shown that the saturated PDA controller typically introduces subcritical Hopf bifurcation into the system, which can also lead to falling for large enough perturbations. The subcriticality becomes stronger as the acceleration feedback gain increases or the saturation torque limit decreases. These explain some features of human balancing failure related to the increased reaction delay of inactive elderly people.

A Tutorial for the Analysis of the Piecewise-Smooth Dynamics of a Constrained Multibody Model of Vertical Hopping

Contradictory demands are present in the dynamic modeling and analysis of legged locomotion: on the one hand, the high degrees-of-freedom (DoF) descriptive models are geometrically accurate, but the analysis of self-stability and motion pattern generation is extremely challenging; on the other hand, low DoF models of locomotion are thoroughly analyzed in the literature; however, these models do not describe the geometry accurately. We contribute by narrowing the gap between the two modeling approaches. Our goal is to develop a dynamic analysis methodology for the study of self-stable controlled multibody models of legged locomotion. An efficient way of modeling multibody systems is to use geometric constraints among the rigid bodies. It is especially effective when closed kinematic loops are present, such as in the case of walking models, when both legs are in contact with the ground. The mathematical representation of such constrained systems is the differential algebraic equation (DAE). We focus on the mathematical analysis methods of piecewise-smooth dynamic systems and we present their application for constrained multibody models of self-stable locomotion represented by DAE. Our numerical approach is demonstrated on a linear model of hopping and compared with analytically obtained reference results.

Microchaos in human postural balance: Sensory dead zones and sampled time-delayed feedback

Models for the stabilization of an inverted pendulum figure prominently in studies of human balance control. Surprisingly, fluctuations in measures related to the vertical displacement angle for quietly standing adults with eyes closed exhibit chaos. Here we show that small-amplitude chaotic fluctuations ("microchaos") can be generated by the interplay between three essential components of human neural balance control, namely time-delayed feedback, a sensory dead zone, and frequency-dependent encoding of force. When the sampling frequency of the force encoding is decreased, the sensitivity of the balance control to changes in the initial conditions increases. The sampled, time-delayed nature of the balance control may provide insights into why falls are more common in the very young and the elderly.

Effect of sensory-motor latencies and active muscular stiffness on stability for an ankle-hip model of balance on a balance board

To achieve human upright posture (UP) and avoid falls, the central nervous system processes visual, vestibular, and proprioceptive information to activate the appropriate muscles to accelerate or decelerate the body's center of mass. In this process, sensory-motor (SM) latencies and muscular deficits, even in healthy older adults, may cause falls. This condition is worse for people with chronic neuromuscular deficits (stroke survivors, patients with multiple sclerosis or Parkinson's disease). One therapeutic approach is to recover or improve quiet UP by utilizing a balance board (BB) (a rotating surface with a tunable stiffness and time delay), where a patient attempts to maintain UP while task difficulty is manipulated. While BBs are commonly used, it is unclear how UP is maintained or how changes in system parameters such as SM latencies and BB time delay affect UP stability. To understand these questions, it is important that mathematical models be developed with enough degrees-of-freedom to capture the many responses evoked during the maintenance of UP on a BB. This paper presents an ankle-hip model of balance on a BB, which is used to study the combined effect of SM latencies and active muscular stiffness of the ankle and hip joints, and the BB stiffness and time delay on UP stability. The analysis predicts that people with proprioceptive, visual, vestibular loss, or increased SM latencies may show either leaning postures or larger body-sway. The results show that the BB time delay and the visual and vestibular feedback have the largest impact on UP stability.

Stabilizing skateboard speed-wobble with reflex delay

A simple mechanical model of the skateboard-skater system is analysed, in which the effect of human control is considered by means of a linear proportional-derivative (PD) controller with delay. The equations of motion of this non-holonomic system are neutral delay-differential equations. A linear stability analysis of the rectilinear motion is carried out analytically. It is shown how to vary the control gains with respect to the speed of the skateboard to stabilize the uniform motion. The critical reflex delay of the skater is determined as the function of the speed. Based on this analysis, we present an explanation for the linear instability of the skateboard-skater system at high speed. Moreover, the advantages of standing ahead of the centre of the board are demonstrated from the viewpoint of reflex delay and control gain sensitivity.

Changes in balance coordination and transfer to an unlearned balance task after slackline training: a self-organizing map analysis

How humans maintain balance and change postural control due to age, injury, immobility or training is one of the basic questions in motor control. One of the problems in understanding postural control is the large set of degrees of freedom in the human motor system. Therefore, a self-organizing map (SOM), a type of artificial neural network, was used in the present study to extract and visualize information about high-dimensional balance strategies before and after a 6-week slackline training intervention. Thirteen subjects performed a flamingo and slackline balance task before and after the training while full body kinematics were measured. Range of motion, velocity and frequency of the center of mass and joint angles from the pelvis, trunk and lower leg (45 variables) were calculated and subsequently analyzed with an SOM. Subjects increased their standing time significantly on the flamingo (average +2.93 s, Cohen's d = 1.04) and slackline (+9.55 s, d = 3.28) tasks, but the effect size was more than three times larger in the slackline. The SOM analysis, followed by a k-means clustering and marginal homogeneity test, showed that the balance coordination pattern was significantly different between pre- and post-test for the slackline task only (χ ² = 82.247; p < 0.001). The shift in balance coordination on the slackline could be characterized by an increase in range of motion and a decrease in velocity and frequency in nearly all degrees of freedom simultaneously. The observation of low transfer of coordination strategies to the flamingo task adds further evidence for the task-specificity principle of balance training, meaning that slackline training alone will be insufficient to increase postural control in other challenging situations.

Hybrid Statistical and Mechanistic Mathematical Model Guides Mobile Health Intervention for Chronic Pain

Nearly a quarter of visits to the emergency department are for conditions that could have been managed via outpatient treatment; improvements that allow patients to quickly recognize and receive appropriate treatment are crucial. The growing popularity of mobile technology creates new opportunities for real-time adaptive medical intervention, and the simultaneous growth of "big data" sources allows for preparation of personalized recommendations. Here we focus on the reduction of chronic suffering in the sickle cell disease (SCD) community. SCD is a chronic blood disorder in which pain is the most frequent complication. There currently is no standard algorithm or analytical method for real-time adaptive treatment recommendations for pain. Furthermore, current state-of-the-art methods have difficulty in handling continuous-time decision optimization using big data. Facing these challenges, in this study, we aim to develop new mathematical tools for incorporating mobile technology into personalized treatment plans for pain. We present a new hybrid model for the dynamics of subjective pain that consists of a dynamical systems approach using differential equations to predict future pain levels, as well as a statistical approach tying system parameters to patient data (both personal characteristics and medication response history). Pilot testing of our approach suggests that it has significant potential to well predict pain dynamics, given patients reported pain levels and medication usages. With more abundant data, our hybrid approach should allow physicians to make personalized, data-driven recommendations for treating chronic pain.

Intermittent Feedback-Control Strategy for Stabilizing Inverted Pendulum on Manually Controlled Cart as Analogy to Human Stick Balancing

The stabilization of an inverted pendulum on a manually controlled cart (cart-inverted-pendulum; CIP) in an upright position, which is analogous to balancing a stick on a fingertip, is considered in order to investigate how the human central nervous system (CNS) stabilizes unstable dynamics due to mechanical instability and time delays in neural feedback control. We explore the possibility that a type of intermittent time-delayed feedback control, which has been proposed for human postural control during quiet standing, is also a promising strategy for the CIP task and stick balancing on a fingertip. Such a strategy hypothesizes that the CNS exploits transient contracting dynamics along a stable manifold of a saddle-type unstable upright equilibrium of the inverted pendulum in the absence of control by inactivating neural feedback control intermittently for compensating delay-induced instability. To this end, the motions of a CIP stabilized by human subjects were experimentally acquired, and computational models of the system were employed to characterize the experimental behaviors. We first confirmed fat-tailed non-Gaussian temporal fluctuation in the acceleration distribution of the pendulum, as well as the power-law distributions of corrective cart movements for skilled subjects, which was previously reported for stick balancing. We then showed that the experimental behaviors could be better described by the models with an intermittent delayed feedback controller than by those with the conventional continuous delayed feedback controller, suggesting that the human CNS stabilizes the upright posture of the pendulum by utilizing the intermittent delayed feedback-control strategy.

A Correlation-Based Framework for Evaluating Postural Control Stochastic Dynamics

The inability to maintain balance during varying postural control conditions can lead to falls, a significant cause of mortality and serious injury among older adults. However, our understanding of the underlying dynamical and stochastic processes in human postural control have not been fully explored. To further our understanding of the underlying dynamical processes, we examine a novel conceptual framework for studying human postural control using the center of pressure (COP) velocity autocorrelation function (COP-VAF) and compare its results to Stabilogram Diffusion Analysis (SDA). Eleven healthy young participants were studied under quiet unipedal or bipedal standing conditions with eyes either opened or closed. COP trajectories were analyzed using both the traditional posturographic measure SDA and the proposed COP-VAF. It is shown that the COP-VAF leads to repeatable, physiologically meaningful measures that distinguish postural control differences in unipedal versus bipedal stance trials with and without vision in healthy individuals. More specifically, both a unipedal stance and lack of visual feedback increased initial values of the COP-VAF, magnitude of the first minimum, and diffusion coefficient, particularly in contrast to bipedal stance trials with open eyes. Use of a stochastic postural control model, based on an Ornstein-Uhlenbeck process that accounts for natural weight-shifts, suggests an increase in spring constant and decreased damping coefficient when fitted to experimental data. This work suggests that we can further extend our understanding of the underlying mechanisms behind postural control in quiet stance under varying stance conditions using the COP-VAF and provides a tool for quantifying future neurorehabilitative interventions.

Characterisation of whisker control in the California sea lion (Zalophus californianus) during a complex, dynamic sensorimotor task

Studies in pinniped whisker use have shown that their whiskers are extremely sensitive to tactile and hydrodynamic signals. While pinnipeds position their whiskers on to objects and have some control over their whisker protractions, it has always been thought that head movements are more responsible for whisker positioning than the movement of the whiskers themselves. This study uses ball balancing, a dynamic sensorimotor skill that is often used in human and robotic coordination studies, to promote sea lion whisker movements during the task. For the first time, using tracked video footage, we show that sea lion whisker movements respond quickly (26.70 ms) and mirror the movement of the ball, much more so than the head. We show that whisker asymmetry and spread are both altered to help sense and control the ball during balancing. We believe that by designing more dynamic sensorimotor tasks we can start to characterise the active nature of this specialised sensory system in pinnipeds.

Multiplicative-cascade dynamics in pole balancing

Pole balancing is a key task for probing the prospective control that organisms must engage in for purposeful action. The temporal structure of pole-balancing behaviors will reflect the on-line operation of control mechanisms needed to maintain an upright posture. In this study, signatures of multifractality are sought and found in time series of the vertical angle of a pole balanced on the fingertip. Comparisons to surrogate time series reveal multiplicative-cascade dynamics and interactivity across scales. In addition, simulations of a pole-balancing model generating on-off intermittency [J. L. Cabrera and J. G. Milton, Phys. Rev. Lett. 89, 158702 (2002)] were analyzed. Evidence of multifractality is also evident in simulations, though comparing simulated and participant series reveals a significantly greater contribution of cross-scale interactivity for the latter. These findings suggest that multiplicative-cascade dynamics are an extension of on-off intermittency and play a role in prospective coordination.

Sensory uncertainty and stick balancing at the fingertip

The effects of sensory input uncertainty, [Formula: see text], on the stability of time-delayed human motor control are investigated by calculating the minimum stick length, [Formula: see text], that can be stabilized in the inverted position for a given time delay, [Formula: see text]. Five control strategies often discussed in the context of human motor control are examined: three time-invariant controllers [proportional-derivative, proportional-derivative-acceleration (PDA), model predictive (MP) controllers] and two time-varying controllers [act-and-wait (AAW) and intermittent predictive controllers]. The uncertainties of the sensory input are modeled as a multiplicative term in the system output. Estimates based on the variability of neural spike trains and neural population responses suggest that [Formula: see text]-13 %. It is found that for this range of uncertainty, a tapped delay-line type of MP controller is the most robust controller. In particular, this controller can stabilize inverted sticks of the length balanced by expert stick balancers (0.25-0.5 m when [Formula: see text] s). However, a PDA controller becomes more effective when [Formula: see text]. A comparison between [Formula: see text] for human stick balancing at the fingertip and balancing on the rubberized surface of a table tennis racket suggest that friction likely plays a role in balance control. Measurements of [Formula: see text], and a variability of the fluctuations in the vertical displacement angle, an estimate of [Formula: see text], may make it possible to study the changes in control strategy as motor skill develops.

Dynamic reweighting of three modalities for sensor fusion

We simultaneously perturbed visual, vestibular and proprioceptive modalities to understand how sensory feedback is re-weighted so that overall feedback remains suited to stabilizing upright stance. Ten healthy young subjects received an 80 Hz vibratory stimulus to their bilateral Achilles tendons (stimulus turns on-off at 0.28 Hz), a ±1 mA binaural monopolar galvanic vestibular stimulus at 0.36 Hz, and a visual stimulus at 0.2 Hz during standing. The visual stimulus was presented at different amplitudes (0.2, 0.8 deg rotation about ankle axis) to measure: the change in gain (weighting) to vision, an intramodal effect; and a change in gain to vibration and galvanic vestibular stimulation, both intermodal effects. The results showed a clear intramodal visual effect, indicating a de-emphasis on vision when the amplitude of visual stimulus increased. At the same time, an intermodal visual-proprioceptive reweighting effect was observed with the addition of vibration, which is thought to change proprioceptive inputs at the ankles, forcing the nervous system to rely more on vision and vestibular modalities. Similar intermodal effects for visual-vestibular reweighting were observed, suggesting that vestibular information is not a “fixed” reference, but is dynamically adjusted in the sensor fusion process. This is the first time, to our knowledge, that the interplay between the three primary modalities for postural control has been clearly delineated, illustrating a central process that fuses these modalities for accurate estimates of self-motion.

Human stick balancing: an intermittent control explanation

There are two issues in balancing a stick pivoting on a finger tip (or mechanically on a moving cart): maintaining the stick angle near to vertical and maintaining the horizontal position within the bounds of reach or cart track. The (linearised) dynamics of the angle are second order (although driven by pivot acceleration), and so, as in human standing, control of the angle is not, by itself very difficult. However, once the angle is under control, the position dynamics are, in general, fourth order. This makes control quite difficult for humans (and even an engineering control system requires careful design). Recently, three of the authors have experimentally demonstrated that humans control the stick angle in a special way: the closed-loop inverted pendulum behaves as a non-inverted pendulum with a virtual pivot somewhere between the stick centre and tip and with increased gravity. Moreover, they suggest that the virtual pivot lies at the radius of gyration (about the mass centre) above the mass centre. This paper gives a continuous-time control-theoretical interpretation of the virtual-pendulum approach. In particular, by using a novel cascade control structure, it is shown that the horizontal control of the virtual pivot becomes a second-order problem which is much easier to solve than the generic fourth-order problem. Hence, the use of the virtual pivot approach allows the control problem to be perceived by the subject as two separate second-order problems rather than a single fourth-order problem, and the control problem is therefore simplified. The theoretical predictions are verified using the data previously presented by three of the authors and analysed using a standard parameter estimation method. The experimental data indicate that although all subjects adopt the virtual pivot approach, the less expert subjects exhibit larger amplitude angular motion and poorly controlled translational motion. It is known that human control systems are delayed and intermittent, and therefore, the continuous-time strategy cannot be correct. However, the model of intermittent control used in this paper is based on the virtual pivot continuous-time control scheme, handles time delays and moreover masquerades as the underlying continuous-time controller. In addition, the event-driven properties of intermittent control can explain experimentally observed variability.

Acceleration feedback improves balancing against reflex delay

A model for human postural balance is considered in which the time-delayed feedback depends on position, velocity and acceleration (proportional-derivative-acceleration (PDA) feedback). It is shown that a PDA controller is equivalent to a predictive controller, in which the prediction is based on the most recent information of the state, but the control input is not involved into the prediction. A PDA controller is superior to the corresponding proportional-derivative controller in the sense that the PDA controller can stabilize systems with approximately 40 per cent larger feedback delays. The addition of a sensory dead zone to account for the finite thresholds for detection by sensory receptors results in highly intermittent, complex oscillations that are a typical feature of human postural sway.

Balancing on tightropes and slacklines

Balancing on a tightrope or a slackline is an example of a neuromechanical task where the whole body both drives and responds to the dynamics of the external environment, often on multiple timescales. Motivated by a range of neurophysiological observations, here we formulate a minimal model for this system and use optimal control theory to design a strategy for maintaining an upright position. Our analysis of the open and closed-loop dynamics shows the existence of an optimal rope sag where balancing requires minimal effort, consistent with qualitative observations and suggestive of strategies for optimizing balancing performance while standing and walking. Our consideration of the effects of nonlinearities, potential parameter coupling and delays on the overall performance shows that although these factors change the results quantitatively, the existence of an optimal strategy persists.

The delayed and noisy nervous system: implications for neural control

Recent advances in the study of delay differential equations draw attention to the potential benefits of the interplay between random perturbations ('noise') and delay in neural control. The phenomena include transient stabilizations of unstable steady states by noise, control of fast movements using time-delayed feedback and the occurrence of long-lived delay-induced transients. In particular, this research suggests that the interplay between noise and delay necessitates the use of intermittent, discontinuous control strategies in which corrective movements are made only when controlled variables cross certain thresholds. A potential benefit of such strategies is that they may be optimal for minimizing energy expenditures associated with control. In this paper, the concepts are made accessible by introducing them through simple illustrative examples that can be readily reproduced using software packages, such as XPPAUT.

Time delay in physiological systems: analyzing and modeling its impact

This article examines the functional and clinical impact of time delays that arise in human physiological systems, especially control systems. An overview of the mathematical and physiological contexts for considering time delays will be illustrated, from the system level to cell level, by examining models that incorporate time delays. This examination will highlight how such delays in combination with other system structures and parameters influence system dynamics. Model analysis that reveals the influence of delays can also reveal related physiological effects which may have medical consequences and clinical applications.

Stochastic two-delay differential model of delayed visual feedback effects on postural dynamics

We report on experiments and modelling involving the 'visuo-postural control loop' in the upright stance. We experimentally manipulated an artificial delay to the visual feedback during standing, presented at delays ranging from 0 to 1 s in increments of 250 ms. Using stochastic delay differential equations, we explicitly modelled the centre-of-pressure (COP) and centre-of-mass (COM) dynamics with two independent delay terms for vision and proprioception. A novel 'drifting fixed point' hypothesis was used to describe the fluctuations of the COM with the COP being modelled as a faster, corrective process of the COM. The model was in good agreement with the data in terms of probability density functions, power spectral densities, short- and long-term correlations (Hurst exponents) as well the critical time between the two ranges.

Delayed feedback of sampled higher derivatives

Systems where the present rate of change of the state depends on the past values of the higher rates of change of the state are described by so-called advanced functional differential equations (AFDEs). In an AFDE, the highest derivative of the state-space coordinate appears with delayed argument only. The corresponding linearized equations are always unstable with infinitely many unstable poles, and are rarely related to practical applications due to their inherently implicit nature. In this paper, one of the simplest AFDEs, a linear scalar first-order system, is considered with the delayed feedback of the second derivative of the state in the presence of sampling in the feedback loop (i.e. in the case of digital control). It is shown that sampling of the feedback may stabilize the originally infinitely unstable system for certain parameter combinations. The result explains the stable behaviour of certain dynamical systems with feedback delay in the highest derivative.

Balancing with vibration: a prelude for "drift and act" balance control

Stick balancing at the fingertip is a powerful paradigm for the study of the control of human balance. Here we show that the mean stick balancing time is increased by about two-fold when a subject stands on a vibrating platform that produces vertical vibrations at the fingertip (0.001 m, 15-50 Hz). High speed motion capture measurements in three dimensions demonstrate that vibration does not shorten the neural latency for stick balancing or change the distribution of the changes in speed made by the fingertip during stick balancing, but does decrease the amplitude of the fluctuations in the relative positions of the fingertip and the tip of the stick in the horizontal plane, A(x,y). The findings are interpreted in terms of a time-delayed "drift and act" control mechanism in which controlling movements are made only when controlled variables exceed a threshold, i.e. the stick survival time measures the time to cross a threshold. The amplitude of the oscillations produced by this mechanism can be decreased by parametric excitation. It is shown that a plot of the logarithm of the vibration-induced increase in stick balancing skill, a measure of the mean first passage time, versus the standard deviation of the A(x,y) fluctuations, a measure of the distance to the threshold, is linear as expected for the times to cross a threshold in a stochastic dynamical system. These observations suggest that the balanced state represents a complex time-dependent state which is situated in a basin of attraction that is of the same order of size. The fact that vibration amplitude can benefit balance control raises the possibility of minimizing risk of falling through appropriate changes in the design of footwear and roughness of the walking surfaces.

Delay effects in brain dynamics. Introduction

This brief introductory paper reviews the methods and the results presented in the special issue. The general destabilizing effects of time delays in nonlinear dynamical systems are summarized and some similarities in the philosophical approaches of neural systems research in distinct disciplines are pointed out. All the invited papers focus on the central role of time delays in the dynamics of neural systems. The research contributions are set in order according to the increasing number of neurons involved in the corresponding study from a couple of neurons through neural fields to populations and clusters of neurons.