Taishin Nomura

A model of postural control in quiet standing: robust compensation of delay-induced instability using intermittent activation of feedback control.

The main purpose of this study is to compare two different feedback controllers for the stabilization of quiet standing in humans, taking into account that the intrinsic ankle stiffness is insufficient and that there is a large delay inducing instability in the feedback loop: 1) a standard linear, continuous-time PD controller and 2) an intermittent PD controller characterized by a switching function defined in the phase plane, with or without a dead zone around the nominal equilibrium state. The stability analysis of the first controller is carried out by using the standard tools of linear control systems, whereas the analysis of the intermittent controllers is based on the use of Poincaré maps defined in the phase plane. When the PD-control is off, the dynamics of the system is characterized by a saddle-like equilibrium, with a stable and an unstable manifold. The switching function of the intermittent controller is implemented in such a way that PD-control is ‘off’ when the state vector is near the stable manifold of the saddle and is ‘on’ otherwise. A theoretical analysis and a related simulation study show that the intermittent control model is much more robust than the standard model because the size of the region in the parameter space of the feedback control gains (P vs. D) that characterizes stable behavior is much larger in the latter case than in the former one. Moreover, the intermittent controller can use feedback parameters that are much smaller than the standard model. Typical sway patterns generated by the intermittent controller are the result of an alternation between slow motion along the stable manifold of the saddle, when the PD-control is off, and spiral motion away from the upright equilibrium determined by the activation of the PD-control with low feedback gains. Remarkably, overall dynamic stability can be achieved by combining in a smart way two unstable regimes: a saddle and an unstable spiral. The intermittent controller exploits the stabilizing effect of one part of the saddle, letting the system evolve by alone when it slides on or near the stable manifold; when the state vector enters the strongly unstable part of the saddle it switches on a mild feedback which is not supposed to impose a strict stable regime but rather to mitigate the impending fall. The presence of a dead zone in the intermittent controller does not alter the stability properties but improves the similarity with biological sway patterns. The two types of controllers are also compared in the frequency domain by considering the power spectral density (PSD) of the sway sequences generated by the models with additive noise. Different from the standard continuous model, whose PSD function is similar to an over-damped second order system without a resonance, the intermittent control model is capable to exhibit the two power law scaling regimes that are typical of physiological sway movements in humans.

Bounded stability of the quiet standing posture: an intermittent control model.

The paper presents a control model of body sway in quiet standing, which aims at achieving bounded stability by means of an intermittent control mechanism. Control bursts are generated when the current state vector exits an area of uncertainty around the reference point in the phase plane. This area is determined by the limited resolution of proprioceptive signals and the burst generation mechanism is predictive in the sense that it incorporates a rough, but working knowledge (internal model) of the biomechanics of the human inverted pendulum. We show that such a model, in spite of its simplicity and of the fact that it relies on very noisy measurements, is robust and can explain in a detailed way the measured sway patterns.

CellML metadata standards, associated tools and repositories

The development of standards for encoding mathematical models is an important component of model building and model sharing among scientists interested in understanding multi-scale physiological processes. CellML provides such a standard, particularly for models based on biophysical mechanisms, and a substantial number of models are now available in the CellML Model Repository. However, there is an urgent need to extend the current CellML metadata standard to provide biological and biophysical annotation of the models in order to facilitate model sharing, automated model reduction and connection to biological databases. This paper gives a broad overview of a number of new developments on CellML metadata and provides links to further methodological details available from the CellML website.

Possible functional roles of phase resetting during walking

Abstract. The walking rhythm is known to show phase shift or “reset” in response to external impulsive perturbations. We tried to elucidate functional roles of the phase reset possibly used for the neural control of locomotion. To this end, a system with a double pendulum as a simplified model of the locomotor control and a model of bipedal locomotion were employed and analyzed in detail. In these models, a movement corresponding to the normal steady-state walking was realized as a stable limit cycle solution of the system. Unexpected external perturbations applied to the system can push the state point of the system away from its limit cycle, either outside or inside the basin of attraction of the limit cycle. Our mathematical analyses of the models suggested functional roles of the phase reset during walking as follows. Function 1: an appropriate amount of the phase reset for a given perturbation can contribute to relocating the systems state point outside the basin of attraction of the limit cycle back to the inside. Function 2: it can also be useful to reduce the convergence time (the time necessary for the state point to return to the limit cycle). In experimental studies during walking of animals and humans, the reset of walking rhythm induced by perturbations was investigated using the phase transition curve (PTC) or the phase resetting curve (PRC) representing phase-dependent responses of the walking. We showed, for the simple double-pendulum model, the existence of the optimal phase control and the corresponding PTC that could optimally realize the aforementioned functions in response to impulsive force perturbations. Moreover, possible forms of PRC that can avoid falling against the force perturbations were predicted by the biped model, and they were compared with the experimentally observed PRC during human walking. Finally, physiological implications of the results were discussed.

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

Abstract. The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with Iext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations.

Specifications of insilicoML 1.0: a multilevel biophysical model description language.

An extensible markup language format, insilicoML (ISML), version 0.1, describing multi-level biophysical models has been developed and available in the public domain. ISML is fully compatible with CellML 1.0, a model description standard developed by the IUPS Physiome Project, for enhancing knowledge integration and model sharing. This article illustrates the new specifications of ISML 1.0 that largely extend the capability of ISML 0.1. ISML 1.0 can describe various types of mathematical models, including ordinary/partial differential/difference equations representing the dynamics of physiological functions and the geometry of living organisms underlying the functions. ISML 1.0 describes a model using a set of functional elements (modules) each of which can specify mathematical expressions of the functions. Structural and logical relationships between any two modules are specified by edges, which allow modular, hierarchical, and/or network representations of the model. The role of edge-relationships is enriched by key words in order for use in constructing a physiological ontology. The ontology is further improved by the traceability of history of the models development and by linking between different ISML models stored in the models database using meta-information. ISML 1.0 is designed to operate with a model database and integrated environments for model development and simulations for knowledge integration and discovery.

Classifying lower limb dynamics in Parkinson’s disease

To classify lower limb dynamics in patients with Parkinsons disease (PD), we conducted a clinical study by using pedaling exercise.Twenty-seven patients with idiopathic PD were included in this study. We measured rotational velocities of pedals during pedaling movements with a newly developed ergometer. The velocity waveforms exhibited different characteristics among patients, which could be categorized into four different clusters. In cluster 1, the amplitude on each side was constant and the relative phase was locked at 180 degrees. The pattern was the same as seen in normal subjects. In cluster 2, the amplitude on each side was constant, but the relative phase was locked at 90 degrees. In cluster 3, the amplitude on each side was modulated, and the relative phase drifted monotonously from 0 to 360 degrees during pedaling cycles. In cluster 4, the amplitude on each side was synchronously and irregularly modulated, and the relative phase fluctuated with intermittent spike-like decrement. In order to evaluate, the correlation between pattern and severity of PD, we divided 13 patients, who underwent measurement of pedaling patterns more than three times, into three groups, and found that the abnormal coordination pattern correlated with the presence of freezing phenomenon in patients with PD. Our clinical analysis may contribute in analyzing and classifying the dynamics of PD.

GLOBAL BIFURCATION STRUCTURE OF A BONHOEFFER-VAN DER POL OSCILLATOR DRIVEN BY PERIODIC PULSE TRAINS - COMPARISON WITH DATA FROM A PERIODICALLY INHIBITED BIOLOGICAL PACEMAKER

The Bonhoeffer-van der Pol (BVP) oscillator is a valuable dynamical system model of pacemaker neurons. Isochrons, phase transition curves (PTC), and two dimensional bifurcation diagrams served to analyze the neurons response to periodic pulse stimuli. Responses are described and explained in terms of the nonlinear dynamical system theory. An important issue in the generation of spikes by pacemaker neurons is the existence of both slow and fast dynamics in the state points trajectory in the phase plane. It is this feature in particular that makes the BVP oscillator a faithful model of living pacemaker neurons. Comparison of the models responses with those of a living pacemaker was based also on return maps of interspike intervals. Analyzed in detail were the complex discharges called ‘stammering’ which involve interspike intervals that arise unpredictably and exhibit histograms with several modes separated by the equal intervals.

Dynamic stability and phase resetting during biped gait

Dynamic stability during periodic biped gait in humans and in a humanoid robot is considered. Here gait systems of human neuromusculoskeletal system and a humanoid are simply modeled while keeping their mechanical properties plausible. We prescribe periodic gait trajectories in terms of joint angles of the models as a function of time. The equations of motion of the models are then constrained by one of the prescribed gait trajectories to obtain types of periodically forced nonlinear dynamical systems. Simulated gait of the models may or may not fall down during gait, since the constraints are made only for joint angles of limbs but not for the motion of the body trunk. The equations of motion can exhibit a limit cycle solution (or an oscillatory solution that can be considered as a limit cycle practically) for each selected gait trajectory, if an initial condition is set appropriately. We analyze the stability of the limit cycle in terms of Poincaré maps and the basin of attraction of the limit cycle in order to examine how the stability depends on the prescribed trajectory. Moreover, the phase resetting of gait rhythm in response to external force perturbation is modeled. Since we always prescribe a gait trajectory in this study, reacting gait trajectories during the phase resetting are also prescribed. We show that an optimally prescribed reacting gait trajectory with an appropriate amount of the phase resetting can increase the gait stability. Neural mechanisms for generation and modulation of the gait trajectories are discussed.

A coupled oscillator model of disordered interlimb coordination in patients with Parkinson's disease

Abstract. Coordination between the left and right limbs during cyclic movements, which can be characterized by the amplitude of each limbs oscillatory movement and relative phase, is impaired in patients with Parkinsons disease (PD). A pedaling exercise on an ergometer in a recent clinical study revealed several types of coordination disorder in PD patients. These include an irregular and burst-like amplitude modulation with intermittent changes in its relative phase, a typical sign of chaotic behavior in nonlinear dynamical systems. This clinical observation leads us to hypothesize that emergence of the rhythmic motor behaviors might be concerned with nonlinearity of an underlying dynamical system. In order to gain insight into this hypothesis, we consider a simple hard-wired central pattern generator model consisting of two identical oscillators connected by reciprocal inhibition. In the model, each oscillator acts as a neural half-center controlling movement of a single limb, either left or right, and receives a control input modeling a flow of descending signals from higher motor centers. When these two control inputs are tonic-constant and identical, the model has left-right symmetry and basically exhibits ordered coordination with an alternating periodic oscillation. We show that, depending on the intensities of these two control inputs and on the difference between them that introduces asymmetry into the model, the model can reproduce several behaviors observed in the clinical study. Bifurcation analysis of the model clarifies two possible mechanisms for the generation of disordered coordination in the model: one is the spontaneous symmetry-breaking bifurcation in the model with the left-right symmetry. The other is related to the degree of asymmetry reflecting the difference between the two control inputs. Finally, clinical implications by the models dynamics are briefly discussed.