The generation of handwriting with delta-lognormal synergies

This paper presents a handwriting generation model that takes advantage of the asymptotic impulse response of neuromuscular networks to produce and control complex two-dimensional synergistic movements. A parametric definition of a ballistic stroke in the context of the kinematic theory of rapid human movements is given. Two types of parameters are used: command and system parameters. The first group provides a representation of the action plan while the second takes into account the temporal properties of the neuromuscular systems executing that plan. Handwriting is described as the time superimposition of basic discontinuous strokes that results in a continuous summation of delta-lognormal velocity vectors. The model leads to trajectory reconstruction, both in the spatial and in the kinematic domain. According to this new paradigm, the angular velocity does not have to be controlled independently and continuously; it naturally emerges from the vectorial summation process. Several psychophysical phenomena related to two-dimensional movements are explained and analyzed in the context of the model: the speed/accuracy trade-offs, spatial scaling, the isochrony principle, the two-thirds power law, effector independence, etc. The overall approach also shows how basic handwriting characteristics (dimension, slant, baseline, shape, etc.) are affected and controlled using an action plan made up of virtual targets fed into a neuromuscular synergy that is governed by a delta-lognormal law. Received: 22 July 1996 / Accepted in revised form: 15 September 1997     

A kinematic theory of rapid human movements: Part III. Kinetic outcomes

This paper describes the kinematic and kinetic properties of simple rapid movements using a single and unique framework based on a delta-lognormal law (Plamondon 1993a,b, 1995a,b). Predictions concerning isotonic measurements are made using the properties of acceleration profiles, as described by the first time derivative of the delta-lognormal law. Predictions dealing with isometric measurements are directly analyzed using the delta-lognormal law, after demonstrating the experimental equivalence between isometric forces and virtual velocity profiles. The theory is also used to make statistical predictions about the variability of numerous kinematic and kinetic variables. The overall approach can be viewed as if, at some level of representation, the central nervous system were planning, executing and evaluating simple rapid movements in terms of momentum and energy instead of forces. The unifying perspective provided by the theory constitutes a powerful tool with which to study and analyze movements under numerous experimental conditions, using a single analytical law. Received: 13 November 1996 / Accepted in revised form: 6 November 1997     

Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays

This paper studies a non-autonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are "profitless". Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka-Volterra systems.     

非自治Lotka-Volterra型捕食扩散系统中有害时滞对该系统持久性的影响

研究了时滞对一类非自治Lotka-Volterra型捕食扩散系统的影响,该系统由n个斑块组成,食饵种群可以在斑块间迁移,而摘食者限制在某一个斑块不能扩散．我们假设密度制约项系数并不总是严格正的．通过运用比较定理及时滞泛函微分方程的基本原理,分两种情况表明了在一定条件下系统是一致持久的．两种情况的结果表明时滞的引入和变化即可能是“有害”,也可能是”无害”．进一步还说明了系统在一致持久性的条件下至少存在一个正周期解．这些结果是对已知的非自治Lotka-Volterra系统的一些结果的推广与改进．     

Neural Computation via Neural Geometry: A Place Code for Inter-whisker Timing in the Barrel Cortex?

The place theory proposed by Jeffress (1948) is still the dominant model of how the brain represents the movement of sensory stimuli between sensory receptors. According to the place theory, delays in signalling between neurons, dependent on the distances between them, compensate for time differences in the stimulation of sensory receptors. Hence the location of neurons, activated by the coincident arrival of multiple signals, reports the stimulus movement velocity. Despite its generality, most evidence for the place theory has been provided by studies of the auditory system of auditory specialists like the barn owl, but in the study of mammalian auditory systems the evidence is inconclusive. We ask to what extent the somatosensory systems of tactile specialists like rats and mice use distance dependent delays between neurons to compute the motion of tactile stimuli between the facial whiskers (or 'vibrissae'). We present a model in which synaptic inputs evoked by whisker deflections arrive at neurons in layer 2/3 (L2/3) somatosensory 'barrel' cortex at different times. The timing of synaptic inputs to each neuron depends on its location relative to sources of input in layer 4 (L4) that represent stimulation of each whisker. Constrained by the geometry and timing of projections from L4 to L2/3, the model can account for a range of experimentally measured responses to two-whisker stimuli. Consistent with that data, responses of model neurons located between the barrels to paired stimulation of two whiskers are greater than the sum of the responses to either whisker input alone. The model predicts that for neurons located closer to either barrel these supralinear responses are tuned for longer inter-whisker stimulation intervals, yielding a topographic map for the inter-whisker deflection interval across the surface of L2/3. This map constitutes a neural place code for the relative timing of sensory stimuli.     

Dynamic stability of a human standing on a balance board

The neuromuscular system used to stabilize upright posture in humans is a nonlinear dynamical system with time delays. The analysis of this system is important for improving balance and for early diagnosis of neuromuscular disease. In this work, we study the dynamic coupling between the neuromuscular system and a balance board—an unstable platform often used to improve balance in young athletes, and older or neurologically impaired patients. Using a simple inverted pendulum model of human posture on a balance board, we describe a surprisingly broad range of divergent and oscillatory CoP/CoM responses associated with instabilities of the upright equilibrium. The analysis predicts that a variety of sudden changes in the stability of upright postural equilibrium occurs with slow continuous deterioration in balance board stiffness, neuromuscular gain, and time delay associated with the changes in proprioceptive/vestibular/visual-neuromuscular feedback. The analysis also provides deeper insight into changes in the control of posture that enable stable upright posture on otherwise unstable platforms.     

Adjustment of the human arm viscoelastic properties to the direction of reaching

The viscoelastic properties of the human arm were measured by means of short force perturbations during fast reaching movements in two orthogonal directions. A linear spring model with time delay described the neuromuscular system of the human arm. The obtained viscoelastic parameters ensured movement stability in spite of the time delay of 50 ms. The stiffness and viscosity ellipses appeared to be predominantly orthogonal to the movement direction, which reduced the effect of force perturbation in the direction orthogonal to the reaching movement. Thus, it can be argued that the viscoelastic properties of the neuromuscular system of the human arm are adjusted to the direction of movement according to a “path preserving” strategy, which minimizes the deviation of the movement path from a straight line, when exposed to an unexpected external force.     

Effects of reflex delays on postural control during unstable seated balance

Patients with low-back pain (LBP) exhibit longer trunk muscle reflex latencies and poorer postural control than healthy individuals. We hypothesized that balance during a simulated postural control task would become impaired when the delays exhibited by LBP patients were incorporated into neuromuscular control. The task chosen for this investigation was seated balancing, which emphasizes trunk muscles’ contribution in postural control. This task was modeled in Simulink? as a fourth order linearized dynamic system with feedback delays. Optimization (minimizing error between experimental and model data) of state variables was used to determine neuromuscular control parameters. Experimental data were obtained from 7 subjects during 5 perturbation trials while balancing on the seat with eyes closed. Model accuracy, reflecting the ability of the model to capture the dynamics of seated balance, was correlated with seated balance performance (r=0.91, p<0.001). To minimize the risk of erroneous findings from inaccurate modeling, only the best five balancers’ data were used for hypothesis testing. In these five subjects, feedback delays in modeled neuromuscular control were increased to determine their effect on task stability, trunk displacement and trunk moment. Simulations showed that longer delays found in LBP, in general, did not produce unstable balancing, but did result in increased trunk displacement (p<0.001) and trunk moment (p=0.001). This impairment in neuromuscular control in chronic LBP patients could possibly exacerbate their condition by increasing tissue strain (more spinal displacement) and stress (more spinal loading).     

Concurrent adaptation of force and impedance in the redundant muscle system

This article examines the validity of a model to explain how humans learn to perform movements in environments with novel dynamics, including unstable dynamics typical of tool use. In this model, a simple rule specifies how the activation of each muscle is adapted from one movement to the next. Simulations of multijoint arm movements with a neuromuscular plant that incorporates neural delays, reflexes, and signal-dependent noise, demonstrate that the controller is able to compensate for changing internal or environment dynamics and noise properties. The computational model adapts by learning both the appropriate forces and required limb impedance to compensate precisely for forces and instabilities in arbitrary directions with patterns similar to those observed in motor learning experiments. It learns to regulate reciprocal activation and co-activation in a redundant muscle system during repeated movements without requiring any explicit transformation from hand to muscle space. Independent error-driven change in the activation of each muscle results in a coordinated control of the redundant muscle system and in a behavior that reduces instability, systematic error, and energy.     

A kinematic theory of rapid human movements

 This paper proposes a kinematic theory that can be used to study and analyze rapid human movements. It describes a synergy in terms of the agonist and antagonist neuromuscular systems involved in the production of these movements. It is shown that these systems have a log-normal impulse response that results from the limiting behavior of a large number of interdependent neuromuscular networks, as predicted by the central limit theorem. The delta log-normal law that follows from this model is very general and can reproduce almost perfectly the complete velocity patterns of an end-effector. The theory accounts for the invariance and rescalability of these patterns, as well as for the various observations that have been reported concerning the change in maximum and mean velocities, time to maximum velocity, etc., under different experimental conditions. Movement time, load effects, and control strategies are discussed in a companion paper. Received: 15 February 1993/Accepted in revised form: 15 September 1994     

Steady states and outbreaks of two-phase nonlinear age-structured model of population dynamics with discrete time delay

In this paper we study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality. Both phases are modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE. The obtained system is studied both theoretically and numerically. Three different regimes of population dynamics for asymptotically stable states of autonomous systems are obtained in numerical experiments for the different initial values of population density. The quasi-periodical travelling wave solutions are studied numerically for the autonomous system with the different values of time delays and for the system with oscillating death rate and birth modulus. In both cases it is observed three types of travelling wave solutions: harmonic oscillations, pulse sequence and single pulse.     

Modeling stochastic noise in gene regulatory systems

The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation and simulation methods are essential. However, there is still a lack of consensus about the best way to carry these out. To help clarify the situation, we review Master equation models of gene regulation, theoretical approximations based on an expansion method due to N.G. van Kampen and R. Kubo, and simulation algorithms due to D.T. Gillespie and P. Langevin. Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation. Additional theory, also due to van Kampen, describes the asymptotic behavior of multistable systems. To support and illustrate the theory and provide further insight into the complex behavior of multistable systems, we perform a detailed simulation study comparing the various approximation and simulation methods applied to synthetic gene regulatory systems with various qualitative characteristics. The simulation studies show that for large stochastic systems with a single steady-state, deterministic models are quite accurate, since the probability distribution of the solution has a single peak tracking the deterministic trajectory whose variance is inversely proportional to the system size. In multistable stochastic systems, large fluctuations can cause individual trajectories to escape from the domain of attraction of one steady-state and be attracted to another, so the system eventually reaches a multimodal probability distribution in which all stable steady-states are represented proportional to their relative stability. However, since the escape time scales exponentially with system size, this process can take a very long time in large systems.     

Feedback and delays in neurological diseases: A modeling study using gynamical systems

Numerous regulatory mechanisms in motor control involve the presence of time delays in the controlled behavior of the system. Experimentally, we have shown that an increase of the time delay in visual feedback induces different oscillations in control subjects and in patients with neurological diseases during the performance of a simple compensatory tracking task. A preliminary model is proposed to describe the oscillations observed in control subjects and in patients with neurological diseases. The influence of delays in two feedback loops are the main components of the motor control circuitry involved in this task and are studied from an analytical and physiological perspective. We analytically determine the influence in the model of each of these delays on the stability of the finger position. In addition, the influence of stochastic elements (“noise”) in the modeling equation is seen to contribute qualitatively to a more accurate reproduction of experimental traces in patients with Parkinson's disease but not in patients with cerebellar disease.     

Learning combined feedback and feedforward control of a musculoskeletal system

The goal of this paper is the learning of neuromuscular control, given the following necessary conditions: (1) time delays in the control loop, (2) non-linear muscle characteristics, (3) learning of feedforward and feedback control, (4) possibility of feedback gain modulation during a task. A control system and learning methodology that satisfy those conditions is given. The control system contains a neural network, comprising both feedforward and feedback control. The learning method is backpropagation through time with an explicit sensitivity model. Results will be given for a one degree of freedom arm with two muscles. Good control results are achieved which compare well with experimental data. Analysis of the controller shows that significant differences in controller characteristics are found if the loop delays are neglected. During a control task the system shows feedback gain modulation, similar to experimentally found reflex gain modulation during rapid voluntary contraction. If only limited feedback information is available to the controller the system learns to co-contract the antagonistic muscle pair. In this way joint stiffness increases and stable control is more easily maintained. Received: 7 November 1995 / Accepted in revised form: 13 February 1996     

Entrainment to Periodic Initiation and Transition Rates in a Computational Model for Gene Translation

Periodic oscillations play an important role in many biomedical systems. Proper functioning of biological systems that respond to periodic signals requires the ability to synchronize with the periodic excitation. For example, the sleep/wake cycle is a manifestation of an internal timing system that synchronizes to the solar day. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. Entrainment is also important in synthetic biology. For example, connecting several artificial biological systems that entrain to a common clock may lead to a well-functioning modular system. The cell-cycle is a periodic program that regulates DNA synthesis and cell division. Recent biological studies suggest that cell-cycle related genes entrain to this periodic program at the gene translation level, leading to periodically-varying protein levels of these genes. The ribosome flow model (RFM) is a deterministic model obtained via a mean-field approximation of a stochastic model from statistical physics that has been used to model numerous processes including ribosome flow along the mRNA. Here we analyze the RFM under the assumption that the initiation and/or transition rates vary periodically with a common period . We show that the ribosome distribution profile in the RFM entrains to this periodic excitation. In particular, the protein synthesis pattern converges to a unique periodic solution with period . To the best of our knowledge, this is the first proof of entrainment in a mathematical model for translation that encapsulates aspects such as initiation and termination rates, ribosomal movement and interactions, and non-homogeneous elongation speeds along the mRNA. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and may suggest a new approach for re-engineering genetic systems to obtain a desired, periodic, protein synthesis rate.     

Optimal coordination and control of posture and locomotion.

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.     

Duration time of a one-dimensional random walk as a function of the energies of the intermediate states: application for dissociation and relaxation processes in DNA hybrids.

Kinetic parameters of macromolecular systems are important for their function in vitro and in vivo. These parameters describe how fast the system dissociates (the characteristic dissociation time), and how fast the system reaches equilibrium (characteristic relaxation time). For many macromolecular systems, the transitions within the systems are described as a random walk through a number of states with various free energies. The rate of transition between two given states within the system is characterized by the average time which passes between starting the movement from one state, and reaching the other state. This time is referred to as the mean first-passage time between two given states. The characteristic dissociation and relaxation times of the system depend on the first-passages times between the states within the system. Here, for a one-dimensional random walk we derived an equation, which connects the mean first-passage time between two states with the free energies of the states within the system. We also derived the general equation, which is not restricted to one-dimensional systems, connecting the relaxation time of the system with the first-passage times between states. The application of these equations to DNA branch migration, DNA structural transitions and other processes is discussed.     

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.     

Chemical Organisation Theory

Complex dynamical reaction networks consisting of many components that interact and produce each other are difficult to understand, especially, when new component types may appear and present component types may vanish completely. Inspired by Fontana and Buss (Bull. Math. Biol., 56, 1–64) we outline a theory to deal with such systems. The theory consists of two parts. The first part introduces the concept of a chemical organisation as a closed and self-maintaining set of components. This concept allows to map a complex (reaction) network to the set of organisations, providing a new view on the system’s structure. The second part connects dynamics with the set of organisations, which allows to map a movement of the system in state space to a movement in the set of organisations. The relevancy of our theory is underlined by a theorem that says that given a differential equation describing the chemical dynamics of the network, then every stationary state is an instance of an organisation. For demonstration, the theory is applied to a small model of HIV-immune system interaction by Wodarz and Nowak (Proc. Natl. Acad. USA, 96, 14464–14469) and to a large model of the sugar metabolism of E. Coli by Puchalka and Kierzek (Biophys. J., 86, 1357–1372). In both cases organisations where uncovered, which could be related to functions. Both authors contributed equally.