Traveling-wave pattern generator controls movement and organization of sensory feedback in a spinal cord model

 A traveling wave in a two-dimensional spinal cord model constitutes a stable pattern generator for quadruped gaits. In the context of the somatotopic organization of the spinal cord, this pattern generator is sufficient to generate stable locomotive limb trajectories. The elastic properties of muscles alone, providing linear negative feedback, are sufficient to stabilize stance and locomotion in the presence of perturbative forces. We further show that such a pattern generator is capable of organizing sensory processing in the spinal cord. A single-layer perceptron was trained to associate the sensory feedback from the limb (coding force, length, and change of length for each muscle) with the two-dimensional activity profile of the traveling wave. This resulted in a well-defined spatial organization of the connections within the spinal network along a rostrocaudal axis. The spinal network driven by peripheral afferents alone supported autonomous locomotion in the positive feedback mode, whereas in the negative feedback mode stance was stabilized in response to perturbations. Systematic variation of a parameter representing the effect of gamma-motor neurons on muscle spindle activity in our model led to a corresponding shift of limb position during stance and locomotion, resulting in a systematic displacement alteration of foot positions. Received: 30 July 2001 / Accepted in revised form: 17 April 2002 Correspondence to: A. Kaske (e-mails: alexander.kaske@mtc.ki.se, alexander.kaske@vglab.com)     

Bifurcation and stability analysis in musculoskeletal systems: a study in human stance

Reflexes are important in the control of such daily activities as standing and walking. The goal of this study is to establish how reflexive feedback of muscle length, velocity, and force can lead to stable equilibria (i.e., posture) and limit cycles (e.g., ankle clonus and gait). The influence of stretch reflexes on the behavior and stability of musculoskeletal systems was examined using a model of human stance. We computed branches of fold and Hopf bifurcations by numerical bifurcation analysis of the model. These fold and Hopf branches divide the parameter space, constructed by the reflexive feedback gains, into regions of different behavior: unstable posture, stable posture, and stable limit cycles. These limit cycles correspond to a neural deficiency, termed ankle clonus. We also linked bifurcation analysis to known biomechanical concepts by linearizing the model: the fold branch corresponds to zero ankle stiffness and defines the minimal muscle length feedback necessary for stable posture; the Hopf branch is related to unstable reflex loops. Crossing the Hopf branch can lead to the above-mentioned stable limit cycles. The Hopf branch reduces with increasing time delays, making the subjects posture more susceptible to unstable reflex loops. This might be one of the reasons why elderly people, or those with injuries to the central nervous system, often have trouble with standing and other posture tasks. The influence of cocontraction and force feedback on the behavior of the posture model was also investigated. An increase in cocontraction leads to an increase in ankle stiffness (i.e., intrinsic muscle stiffness) and a decrease in the effective reflex loop gain. On the one hand, positive force feedback increases the ankle stiffness (i.e., intrinsic and reflexive muscle stiffness); on the other hand it makes the posture more susceptible to unstable reflex loops. For negative force feedback, the opposite is true. Finally, we calculated areas of reflex gains for perturbed stance and quiet stance in healthy subjects by fitting the model to data from the literature. The overlap of these areas of reflex gains could indicate that stretch reflexes are the major control mechanisms in both quiet and perturbed stance. In conclusion, this study has successfully combined bifurcation analysis with the more common biomechanical concepts and tools to determine the influence of reflexes on the stability and quality of stance. In the future, we will develop this line of research to look at rhythmic tasks, such as walking.     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

Stabilizing function of antagonistic neuromusculoskeletal systems: an analytical investigation

 Under normal conditions human walking or running consists of stable cyclic movements. Minor perturbances such as a stone or a pothole do not disrupt the cycle, and the system returns to its prescribed trajectory. We investigated whether a pair of antagonistic muscles is able to stabilize the movement without neuronal feedback. The human is represented by a model consisting of a massless two-segment linkage system (leg) topped by a point mass. Both the extensor and flexor muscles are described by a Hill-type muscle model. Conditions for stability are calculated analytically based on the Ljapunov Theory and the results are illustrated by numerical examples. The activation functions of both the extensor and flexor muscles can be calculated for a prescribed trajectory to maintain the self-stabilizing ability of such a system. Experimental evidence supports the prediction. Our investigation shows that a moving center of rotation of the kneejoint, a biarticular flexor muscle group, the force-velocity relation, and the ascending limb of the force-length relation improves the self-stabilizing ability of human movement. Received: 24 March 1999 / Accepted: 20 February 2003 / Published online: 22 May 2003 Correspondence to: H. Wagner (e-mail: heiko.wagner@uni-jena.de, Tel.:+49-3641-945706) Acknowledgements. The authors wish to express their gratitude to Arnd Friedrichs and Lars Wendrock for their contribution to data acquisition and analysis. Veit Wank is acknowledged for making available for our use the x-ray photos of several knee joints. We would like to thank Anna N. Ahn for her very detailed and helpful comments. With support from DFG (Innovationskolleg “Motion Systems”).     

Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.     

Stabilizing PID controllers for a single-link biomechanical model with position, velocity, and force feedback

In this paper we address the problem of PID stabilization of a single-link inverted pendulum-based biomechanical model with force feedback, two levels of position and velocity feedback, and with delays in all the feedback loops. The novelty of the proposed model lies in its physiological relevance, whereby both small and medium latency sensory feedbacks from muscle spindle (MS), and force feedback from Golgi tendon organ (GTO) are included in the formulation. The biomechanical model also includes active and passive viscoelastic feedback from Hill-type muscle model and a second-order low-pass function for muscle activation. The central nervous system (CNS) regulation of postural movement is represented by a proportional-integral-derivative (PID) controller. Padé approximation of delay terms is employed to arrive at an overall rational transfer function of the biomechanical model. The Hermite-Biehler theorem is then used to derive stability results, leading to the existence of stabilizing PID controllers. An algorithm for selection of stabilizing feedback gains is developed using the linear matrix inequality (LMI) approach.     

Stimulus-dependent onset latency of inhibitory recurrent activity

 This paper gives an explanation for the experimentally observed onset latencies of the inhibitory responses that vary from a few milliseconds to hundreds of milliseconds in systems where the conduction delays are only several milliseconds in the feedback pathways. To do this we use a simple mathematical model. The model consists of two delay differential equations (DDE) where the nonlinear relation between the postsynaptic potential and the firing frequency of the neuron population arises from the stoichiometry of the transmitter-receptor kinetics. The parameters of the model refer to the hippocampal feedback system, and the modeling results are compared with corresponding experiments. Received: 31 May 2002 / Accepted: 5 February 2003 / Published online: 20 May 2003 Correspondence to: C. Hauptmann (e-mail: chauptma@cnd.mcgill.ca) Acknowledgements. We thank Prof. Krnjevic and Prof. Glavinovic for helpful and extensive discussions about this problem. This work was supported by MITACS (Canada), the Natural Sciences and Engineering Research Council (NSERC grant OGP-0036920, Canada), the Alexander von Humboldt Stiftung, Le Fonds pour la Formation de Chercheurs et l'Aide à la Recherche (FCAR grant 98ER1057, Québec), and the Leverhulme Trust (U.K.).     

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).     

Muscle fatigue does not lead to increased instability of upper extremity repetitive movements

Muscle fatigue alters neuromuscular responses. This may lead to increased sensitivity to perturbations and possibly to subsequent injury risk. We studied the effects of muscle fatigue on movement stability during a repetitive upper extremity task. Twenty healthy young subjects performed a repetitive work task, similar to sawing, synchronized with a metronome before and after performing each of two fatiguing tasks. The first fatigue task (LIFT) primarily fatigued the shoulder flexor muscles, while the second fatigue task (SAW) fatigued all of the muscles of the arm. Subjects performed each task in random order on two different days at least seven days apart. Instantaneous mean EMG frequencies (IMNF) decreased over both fatiguing tasks indicating that subjects did experience significant muscle fatigue. The slopes of the IMNF over time and the decreases in maximum force measurements demonstrated that the LIFT fatigue task successfully fatigued the shoulder flexors to a greater extent than any other muscle. On average, subjects exhibited more locally stable shoulder movements after the LIFT fatigue task (p=0.035). They also exhibited more orbitally stable shoulder (p=0.021) and elbow (p=0.013) movements after the SAW fatigue task. Subjects also had decreased cocontraction at the wrist post-fatigue for both tasks (p=0.001) and at the shoulder (p<0.001) for the LIFT fatigue task. Therefore, increased dynamic stability of these repeated movements cannot be explained by increased muscle cocontraction. Possible alternative mechanisms are discussed.     

Mechanical models for insect locomotion: active muscles and energy losses

 We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion in the horizontal plane developed in Schmitt and Holmes (2000a,b, 2001), where gaits characteristic of steady cockroach running and turning were evoked. In this paper, we include dissipation and energy inputs via active “muscles” in three forms: via prescribed torques at the “hip” pivot, via an active spring element of variable length, and via a pair of Hill-type muscle models representing an extensor/flexor system. Due to mechanical feedback of passive elastic forces, the stable gaits of the conservative models are preserved, and now energy input and absorption balances to additionally stabilize a preferred speed, with only modest neural sensing and feedback being required. However, these bipedal models still cannot simultaneously match observed moment-yaw magnitudes and fore-aft dynamics. Received: 17 September 2001 / Accepted: 20 February 2003 / Published online: 20 May 2003 Correspondence to: P. Holmes (e-mail: pholmes@math.Princeton.EDU) Acknowledgements. This work was supported by DARPA/ONR: N00014-98-1-0747 and DoE: DE-FG02-95ER25238. John Schmitt was partially supported by a DoD Graduate Fellowship, a Wu Fellowship of the School of Engineering and Applied Science, and a George Van Ness Lothrop Honorific Fellowship of the Graduate School at Princeton University. We thank Kenneth Meijer for allowing us to use his muscle model in Sect. 4 and Bob Full and Dan Koditschek for numerous helpful suggestions.     

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     

Feedback-mediated dynamics in two coupled nephrons

Previously, we developed a dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semi-linear hyperbolic partial differential equation was used to represent two fundamental processes of solute transport in the nephron’s thick ascending limb (TAL): chloride advection by fluid flow along the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that stable limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for sufficiently large feedback gain magnitude and time delay. In this study, the single-nephron model was extended to two nephrons, which were coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time delays but differing feedback gains, and (2) identical gains but differing delays. Similar to the case of a single nephron, our analysis indicates that stable LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO.     

Delay-induced oscillations in Wilson and Cowan’s model: an analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects

The model proposed by Wilson and Cowan (1972) describes the dynamics of two interacting subpopulations of excitatory and inhibitory neurons. It has been used to model neural structures like the olfactory bulb, whisker barrels, and the subthalamo-pallidal system. It is well-known that this system can exhibit an oscillatory behavior that is amplified by the presence of delays. In the absence of delays, the conditions for stability are well-known. The aim of our paper is to clarify these conditions when delays are included in the model. The first ingredient of our methods is a new necessary and sufficient condition for the existence of multiple equilibria. This condition is related to those for local asymptotic stability. In addition, a sufficient condition for global stability is also proposed. The second and main ingredient is a stability analysis of the system in the frequency-domain, based on the Nyquist criterion, that takes the four independent delays into account. The methods proposed in this paper can be applied to analyse the stability of the subthalamo-pallidal feedback loop, a deep brain structure involved in Parkinson’s disease. Our stability conditions are easy to compute and characterize sharply the system’s parameters for which spontaneous oscillations appear.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

Feedback control for chemostat models

 It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant. Received: 1 November 2001 / Revised version: 19 August 2002 / Published online: 19 November 2002 Mathematics Subject Classification (2000): 34H05, 34D23, 93C10 Keywords or phrases: Chemostat – Feedback control – Coexistence     

Winner-take-all selection in a neural system with delayed feedback

We consider the effects of temporal delay in a neural feedback system with excitation and inhibition. The topology of our model system reflects the anatomy of the avian isthmic circuitry, a feedback structure found in all classes of vertebrates. We show that the system is capable of performing a 'winner-take-all' selection rule for certain combinations of excitatory and inhibitory feedback. In particular, we show that when the time delays are sufficiently large a system with local inhibition and global excitation can function as a 'winner-take-all' network and exhibit oscillatory dynamics. We demonstrate how the origin of the oscillations can be attributed to the finite delays through a linear stability analysis.     

A ternary logic model for recurrent neuromime networks with delay

 In contrast to popular recurrent artificial neural network (RANN) models, biological neural networks have unsymmetric structures and incorporate significant delays as a result of axonal propagation. Consequently, biologically inspired neural network models are more accurately described by nonlinear differential-delay equations rather than nonlinear ordinary differential equations (ODEs), and the standard techniques for studying the dynamics of RANNs are wholly inadequate for these models. This paper develops a ternary-logic based method for analyzing these networks. Key to the technique is the realization that a nonzero delay produces a bounded stability region. This result significantly simplifies the construction of sufficient conditions for characterizing the network equilibria. If the network gain is large enough, each equilibrium can be classified as either asymptotically stable or unstable. To illustrate the analysis technique, the swim central pattern generator (CPG) of the sea slug Tritonia diomedea is examined. For wide range of reasonable parameter values, the ternary analysis shows that none of the network equilibria are stable, and thus the network must oscillate. The results show that complex synaptic dynamics are not necessary for pattern generation. Received: 15 June 1994/Accepted in revised form: 10 February 1995     

Can spectral cues contribute to species separation in closely related grasshoppers?

 The acridid grasshoppers Chorthippus biguttulus and Ch. mollis, which are closely related and often sympatric species, were compared intra- and interspecifically with regard to the spectra of their calling and courtship songs and of the sound-induced vibrations of the tympanal membrane, as well as the threshold curves of the tympanal nerve. In the low-frequency range but not in the ultrasound region, the maxima of these various curves fall at distinctly different frequencies in the two species. It is shown that the low-frequency sensitivity of the auditory system in both species, especially in females, is well matched to the conspecific song spectra but not to those of the heterospecific songs. Whether these characteristics actually contribute to species discrimination remains to be determined by behavioural tests. Accepted: 23 August 1996     

Cell division and the stability of cellular populations

 This paper couples a general d-dimensional (d arbitrary) model for the intracellular biochemistry of a generic cell with a probabilistic division hypothesis and examines the consequence of division for stability of cell function and structure. We show rather surprisingly that cell division is capable of giving rise to a stable population of cells with respect to function and structure even if, in the absence of cell division, the underlying biochemical dynamics are unstable. In the context of a simple example, our stability condition suggests that rapid cell proliferation plays a stabilizing role for cellular populations. Received: 15 January 1996 / Revised version: 31 July 1998     

A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size