Nonlinear characteristics of changes in active muscle length

The principle nonlinear characteristics of changes in the length of active (soleus, gastrocnemius, and plantaris) muscle resulting from controlled changes in external load were examined during acute experiments on anesthetized cats. Summation of successive muscle responses to repetitive phased changes in load was shown to be absent due to hysteresis effects; this does not satisfy the principles of superposition and leads to an important functional result: the muscle exerts a stabilizing effect on overall motor system dynamics, limiting unwanted shifts in joint angles during variation in external load. A relationship between the trajectory profile of change in muscle length and the lead-up to the movement arises due to muscle contraction hysteresis. Velocity at the initial stage of movement was always higher when the latter was preceded by motion in the same direction. The functional significance of the nonlinear properties of active muscle movement accompanying changing external load is discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 20, No. 6, pp. 736–743, November–December, 1988.     

Quantitative analysis of change in active muscle length with varying external load

A steady rate of efferent stimulation was applied to anesthetized cats during acute experiments. Changes in the length of m.soleus, m. gastrocnemius, and m. plantaris of the hindlimb were studied during linear changes in the external load (ramp-and-hold alterations). A servocontrolled linear motor was used to provide mechanical stimulation. It was found that muscular contraction produced by load may be extremely accurately approximated by reaction of a dynanic system consisting of a sequence with nonlinear statistics and linear dynamics. Nonlinear statistics of muscle properties are determined according to the nonlinear statistical pattern of the power itself on the one hand and by hysteresis effects on the other. A hypothesis is presented regarding connections between the latter and function of the troponin-tropomyosin regulatory complex. A first order linear differential equation is used to describe the dynamic sequence corresponding to the familiar three component viscoelastic muscle model. The proposed model could be used to help analyze processes of muscle stretch in a yield situation, although satisfactory indication was only obtained for fairly slow load increments.A. A. Bogomolets Institute of Physiology, Academy of Scences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 20, No. 5, pp. 694–702, September–October, 1988.     

Muscle function in animal movement: passive mechanical properties of leech muscle

We investigated passive properties of leech body wall as part of a larger project to understand better mechanisms that control locomotion and to establish mathematical models that predict such dynamical behavior. In tests of length-tension relationships in 2-segment-long preparations of body wall through step-stretch manipulations (step size = 1 mm), we discovered that these relationships are nonlinear, with significant hysteresis, even for the relatively small changes in length that occur during swimming. We developed a mathematical model comprising three nonlinear springs, two in series with nonlinear dashpots that describe well the tension statics and dynamics for step-stretch experiments. This model suggested that body wall dynamics are slow enough to be neglected when predicting the tension generated by imposed sinusoidal length changes (about ±10% of nominal) at 1–3 Hz, mimicking swimming. We derived a static model, comprising one nonlinear spring, which predicts sinusoidal data accurately, even when preparations were exposed to serotonin (0.1–10 μM). Preparations bathed in saline-serotonin had significantly reduced steady-state and peak tensions, without alterations in tension dynamics. Anesthetizing preparations (8% ethanol) reduced body wall tension by 77%, indicating that passive tension in the obliquely striated longitudinal muscles of leeches results primarily from a resting tonus.     

Analysis of muscle movement during frequency-modulated activation of efferents

The basic principles governing trajectories of change in muscle length (henceforth referred to as "movement") were analyzed at varying rates of distributed afferent stimulation during experiments on the soleus and plantaris muscles in unanesthetized cats. The theoretical possibility of describing evoked movements within the context of a model having nonlinear hysteresis properties and dependence of dynamic parameters on direction of movement were demonstrated. A difference in static transitions between muscle contraction and lengthening was found and vice versa and retardation of movement at the start of lengthening reaction (induced by a reduced efferent stimulation rate) was more pronounced. Interaction was discovered between two disruptive influences: changes in the rate of efferent stimulation and external load, mainly due to hysteresis effects of muscle contraction. The trajectory of movement produced by alteration in one of the inputs at work (external load or afferent stimulation) is associated with the lead-up to the muscle motion, irrespective of the reason inducing the foregoing movement. Functional implications of the nonlinear dynamics of muscular contraction are discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 21, No. 4, pp. 443–450, July–August, 1989.     

Distributed Representations for Actin-Myosin Interaction in the Oscillatory Contraction of Muscle

In this paper we suggest and test a specific hypothesis relating the attachment-detachment cycle of cross bridges between actin (I) and myosin (A) filaments to the measured length-tension dynamics of active insect fibrillar flight muscle. It is first shown that if local A-filament strain perturbs the rate constants in the cross-bridge cycle appropriately, then exponentially delayed tension changes can follow imposed changes of length; the latter phenomenon is sufficient for the work-producing property of fibrillar muscle, as measured with small-signal forcing of length and at low Ca²⁺ concentration, and possibly for related effects described recently in frog striated muscle. It is not clear a priori that the above explanation of work production by fibrillar muscle will remain tenable when the viscoelastic complexity of the heterogeneous sarcomere is taken into account. However, White's (1967) recent mechanical and electron microscope study of the passive dynamics of glycerinated fibrillar muscle has produced a model of the distributed viscoeleastic structure sufficiently explicit that alternative schemes for cross-bridge force generation in this muscle can now be tested more critically than previously. Therefore, we derive and solve third-order partial-differential equations which relate local interfilament shear forces associated with the perturbed cross-bridge cycles to the over-all length-tension dynamics of an idealized sarcomere. We then show (a) that the starting hypothesis can account approximately for the small-signal dynamics of glycerinated muscle in the work-producing state over two decades of frequency and (b) that the rate constants for cross-bridge formation and breakage, restricted solely by fitting of the model to the mechanical data, determine a cycling rate of cross bridges in the model compatible with recent measurements of ATP hydrolysis rate vs. stretch in this muscle. Finally, the formulation is extended tentatively to the large-signal nonlinear case, and shown to compare favorably with previous suggestions for the origin of the work-producing dynamics of fibrillar flight muscle.     

Characterization of load-length-velocity relationships of nine different skeletal muscles

A three-dimensional characterization of muscle load, length and velocity was obtained from nine muscles in the cat's hind limb through contractions where the muscles shortened against inertial-gravitational loads. A model based on the load-length characteristic and second-order dynamics describes shortening velocity related to load and length under these conditions. We conclude that this model describes well contraction velocity as function of length and load under inertial-gravitational load conditions, with correlation coefficients higher than 0.9 in most of the tested muscles.     

A biomechanical model to simulate the effect of a high vertical loading on trunk flexural stiffness

A human trunk model was developed to simulate the effect of a high vertical loading on trunk flexural stiffness. A force–length relationship is attributed to each muscle of the multi-body model. Trunk stiffness and muscle forces were evaluated experimentally and numerically for various applied loads. Experimental evaluation of trunk stiffness was carried out by measuring changes in reaction force following a sudden horizontal displacement at the T10 level prior to paraspinal reflexes induction. Results showed that the trunk stiffness increases under small applied loads, peaks when the loads were further increased and decreases when higher loads are applied. A sensitivity analysis to muscle force–length relationship is provided to determine the model's limitations. This model pointed out the importance of taking into account the changes in muscle length to evaluate the effect of spinal loads beyond the safe limit that cannot be evaluated experimentally and to predict the trunk instability under vertical load.     

Stability of bipedal stance: the contribution of cocontraction and spindle feedback

 The aim of this study is to assess the contribution of cocontraction and spindle feedback to local stability during bipedal stance. To that aim, an existing nonlinear state space model of the human musculoskeletal system is linearized in a reference equilibrium state. The maximal real part of the eigenvalues of the linearized system matrix A and the low-frequency joint stiffness are used as a measure of local stability. Muscle properties, as represented in a Hill-type muscle model, are shown to improve the behavior, the improvement being larger at high cocontraction. However, even at maximal cocontraction the low-frequency joint stiffness generated by the muscle properties is insufficient to yield a locally stable system. It follows that feedback is necessary to ensure local stability. In this study, the potential contribution of spindle feedback is investigated by optimizing the feedback gains for contractile element length and velocity for each muscle. It is found that in the case of time-delayed negative feedback, it is impossible to stabilize the system on the basis of spindle feedback. When positive time-delayed feedback is allowed, a barely stable system is obtained. When the time delays are removed, the feedback gains can be chosen such that a locally stable system is obtained, indicating the limitations imposed by the presence of time delays. Finally, it is shown that for small perturbations the response of the linear system to an arbitrary perturbation is similar to that of the nonlinear system, indicating the validity of the approach used. It is concluded that the combination of muscle properties and time-delayed spindle feedback is insufficient to obtain a system with reasonable local stability. Received: 10 October 2002 / Accepted: 3 December 2002 / Published online: 3 April 2003 Correspondence to: A. J. van Soest (e-mail: a_j_van_soest@fbw.vu.nl) Acknowledgements. We acknowledge Richard Casius for his help in carrying out the optimizations.     

Variation of muscle stiffness with force at increasing speeds of shortening

Single frog skeletal muscle fibers were attached to a servo motor and force transducer by knotting the tendons to pieces of wire at the fiber insertions. Small amplitude, high frequency sinusoidal length changes were then applied during tetani while fibers contracted both isometrically and isotonically at various constant velocities. The amplitude of the resulting force oscillation provides a relative measure of muscle stiffness. It is shown from an analysis of the transient force responses observed after sudden changes in muscle length applied both at full and reduced overlap and during the rising phase of short tetani that these responses can be explained on the basis of varying numbers of cross bridges attached at the time of the length step. Therefore, the stiffness measured by the high frequency length oscillation method is taken to be directly proportional to the number of cross bridges attached to thin filament sites. It is found that muscle stiffness measured in this way falls with increasing shortening velocity, but not as rapidly as the force. The results suggest that at the maximum velocity of shortening, when the external force is zero, muscle stiffness is still substantial. The findings are interpreted in terms of a specific model for muscle contraction in which the maximum velocity of shortening under zero external load arises when a force balance is attained between attached cross bridges some of which are aiding and others opposing shortening. Other interpretations of these results are also discussed.     

Muscle anatomy is a primary determinant of muscle relaxation dynamics in the lobster (<Emphasis Type="Italic">Panulirus interruptus</Emphasis>) stomatogastric system

We stained sarcomere thin filaments with fluorescently labeled phalloidin, measured sarcomere and muscle length, and calculated sarcomere number in pyloric and gastric mill muscles. A wide range of sarcomere lengths (3.25–12.29 μm), muscle lengths (5.9–21.1 mm), and sarcomere numbers (648–3,036) were observed. Sarcomere number differences occurred both because of changes in sarcomere length and muscle length, and sarcomere and muscle length varied independently. This independence, the wide range of sarcomere numbers present, and the muscles being all ‘slow’, graded muscles allowed us to use these data to test Huxley and Neidergerke’s (1954) hypothesis that muscle dynamics depend on sarcomere number. The time constants of exponential fits to contraction relaxations were used to measure muscle dynamics, and comparison of theoretical predictions and experimental results quantitatively confirm the predicted dependence. The differing dynamics of the various pyloric muscles are likely functionally important, and the dependence of muscle dynamics on sarcomere number implies that sarcomere number is likely closely regulated in these muscles. The stomatogastric system may thus be an excellent model system for studying the mechanisms regulating muscle sarcomere number.     

The nature of phonons and solitary waves in alpha-helical proteins.

A parametric study of the Davydov model of energy transduction in alpha-helical proteins is described. Previous investigations have shown that the Davydov model predicts that nonlinear interactions between phonons and amide-I excitations can stabilize the latter and produce a long-lived combined excitation (the so-called Davydov soliton), which propagates along the helix. The dynamics of this solitary wave are approximately those of solitons described using the nonlinear Schr?dinger equation. The present study extends these previous investigations by analyzing the effect of helix length and nonlinear coupling efficiency on the phonon spectrum in short and medium length alpha-helical segments. The phonon energy accompanying amide-I excitation shows periodic variation in time with fluctuations that follow three different time scales. The phonon spectrum is highly dependent upon chain length but a majority of the energy remains localized in normal mode vibrations even in the long chain alpha-helices. Variation of the phonon-exciton coupling coefficient changes the amplitudes but not the frequencies of the phonon spectrum. The computed spectra contain frequencies ranging from 200 GHz to 6 THz, and as the chain length is increased, the long period oscillations increase in amplitude. The most important prediction of this study, however, is that the dynamics predicted by the numerical calculations have more in common with dynamics described by using the Frohlich polaron model than by using the Davydov soliton. Accordingly, the relevance of the Davydov soliton model was applied to energy transduction in alpha-helical proteins is questionable.(ABSTRACT TRUNCATED AT 250 WORDS)     

Grey-box modeling and hypothesis testing of functional near-infrared spectroscopy-based cerebrovascular reactivity to anodal high-definition tDCS in healthy humans

Transcranial direct current stimulation (tDCS) has been shown to evoke hemodynamics response; however, the mechanisms have not been investigated systematically using systems biology approaches. Our study presents a grey-box linear model that was developed from a physiologically detailed multi-compartmental neurovascular unit model consisting of the vascular smooth muscle, perivascular space, synaptic space, and astrocyte glial cell. Then, model linearization was performed on the physiologically detailed nonlinear model to find appropriate complexity (Akaike information criterion) to fit functional near-infrared spectroscopy (fNIRS) based measure of blood volume changes, called cerebrovascular reactivity (CVR), to high-definition (HD) tDCS. The grey-box linear model was applied on the fNIRS-based CVR during the first 150 seconds of anodal HD-tDCS in eleven healthy humans. The grey-box linear models for each of the four nested pathways starting from tDCS scalp current density that perturbed synaptic potassium released from active neurons for Pathway 1, astrocytic transmembrane current for Pathway 2, perivascular potassium concentration for Pathway 3, and voltage-gated ion channel current on the smooth muscle cell for Pathway 4 were fitted to the total hemoglobin concentration (tHb) changes from optodes in the vicinity of 4x1 HD-tDCS electrodes as well as on the contralateral sensorimotor cortex. We found that the tDCS perturbation Pathway 3 presented the least mean square error (MSE, median <2.5%) and the lowest Akaike information criterion (AIC, median -1.726) from the individual grey-box linear model fitting at the targeted-region. Then, minimal realization transfer function with reduced-order approximations of the grey-box model pathways was fitted to the ensemble average tHb time series. Again, Pathway 3 with nine poles and two zeros (all free parameters), provided the best Goodness of Fit of 0.0078 for Chi-Square difference test of nested pathways. Therefore, our study provided a systems biology approach to investigate the initial transient hemodynamic response to tDCS based on fNIRS tHb data. Future studies need to investigate the steady-state responses, including steady-state oscillations found to be driven by calcium dynamics, where transcranial alternating current stimulation may provide frequency-dependent physiological entrainment for system identification. We postulate that such a mechanistic understanding from system identification of the hemodynamics response to transcranial electrical stimulation can facilitate adequate delivery of the current density to the neurovascular tissue under simultaneous portable imaging in various cerebrovascular diseases.     

Recruitment of single human low-threshold motor units with increasing loads at different muscle lengths.

We investigated the recruitment behaviour of low threshold motor units in flexor digitorum superficialis by altering two biomechanical constraints: the load against which the muscle worked and the initial muscle length. The load was increased using isotonic (low load), loaded dynamic (intermediate load) and isometric (high load) contractions in two studies. The initial muscle position reflected resting muscle length in series A, and a longer length with digit III fully extended in series B. Intramuscular EMG was recorded from 48 single motor units in 10 experiments on five healthy subjects, 21 units in series A and 27 in series B, while subjects performed ramp up, hold and ramp down contractions. Increasing the load on the muscle decreased the force, displacement and firing rate of single motor units at recruitment at shorter muscle lengths (P<0.001, dependent t-test). At longer muscle lengths this recruitment pattern was observed between loaded dynamic and isotonic contractions, but not between isometric and loaded dynamic contractions. Thus, the recruitment properties of single motor units in human flexor digitorum superficialis are sensitive to changes in both imposed external loads and the initial length of the muscle.     

Identification of the nonlinear state-space dynamics of the action-perception cycle for visually induced postural sway

Human subjects standing in a sinusoidally moving visual environment display postural sway with characteristic dynamical properties. We analyzed the spatiotemporal properties of this sway in an experiment in which the frequency of the visual motion was varied. We found a constant gain near 1, which implies that the sway motion matches the spatial parameters of the visual motion for a large range of frequencies. A linear dynamical model with constant parameters was compared quantitatively with the data. Its failure to describe correctly the spatiotemporal properties of the system led us to consider adaptive and nonlinear models. To differentiate between possible alternative structures we directly fitted nonlinear differential equations to the sway and visual motion trajectories on a trial-by-trial basis. We found that the eigenfrequency of the fitted model adapts strongly to the visual motion frequency. The damping coefficient decreases with increasing frequency. This indicates that the system destabilizes its postural state in the inertial frame. This leads to a faster internal dynamics which is capable of synchronizing posture with fast-moving visual environments. Using an algorithm which allows the identification of essentially nonlinear terms of the dynamics we found small nonlinear contributions. These nonlinearities are not consistent with a limit-cycle dynamics, accounting for the robustness of the amplitude of postural sway against frequency variations. We interpret our results in terms of active generation of postural sway specified by sensory information. We derive also a number of conclusions for a behavior-oriented analysis of the postural system.     

Predicting muscle forces in gait from EMG signals and musculotendon kinematics

An EMG-driven muscle model for determining muscle force-time histories during gait is presented. The model, based on Hill's equation (1938), incorporates morphological data and accounts for changes in musculotendon length, velocity, and the level of muscle excitation for both concentric and eccentric contractions. Musculotendon kinematics were calculated using three-dimensional cinematography with a model of the musculoskeletal system. Muscle force-length-EMG relations were established from slow isokinetic calibrations. Walking muscle force-time histories were determined for two subjects. Joint moments calculated from the predicted muscle forces were compared with moments calculated using a linked segment, inverse dynamics approach. Moment curve correlations ranged from r = 0.72 to R = 0.97 and the root mean square (RMS) differences were from 10 to 20 Nm. Expressed as a relative RMS, the moment differences ranged from a low of 23% at the ankle to a high of 72% at the hip. No single reason for the differences between the two moment curves could be identified. Possible explanations discussed include the linear EMG-to-force assumption and how well the EMG-to-force calibration represented excitation for the whole muscle during gait, assumptions incorporated in the muscle modeling procedure, and errors inherent in validating joint moments predicted from the model to moments calculated using linked segment, inverse dynamics. The closeness with which the joint moment curves matched in the present study supports using the modeling approach proposed to determine muscle forces in gait.     

Further study of the model of a human skull as a cupola with a plane base]

An experimental study of the base deformation of isolated human scull under conditions of scull collision with an obstacle has been carried out. The findings are compared with human cerebrospinal traums phenomena. An earlier suggested continuous scull model is modified on the basis of the data obtained. It is shown that the modified model (a part of spherical shell with the flat base) resembles scull behaviour in statics and dynamics better than the scull model in the form of spherical shell.     

A new kinetic model of protein adsorption on suspended anion-exchange resin particles.

The kinetics of adsorption of bovine serum albumin on an anion-exchange resin were measured in a batch system using a flow cell and ultraviolet absorbance, as a function of initial liquid-phase protein concentration and solid-to-liquid phase ratio. A new mathematical model for adsorption kinetics is presented that fits the experimental data to give a highly linear relationship with time, following a short transient period. Numerical integration of the differential form of the new composite nonlinear (CNL) kinetic model, containing three independent parameters, is shown to describe the dynamics of batch adsorption much better than alternative lumped parameter models. Although the new model is phenomenological rather than mechanistic, its principal parameter is shown to be a direct linear function of a physically measurable quantity. This study demonstrates that the model can accurately simulate protein concentration-time profiles using parameter estimates derived from correlations over a wide range of initial protein concentrations and phase ratios. The new CNL model is shown to be considerably superior to the Langmuir and solid-film linear kinetic models in this regard, having the additional advantage that an equilibrium isotherm for the system is not required.     

A nonlinear model for transient responses from light-adapted wolf spider eyes

A quantitative model is proposed to test the hypothesis that the dynamics of nonlinearities in retinal action potentials from light-adapted wolf spider eyes may be due to delayed asymmetries in responses of the visual cells. For purposes of calculation, these delayed asymmetries are generated in an analogue by a time-variant resistance. It is first shown that for small incremental stimuli, the linear behavior of such a resistance describes peaking and low frequency phase lead in frequency responses of the eye to sinusoidal modulations of background illumination. It also describes the overshoots in linear step responses. It is next shown that the analogue accounts for nonlinear transient and short term DC responses to large positive and negative step stimuli and for the variations in these responses with changes in degree of light adaptation. Finally, a physiological model is proposed in which the delayed asymmetries in response are attributed to delayed rectification by the visual cell membrane. In this model, cascaded chemical reactions may serve to transduce visual stimuli into membrane resistance changes.     

The role of the relaxation properties of muscles in the formation of movement

A linear rheology model with memory was constructed theoretically for an activated muscle moving joint of animal body. It was shown that during unidirectional moving the muscle acts like a controlled viscoelastic body with drifting of length.     

Nonlinear smooth orthogonal decomposition of kinematic features of sawing reconstructs muscle fatigue evolution as indicated by electromyography

Tracking or predicting physiological fatigue is important for developing more robust training protocols and better energy supplements and/or reducing muscle injuries. Current methodologies are usually impractical and/or invasive and may not be realizable outside of laboratory settings. It was recently demonstrated that smooth orthogonal decomposition (SOD) of phase space warping (PSW) features of motion kinematics can identify fatigue in individual muscle groups. We hypothesize that a nonlinear extension of SOD will identify more optimal fatigue coordinates and provide a lower-dimensional reconstruction of local fatigue dynamics than the linear SOD. Both linear and nonlinear SODs were applied to PSW features estimated from measured kinematics to reconstruct muscle fatigue dynamics in subjects performing a sawing motion. Ten healthy young right-handed subjects pushed a weighted handle back and forth until voluntary exhaustion. Three sets of joint kinematic angles were measured from the right upper extremity in addition to surface electromyography (EMG) recordings. The SOD coordinates of kinematic PSW features were compared against independently measured fatigue markers (i.e., mean and median EMG spectrum frequencies of individual muscle groups). This comparison was based on a least-squares linear fit of a fixed number of the dominant SOD coordinates to the appropriate local fatigue markers. Between subject variability showed that at most four to five nonlinear SOD coordinates were needed to reconstruct fatigue in local muscle groups, while on average 15 coordinates were needed for the linear SOD. Thus, the nonlinear coordinates provided a one-order-of-magnitude improvement over the linear ones.