A study of the muscle force waveform using a population stochastic model of skeletal muscle

A population stochastic model based on the differing properties and the independent activation of motor units is used to describe the production of force in the contracting skeletal muscle. Detailed force predictions of the model concerning a hand muscle are obtained by computer simulation. General features of the force signal are established analyticaly on the basis of the general properties of the neuromuscular system which the population model takes into account. The results show that the asynchronous activity of motor units and the distribution of their filtering and firing properties at various levels of muscle contraction are esponsible, at least partially, for the main features of the muscle force waveform, including tremor.     

Maximization of learning speed in the motor cortex due to neuronal redundancy

Many redundancies play functional roles in motor control and motor learning. For example, kinematic and muscle redundancies contribute to stabilizing posture and impedance control, respectively. Another redundancy is the number of neurons themselves; there are overwhelmingly more neurons than muscles, and many combinations of neural activation can generate identical muscle activity. The functional roles of this neuronal redundancy remains unknown. Analysis of a redundant neural network model makes it possible to investigate these functional roles while varying the number of model neurons and holding constant the number of output units. Our analysis reveals that learning speed reaches its maximum value if and only if the model includes sufficient neuronal redundancy. This analytical result does not depend on whether the distribution of the preferred direction is uniform or a skewed bimodal, both of which have been reported in neurophysiological studies. Neuronal redundancy maximizes learning speed, even if the neural network model includes recurrent connections, a nonlinear activation function, or nonlinear muscle units. Furthermore, our results do not rely on the shape of the generalization function. The results of this study suggest that one of the functional roles of neuronal redundancy is to maximize learning speed.     

Differences between properties of male and female motor units in the rat medial gastrocnemius muscle.

Differences between motor units in hindlimb locomotor muscles of male and female Wistar rats were studied. The contractile and action potential properties of various types of motor units as well as proportions of these units in the medial gastrocnemius muscle were analyzed. Experiments were based on functional isolation and electrical stimulation of axons of single motor units. Composition of motor units was different for male and female subjects, with higher number of the fast fatigable and lower number of slow type units in male animals. The contraction and the half-relaxation times were significantly longer in male motor units, what might be due to differences in muscle size. Slower contraction of male motor units likely corresponds to lower firing rates of their motoneurons. On the other hand, no significant differences between sexes were observed with respect to force parameters of motor units (the twitch and the maximum tetanus forces), except the fast resistant units (higher force values in male muscles). The mass of the muscle was approximately 1.5 time bigger in male rats. However, the mean ratio of motor unit tetanus force to the muscle mass was almost twice smaller in this group, what indirectly suggests that muscles of male rats are composed of higher number of motor units. Finally, female muscles appeared to have higher fatigue resistance as the effect of higher proportion of resistant units (slow and fast resistant) and higher values of the fatigue index in respective motor unit types. The motor unit action potentials in female rats had slightly lower amplitudes and shorter time parameters although this difference was significant only for fast resistant units.     

A model for neural control of gradation of muscle force

A mathematical muscle model is presented that relates neural control signals linearly to muscle force without violating important known physiological constraints, such as the size-principle (Henneman and Mendell 1981) and non-linear twitch summation (Burke et al. 1976). This linearity implies that the neural control signals (defined as a weighted sum of activities in a nerve bundle) can be interpreted as the internal representation of total muscle force. The model allows for different relative contributions from the two force-grading mechanisms, i.e. the recruitment of motor units and the modulation of their firing frequency. It can therefore be applied to a variety of (distal and proximal) muscles. Furthermore, it permits simple mechanisms for controlling muscle force, e.g. in superposed motor tasks. The model confirms our intuitive notion that a weighted sum of activities in a nerve bundle can directly represent an external controlled variable, which in this case is exerted muscle force.     

The influence of the way the muscle force is modeled on the predicted results obtained by solving indeterminate problems for a fast elbow flexion

A critical point in models of the human limbs when the aim is to investigate the motor control is the muscle model. More often the mechanical output of a muscle is considered as one musculotendon force that is a design variable in optimization tasks solved predominantly by static optimization. For dynamic conditions, the relationship between the developed force, the length and the contraction velocity of a muscle becomes important and rheological muscle models can be incorporated in the optimization tasks. Here the muscle activation can be a design variable as well. Recently a new muscle model was proposed. A muscle is considered as a mixture of motor units (MUs) with different peculiarities and the muscle force is calculated as a sum of the MUs twitches. The aim of the paper is to compare these three ways for presenting the muscle force. Fast elbow flexion is investigated using a planar model with five muscles. It is concluded that the rheological models are suitable for calculation of the current maximal muscle forces that can be used as weight factors in the objective functions. The model based on MUs has many advantages for precise investigations of motor control. Such muscle presentation can explain the muscle co-contraction and the role of the fast and the slow MUs. The relationship between the MUs activation and the mechanical output is more clear and closer to the reality.     

Tetanic depression in fast motor units of the cat gastrocnemius muscle.

Ability of muscle fibers to generate force is decreased when higher frequency of stimulation of motor units immediately follows lower frequency. This phenomenon called tetanic depression was found in rat medial gastrocnemius. However, it was not clear whether tetanic depression occurred only in rat muscle or it concerns all mammals. This study was conducted on motor units of cat medial gastrocnemius. Analyses were made at three successive trains of stimulation: 30 Hz, 20 and 30 Hz and again 30 Hz (the first pattern) or 40 Hz, 25 and 40 Hz and 40 Hz (the second pattern). In all fast units force generated within the middle tetanus was lower than force generated at the same, but constant frequency of stimulation applied earlier or later. The mean tetanic depression in 30 Hz tetani amounted to 10.9% for fast fatigable (FF) and 15.9% for fast resistant (FR) motor units, whereas in 40 Hz tetani mean values were 5.6% and 7.3% for FF and FR motor units, respectively. In slow motor units tetanic depression was not observed. These results proved the existence of tetanic depression in the feline muscle and indicated that its intensity depends on the fusion of tetanus. It has been concluded, that the tetanic depression is a general property of fast motor units in mammals.     

Effects of muscle model parameter dispersion and multi-loop segmental interaction on the neuromuscular system performance

The effects of parameter dispersion among motor units on the neuromuscular system performance as well as interaction between muscle segments and spinal cord mechanisms are investigated. Elementary components of the system are modeled to simulate with simple models their input-output characteristics. A leaky SS-IPFM encoder with a time-dependent threshold simulates the motor-neuron encoding characteristics. An amplitude and time dependent nonlinear model represent the motor unit mechanical output to neuronal input relationship. The dispersion of parameters in the components of the whole muscle control model is investigated in the open loop mode. It is shown that the dispersion of parameters in the multi-efferent channels converging on a common tendon provides a spatial filtration generating a smoother muscle force in addition to extending the linear dynamic range compared to a similar system having identical motor units. Muscle segmental interaction is investigated in this distributed model by closing the loop through a coupling matrix, representing afferent-motorneuron interaction on the spinal cord level. A diagonal matrix represents no segmental interaction and a uniform matrix represents a uniform interaction between segments through the muscle spindles and Golgi tendon feedback elements. The close loop simulation studied shows that (a). The type of segmental interaction has little effect on the overall system performance, i.e., range of linerity and stability, which is the result of having a muscle system with a large number of motor units. (b) There are only minor differences in results between the uniform and normal parameter distributions tested. (c) A loop gain of 4 divided by 8 in the distributed model can provide linearity through the full physiological force range. (d) Type of segmental interaction has significant effects on the individual segment. A uniform matrix provides a more stable segment due to the spatial filtration resulting from the segmental interaction, while the diagonal noninteracting matrix shows instabilities on the local segmental level despite global stability. The more realistic exponentially decaying spatial interaction matrix yields both global neuromuscular and local segmental stability with the same linear dynamic range generated with the uniform or diagonal matrices.     

Response linearity determined by recruitment strategy in detailed model of nictitating membrane control

Many models of eyeblink conditioning assume that there is a simple linear relationship between the firing patterns of neurons in the interpositus nucleus and the time course of the conditioned response (CR). However, the complexities of muscle behaviour and plant dynamics call this assumption into question. We investigated the issue by implementing the most detailed model available of the rabbit nictitating membrane response (Bartha and Thompson in Biol Cybern 68:135-143, 1992a and in Biol Cybern 68:145-154, 1992b), in which each motor unit of the retractor bulbi muscle is represented by a Hill-type model, driven by a non-linear activation mechanism designed to reproduce the isometric force measurements of Lennerstrand (J Physiol 236:43-55, 1974). Globe retraction and NM extension are modelled as linked second order systems. We derived versions of the model that used a consistent set of SI units, were based on a physically realisable version of calcium kinetics, and used simulated muscle cross-bridges to produce force. All versions showed similar non-linear responses to two basic control strategies. (1) Rate-coding with no recruitment gave a sigmoidal relation between control signal and amplitude of CR, reflecting the measured relation between isometric muscle force and stimulation frequency. (2) Recruitment of similar strength motor units with no rate coding gave a sublinear relation between control signal and amplitude of CR, reflecting the increase in muscle stiffness produced by recruitment. However, the system response could be linearised by either a suitable combination of rate-coding and recruitment, or by simple recruitment of motor units in order of (exponentially) increasing strength. These plausible control strategies, either alone or in combination, would in effect present the cerebellum with the simplified virtual plant that is assumed in many models of eyeblink conditioning. Future work is therefore needed to determine the extent to which motor neuron firing is in fact linearly related to the nictitating membrane response.     

A Neuro-Mechanical Model of a Single Leg Joint Highlighting the Basic Physiological Role of Fast and Slow Muscle Fibres of an Insect Muscle System

In legged animals, the muscle system has a dual function: to produce forces and torques necessary to move the limbs in a systematic way, and to maintain the body in a static position. These two functions are performed by the contribution of specialized motor units, i.e. motoneurons driving sets of specialized muscle fibres. With reference to their overall contraction and metabolic properties they are called fast and slow muscle fibres and can be found ubiquitously in skeletal muscles. Both fibre types are active during stepping, but only the slow ones maintain the posture of the body. From these findings, the general hypothesis on a functional segregation between both fibre types and their neuronal control has arisen. Earlier muscle models did not fully take this aspect into account. They either focused on certain aspects of muscular function or were developed to describe specific behaviours only. By contrast, our neuro-mechanical model is more general as it allows functionally to differentiate between static and dynamic aspects of movement control. It does so by including both muscle fibre types and separate motoneuron drives. Our model helps to gain a deeper insight into how the nervous system might combine neuronal control of locomotion and posture. It predicts that (1) positioning the leg at a specific retraction angle in steady state is most likely due to the extent of recruitment of slow muscle fibres and not to the force developed in the individual fibres of the antagonistic muscles; (2) the fast muscle fibres of antagonistic muscles contract alternately during stepping, while co-contraction of the slow muscle fibres takes place during steady state; (3) there are several possible ways of transition between movement and steady state of the leg achieved by varying the time course of recruitment of the fibres in the participating muscles.     

Motor unit recruitment patterns during reflex compensation of muscle yield investigated by computer simulations

 An important function of the stretch reflex in the soleus muscle in the decerebrate cat preparation is to compensate for the tendency of muscle suddenly to yield during ramp increases in length. As the level of background (i.e. pre-stretch) force increases, there is a systematic change in the curvature of the force trajectory during this reflex compensation, from concave to convex with respect to increasing force. The hypothesis that this change in curvature was due to background force-dependent changes in the recruitment pattern of motor units was investigated with a combined computer simulation/ experimental technique. The simulation consisted of 20 model motor units for the soleus muscle, each based on a distributed moment muscle model. The timing of recruitment of the motor units was optimized to allow the simulation outputs to fit a set of experimental data records on the reflex response to stretch initiated at five different levels of pre-stretch force. The resulting recruitment patterns showed that a tendency for recruitment to be concentrated progressively in the early portion of the stretch as pre-stretch force increased could account for the changes in reflex force curvature. These results are consistent with the skewed distribution of intrinsic electrical thresholds of motoneurons, in which low-threshold units are much more frequent than high-threshold ones. Therefore the changes in recruitment pattern and reflex force curvature may be due primarily to the intrinsic properties of motoneurons. Received: 18 September 1995/Accepted in revised form: 21 May 1996     

Comparison of force and EMG measures in normal and reinnervated tibialis anterior muscles of the rat.

The relationship between motor unit force and the recorded voltage produced by activated muscle unit fibres (electromyogram, EMG) was examined in normal and reinnervated rat tibialis anterior muscles. The number, cross-sectional area, and radial distance from the recording electrode of muscle fibres in a given unit, obtained directly from a sample of glycogen-depleted motor units, were analysed in relation to the magnitude of the EMG signal produced by that unit. EMG peak to peak amplitude and area varied as approximately the square root of twitch force in both normal and reinnervated units. Furthermore, the EMG amplitude increased approximately as the total cross-sectional area of the motor unit (number of muscle fibres x the average cross-sectional area of the fibres) and inversely with approximately the square root of the distance of fibres from the recording electrodes on the surface of the muscle.     

The dependence of the twitch course of medial gastrocnemius muscle of the rat and its motor units on stretching of the muscle.

The relationship between the force of a single twitch of the medial gastrocnemius muscle of the rat and contraction and half-relaxation times, on one hand, and the load of the muscle on the other, was studied. Twitches of the whole muscle and its individual motor units were induced. The optimal load, at which the majority of motor units reached the greatest twitch force, was 10 G. Mean optimal loads for twitches of different types of motor units were very similar. Slow motor units reached a slightly greater twitch force at greater loads (12.5 G) than at 10 G. However, the optimal load for the twitch of the whole muscle was much greater. It was 47 G on the average. The contraction and half-relaxation times of motor units, as well as of the whole muscle, became longer as the force stretching the muscle increased. Half-relaxation time changed more rapidly than contraction time. Both parameters were undergoing the greatest changes in slow motor units.     

Force and surface mechanomyogram frequency responses in cat gastrocnemius

Muscle surface displacement is a mechanical event taking place simultaneously with the tension generation at the tendon. The two phenomena can be studied by the surface mechanomyogram signal (MMG) (produced by a laser distance sensor) and the force signal (from a load cell). The aim of this paper was to provide data on the reliability of the laser detected MMG in muscle mechanics research. To this purpose it was verified if the laser detected MMG was suitable to estimate a frequency response in the cat medial gastrocnemius and its frequency response was compared with the one retrieved by the force signal at the tendon level. The force and MMG from the exposed medial gastrocnemius of four cats were analysed. The frequency response was investigated by sinusoidally changing the number of orderly recruited motor units, in different trials, in the 0.4-6 Hz range. It resulted that it was possible to model the force and MMG frequency response by a critically damped second-order system with two real double poles and a pure time delay. On the average, the poles were at 1.83 Hz (with 22.6 ms delay) and at 2.75 Hz (with 38 ms delay) for force and MMG, respectively. It can be concluded that MMG appears to be a reliable tool to investigate the muscle frequency response during stimulated isometric contraction. Even though not statistically significant. the differences in the second-order system parameters suggest that different components of the muscle mechanical model may specifically affect the force or MMG.     

Interspecies differences in the force-frequency relationship of the medial gastrocnemius motor units.

Single, functionally isolated motor units were studied in the medial gastrocnemius (MG) muscle of cats and rats. Axons of their motoneurons were stimulated with trains of pulses at frequencies increasing from 1 to 150 Hz and forces developed by muscle fibers were measured and force-frequency curves were compared between species. The following observations were made: (1) the most steep parts of curves (related to unfused tetani of motor units) begun at lower frequencies of stimulations in all types of feline motor units, (2) for fast motor units, the same relative values of force of unfused tetani were achieved at significantly lower frequencies of stimulations in the cat than in the rat. Twitch time parameters of both species influenced the course of force-frequency curves. It was showed that the contraction times of feline units varied in the wide range (21-81 ms), and these units reached 60% of the maximum force at stimulation frequencies between 10 and 38 Hz. On the other hand, contraction times of rat units ranged from 10 to 34 ms, whereas stimulation frequencies necessary to reach 60% of the maximum force varied considerably, from 12 to 65 Hz. The correlations between the above parameters were found for motor units of each species. However, the regression lines drown for the collected population of cat and rat units did not form linear continuity. Thus it seems that interspecies differences in the twitch contraction times do not fully explain different force-frequency relationships in mammalian skeletal muscles.     

The resilience of the size principle in the organization of motor unit properties in normal and reinnervated adult skeletal muscles

Henneman's size principle relates the input and output properties of motoneurons and their muscle fibers to size and is the basis for size-ordered activation or recruitment of motor units during movement. After nerve injury and surgical repair, the relationship between motoneuron size and the number and size of the muscle fibers that the motoneuron reinnervates is initially lost but returns with time, irrespective of whether the muscles are self- or cross-reinnervated by the regenerated axons. Although the return of the size relationships was initially attributed to the recovery of the cross-sectional area of the reinnervated muscle fibers and their force per fiber, direct enumeration of the innervation ratio and the number of muscle fibers per motoneuron demonstrated that a size-dependent branching of axons accounts for the size relationships in normal muscle, as suggested by Henneman and his colleagues. This same size-dependent branching accounts for the rematching of motoneuron size and muscle unit size in reinnervated muscles. Experiments were carried out to determine whether the daily amount of neuromuscular activation of motor units accounts for the size-dependent organization and reorganization of motor unit properties. The normal size-dependent matching of motoneurons and their muscle units with respect to the numbers of muscle fibers per motoneuron was unaltered by synchronous activation of all of the motor units with the same daily activity. Hence, the restored size relationships and rematching of motoneuron and muscle unit properties after nerve injuries and muscle reinnervation sustain the normal gradation of muscle force during movement by size-ordered recruitment of motor units and the process of rate coding of action potentials. Dynamic modulation of size of muscle fibers and their contractile speed and endurance by neuromuscular activity allows for neuromuscular adaptation in the context of the sustained organization of the neuromuscular system according to the size principle.     

A dynamical model of muscle activation, fatigue, and recovery

A dynamical model is presented as a framework for muscle activation, fatigue, and recovery. By describing the effects of muscle fatigue and recovery in terms of two phenomenological parameters (F, R), we develop a set of dynamical equations to describe the behavior of muscles as a group of motor units activated by voluntary effort. This model provides a macroscopic view for understanding biophysical mechanisms of voluntary drive, fatigue effect, and recovery in stimulating, limiting, and modulating the force output from muscles. The model is investigated under the condition in which brain effort is assumed to be constant. Experimental validation of the model is performed by fitting force data measured from healthy human subjects during a 3-min sustained maximal voluntary handgrip contraction. The experimental results confirm a theoretical inference from the model regarding the possibility of maximal muscle force production, and suggest that only 97% of the true maximal force can be reached under maximal voluntary effort, assuming that all motor units can be recruited voluntarily. The effects of different motor unit types, time-dependent brain effort, sources of artifacts, and other factors that could affect the model are discussed. The applications of the model are also discussed.     

The effect of activity during early postnatal development on motor unit size

At early stages of neuromuscular development, motor unit territory is expanded, with each muscle fibre being supplied by several axons. During postnatal development, some synapses are eliminated, motor unit size decreases, and the adult distribution of motor unit sizes emerges. This process depends on activity, since it proceeds more rapidly when the nerve is activated and is slower when activity is reduced. Here we studied whether, in addition to influencing the rate of retraction of motor unit territory, activity during the critical period of development affects the final outcome of the distribution of motor unit sizes. The sciatic nerve of 8- to 12-day-old rats was stimulated daily. One week later the tension of the extensor digitorum longus muscle and that of its individual motor units was recorded. The sizes of individual motor units were calculated and compared with those from animals that received no stimulation. The distribution of motor unit sizes from stimulated muscles was not significantly different from those from control muscles. Therefore, we conclude that although activity increases the rate at which motor units attain their adult size, it does not influence the final outcome of motor unit size distribution.     

Influence of motor units recruitment and firing rate on the soundmyogram and EMG characteristics in cat gastrocnemius

The separate contributions of the recruitment level and of the firing rate of the motor units on the soundmyogram and electromyogram time domain parameters were investigated during stimulation of the motor nerve of the cat gastrocnemius muscle. Upon orderly increase in the number of active motor units at a fixed firing rate, both the peak to peak amplitude (P-P_max) and the root mean square (RMS) of the sound myogram increased. At full recruitment the increase in firing rate from 2.5 to 50 Hz induced an exponential decline in the P-P_max. The RMS, however, followed this trend only from 15 to 50 Hz while showing an increase from 2.5 to 10 Hz. During simultaneous changes of recruitment and firing rate, the effect of increasing the number of motor units on the P-P_max and RMS is dampened by the increasing firing rate. The peak to peak amplitude of the EMG compound action potential increased with the number of active motor units. Moreover, its amplitude was not influenced by the firing rate. The EMG RMS, however, increases as a function of the firing rate. The results indicate that both the number and the firing rate of the active motor units contribute to the determination of the soundmyogram characteristics. Moreover, the peculiar changes of the soundmyogram time domain properties, compared to the ones of the EMG, allow one to differentiate the influence of the motor units number and firing rate on the electrical and mechanical performance of the muscle when stimulated.     

Dynamic performance of a load-moving skeletal muscle

The dynamic response of the tibialis anterior muscle of the cat was determined while it was subjected to sinusoidally varying orderly stimulation of motor units and to different isotonic loads in the range of 14-85% of the maximal isometric force. The dynamic response consisted of three major components: the displacement gain, the displacement attenuation, and a pure time delay. The displacement gain was dependent on the passive load applied to the muscle and the active force generated during contraction, which could be determined from the length-tension relationships and the corresponding shortening velocity. In general, the load displacement decreased as the load mass increased from 25 to 85% of the maximal isometric force. For loads less than 25% of the maximal isometric force, slight decrease in displacement was consistently observed. The displacement attenuation was dependent on the contraction frequency but uniform for all the load masses applied to the muscle. A pure time delay of 5 ms was present and accounted for various physiological processes such as conduction time in nerve and muscle, neuromuscular junction transmission, and excitation-contraction coupling. A quantitative equation was developed to describe the muscle's dynamic response under isotonic conditions and for a wide range of loads for use in various applications.     

The effects of four time-varying factors on the mean frequency of a myoelectric signal.

Daily activities involve dynamic muscle contractions that yield nonstationary myoelectric signals (MESs). The purpose of this work was to determine the individual effects of four time-varying factors (the number and firing rate of active motor units, muscle force and joint angle) on the mean frequency of a MES. Previous theoretical and experimental work revealed that although changes in the number and firing rate of active motor units contribute to the nonstationarities of the signal, they do not significantly affect the mean frequency. In the experimental work, 12 subjects performed 25 static contractions, one for each force (20, 30, 40, 50 and 60% of maximum voluntary contraction) and elbow joint angle (50, 70, 90, 110 and 130 degrees extension) combination. A MES was recorded from the surface of the biceps brachii during each contraction. The results indicated that muscle force only weakly affects the mean frequency. Also shown was that alteration in muscle geometry resulting from changes in elbow joint angle does significantly affect the mean frequency. Knowing this is important for the assessment of muscle fatigue during dynamic contractions.