A physiologically based model of muscle activation verified by electrical stimulation

Adaptive assistive robotics: a framework for triadic collaboration between humans and robots

Robots and other assistive technologies have a huge potential to help society in domains ranging from factory work to healthcare. However, safe and effective control of robotic agents in these environments is complex, especially when it involves close interactions and multiple actors. We propose an effective framework for optimizing the behaviour of robots and complementary assistive technologies in systems comprising a mix of human and technological agents with numerous high-level goals. The framework uses a combination of detailed biomechanical modelling and weighted multi-objective optimization to allow for the fine tuning of robot behaviours depending on the specification of the task at hand. We illustrate our framework via two case studies across assisted living and rehabilitation scenarios, and conduct simulations and experiments of triadic collaboration in practice. Our results indicate a marked benefit to the triadic approach, showing the potential to improve outcome measures for human agents in robot-assisted tasks.

MODELING AND SIMULATION OF SKELETAL MUSCLE BASED ON METABOLISM PHYSIOLOGY

Skeletal muscle energy metabolism plays a very important role in controlling movement of the whole body and has important theoretical guidance for making exercise training plans and losing weight. In this paper, we developed a mathematical model of skeletal muscle excitation–contraction pathway based on the energy metabolism that links excitation to contraction to explore the effects of different metabolic energy systems on calcium ion changes and the force during skeletal muscle contraction. In this paper, a membrane potential model, a calcium cycle model, a cross-bridge dynamics model and an energy metabolism model were established. Finally, the physiological phenomenon of calcium ion transport and calcium ion concentration change of the sarcoplasm was simulated. The results show that the phosphagen system has the fastest metabolic rate and the phosphagen system has the largest impact on the explosive power of skeletal muscle exercise. The specific characteristics of the three metabolic energy systems supporting skeletal muscle movement in vivo were also analyzed in detail.

Multiscale Hill-type modeling of the mechanical muscle behavior driven by the neural drive in isometric conditions

In this study, we present a new model describing the mechanical behavior of the skeletal muscle during isometric contraction. This model is based on a former Hill-inspired model detailing the electromechanical behavior of the muscle based on the Huxley formulation. However, in this new multiscale model the muscle is represented at the Motor Unit (MU) scale. The proposed model is driven by a physiological input describing the firing moments of the activated MUs. Definition of both voluntary and evoked MU recruitment schemes are described, enabling the study of both contractions in isometric conditions. During this type of contraction, there is no movement of the joints and the tendon-muscle complex remains at the same length. Moreover, some well-established macroscopic relationships such as force-length or force-velocity properties are considered. A comparison with a twitch model using the same input definition is provided with both recruitment schemes exhibiting limitations of twitch type models. Finally, the proposed model is validated with a comparison between simulated and recorded force profiles following eight electrical stimulations pulses in isometric conditions. The simulated muscle force was generated to mimic the one recorded from the quadriceps of a patient implanted with a functional electrical stimulation neuroprosthesis. This validation demonstrates the ability of the proposed model to reproduce realistically the skeletal muscle contractions and to take into account subject-specific parameters.

An approach to a muscle force model with force-pulse amplitude relationship of human quadriceps muscles

BACKGROUND

Recent advanced applications of the functional electrical stimulation (FES) mostly used closed-loop control strategies based on mathematical models to improve the performance of the FES systems. In most of them, the pulse amplitude was used as an input control. However, in controlling the muscle force, the most popular force model developed by Ding et al. does not take into account the pulse amplitude effect. The purpose of this study was to include the pulse amplitude in the existing Ding et al. model based on the recruitment curve function.

METHODS

Quadriceps femoris muscles of eight healthy subjects were tested. Forces responses to stimulation trains with different pulse amplitudes (30-100 mA) and frequencies (20-80 Hz) were recorded and analyzed. Then, specific model parameter values were identified by fitting the measured forces for one train (50 Hz, 100 mA). The obtained model parameters were then used to identify the recruitment curve parameter values by fitting the force responses for different pulse amplitudes at the same frequency train. Finally, the extended model was used to predict force responses for a range of stimulation pulse amplitudes and frequencies.

RESULTS

The experimental results indicated that our adapted model accurately predicts the force-pulse amplitude relationship with an excellent agreement between measured and predicted forces (R²=0.998, RMSE = 6.6 N).

CONCLUSIONS

This model could be used to predict the pulse amplitude effect and to design control strategies for controlling the muscle force in order to obtain precise movements during FES sessions using intensity modulation.

A New Mathematical Force Model that Predicts the Force-pulse Amplitude Relationship of Human Skeletal Muscle

Current functional electrical stimulation (FES) systems vary the stimulation intensity to control the muscle force in order to produce precise functional movements. However, mathematical model that predicts the intensity effect on the muscle force is required for model-based controller design. The most previous force model designed by Ding et al was validated only for a standardized stimulation pulse amplitude (intensity). Thus, the purpose of this study was to adapt the Ding et al model to be able to predict the force-pulse amplitude relationship. The experimental results tested on quadriceps femoris muscles of healthy subjects (N=5) show that our adapted model accurately predicts the force response for trains of a wide range of stimulation intensities (30-100 mA). The accurate predictions indicate that our adapted model could be used for designing model-based control strategies to control the muscle force through FES.

Functionality indexes assessed through a simple model of muscle activation, fatigue and recovery

A nonlinear dynamical system is proposed as a qualitative mathematical model with the twofold aim to reasonably describe the force behavior in a fatiguing sub-maximal contraction and to be possibly employed in assessing muscular activation indexes. The model's properties are studied in terms of its equilibria and their stability properties and the existence of the fatigue equilibrium is ensured as the only system's attractor in the feasibility range of the parameters. Suitable mathematical indicators — related to the dynamical properties of resilience and reactivity — are introduced to characterize the asymptotic and the transient system's behavior. The practical impact of the analytical results is elucidated and a connection is established between the introduced mathematical indicators and muscle functionality indexes as rate of force development, task failure time and complete restore time. Experimental validation with handgrip force signal at high load and possible practical applications are also presented.

Experimental parameter identification of a multi-scale musculoskeletal model controlled by electrical stimulation: application to patients with spinal cord injury

We investigated the parameter identification of a multi-scale physiological model of skeletal muscle, based on Huxley's formulation. We focused particularly on the knee joint controlled by quadriceps muscles under electrical stimulation (ES) in subjects with a complete spinal cord injury. A noninvasive and in vivo identification protocol was thus applied through surface stimulation in nine subjects and through neural stimulation in one ES-implanted subject. The identification protocol included initial identification steps, which are adaptations of existing identification techniques to estimate most of the parameters of our model. Then we applied an original and safer identification protocol in dynamic conditions, which required resolution of a nonlinear programming (NLP) problem to identify the serial element stiffness of quadriceps. Each identification step and cross validation of the estimated model in dynamic condition were evaluated through a quadratic error criterion. The results highlighted good accuracy, the efficiency of the identification protocol and the ability of the estimated model to predict the subject-specific behavior of the musculoskeletal system. From the comparison of parameter values between subjects, we discussed and explored the inter-subject variability of parameters in order to select parameters that have to be identified in each patient.

ON THE COMPLEXITY OF BIOMECHANICAL MODELS USED FOR NEUROPROSTHESES DEVELOPMENT

The use of mathematical models has the potential to enhance the development of lower extremity neuroprostheses (NP) based on Functional Electrical Simulation (FES). The choice of model complexity is not trivial when building a model for FES control design. On the one hand, a comprehensive model might be useful to account for the many different biomechanical and neurophysiological effects that can be observed during FES-induced movements. On the other hand, too complex models are difficult to be utilized in and identified for NP applications. In this paper we discuss the disadvantages of too complex models, and propose potential simplifications on the basis of existing models that are commonly used to describe muscle activation, muscle contraction and body-segmental motion. The obtained model approach is simple enough to be identified, and sufficiently comprehensive to describe most of the relevant effects that occur during FES-induced locomotion.

A comparison of models of the isometric force of locust skeletal muscle in response to pulse train inputs

Muscle models are an important tool in the development of new rehabilitation and diagnostic techniques. Many models have been proposed in the past, but little work has been done on comparing the performance of models. In this paper, seven models that describe the isometric force response to pulse train inputs are investigated. Five of the models are from the literature while two new models are also presented. Models are compared in terms of their ability to fit to isometric force data, using Akaike's and Bayesian information criteria and by examining the ability of each model to describe the underlying behaviour in response to individual pulses. Experimental data were collected by stimulating the locust extensor tibia muscle and measuring the force generated at the tibia. Parameters in each model were estimated by minimising the error between the modelled and actual force response for a set of training data. A separate set of test data, which included physiological kick-type data, was used to assess the models. It was found that a linear model performed the worst whereas a new model was found to perform the best. The parameter sensitivity of this new model was investigated using a one-at-a-time approach, and it found that the force response is not particularly sensitive to changes in any parameter.

A coupled electromechanical model for the excitation-dependent contraction of skeletal muscle

This work deals with the development and implementation of an electromechanical skeletal muscle model. To this end, a recently published hyperelastic constitutive muscle model with transversely isotropic characteristics, see Ehret et al. (2011), has been weakly coupled with Ohm's law describing the electric current. In contrast to the traditional way of active muscle modelling, this model is rooted on a non-additive decomposition of the active and passive components. The performance of the proposed modelling approach is demonstrated by the use of three-dimensional illustrative boundary-value problems that include electromechanical analysis on tissue strips. Further, simulations on the biceps brachii muscle document the applicability of the model to realistic muscle geometries.

Isometric force generated by locust skeletal muscle: responses to single stimuli

A mathematical model of the locust hind leg extensor muscle is described. The model accounts for the force response of the muscle to well-separated input stimuli under isometric conditions. Experimental data was collected by stimulating the extensor muscle and measuring the force generated at the tibia. In developing a model it was assumed that the response to a single isolated stimulus was linear. A linear model was found to fit well to the response to an isolated stimulus. No assumptions were made about the model order and models of various order were fitted to data in the frequency domain, using a least squares fit. The stimulus can be approximated as an impulse, with the response to each stimulus well described by a linear second-order system. Using a third-order model provided a better fit to data, but the improvement in fit was marginal and the model uses one extra parameter. A fourth-order model, which is often used to describe the behaviour of isometric muscle was found to overfit the data. Using a second-order model provides a simpler way of describing the behaviour of an isometric twitch.

A model for insect locomotion in the horizontal plane: feedforward activation of fast muscles, stability, and robustness

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions of the body. Muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using an activation function and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs and muscle force levels are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown and lift-off timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.

A mathematical model of fatigue in skeletal muscle force contraction

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation-contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation-contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

Oviduct contraction in Drosophila is modulated by a neural network that is both, octopaminergic and glutamatergic

Fertility is a highly complex and regulated phenomenon essential for the survival of any species. To identify Drosophila fertility-specific neural networks, we used a GAL4/UAS enhancer trap genetic screen that selectively inactivates groups of neurons. We identified a GAL4 line (bwktqs) that has a female sterile phenotype only when it expresses the tetanus toxin light chain (TeTxLC). These flies lack oviduct contraction, lay almost no eggs, sperm accumulate in the oviducts, and fewer than normal are seen in the storage organs. In insects, two neuroactive substances are important for oviduct contraction: octopamine (OA), a monoamine that inhibits oviduct contraction, and glutamate (Glu), a neurotransmitter that induces contraction. It is known that octopaminergic neurons of the thoracic abdominal ganglion (TAG) modulate oviduct contraction, however, the glutamatergic neurons that innervate the oviduct have not been identified yet and the interaction between these two neuroactive substances is not well understood. Immunostaining experiments revealed that the bwktqs line trapped an octopaminergic neural network that innervates the genital tract. We show that wt like oviduct contraction in TeTxLC-inactivated flies can only be rescued by simultaneous application of Glu and OA suggesting that the abdominal bwktqs neurons are both octopaminergic and glutamatergic, the use of an agonist and an antagonist for Glu receptors as well as their direct visualization confirmed its participation in this phenomenon. Our work provides the first evidence that adult abdominal type II visceral innervations co-express Glu and OA and allows us to re-evaluate the previous model of neuronal network controlling insect oviduct contraction.

An elastic rod model for anguilliform swimming

We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158-183, 1952). We employ a eometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor's assumption of prescribed shape to fail, leading to time-periodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158-183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.

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.

Gradual potentiation of isometric muscle force during constant electrical stimulation

An investigation was carried out into how stimulation frequency and stimulation history affect the potentiation of muscle force during 20s of constant stimulation of the two knee extensors in isometric conditions. Stimulation frequency significantly affected the potentiation pattern: low-frequency (2.5-10 Hz) stimulation showed a reduction and subsequent enhancement of force, and high-frequency (14.3-25 Hz) stimulation showed only enhancement of force. The degree of enhancement in force and time-to-peak decreased with the stimulation frequency. Whereas 40 Hz conditioning stimulation enhanced the muscle force, 14 Hz stimulation after 10s of rest induced little force enhancement (8% in both muscles). When the frequency of the conditioning stimulation was 14 Hz and the main 14 Hz stimulation was applied after 50s of rest, the initial force at the main stimulation was similar to the final force value of the conditioning stimulation (above 90% similarity). The potentiated twitch force slowly decayed during rest, with an average time constant of 2.4 min. These observations indicate that muscle potentiation depends on the stimulation frequency and stimulation history, and therefore a computer model of potentiation can play an important role in predicting muscle force and body movement induced by electrical stimulation.

Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains

Because muscles must be repetitively activated during functional electrical stimulation, it is desirable to identify the stimulation pattern that produces the most force. Previous experimental work has shown that the optimal pattern contains an initial high-frequency burst of pulses (i.e., an initial doublet or triplet) followed by a low, constant-frequency portion. Pattern optimization is particularly challenging, because a muscle's contractile characteristics and, therefore, the optimal pattern change under different physiological conditions and are different for each person. This work describes the continued development and testing of a mathematical model that predicts isometric forces from fresh and fatigued muscles in response to brief trains of electrical pulses. By use of this model and an optimization algorithm, stimulation patterns that produced maximum forces from each subject were identified.

Adaptive fuzzy control of electrically stimulated muscles for arm movements

A modified adaptive Takagi-Sugeno (TS) fuzzy logic controller (FLC) is proposed that allows a simulated elbow-like biomechanical system to accurately track sigmoidal and sinusoidal trajectories in the sagittal plane. The work is a first effort towards the implementation of a system to restore elbow movements in quadriplegics using functional neuromuscular stimulation. The single-joint musculo-skeletal system is composed of a co-contractable pair of electrically stimulated muscles; the muscle model accounts for the increase in fatigue during the tracking exercise. In the proposed controller structure, a reinforcement learning scheme is used to accomplish the parameter tuning, and the parameter projection algorithm guarantees the system stability during the adaptation process. The controller performance is evaluated using computer simulation experiments and compared with the performance achievable when a standard proportional-integrative-derivative (PID) controller is employed for the same application. The modified adaptive TSFLC outperforms the PID controller in all tested situations, with a clear-cut advantage in the case of high-frequency sinusoidal trajectories (angular frequencies spanning the interval 8-12 rad s-1). The standard controller suffers from a dramatic increase in root mean square (RMS) tracking error above the value at 8 rad s-1, e.g. ERMS > or = 0.013, whereas the correlation coefficient between the actual and desired trajectory falls almost to zero, starting from the value rho approximately equal to 0.97 at 8 rad s-1. On the other hand, the adaptive TSFLC yields ERMS < or = 0.015, with rho > or = 0.78, over the whole range of tested angular frequencies.

An approach to a muscle model with a stimulus frequency-force relationship for FES applications

A simplified model of electrically stimulated muscle for use in applications of functional electrical stimulation (FES) is discussed in this paper. The muscle model was required to have both stimulus frequency and stimulus intensity (amplitude/width) inputs. The stimulus frequency versus force relationship of rabbit muscle was modeled first with a small number of model parameters that could be identified by simple experiments in a short time. The model identified was found to be applicable to human muscles. The frequency-force relationships of electrically stimulated fast and slow type muscles were also predicted by the model. The frequency-force model and a simplified model of muscle activation dynamics were used to construct a muscle model that described the summation of muscle contraction. The use of this model decreased the time burden on patients during parameter identification at the clinical site. The clinical applicability of these new model descriptions was suggested through computer simulations.

A mathematical model for skeletal muscle activated by N-let pulse trains

A physiologically based mathematical model for skeletal muscle activated by neural impulses is presented. This model is developed specifically to capture the behavior for mammalian skeletal muscle activated by N-lets (sets of N high-frequency pulses with variable interpulse intervals). N-let pulse trains have been demonstrated as a possible means of producing contractions with reduced fatigue and fiber-type transformation, while maximizing the force-time integral per pulse (FTIpP) of electrically stimulated muscle. This model is developed by modeling the underlying biophysical processes responsible for the initiation and maintenance of force generation in muscle. The release and reaccumulation dynamics of calcium ions from the sarcoplasmic reticulum are modeled and proposed as the governing mechanism for the observed N-let effects. It is found that the new model is robust, numerically stable and easily implemented. Simulation results are presented that demonstrate the model's ability to capture a variety of the nonlinear summation, force and stiffness variation effects seen experimentally when activating skeletal muscle with N-lets. General properties of FES muscle are also predicted by the model. The significant insight provided by this model into the internal dynamics of skeletal muscle is used to assess a variety of mechanisms proposed for N-let behavior. It is postulated that the calcium release and reaccumulation dynamics, as incorporated in this model, are responsible for the N-let effects found in experiment.