A nonlinear mathematical model of electrically stimulated skeletal muscle

Closed-loop optogenetic neuromodulation enables high-fidelity fatigue-resistant muscle control

Closed-loop neuroprostheses show promise in restoring motion in individuals with neurological conditions. However, conventional activation strategies based on functional electrical stimulation (FES) fail to accurately modulate muscle force and exhibit rapid fatigue because of their unphysiological recruitment mechanism. Here, we present a closed-loop control framework that leverages physiological force modulation under functional optogenetic stimulation (FOS) to enable high-fidelity muscle control for extended periods of time (>60 minutes) in vivo. We first uncovered the force modulation characteristic of FOS, showing more physiological recruitment and significantly higher modulation ranges (>320%) compared with FES. Second, we developed a neuromuscular model that accurately describes the highly nonlinear dynamics of optogenetically stimulated muscle. Third, on the basis of the optogenetic model, we demonstrated real-time control of muscle force with improved performance and fatigue resistance compared with FES. This work lays the foundation for fatigue-resistant neuroprostheses and optogenetically controlled biohybrid robots with high-fidelity force modulation.

Estimation of the firing behaviour of a complete motoneuron pool by combining electromyography signal decomposition and realistic motoneuron modelling

Our understanding of the firing behaviour of motoneuron (MN) pools during human voluntary muscle contractions is currently limited to electrophysiological findings from animal experiments extrapolated to humans, mathematical models of MN pools not validated for human data, and experimental results obtained from decomposition of electromyographical (EMG) signals. These approaches are limited in accuracy or provide information on only small partitions of the MN population. Here, we propose a method based on the combination of high-density EMG (HDEMG) data and realistic modelling for predicting the behaviour of entire pools of motoneurons in humans. The method builds on a physiologically realistic model of a MN pool which predicts, from the experimental spike trains of a smaller number of individual MNs identified from decomposed HDEMG signals, the unknown recruitment and firing activity of the remaining unidentified MNs in the complete MN pool. The MN pool model is described as a cohort of single-compartment leaky fire-and-integrate (LIF) models of MNs scaled by a physiologically realistic distribution of MN electrophysiological properties and driven by a spinal synaptic input, both derived from decomposed HDEMG data. The MN spike trains and effective neural drive to muscle, predicted with this method, have been successfully validated experimentally. A representative application of the method in MN-driven neuromuscular modelling is also presented. The proposed approach provides a validated tool for neuroscientists, experimentalists, and modelers to infer the firing activity of MNs that cannot be observed experimentally, investigate the neuromechanics of human MN pools, support future experimental investigations, and advance neuromuscular modelling for investigating the neural strategies controlling human voluntary contractions.

Our experimental understanding of the firing behaviour of motoneuron (MN) pools during human voluntary muscle contractions is currently limited to the observation of small samples of active MNs obtained from EMG decomposition. EMG decomposition therefore provides an important but incomplete description of the role of individual MNs in the firing activity of the complete MN pool, which limits our understanding of the neural strategies of the whole MN pool and of how the firing activity of each MN contributes to the neural drive to muscle. Here, we combine decomposed high-density EMG (HDEMG) data and a physiologically realistic model of MN population to predict the unknown recruitment and firing activity of the remaining unidentified MNs in the complete MN pool. In brief, an experimental estimation of the synaptic current is input to a cohort of MN models, which are calibrated using the available decomposed HDEMG data, and predict the MN spike trains fired by the entire MN population. This novel approach is experimentally validated and applied to muscle force prediction from neuromuscular modelling.

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.

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.

An FES cycling control system based on CPG

This paper presents a scientific strategy for cycling induced by the functional electrical stimulation. In order to simulate the FES-cycling movement produced by human body, a neuro-musculo-skeletal model containing 16 segments and 186 muscles is developed, which can simulate human movements precisely. This model contains mathematical model of electrically stimulated skeletal muscles. Having known the kinematics and dynamics of the model, we design an FES-cycling control system based on the central pattern generator (CPG), which can produce rhythm stimulus to produce desired torque and generate rhythm cycling movements. And an approach to control multiple muscles is proposed. In the end of this paper, the simulation results are provided.

Biomechanics of the vibrissa motor plant in rat: rhythmic whisking consists of triphasic neuromuscular activity

The biomechanics of a motor plant constrain the behavioral strategies that an animal has available to extract information from its environment. We used the rat vibrissa system as a model for active sensing and determined the pattern of muscle activity that drives rhythmic exploratory whisking. Our approach made use of electromyography to measure the activation of all relevant muscles in both head-fixed and unrestrained rats and two-dimensional imaging to monitor the position of the vibrissae in head-fixed rats. Our essential finding is that the periodic motion of the vibrissae and mystacial pad during whisking results from three phases of muscle activity. First, the vibrissae are thrust forward as the rostral extrinsic muscle, musculus (m.) nasalis, contracts to pull the pad and initiate protraction. Second, late in protraction, the intrinsic muscles pivot the vibrissae farther forward. Third, retraction involves the cessation of m. nasalis and intrinsic muscle activity and the contraction of the caudal extrinsic muscles m. nasolabialis and m. maxillolabialis to pull the pad and the vibrissae backward. We developed a biomechanical model of the whisking motor plant that incorporates the measured muscular mechanics along with movement vectors observed from direct muscle stimulation in anesthetized rats. The results of simulations of the model quantify how the combination of extrinsic and intrinsic muscle activity leads to an enhanced range of vibrissa motion than would be available from the intrinsic muscles alone.

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.

Predicting human chronically paralyzed muscle force: a comparison of three mathematical models

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.

Mathematical models use varying parameter strategies to represent paralyzed muscle force properties: a sensitivity analysis

Background

Mathematical muscle models may be useful for the determination of appropriate musculoskeletal stresses that will safely maintain the integrity of muscle and bone following spinal cord injury. Several models have been proposed to represent paralyzed muscle, but there have not been any systematic comparisons of modelling approaches to better understand the relationships between model parameters and muscle contractile properties. This sensitivity analysis of simulated muscle forces using three currently available mathematical models provides insight into the differences in modelling strategies as well as any direct parameter associations with simulated muscle force properties.

Methods

Three mathematical muscle models were compared: a traditional linear model with 3 parameters and two contemporary nonlinear models each with 6 parameters. Simulated muscle forces were calculated for two stimulation patterns (constant frequency and initial doublet trains) at three frequencies (5, 10, and 20 Hz). A sensitivity analysis of each model was performed by altering a single parameter through a range of 8 values, while the remaining parameters were kept at baseline values. Specific simulated force characteristics were determined for each stimulation pattern and each parameter increment. Significant parameter influences for each simulated force property were determined using ANOVA and Tukey's follow-up tests (α ≤ 0.05), and compared to previously reported parameter definitions.

Results

Each of the 3 linear model's parameters most clearly influence either simulated force magnitude or speed properties, consistent with previous parameter definitions. The nonlinear models' parameters displayed greater redundancy between force magnitude and speed properties. Further, previous parameter definitions for one of the nonlinear models were consistently supported, while the other was only partially supported by this analysis.

Conclusion

These three mathematical models use substantially different strategies to represent simulated muscle force. The two contemporary nonlinear models' parameters have the least distinct associations with simulated muscle force properties, and the greatest parameter role redundancy compared to the traditional linear model.

A mathematical model that predicts the force-frequency relationship of human skeletal muscle

In previous work we developed and validated a mathematical model that predicted force output from skeletal muscles subjected to six-pulse stimulation trains under isometric condition. The current study investigated the model's ability to predict force responses to longer stimulation trains under both nonfatigued and fatigued conditions. Using the six-pulse train model to predict the force produced by longer stimulation trains showed that the model was successful, but a modified parameter identification scheme was required. For most of the trains tested the model accounted for 95% of the variance in the experimental forces produced by stimulation trains, with mean frequencies from 12.5 to 100 HZ, train durations from 485 to 1000 ms, and number of pulses from 14 to 50 for both nonfatigued and fatigued muscles. The success of our mathematical model in predicting forces produced by stimulations with a wide range of frequencies, durations, and number of pulses implies great potential of the model for the identification of optimal activation patterns that should be used during functional electrical stimulation.

A nonlinear approach to modeling of electrically stimulated skeletal muscle

This paper is concerned with the development and analysis of a nonlinear approach to modeling of the contraction of electrically stimulated skeletal muscle. The model structure is based on a network of locally valid linear models which are blended together by a scheduler. Data are from experiments with rabbit tibialis anterior muscles in which the muscles contracted isometrically while being stimulated by supramaximal impulses with randomly varying inter-pulse intervals. The model accounts for nonlinear effects due to variations of the stimulation frequency, such as the "catch-like" effect. It is shown that this modeling technique is suitable for modeling the contraction of muscles with very different characteristics, such as muscle with a majority of fast motor units and muscle with mainly slow motor units. The approach is also suitable as a basis for the design of muscle stimulation controllers. Index Terms-Functional electrical stimulation, local model network, muscle modeling, nonlinear system identification.

The relationship between electrical stimulus and joint torque: a dynamic model

The knowledge of the behavior of electrically activated muscles is an important requisite for the development of functional electrical stimulation (FES) systems to restore mobility to persons with paralysis. The aim of this work was to develop a model capable of relating electrical parameters to dynamic joint torque for FES applications. The knee extensor muscles, stimulated using surface electrodes, were used for the experimental preparation. Both healthy subjects and people with paraplegia were tested. The dynamics of the lower limb were represented by a nonlinear second order model, which took account of the gravitational and inertial characteristics of the anatomical segments as well as the damping and stiffness properties of the knee joint. The viscous-elastic parameters of the system were identified experimentally through free pendular movements of the leg. Leg movements induced by quadriceps stimulation were acquired too, using a motion analysis system. Results showed that, for the considered experimental conditions, a simple one-pole transfer function is able to model the relationship between stimulus pulsewidth (PW) and active muscle torque. The time constant of the pole was found to depend on the stimulus pattern (ramp or step) while gain was directly dependent on stimulation frequency.

A model for human skin impedance during surface functional neuromuscular stimulation

A new mathematical model for the bulk electrical impedance of human skin is presented. In particular this model describes the impedance of skin during surface functional neuromuscular stimulation (FNS) with square stimulation pulses. Experimental data are presented that illustrate the nonlinear dynamic properties of human skin during current and voltage controlled stimulation. Model predictions are compared to experimental data, measured under both constant voltage and constant current transcutaneous stimulation. It is found that this model captures a variety of nonlinear time-varying effects observed in the skin impedance when stimulating with either protocol. This model may be used as part of large neuromusculoskeletal models or in the more accurate modeling of transcutaneous FNS, which is currently the most common clinical implementation of FNS.

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.