Computational Models for Neuromuscular Function

Abstract

Computational models of the neuromuscular system hold the potential to allow us to reach a deeper understanding of neuromuscular function and clinical rehabilitation by complementing experimentation. By serving as a means to distill and explore specific hypotheses, computational models emerge from prior experimental data and motivate future experimental work. Here we review computational tools used to understand neuromuscular function including musculoskeletal modeling, machine learning, control theory, and statistical model analysis. We conclude that these tools, when used in combination, have the potential to further our understanding of neuromuscular function by serving as a rigorous means to test scientific hypotheses in ways that complement and leverage experimental data.

Fig. 1

Simple model of the human arm consisting of two planar joints and six muscles.

Fig. 2

Schematic description of the interactions among machine learning, control theory, and estimation-detection theory.

Fig. 3

Block diagram representation of data collection and supervised learning schemes (see text for detailed description of each case). In every case, data is collected in the real world by feeding joint torques to the real-world Plant (gray block). These torques can be: (A) selected at random, (B) based on a preliminary inverse model that may (C) include noise and selective use of training data, or (D) selected with the benefit of a demonstrator. For simplicity of illustration, the dependence of the inverse model and controller on state, x, ẋ, is omitted. (A) Direct inverse modeling. (B) Feedback-error learning. (C) Staged learning. (D) Learning from demonstration.

Fig. 4

Illustration that an exploration in input space (here, torque) may not sample a desired output (acceleration). Sampling in input space is limited to the range ±0.5—any practical setting requires limits of exploration.

Fig. 5

Mapping from spring resting lengths (s_i) to hand positions (x, y). Several redundant resting lengths are solutions for one desired hand position (red). The graph on the left shows a two-dimensional projection of a cross section of the six-dimensional nullspace of spring resting lengths: s₁ and s₂ were set to constant values; s₃ and s₄ were randomly drawn (within dashed box), and all values of s₅ and s₆ were projected onto the displayed plane. The original six-dimensional nullspace in rest-length space is, therefore, nonconvex. Thus, the average of all rest-lengths solutions does not map onto the desired hand position.

Fig. 6

Simulation results for our two-link arm model using an optimal feedback controller. The task is to move the two-link arm from the initial configuration of (θ₁, θ₂) = (0, 0) to (θ₁, θ₂) = (60°, 90°) in the time horizon of 1 s and with 0 terminal velocity (ω₁(T), ω₂(T)) = (0, 0)). The lower left panel illustrates the reduction of the cost function for every iteration of the ILQG algorithm. The algorithm convergences quickly (after about 15 iterations), and yields smooth joint-space trajectories with close to bell-shaped velocity profiles.

Fig. 7

Monte Carlo approach to model evaluation and hypothesis testing. An experiment is performed that produces some data, from which a test statistic is calculated. A computer model is coded that generates an output comparable to the statistics of the experimental data (or target test statistic). All parameters are varied stochastically within their feasible range, and a distribution of possible test statistics are generated for that model. One can then determine whether there exist sets of parameter values for the model that can replicate the distribution of the experimental data. If possible predictions of the model cannot replicate the experimental data, the hypothesis encoded in the model is likely untrue and a new hypothesis needs to be developed and encoded. In addition, by investigating the sensitivity of model predictions to specific subsets of parameters, the components of the model of particular importance can be identified.

Fig. 8

Example of Monte Carlo analysis of possible muscle activation patterns for the Planar Arm Example. 100 000 muscle force vectors that produced 50% maximal force in the forward direction were calculated, and then histograms were made of the valid solutions in each muscle for each of two postures. Notice that in both postures, some muscles are necessary (zero force is not a valid solution). Notice also that some muscles switch from being necessary in one posture to redundant (zero force is an allowed solution) in other postures (e.g., muscle 5). A similar example of this approach is presented in [182].

Fig. 9

Monte Carlo analysis of the Fuglevand Model. (A) Each line shows the force/force-variability relation generated by different parameter sets. (B) Each line shows the EMG/force relation generated by the same parameter sets shown in (A). (C) Relations found in (A) and (B) are evaluated by test statistics that are regression slopes [log-log in the case of (A)]. Good fits to experimental data are force/force-variability slopes of greater than 0.75 and EMG/force slopes of less than 1.05; thus, very few parameter sets are able to reproduce experimental data. Adapted from [97].

Fig. 10

Example of Monte Carlo hypothesis testing. (A) Illustration of two hypotheses and sources of noise. (B)–(D) Monte Carlo distributions of test statistics (target-directedness) generated by the two models, as compared with the experimentally observed value. The synergistic hypothesis can only replicate the data under specific conditions, and induces muscle force correlations that are unrealistic. Adapted from [231]. (A) Hypotheses and noise sources. (B) Both hypotheses have Sig-Indep. Noise only. (C) Both hypotheses have Muscle SDN only. (D) Flexible hypothesis has only Muscle SD, synergistic hypothesis has Muscle SDN and Synergy SDN equally. (E) Flexible hypothesis has only Muscle SDN, synergistic hypothesis has Synergy SDN ten times Muscle SDN.