A neuromuscular model was developed to produce a valid mapping from residual muscle activations to the physical domain of subtalar IE kinematics. The IE axis, as opposed to the DP axis, was chosen to facilitate comparison of bilaterally symmetric and parallel coordination between subjects with amputation and the non-amputee control group.

Consisting of only an agonist-antagonist muscle pair acting upon a second-order rotational mechanical system, the model is physically capable of generating modulated oscillatory kinematic trajectories from neural excitation inputs (Fig. (Fig.3).3). Importantly, this simple model was designed with deference to the idea that needless complexity obfuscates insight into underlying phenomena [45]. The model’s design does not replicate the exact morphology of an intact subtalar joint with accompanying tissue. Rather, it is specific enough to limit behavior to IE-like kinematics while retaining a sufficient number of tunable morphological parameters to fit a wide range of subjects. A bond graph representing detailed energetic relationships between model components is provided (Additional file 1: Fig. S2).

Neuromuscular subtalar model. Mechanical diagram of the base planar IE dynamic neuromuscular model with passive rotational stiffness (k), rotational damping (b), mass (m), and two Hill-type muscle-tendon units (MTUs) in agonist-antagonist configuration. Parameter value ranges are provided in Additional file 1: Table S1. Figure not drawn to scale

Referring to the dimensions of the model in Fig. Fig.3,3, l1 was chosen to be 0.7 m, equivalent to the horizontal span of each MTU. This is a relatively large value compared to other dimensions due to the finite stroke length of each MTU; the actuators need to be effective over the entire natural IE range of motion considering force generation limitations from optimal contractile element fascicle length and active force curve shape. A rotational domain was chosen over a linear one to allow effective MTU moment arms to vary over the range of motion in a physiologically relevant manner [46]. As θIE deviates from zero, effective MTU moment arm lengths decrease from the maximum value of l3=0.1 m, and the relative effectiveness of passive restoring stiffness k, modeled after ligament contributions, increases. Damping b is modeled to represent energetic losses from tissue sliding surfaces. Point mass m=0.4 kg is located at a distance l2=0.2 m from the pivot and thus has a moment of inertia J=0.016 kg·m2.

Including state variables for the MTUs, dynamics of the planar subtalar model are given by

where F(μ,l,v) is the force generated by an MTU through the nonlinear dynamics of a Hill-type muscle model with μ, l, and v being the excitation, length, and velocity of the MTU’s contractile element as a function of time. A thorough explanation of the specific Hill-type muscle model implemented is given by Thelen [44].

A genetic algorithm was used to optimize the model’s morphological parameters for each subject to minimize the residual sum of squares between the mirrored subtalar trajectory and the model’s output trajectory within a three-second window of training data. Because slow movements are difficult for humans to produce smoothly [47], and fast movements are difficult to produce accurately [23], the three-second window of training data was taken from the intermediate 1.4 Hz section of each subject’s blind symmetry trials to minimize trajectory error generated by the subject. From this window, raw sEMG data from residual inverter and everter muscles were processed using techniques described previously to produce estimated neural excitations μ(t). Mirrored IE trajectories from the contralateral intact subtalar were used as the desired reference trajectory.

The subtalar model was implemented in OpenSim 4.0, an open source dynamics environment widely used for neuromuscular modeling [48, 49]. MATLAB’s genetic algorithm (GA) toolbox was used to find the global optimum values of the MTU morphological and dynamic model parameters (Additional file 1: Table S1). MTU variables which belong to OpenSim’s Thelen2003Muscle class not explicitly mentioned here were fixed to their default values. A total of 18 MTU and 2 model parameters were investigated over the ranges specified in the table. The optimal parameter set minimized the residual sum of squares between generated trajectory and reference trajectory using the single objective cost function

where t is the discretized time index for trajectories n ms long. A full process diagram is provided (Additional file 1: Fig. S3).

Every GA run was uniformly initialized across the solution space with 5,000 members and set to run for 100 generations or until stalling for 3 generations. Two elite members were preserved per generation. Each iteration within the GA performed an OpenSim forward simulation with the specified AMI neural excitation inputs into a neuromuscular subtalar model initialized to a specific population member’s parameters and the initial position of the reference trajectory. Three separate GA runs were performed for each subject, each taking approximately 12 hours on an AMD RyzenTM 2950X CPU with 16 parallel processing threads. The optimized parameter set from the three runs with the best performance was used for all subsequent analysis per subject.

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