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].

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