All subjects were instructed to move their ankle and subtalar joints (both intact and perceived phantom) in time with a metronome under several bilateral coordination conditions. Joint kinematics were recorded from intact joints, and surface electromyography (sEMG) signals were recorded from residual limbs (Additional file 1: Fig. S1).

The first set of binary coordination conditions addressed the presence of visual feedback. Tasks with vision required participants to look toward their feet while blind tasks were performed with eyes closed and blindfolded. The second set of conditions determined the patterns of ankle-subtalar movement. In the symmetric condition, subjects were instructed to fully invert both subtalars on the downbeat and fully evert both subtalars on the (silent) offbeat (Fig. (Fig.2a).2a). In the parallel condition, subjects were instructed to fully evert the left subtalar and invert the right subtalar on the downbeat, mirroring the motion on the offbeat (Fig. (Fig.2b).2b). In the sagittal condition, subjects were instructed to plantarflex both ankles on the downbeat and dorsiflex on the offbeat (Fig. (Fig.2c).2c). All combinations of conditions with coded names are listed in Table Table22.

Bilateral coordination modes. a Coordinated subtalar joints under symmetry instruction. b Coordinated subtalar joints under parallel instruction. c Coordinated ankle joints under sagittal instruction. d Relative phase (θrel) between two joints at any given point in time is defined as the difference in angles formed by their normalized velocities (θ˙) and normalized angular positions (θ) during the movement according to Eqn. 3 [30]

Bilateral coordination test conditions with coded names

Subjects completed three trials for each combination of conditions in random order, with a two-minute break between trials to limit effects from fatigue. Each trial required subjects to perform one cyclical motion per downbeat of a pre-recorded metronome audio track. Subjects with amputation were asked to volitionally activate their residual muscles as if to move their phantom joints. The minute-long audio track consisted of 5 sections with equal duration starting at a tempo of 1 Hz in the first section and incrementing by 0.4 Hz until a final tempo of 2.6 Hz (Additional files 2, 3, and 4).

Subjects were instructed to prioritize maintaining pace with the metronome track, sacrificing full range of motion of movements if necessary. Subjects were asked to repeat their first trial if they misunderstood the instructions (e.g., moving at half tempo). However, all subjects demonstrated a sense of rhythm and were observed to follow the tempo track to a reasonable degree. Inversion-eversion (IE) movements naturally involved some foot rotation in the transverse plane, though this degree of freedom was not explicitly measured. At the end of every trial, subjects performed a slow, controlled range of motion calibration test consisting of two dorsiflesion-plantarflexion (DP) cycles followed by two IE cycles to ensure invariant goniometer placement. Though subjects with amputation performed tests under sagittal instruction, the data are not reported here. All subjects were naive to the nature of the study .

Ankle-subtalar kinematics were collected from all subjects. An electromechanical relay was used to simultaneously trigger all related systems to synchronize each trial’s recorded data in time. A pair of commercial two-axis goniometers (Biometrics Ltd., Newport, UK) was used to measure all intact ankle-subtalar kinematics. Subjects were seated on the edge of a patient bed with knees at 90 flexion. Subjects wore well-fitted athletic shoes. Goniometers were attached with one end adhered to the shoe heel and the other adhered to skin superior to the Achilles tendon. Goniometers were calibrated for subtalar axis and measurement axis parallelism by instructing subjects to perform slow, controlled subtalar motions while adjusting the positioning of the sensors. Afterward, goniometer ends were further secured to the shoe and shank using porous medical tape.

All ankle-subtalar position trajectories were recorded at a sampling frequency of 1,000 Hz. A 4th order 10 Hz low pass IIR Butterworth filter was applied forward and backward over the raw trajectory data. Velocity trajectories were generated from the filtered position trajectories by two-point forward finite differentiation.

Surface electromyography signals were collected from the residual limbs of subjects with amputation. A commercial Refa (TMSi, Oldenzaal, Netherlands) 128-channel amplifier was used to collect sEMG signals. The skin was cleaned with isopropyl alcohol, and adhesive wet Ag/AgCl surface electrodes were placed in clusters to capture activity from the following residual muscles: tibialis anterior (TA), lateral gastrocnemius (LGAS), peroneus longus (PL), and tibialis posterior (TP). The residuum was palpated while asking subjects to perform ankle-subtalar movements to identify likely muscle locations. An additional ground reference electrode was placed on the patella of the affected side. 1.5 m shielded cables connected each electrode terminal to the Refa amplifier.

Monopolar sEMG signals were recorded at a sampling frequency of 2,048 Hz. From these, bipolar signals were reconstructed for each residual muscle after data collection based on the set of pairs which demonstrated the least amount of cocontraction during the calibration portion of the bilateral coordination tasks. A 4th order 10–500 Hz band pass infinite impulse response (IIR) Butterworth filter was designed in MATLAB R2019b (The MathWorks) and applied forward and backward over each recording. Filtered data were then rectified and normalized against the maximum voltage within each recording, assumed to represent maximum voluntary contraction. A 4th order 10 Hz low pass IIR Butterworth filter was then applied forward and backward to produce a record of neural excitation μ(t) [43]. Neural excitations were level shifted such that the minimum amplitude across each trial was 0.01 to avoid Thelen Hill-type muscle tendon unit (MTU) numerical singularities [44].

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