Figure 1
Tremor propagation. A. Overview: tremorogenic activity in multiple muscles generates tremorogenic forces in the same muscles before combining into tremorogenic torques and finally tremor in joint DOF throughout the upper limb. B. Model: this process is modeled using three sub-models that transform activity in 50 muscles (u) into force in the same muscles (f), then joint torque (τ) and finally joint rotation (q) in the 7 joint DOF from shoulder to wrist. Variables u and f are 50-element time-varying vectors, and τ and q are 7-element time-varying vectors. Parameters t1, t2, and C are 50-by-50 diagonal matrices representing time constants of excitation-contraction coupling (t1 and t2) and maximum muscle force (C). Parameter M is a 7-by-50 matrix of muscle moment arms, and I, D, and K are 7-by-7 impedance matrices representing coupled joint inertia, damping, and stiffness, respectively. C. Example: simulated tremorogenic activity in the triceps longus muscle generates tremorogenic force in the triceps longus muscle, tremorogenic torque in the DOF of the shoulder and elbow, and finally tremor in all 7 DOF. These simulations include both the transient and steady-state responses, but the rest of the paper considers only steady-state responses.
Figure 2
Tremor propagation was simulated in these seven postures. Postures 1–5 are representative of activities of daily living: anatomical posture, but with the elbow flexed at 90° and the forearm midway between pronation and supination (Posture 1); hand held in front of the mouth (Posture 2); hand held in front of the abdomen (Posture 3); upper limb partially extended in front of the torso (Posture 4); and upper limb partially extended to the side and in front of the torso, roughly midway between the parasagittal and frontal planes (Posture 5). Postures 6 and 7 are common in clinical evaluation of tremor using the TETRAS [75]: wing beating posture (Posture 6) and forward horizontal reach posture (Posture 7). For exact joint angles, see Table 2.
Table 1
List of muscles included in the model, together with abbreviations and peak forces, ordered roughly from proximal to distal.
| MUSCLE | ABBREVIATION | PEAK FORCE (N) |
|---|---|---|
| Deltoid Anterior | DELT1 | 1218.9 |
| Deltoid Middle | DELT2 | 1103.5 |
| Deltoid Posterior | DELT3 | 201.6 |
| Supraspinatus | SUPSP | 499.2 |
| Infraspinatus | INFSP | 1075.8 |
| Subscapularis | SUBSC | 1306.9 |
| Teres minor | TMIN | 269.5 |
| Teres major | TMAJ | 144.0 |
| Pectoralis major Clavicular | PECM1 | 444.3 |
| Pectoralis major Sternal | PECM2 | 658.3 |
| Pectoralis major Ribs | PECM3 | 498.1 |
| Latissimus dorsi Thoracic | LAT1 | 290.5 |
| Latissimus dorsi Lumbar | LAT2 | 317.5 |
| Latissimus dorsi Iliac | LAT3 | 189.0 |
| Coracobrachialis | CORB | 208.2 |
| Triceps Long | TRIlong | 771.8 |
| Triceps Lateral | TRIlat | 717.5 |
| Triceps Medial | TRImed | 717.5 |
| Anconeus | ANC | 283.2 |
| Supinator | SUP | 379.6 |
| Biceps Long | BIClong | 525.1 |
| Biceps Short | BICshort | 316.8 |
| Brachialis | BRA | 1177.4 |
| Brachioradialis | BRD | 276.0 |
| Extensor carpi radialis longus | ECRL | 337.3 |
| Extensor carpi radialis brevis | ECRB | 252.5 |
| Extensor carpi ulnaris | ECU | 192.9 |
| Flexor Carpi radialis | FCR | 407.9 |
| Flexor capri ulnaris | FCU | 479.8 |
| Palmaris longus | PL | 101.0 |
| Pronator teres | PT | 557.2 |
| Pronator quadratus | PQ | 284.7 |
| Flexor digitorum superficialis Digit 5 | FDSL | 75.3 |
| Flexor digitorum superficialis Digit 4 | FDSR | 171.2 |
| Flexor digitorum superficialis Digit 3 | FDSM | 258.8 |
| Flexor digitorum superficialis Digit 2 | FDSI | 162.5 |
| Flexor digitorum produndus Digit 5 | FDPL | 236.8 |
| Flexor digitorum produndus Digit 4 | FDPR | 172.9 |
| Flexor digitorum produndus Digit 3 | FDPM | 212.4 |
| Flexor digitorum produndus Digit 2 | FDPI | 197.3 |
| Extensor digitorum communis Digit 5 | EDCL | 39.4 |
| Extensor digitorum communis Digit 4 | EDCR | 109.2 |
| Extensor digitorum communis Digit 3 | EDCM | 94.4 |
| Extensor digitorum communis Digit 2 | EDCI | 48.8 |
| Extensor digiti minimi | EDM | 72.4 |
| Extensor indicis propius | EIP | 47.3 |
| Extensor pollicis longus | EPL | 88.3 |
| Extensor pollicis brevis | EPB | 46.0 |
| Flexor pollicis longus | FPL | 201.0 |
| Abductor pollicis longus | APL | 116.7 |
Table 2
Joint angles for the 7 postures considered in simulation, listed according to the Denavit-Hartenberg parameterization set forth in [39]. The postures are meant to reflect a range of activities of daily living (1–5) in addition to postures commonly used for clinical diagnosis (6–7). See Figure 2 for a description and visualization of each posture.
| POSTURE 1 | POSTURE 2 | POSTURE 3 | POSTURE 4 | POSTURE 5 | POSTURE 6 | POSTURE 7 | |
|---|---|---|---|---|---|---|---|
| θ1 | 0 | π/4 | π/16 | π/5 | π/3 | π/2 | π/2 |
| θ2 | 0 | 0 | –π/16 | π/8 | –π/3 | –5π/18 | 0 |
| θ3 | 0 | π/4 | π/3 | π/8 | π/6 | 5π/12 | π/2 |
| θ4 | π/2 | 3π/4 | π/2 | π/3 | π/6 | 25π/36 | 0 |
| θ5 | π/2 | π/4 | π/2 | π/4 | π/4 | 7π/16 | π/2 |
| θ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| θ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Figure 3
Gains from neural drive to muscle force, neural drive to joint torque, and neural drive to joint rotation, averaged across all postures (evaluated at a tremor frequency of 6 Hz). Alternatively, the heatmaps can be interpreted as the muscle force, joint torque, and joint rotation that would be result if all muscles received the same amount of tremorogenic activity (relative to each muscle’s maximum voluntary contraction). Muscles are ordered proximal (top) to distal (bottom; for muscle names, see Table 1). Similarly, joint DOF are listed proximal (left) to distal (right): shoulder flexion-extension (SFE), abduction-adduction (SAA), and internal-external rotation (SIER); elbow flexion-extension (EFE) and forearm pronation-supination (FPS); and wrist flexion-extension (WFE) and radial-ulnar deviation (WRUD), respectively. Equivalent heatmaps for individual postures are given in Figures SM1-SM7 in Supplemental Materials.
Figure 4
Tremor propagation patterns are quite robust to changes in tremor frequency, muscle dynamics, subject size, and co-contraction. Each cell represents the average of the gains from one muscle to joint rotation in all 7 DOF, averaged across all postures (i.e. the average of a row in the mean neural-drive-to-joint-rotation gain heatmap in Figure 3). Gains within a muscle (row) are generally more stereotyped than gains between muscles (column), demonstrated by the dominance of horizontal over vertical patterns.
Figure 5
Normalized sensitivity of gains to each element of the inertia, damping, and stiffness matrices (averaged across all 350 input-output relationships and all postures). Left column: mean normalized sensitivity to each matrix element. Right column: fraction of the input-output relationships whose sensitivity is positive, indicating that an increase in the indicated element of inertia, damping, or stiffness matrices causes an increase in gain. Sensitivities associated with matrix elements assumed to be zero were left blank. Different inertia values were non-zero in different postures, so the elements of the average sensitivity and fraction positive heatmaps for inertia represent the postures in which that matrix element was non-zero.