Table 1.
Gain precursor values. The gain precursor from the source muscle to the
target muscle, denoted as
, is assigned to the
row and
column of
. Values of the same color are on the same scale and can be compared to each other. References are indicated in superscripted brackets. Superscripted Greek letters indicate that the value must be divided between muscle pairs. Entries along the diagonal represent homonymous feedback [25–48].
Table 2.
The unscaled matrix of muscle spindle feedback gains, .
Fig 1.
Closed-loop MIMO model of the upper limb.
Feedback mediated by Golgi tendon organs (GTO) is modeled as force feedback, and feedback mediated by muscle spindles (MS) is modeled as proportional-derivative feedback. : vector of neural drives from central nervous system to muscles;
and
: vectors of efferent and afferent delays, respectively;
: diagonal matrix of maximum muscle forces;
and
: vectors of muscle activity and force;
and
: diagonal matrices of muscle time constants;
: matrix of muscle moment arms representing the musculoskeletal geometry (MSK-G) of the upper limb;
and
: vectors of muscle force and joint torque;
,
, and
: matrices of joint inertia, damping, and stiffness representing the mechanical impedance of the upper limb;
: vector of muscle elongation;
and
: matrices of proportional and derivative feedback gains;
and
: vectors of neural outputs from MS and GTO, respectively;
: matrix representing the force feedback gain.
and
represent subsystems referenced in Methods.
Table 3.
Sensitivity of the system-wide gain margin, defined as the maximum value of the muscle spindle gain scaling factor, (before reaching instability). The values indicate the system-wide margin resulting from halving or doubling the default values of the indicated forward-path parameters.
Table 4.
The 13 major superficial muscles of the upper limb included in this study, with their associated abbreviations and maximum muscle force values from the literature, as well as our estimates of homonymous GTO feedback gains and afferent and efferent delays. Note that the GTO feedback gains are expressed in units of 10-3 MVC/N (e.g., the gain for Delt Ant is 1.04·10-3 MVC/N).
Table 5.
Round trip delay times (in milliseconds) reported in the literature. References are indicated in superscripted brackets [26–28,30–35,37–42,44,46,47,51,52].
Table 6.
Excess of Central Delay values (in milliseconds) reported in the literature. Cells are color-coded according to the sign of the feedback gain estimated above in Table 2). Blue and red represent positive (excitatory) and negative (inhibitory) feedback gains, respectively. By definition, there is no excess of central delay for homonymous reflex loops (depicted in gray). White cells do not have a corresponding entry in Table 4.
Table 7.
Locations and characteristics of poles associated with the open-loop system and a third-order Padé approximation of the closed-loop system. Summary statistics of the imaginary portion of poles were calculated using the absolute value of the imaginary portion. In calculating summary statistics of natural frequency and damping ratio, real poles were excluded.
Fig 2.
A) Step responses from descending neural drive to joint displacement (i.e., response in due to step in
), with equal time and joint-displacement ranges (on the horizontal and vertical axes, respectively) for all subplots.
The very large step responses in SAA (~20 rad, subplot A) are caused by large inputs (100% MVC), large maximum muscle forces and moment arms in Delt Lat and Pec Maj, and low joint stiffness in SAA. In actuality, tremorogenic inputs are on the order of 5-10% MVC, so responses would be scaled by 0.05-0.1, and joint stiffness increases significantly toward the ends of joint ranges of motion, further reducing motion amplitude. B) Same as A, but zoomed in, such that each subplot has its own distinct joint-displacement range (along the vertical axis).
Fig 3.
A) Step responses from external joint torque to muscle activity (i.e., response in due to step in
), with equal time and muscle-activity ranges (on the horizontal and vertical axes, respectively) for all subplots. B) Same as A, but zoomed in, such that each subplot has its own distinct muscle-activity displacement range (along the vertical axis).
Fig 4.
A) Step responses from external joint torque to joint displacement (i.e., response in due to step in
), with equal time and joint-displacement ranges (on the horizontal and vertical axes, respectively) for all subplots. B) Same as A, but zoomed in, such that each subplot has its own distinct joint-displacement range (along the vertical axis).
Fig 5.
Round-trip neural delay times plotted against innervation length, together with the equations and correlation coefficients of their least-square line fits.
The inverse of the slopes represent conduction velocities.
Table 8.
The 7 main degrees of freedom of the upper limb from shoulder to wrist, with their abbreviations.