The scaling of motor noise with muscle strength and motor unit number in humans

Abstract

Understanding the origin of noise, or variability, in the motor system is an important step towards understanding how accurate movements are performed. Variability of joint torque during voluntary activation is affected by many factors such as the precision of the descending motor commands, the number of muscles that cross the joint, their size and the number of motor units in each. To investigate the relationship between the peripheral factors and motor noise, the maximum voluntary torque produced at a joint and the coefficient of variation of joint torque were recorded from six adult human subjects for four muscle/joint groups in the arm. It was found that the coefficient of variation of torque decreases systematically as the maximum voluntary torque increases. This decreasing coefficient of variation means that a given torque or force can be more accurately generated by a stronger muscle than a weaker muscle. Simulations demonstrated that muscles with different strengths and different numbers of motor units could account for the experimental data. In the simulations, the magnitude of the coefficient of variation of muscle force depended primarily on the number of motor units innervating the muscle, which relates positively to muscle strength. This result can be generalised to the situation where more than one muscle is available to perform a task, and a muscle activation pattern must be selected. The optimal muscle activation pattern required to generate a target torque using a group of muscles, while minimizing the consequences of signal dependent noise, is derived.

Appendix

Two cost functions describing how muscles should be used to minimise the consequences of signal-dependent noise can be derived from the basic TOPS cost function given in Eq. 8. Using the simulation results given in Eq. 5 and substituting:

$$ CV{\text{ = }}\sigma {\text{/}}f{\text{ }} $$

(10)

gives:

$$ \sigma {\text{ = exp(}}d{\text{) }}f{\text{ MUN }}^{{{\text{ - 0}}{\text{.5}}}} {\text{ }} $$

(11)

As this holds for every muscle, the constant exp(d) can be ignored and σ can be substituted into the TOPS cost function (Eq. 8) to give:

$$ C_{{{\text{TOPS}}}} = {\sum\limits_{i = 1}^n ( }{f_{i} ^{2} } \mathord{\left/ {\vphantom {{f_{i} ^{2} } {{\text{MUN}}_{i} }}} \right. \kern-\nulldelimiterspace} {{\text{MUN}}_{i} }) $$

(12)

Despite the interesting predictions of the MUN cost function, the current paucity of data on motor unit numbers means that this cost function is not easy to apply in practice. For this reason, we also derive a second, more practical cost function, based on the experimental results. Taking the mean parameters for the six subjects, we have shown that:

$$ {\text{CV = exp( - 2}}{\text{.76) MVT }}^{{{\text{ - 0}}{\text{.25}}}} {\text{ }} $$

(13)

Substituting Eq. 10 gives:

$$ \sigma {\text{ = exp( - 2}}{\text{.76) }}f{\text{ MVT }}^{{{\text{ - 0}}{\text{.25}}}} {\text{ }} $$

(14)

Again, the constant exp(−2.76) can be ignored and σ can be substituted into Eq. 8 to give:

$$ C_{{{\text{TOPS}}}} = {\sum\limits_{i = 1}^n ( }{f_{i} ^{2} } \mathord{\left/ {\vphantom {{f_{i} ^{2} } {{\text{MVT}}_{i} ^{{0.5}} }}} \right. \kern-\nulldelimiterspace} {{\text{MVT}}_{i} ^{{0.5}} }) $$

(15)

This cost function can be used to determine the optimal control strategy for minimising noise over multiple muscles and requires only knowledge of the maximum voluntary torque produced by each muscle, which is available in standard tables (for example, Winters and Stark 1988).