Enslaving in a serial chain: interactions between grip force and hand force in isometric tasks

Abstract

This study was motivated by the double action of extrinsic hand muscles that produce grip force and also contribute to wrist torque. We explored interactions between grip force and wrist torque in isometric force production tasks. In particular, we tested a hypothesis that an intentional change in one of the two kinetic variables would produce an unintentional change in the other (enslaving). When young healthy subjects produced accurate changes in the grip force, only minor effects on the force produced by the hand (by wrist flexion/extension action) were observed. In contrast, a change in the hand force produced consistent changes in grip force in the same direction. The magnitude of such unintentional grip force change was stronger for intentional hand force decrease as compared to hand force increase. These effects increased with the magnitude of the initial grip force. When the subjects were asked to produce accurate total force computed as the sum of the hand and grip forces, strong negative covariation between the two forces was seen across trials interpreted as a synergy stabilizing the total force. An index of this synergy was higher in the space of “modes,” hypothetical signals to the two effectors that could be changed by the controller one at a time. We interpret the complex enslaving effects (positive force covariation) as conditioned by typical everyday tasks. The presence of synergic effects (negative, task-specific force covariation) can be naturally interpreted within the referent configuration hypothesis.

Appendix

The following is a proof for the equality of variance in the orthogonal subspace defined in the Force and Mode spaces. We denote vector quantities by a lower-case, bold font letters ($\vec{\varvec{f}}$). For unit vectors, the overhead bar is replaced with a hat ($\hat{\varvec{e}}$). The notation $\vec{\varvec{a}} \cdot \vec{\varvec{b}}$ denotes the dot product of vectors $\vec{\varvec{a}}$ and $\vec{\varvec{b}}$, and the symbol d(.) indicates the derivative.

Consider a system of n elemental variables $\varvec{f}_{\varvec{i}}$ constrained by a single, linear equation: $f_{tot} = \sum\nolimits_{i = 1}^{i = n} {f_{i} }$, and an enslaving matrix $E \in n \times n$ such that all of the n columns add to 1. The matrix E has the form of a left stochastic matrix or a Markov matrix.

The mode is defined as:

$$\bar{\varvec{m}} := E^{ - 1} {\varvec{f}}$$

where $\bar{\varvec{m}}$ is a n-dimensional column vector. Now assuming that the enslaving matrix is constant,

$${\text{d}}\bar{\varvec{m}} = E^{ - 1} {\text{d}}\varvec{f}, \Rightarrow {\text{d}}\varvec{f} = E\text{d}\bar{\varvec{m}}$$

$${\text{Therefore}}, {\text{d}}f_{tot} = J{\text{d}}\vec{\varvec{f}} = JE{\text{d}}\bar{\varvec{m}} = J{\text{d}}\bar{\varvec{m}},$$

(13)

where J is the task Jacobian given by $\left[ {\begin{array}{*{20}c} 1 & 1 & \cdots & 1 \\ \end{array} } \right]$. Equation 13 shows that the task Jacobian is identical in the Force and Mode spaces. Therefore, the orthogonal subspace of the task Jacobian in the two spaces is:

$$\hat{\varvec{o}} = \frac{1}{\sqrt n } \left[ {\begin{array}{*{20}c} 1 \\ {\begin{array}{*{20}c} 1 \\ \vdots \\ \end{array} } \\ 1 \\ \end{array} } \right]$$

It is evident that the vector $\bar{\varvec{v}}: = \left[ {\begin{array}{*{20}c} 1 & 1 & \cdots & 1 \\ \end{array} } \right]$ ∈ 1 × n is a left eigenvector of E, and $\bar{\varvec{v}}^{T}$ is a right eigenvector of E ^T, each with the eigenvalue 1 (i.e., $\bar{\varvec{v}}E = \bar{\varvec{v}}$ and $E^{T} \bar{\varvec{v}}^{T} = \bar{\varvec{v}}^{T}$). Also, $\bar{\varvec{v}}^{T} = \sqrt n \hat{\varvec{o}}$, and therefore,

$$E^{T} \hat{\varvec{o}} = \hat{\varvec{o}}$$

Here we show that

$$\bar{\varvec{f}} \cdot \hat{\varvec{o}} = \left( {E\bar{\varvec{m}}} \right) \cdot \hat{\varvec{o}} = \bar{\varvec{m}} \cdot \hat{\varvec{o}}$$

It is well known that for two arbitrary vectors $\bar{\varvec{p}},\bar{\varvec{q}} \in {\mathbb{R}}^{n}$, and a matrix $A \in n \times n$,

$$A\bar{\varvec{p}} \cdot \bar{\varvec{q}} = \bar{\varvec{p}} \cdot A^{T} \bar{\varvec{q}}$$

Therefore, for any vector $\bar{\varvec{m}}$ in the Mode space,

$$\begin{aligned} \left( {E\bar{\varvec{m}}} \right) \cdot \hat{\varvec{o}}& = \bar{\varvec{m}}\left( {E^{T} \hat{\varvec{o}}} \right)\\ \left({E\bar{\varvec{m}}} \right) \cdot \hat{\varvec{o}}&= \bar{\varvec{m}} \cdot \hat{\varvec{o}}\\ \Rightarrow\bar{\varvec{f}} \cdot \hat{\varvec{o}}& = \bar{\varvec{m}} \cdot \hat{\varvec{o}} = \frac{1}{\sqrt n } \mathop\sum \limits_{i = 1}^{i = n} f_{i}\end{aligned}$$

(14)

Equation (14) demonstrates that the transformation of $\vec{\varvec{f}}$ into ${\bar{\varvec{m}}}$ leaves the projection of $\vec{\varvec{f}}$ onto ${\hat{\varvec{o}}}$ unaltered. It follows that the variances in the orthogonal subspaces of the Mode and Force spaces for such systems are equal.