Motor equivalence during multi-finger accurate force production

Abstract

We explored stability of multi-finger cyclical accurate force production action by analysis of responses to small perturbations applied to one of the fingers and inter-cycle analysis of variance. Healthy subjects performed two versions of the cyclical task, with and without an explicit target. The “inverse piano” apparatus was used to lift/lower a finger by 1 cm over 0.5 s; the subjects were always instructed to perform the task as accurate as they could at all times. Deviations in the spaces of finger forces and modes (hypothetical commands to individual fingers) were quantified in directions that did not change total force (motor equivalent) and in directions that changed the total force (non-motor equivalent). Motor equivalent deviations started immediately with the perturbation and increased progressively with time. After a sequence of lifting–lowering perturbations leading to the initial conditions, motor equivalent deviations were dominating. These phenomena were less pronounced for analysis performed with respect to the total moment of force with respect to an axis parallel to the forearm/hand. Analysis of inter-cycle variance showed consistently higher variance in a subspace that did not change the total force as compared to the variance that affected total force. We interpret the results as reflections of task-specific stability of the redundant multi-finger system. Large motor equivalent deviations suggest that reactions of the neuromotor system to a perturbation involve large changes in neural commands that do not affect salient performance variables, even during actions with the purpose to correct those salient variables. Consistency of the analyses of motor equivalence and variance analysis provides additional support for the idea of task-specific stability ensured at a neural level.

Appendix: Variance and motor equivalence analyses

The force data f were converted into a mode vector m by using the enslaving matrix E, where f = [f _I, f _M, f _R, f _L]^T (T represents matrix transpose).

$$m = \left[ {\mathbf{E}} \right]^{ - 1} \cdot f$$

(5)

The Jacobian matrix J defining the linear map between changes in finger forces (df) modes (dm) and changes in total force dF _TOT was defined:

$${\text{d}}F_{\text{TOT}} = \, \left[ { 1 { 1 1 1}} \right] \, \cdot {\text{ d}}f = \, \left[ { 1 { 1 1 1}} \right] \, \cdot {\mathbf{E}} \cdot {\text{ d}}m$$

(6)

$$\therefore {\mathbf{J}}_{{\mathbf{F}}} = \left[ { 1 { 1 1 1}} \right]{\kern 1pt} \quad {\text{and}}\quad {\mathbf{J}}_{{\mathbf{M}}} = \, \left[ { 1 { 1 1 1}} \right] \, \cdot {\mathbf{E}}$$

(7)

The J matrix defining the changes between the finger force and modes (ƒ/m) and changes in total moment about the longitudinal axis of the forearm/hand with respect to the midpoint of the hand is:

$${\mathbf{J}}_{{\mathbf{F}}} = \, \left[ {d_{\text{I}} , \, d_{\text{M}} , \, d_{\text{R}} , \, d_{\text{L}} } \right]\quad {\text{and}}\quad {\mathbf{J}}_{{\mathbf{M}}} = \left[ {d_{\text{I}} , \, d_{\text{M}} , \, d_{\text{R}} , \, d_{\text{L}} } \right] \, \cdot {\mathbf{E}}$$

(8)

where the d _i entries representing the lever arm of fingers, d _I = 4.5 cm, d _M = 1.5 cm, d _R = 1.5 cm, and d _L = −4.5 cm. The UCM is the null-space of the Jacobian matrix J, spanned by the basis vectors ε _i , solving:

$${\mathbf{J}} \cdot \varepsilon_{i} = \, 0$$

(9)

For the variance analysis, the mean-free ƒ/m (∆x _jk ) for a given j trial and k phase (pre-, during- and post-perturbation) was computed:

$$\Delta x_{j} = \, x_{j} {-}\bar{x}_{0}$$

(10)

where x was either force or mode. The ∆x was projected into the null-space and orthogonal space of J as follows:

$$f_{\parallel } = \mathop \sum \limits_{i = 1}^{n - p} \left( {\varepsilon_{i}^{T} \cdot \Delta x} \right) \cdot \varepsilon_{i}$$

(11)

$$f_{ \bot } =\Delta x - f_{\parallel }$$

(12)

where $\varvec{f}_{\parallel }$ is the ƒ parallel component and $\varvec{f}_{ \bot }$ is the orthogonal component, n is the number of elemental variables (ƒ/m), and p is the number of constraints defined by the performance variable. There are n–p basis vectors, so that the null-space has n–p dimensions.

The variance across trials per degree of freedom along V _ucm and orthogonal V _ort to the UCM was computed.

$$\varvec{V}_{{\varvec{ucm}}} = \frac{{\mathop \sum \nolimits \left| {\varvec{f}_{\parallel } } \right|^{2} }}{{\left( {n - p} \right) N_{\text{trials}} }}$$

(13)

$$\varvec{V}_{{\varvec{ort}}} = \frac{{\mathop \sum \nolimits \left| {\varvec{f}_{ \bot } } \right|^{2} }}{{p N_{trials} }}$$

(14)

For the motor equivalence analysis, the force/mode deviation vectors ∆x _j were computed for each j trial by subtracting the mean pre-perturbed force/mode x _0,AV.

$$\Delta x_{j} = \, x_{j} {-}x_{{0,{\text{AV}}}}$$

(15)

The alignment between x _0,AV and the x _j involved temporal normalization of x _0,AV for each cycle of j trial. The ∆x _j was projected along and orthogonal to the UCM according to Eqs. 11 and 12. The motor equivalence (ME) and non-motor equivalence components (nME) were computed as the length of the projection vector in each subspace, respectively, and normalized by the square root of the degrees of freedom of each space:

$$\varvec{ME}_{\varvec{j}} = \frac{{\left| {\varvec{f}_{\parallel } } \right|}}{{\sqrt {n - p} }}$$

(16)

$$\varvec{nME}_{\varvec{j}} = \frac{{\left| {\varvec{f}_{ \bot } } \right|}}{\sqrt p }$$

(17)