Motor variability within a multi-effector system: experimental and analytical studies of multi-finger production of quick force pulses

Abstract

The purpose of the study was to develop a model of force variability for a fast action performed by a multi-effector system and to verify it for multi-finger quick force production. The experiments involved quick isometric contractions to different target force levels using different finger combinations. Force variance calculated over sets of trials for a multi-finger force production task showed non-monotonic single-peak profiles of force variance with a peak at a time between the times of the maxima of the force rate and of the total force. When analyzed in the four-dimensional space of finger forces, the variance peak was mostly expressed in the direction of the force rate, and was absent in the directions orthogonal to it. The non-monotonic time profile of the force variance could be reproduced by a model of force production, which assumes that each finger force profile is based on a template function scaled in duration and magnitude with two parameters assigned prior to each trial with some variability. The model allows decomposition of the force variance into two fractions related to variability in setting the magnitude and duration scaling parameters. The former fraction changes monotonically with time, while the latter shows a transient peak in the middle of the action. The model was able to reproduce experimental variance time profiles across conditions with the total error of under 8%. The results demonstrate, in particular, that fast multi-finger actions may show transient changes in motor variability in certain directions of the finger force space, particularly in the direction of the first force derivative, without any task-specific coordinating action by the controller. These findings require a reconsideration of some of the conclusions drawn in recent studies on the structure of motor variability in redundant multi-effector systems.

Appendices

Appendices

Appendix A. Model of parametric force control

The equation of parametrical multi-finger force control is f _kn (t)=g(b _kn ,u _n , t,τ _kn )=b _kn u _n (t/τ _kn ) where f _kn (t) is the actual force time profile of the finger n∈{I,M,R,L} in the kth trial, u _n (t) is a template, and b _kn and τ _kn are scaling parameters. For small changes of the scaling parameters

$$\Delta f_n \left( t \right) = {{\partial g_n (t)} \over {\partial b_n }}\Delta b_n + {{\partial g_n (t)} \over {\partial \tau _n }}\Delta \tau _n $$

(11)

where Δf _n (t), Δb _n , and Δτ _n are increments of the force and its scaling parameters, $\frac{{\partial g_{n} (t)}} {{\partial b_{n} }}$ and $\frac{{\partial g_{n} (t)}} {{\partial \tau _{n} }}$ are partial derivatives of the force with respect to the parameters. The two summands of the force increment are:

$${{\partial g_n (t)} \over {\partial b_n }}\Delta b_n = b_n u_n \left( {t/\tau _n } \right){{\Delta b_n } \over {b_n }} = f_n \left( t \right){{\Delta b_n } \over {b_n }}$$

$$ {{\partial g_n (t)} \over {\partial \tau _n }}\Delta \tau _n = - {t \over {\tau _k }}b_n u'_n \left( {t/\tau _n } \right){{\Delta \tau _n } \over {\tau _n }} = - tf'_n \left( t \right){{\Delta \tau _n } \over {\tau _n }}$$

(because f′ _n (t)=$\frac{1} {{\tau _{n} }}$ b _n u′ _n (t/τ _n )). So, Eq. (11) can be written:

$$ \Delta f_n \left( t \right) = f_n \left( t \right){{\Delta b_n } \over {b_n }} - tf'_n \left( t \right){{\Delta \tau _n } \over {\tau _n }}$$

(12)

(see also Gutman and Gottlieb 1992).

Appendix B. Two components of the model force variance

Assume that changes in the finger force time profile are defined by the variability of scaling parameters b _kn and τ _kn , which are random variables with means $\overline{b} _{n}$ and $\bar{\tau }_{n}$ and variances Var(b _n ) and Var(τ _n ) (or standard deviations SD(b _n ) and SD(τ _n )). Assume also that SD(b _n )<<$\overline{b} _{n}$ and SD(τ _n ))<<$\bar{\tau }_{n}$; covariances Cov(b _n ,b _m )≠0, Cov(τ _n ,τ _m )≠0, Cov(b _n ,b _m )=0; and templates u _n (t) are deterministic functions. The covariance between forces of fingers n and m at a time t can be estimated as:

$$ {\text{Cov}}\left( {f_n \left( t \right),f_m \left( t \right)} \right) = \overline {\Delta f_n (t)\Delta f_m (t)} $$

(13)

Here and further, $\overline{{( * )}}$ is the sign of mean. Substituting the expression of $\overline{{\Delta f_{n} (t)\Delta f_{m} (t)}}$ via increments of the scaling parameters (Eq. 12) yields:

$$\eqalign{ & {\rm Cov}\left( {f_n \left( t \right),f_m \left( t \right)} \right) \cr = & \overline {f_n {\rm (}t{\rm )}f_m {\rm (}t{\rm )}} {{{\rm SD}(b_n )} \over {\bar b_n }}{{{\rm SD}(b_m )} \over {\bar b_m }}r(b_n ,b_m ) \cr & + t^2 \overline {f'_n {\rm (}t{\rm )}f'_m {\rm (}t{\rm )}} {{{\rm SD}(\tau _n )} \over {\bar \tau _n }}{{{\rm SD}(\tau _m )} \over {\bar \tau _m }}r(\tau _n ,\tau _m ) \cr} $$

(14)

where r is coefficient of correlation. According to Eq. (14), force covariance is the weighted sum of two functions, $\overline{{f_{n} {\text{(}}t{\text{)}}f_{m} {\text{(}}t{\text{)}}}}$ and $t^{2} \overline{{{f}'_{n} {\text{(}}t{\text{)}}{f}'_{m} {\text{(}}t{\text{)}}}}$, with weights x _nm =$\frac{{{\text{SD}}(b_{n} )}} {{\bar{b}_{n} }}\frac{{{\text{SD}}(b_{m} )}} {{\bar{b}_{m} }}r(b_{n} ,b_{m} )$ and y _nm =$\frac{{{\text{SD}}(\tau _{n} )}} {{\bar{\tau }_{n} }}\frac{{{\text{SD}}(\tau _{m} )}} {{\bar{\tau }_{m} }}r(\tau _{n} ,\tau _{m} )$. Using CV for coefficient of variation, one gets:

$$ x_{{nm}} = {\text{CV}}{\left( {b_{n} } \right)}{\text{CV}}{\left( {b_{m} } \right)}r{\left( {b_{n} ,b_{m} } \right)},\;\;\;\;y_{{nm}} = {\text{CV}}{\left( {\tau _{n} } \right)}{\text{CV}}{\left( {\tau _{m} } \right)}r{\left( {\tau _{n} ,\tau _{m} } \right)}. $$

(15)

If force variability is small, approximately $\overline{{f_{n} {\text{(}}t{\text{)}}f_{m} {\text{(}}t{\text{)}}}}$=$\overline{{f_{n} {\text{(}}t{\text{)}}}} {\kern 1pt} \,\overline{{f_{m} {\text{(}}t{\text{)}}}}$; assuming that force rate variability is also small, the scalar expressions for all n and m can be written as a matrix equation:

$$ {\bf{Cov}}\left( {{\bf{f}}\left( t \right)} \right) = {\bf{D}}(\overline {{\bf{f}}(t)} ){\bf{XD}}(\overline {{\bf{f}}(t)} ) + t^2 {\bf{D}}(\overline {{\bf{f'}}(t)} ){\bf{Y}}{\bf{D}}(\overline {{\bf{f'}}(t)} ) $$

(16)

where ${\bf{D}}(\overline{{{\bf{f}}(t)}} )$ and ${\bf{D}}(\overline{{{\bf{{f}'}}(t)}} )$ are diagonal matrices, in which diagonal entries are mean force {$\overline{{f_{n} {\text{(}}t{\text{)}}}}$} and force rate {$\overline{{{f}'_{n} {\text{(}}t{\text{)}}}}$} time profiles, and {x _mn } and {y _mn } from Eq. (15) are entries of the matrices X and Y. Thus, the covariance matrix consists of two terms, related to setting the magnitude and duration scaling parameters, respectively.

Appendix C. Dependence of force variance upon time and direction in force space.

Values of CV(τ) estimated from experimental data differ for different fingers. However, for simplicity, consider a common timing scaling parameter for all fingers with CV(τ _n )=ξ _τ for all n, and r(τ _n , τ _m )=1 for all n and m. In this case, Y _ss=ξ _τ E, where E is a matrix with unit entries.

The expression for force variance in the direction of a vector w via Rayleigh fraction with a covariance matrix Cov(f(t)) is:

$$ {\bf{w}}^{\text{T}} \left( t \right){\bf{Cov}}\left( {{\bf{f}}\left( t \right)} \right){\bf{w}}\left( t \right) = {\bf{w}}^{\text{T}} (t){\bf{D}}(\overline {{\bf{f}}(t)} ){\bf{X}}_{ss} {\bf{D}}(\overline {{\bf{f}}(t)} ){\bf{w}}(t) + t^2 {\bf{w}}^{\text{T}} (t){\bf{D}}(\overline {{\bf{f'}}(t)} ){\bf{Y}}_{ss} {\bf{D}}(\overline {{\bf{f'}}(t)} ){\bf{w}}(t) $$

(17)

Here, we assume w a vector with unit norm. Let v be orthogonal to the direction of the force rate vector $\overline{{{\mathbf{{f}'}}(t)}}$. The second term of the right side of Eq. (17), Var _τ (t) is:

$$ {\text{Var}}_\tau = \xi _\tau t^2 {\mathbf{v}}^{\text{T}} (t){\bf{D}}(\overline {{\bf{f'}}(t)} ){\bf{ED}}(\overline {{\bf{f'}}(t)} ){\bf{v}}(t) $$

The factor ${\bf{D}}(\overline{{{\bf{{f}'}}(t)}} ){\bf{v}}$ is a vector:

$$ {\bf{D}}(\overline{{{\bf{{f}'}}(t)}} ){\bf{v}} = {\left( {\overline{{{f}'_{{\text{I}}} (t)}} v_{{\text{I}}} (t),\;\overline{{{f}'_{{\text{M}}} (t)}} v_{{\text{M}}} (t),\;\overline{{{f}'_{{\text{R}}} (t)}} v_{{\text{R}}} (t),\;\overline{{{f}'_{{\text{L}}} (t)}} v_{{\text{L}}} (t)} \right)}^{{\text{T}}} . $$

${\mathbf{D}}(\overline{{{\mathbf{{f}'}}(t)}} ){\mathbf{v}}$ multiplied by E is a zero vector, because each of its components

$$ \xi _\tau \left( {\overline {f'_{\text{I}} (t)} v_{\text{I}} (t) + \overline {f'_{\text{M}} (t)} v_{\text{M}} (t) + \overline {f'_{\text{R}} (t)} v_{\text{R}} (t) + \overline {f'_{\text{L}} (t)} v_{\text{L}} (t)} \right) = 0 $$

is a scalar product of orthogonal vectors $\overline{{{\mathbf{{f}'}}_{{}} (t)}}$ and v(t). Therefore, Var _τ (t) is zero. This means, that variance time profile in directions orthogonal to the direction of the force rate vector is defined only by the other term of Eq. (17). In our case, this term is monotonic, and hence the force variance time profile is not expected to show transient peaks (such as the V-peak).