Age-related changes in optimality and motor variability: an example of multifinger redundant tasks

Abstract

We used two methods, analytical inverse optimization (ANIO) and uncontrolled manifold (UCM) analysis of synergies, to explore age-related changes in finger coordination during accurate force and moment of force production tasks. The two methods address two aspects of the control of redundant systems: Finding an optimal solution (an optimal sharing pattern) and using variable solutions across trials (covarying finger forces) that are equally able to solve the task. Young and elderly subjects produced accurate combinations of total force and moment by pressing with the four fingers of the dominant hand on individual force sensors. In session-1, single trials covered a broad range of force–moment combinations. Principal component (PC) analysis showed that the first two PCs explained about 90% and 75% of finger force variance for the young and elderly groups, respectively. The magnitudes of the loading coefficients in the PCs suggested that the young subjects used mechanical advantage to produce moment while elderly subjects did not (confirmed by analysis of moments produced by individual digits). A co-contraction index was computed reflecting the magnitude of moment produced by fingers acting against the required direction of the total moment. This index was significantly higher in the young group. The ANIO approach yielded a quadratic cost function with linear terms. In the elderly group, the contribution of the forces produced by the middle and ring fingers to the cost function value was much smaller than in the young group. The angle between the plane of experimental observations and the plane of optimal solutions (D-angle), was very small (about 1.5°) in the young group and significantly larger (about 5°) in the elderly group. In session-2, four force–moment combinations were used with multiple trials at each. Covariation among finger forces (multifinger synergies) stabilizing total force, total moment, and both was seen in both groups with larger synergy indices in the young group. Multiple regression analysis has shown that, at higher force magnitudes, the synergy indices defined with the UCM method were significantly related to the percent of variance accounted by the first two PCs and to the D-angle computed using the ANIO method. We interpret the results as pointing at a transition with age from synergic control to element-based control (back-to-elements hypothesis). Optimization and analysis of synergies are complementary approaches that focus on two aspects of multidigit coordination, sharing and covariation, respectively.

Appendices

Appendix 1

Analytical Inverse Optimization (ANIO) approach

The optimization problem in the current study was defined as

$$ \begin{array}{*{20}c} {\text{Min}} & {J = \sum\limits_{i = 1}^{4} {g_{i} (F_{i} )} } \\ \end{array} $$

(8)

$$ \begin{array}{*{20}c} {\text{Subject to}} & {F_{i} + F_{m} + F_{r} + F_{l} = \,a \cdot {\text{MVC}}_{\text{IMRL}} } \\ {} & {d_{i} \cdot F_{i} + d_{m} \cdot F_{m} + d_{r} \cdot F_{r} + d_{l} \cdot F_{l} = \,b \cdot 0.07 \cdot d_{i} \cdot {\text{MVC}}_{I} } \\ \end{array} $$

The two linear constraints are expressed as

$$ CF^{T} = B $$

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$$ F = \left[ {\begin{array}{*{20}c} {F_{i} } & {F_{m} } & {F_{r} } & {F_{l} } \\ \end{array} } \right] $$

$$ C = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ {d_{i} } & {d_{m} } & {d_{r} } & {d_{l} } \\ \end{array} } \right] $$

$$ B = \left[ {\begin{array}{*{20}c} {F_{\text{TOT}} } \\ {M_{\text{TOT}} } \\ \end{array} } \right] $$

The task involved two constraints (F _TOT and M _TOT values) and four elemental variables (finger forces). Thus, the solutions of this undetermined system were expected to be confined to a two-dimensional surface in the four-dimensional force space. Planarity of this surface was checked using the PCA. The following computational procedure explains how the optimization cost function is obtained.

First, we identify whether the optimization problem is splittable or not by observing the (4 × 4) matrix:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} = I - C^{T} (CC^{T} )^{ - 1} C $$

(10)

Second, we check whether the experimental data actually lie on a hyperplane (and not for instance on a curved hypersurface) and then define the observed hyperplane mathematically as

$$ A \cdot F^{T} = b $$

(11)

where A is a 2 × 4 matrix composed of the transposed vectors of the two lesser principal components obtained from the PCA from the finger force data in session-1. A large percentage of the total variance explained by the two first principal components was considered an indicator that the data lie on a hyperplane. However, the data points were not perfectly confined to a plane due to the variability of performance and instrumental noise. Also, the plane computed from Eq. 11 is affected by experimental errors.

Third, we compare the experimentally determined hyperplane to the theoretical plane derived from the Uniqueness Theorem. The experimental data must be fitted by the following equation:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{'} (F) = 0, $$

(12)

where $f^{\prime } (F) = \left( {f_{1}^{\prime } (F_{i} ),f_{2}^{\prime } (F_{m} ),f_{3}^{\prime } (F_{r} ),f_{4}^{\prime } (F_{l} )} \right)^{T}$ f _i are arbitrary continuously differentiable functions. At the second step, the data are discovered to lie on the plane and hence the functions fi′(·) are linear:

$$ f_{i}^{\prime } (F_{i} ) = k_{i} F_{i} + w_{i} $$

(13)

where i = {index, middle, ring, and little}. Therefore,

$$ f_{i} (F_{i} ) = {\frac{{k_{i} }}{2}}(F_{i} )^{2} + w_{i} F_{i} $$

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The values of the coefficients of the second-order terms k _i can be determined by minimizing the dihedral angle between the two planes: the plane of optimal solutions $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (F) = 0$ and the plane of experimental observations ($A \cdot F^{T} = 0$). The values of the coefficients of the first-order terms w _i were found to correspond to a minimal vector length (w = (w _index, w _middle, w _ring, w _little)^T) bringing the theoretical and the experimental plane as close to each other as possible. Vector w satisfy the following equation:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (F) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} (KF_{i} + w) $$

(15)

where K = (k _index, k _middle, k _ring, k _little)^T and w = (w _index, w _middle, w _ring, w _little)^T.

Then, the functions g _i in Eq. 10 are the following:

$$ g_{i} (x_{i} ) = rf_{i} (F_{i} ) + q_{i} F_{i} + {\text{const}}_{i} $$

(16)

where r is a non-zero number, const _i can be any real number, and q _i is any real number satisfying the equation $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} q = 0$ (Terekhov et al. 2010). Multiplication of the cost function by a constant value or adding a constant value to it does not change the cost function essentially. Hence, we can arbitrary assume that r = 1 and const_i = 0. According to the Uniqueness Theorem, identification of the cost function can be performed only up to unknown linear terms which are parameterized by the values q _i . We assume that q _i  = 0 in order to simplify g _i (x _i ). It must be kept in mind, however, that the true cost function used by the CNS might have these terms.

Uniqueness theorem (for the mathematical proof see Terekhov et al. 2010)

The core of the ANIO approach is the theorem of uniqueness that specifies conditions for unique (with some restrictions) estimation of the objective functions. The main idea of the theorem of uniqueness is to find necessary conditions for the uniqueness of solutions in an inverse optimization problem. An optimization problem (i.e., direct optimization problem) with an additive objective function and linear constraints are defined as:

$$ \begin{gathered} {\text{Let}}\,{\text{J}}:R^{n} \to R^{1} \hfill \\ {\text{Min}}:J(x) = g_{1} (x_{1} ) + g_{2} (x_{2} ) + \cdots + g_{n} (x_{n} ) \hfill \\ {\text{Subject to}}:CX^{T} = B \hfill \\ \end{gathered} $$

(17)

where X = (x ₁, x ₂,…, x _n )\,$\in$ R ⁿ, g _i is an unknown scalar differentiable function with g′(∙) > 0. g _i came from the Lagrange minimum principle, which has a unique solution. On the contrary, the functions of g _i can be computed from the set of solutions X* (e.g., experimental data). This inverse procedure is called the inverse optimization problem. C is a k × n matrix and B is a k-dimension vector, k < n.

First, assume that the optimization problem (Eq. 17) with k ≥ 2 is non-splittable. If the inverse optimization is splittable, the preliminary step is to split it until a non-splittable subproblem is acquired. If the functions g _i (x _i ) in problem (Eq. 17) are twice continuously differentiable (i.e., twice continuously differentiable functions f _i ) and $f_{i}^{\prime }$ is not identically constant, complying $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime}(X) = 0$ for all $X \in X^{*}$,

$$ f^{\prime } (X) = \left( {f_{1}^{\prime } (x_{1} ), \cdots ,f_{n}^{\prime } (x_{n} )} \right)^{T} $$

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$$ {\text{and}}\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} = I - C^{T} (CC^{T} )^{ - 1} C $$

(19)

$$ {\text{then}}\,g_{i} (x_{i} ) = rf_{i} (x_{i} ) + q_{i} x_{i} + {\text{const}}_{i} $$

(20)

for every $x_{i} \in X_{i}^{*}$, where $X_{i}^{*}$={s| there is $X \in X^{*}$: $x_{i} \in s$} and X* is the set of the solutions for all $B \in R^{k}$. The constants q _i satisfy the equation $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} q = 0$ where $q = (q_{1} , \cdots ,q_{n} )^{T}$. Primes designate derivatives.

If the experimental data correspond to solutions of an inverse optimization problem with additive objective function (g _i ) and linear constraints, equation $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (X) = 0$($X \in X^{*}$) must be satisfied (i.e., the Lagrange principle). The Uniqueness Theorem provides sufficient condition (i.e., $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime}(X) = 0$) for solving the inverse optimization problem in a unique way up to linear terms.

Appendix 2

Uncontrolled manifold (UCM) analysis (see Latash et al. 2002; Scholz et al. 2002 for details)

For F _TOT, changes in the elemental variables (finger forces) sum up to produce a change in F _TOT:

$$ dF_{\text{TOT}} = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {dF_{i} } & {dF_{m} } & {dF_{r} } & {dF_{l} } \\ \end{array} } \right]^{T} $$

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The UCM was defined as an orthogonal set of the vectors e _i in the space of the elemental forces that did not change the net normal force, i.e.,:

$$ 0 = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} } \right]e_{i} $$

(22)

These directions were found by taking the null-space of the Jacobian of this transformation ([1 1 1 1] e _i ). The mean-free forces were then projected onto these directions and summed to produce:

$$ f_{||} = \sum\limits_{i}^{n - p} {\left( {e_{i}^{T} \cdot df} \right)e_{i} } , $$

(23)

where n = 4 is the number of degrees-of-freedom of the elemental variables, and p = 1 is the number of degrees-of-freedom of the performance variable (F _TOT). The component of the de-meaned forces orthogonal to the null-space is given by:

$$ f_{ \bot } = df - f_{||} $$

(24)

The amount of variance per degree of freedom parallel to the UCM is:

$$ V_{\text{UCM}} = {\frac{{\sum {\left| {f_{||} } \right|^{2} } }}{{(n - p)N_{\text{trials}} }}} $$

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The amount of variance per degree of freedom orthogonal to the UCM is :

$$ V_{\text{ORT}} = {\frac{{\sum {\left| {f_{ \bot } } \right|^{2} } }}{{pN_{\text{trials}} }}} $$

(26)

The normalized difference between these variances is quantified by a variable ΔV:

$$ \Updelta V = {\frac{{V_{\text{UCM}} - V_{\text{ORT}} }}{{V_{\text{TOT}} }}} $$

(27)

where V _TOT is the total variance, also quantified per degree-of-freedom. If ΔV is positive, V _UCM > V _ORT, caused by negative covariation of the finger forces, which we interpret as evidence for a force-stabilizing synergy. In contrast, ΔV = 0 indicates independent variation of the finger forces, while ΔV < 0 indicates positive covariation of the individual finger forces, which contributes to variance of F _TOT.

A similar procedure was used to compute the two variance components related to stabilization of M _TOT. The only difference was in using a different Jacobian corresponding to the lever arms of individual finger forces, [d _i d _m d _r d _l ].

We also analyzed the data with respect to stabilization of both F _TOT and M _TOT simultaneously. In that case, the Jacobian was [1 1 1 1 d _i d _m d _r d _l ]. The dimensionality of V _UCM for the analysis with respect to F _TOT and M _TOT separately is three (one constraint), while the dimensionality of V _UCM with respect to F _TOT and M _TOT simultaneously is two (two constraints).