Uncontrolled Manifold (UCM) analysis of motor variability

Decomposes trial-to-trial variance of the elemental variables into the part lying in the uncontrolled manifold (the null space of the task Jacobian – variability that does not change the task variable, "good") and the part orthogonal to it (variability that does, "bad"). A positive synergy index means the elements co-vary to stabilise the task.

Usage

uncontrolledManifold(theta, jacobian = NULL, task = NULL)

Arguments

theta

An N x n matrix: N trials (repetitions), n elemental variables (e.g. joint angles).

jacobian

The d x n task Jacobian at the mean configuration (d = dimension of the task variable). Provide this, or task.

task

Optional task function R^n -> R^d; its Jacobian is computed numerically at colMeans(theta) when jacobian is not given.

Value

a ucm_result list: v_ucm, v_ort (variance per DOF, parallel and orthogonal to the manifold), v_total, delta_v (synergy index, (v_ucm - v_ort) / v_total; > 0 = task-stabilising synergy), n, d, N.

References

Scholz JP, Schoner G (1999) Exp Brain Res 126:289-306; Latash ML, et al. (2002) Exerc Sport Sci Rev 30:26-31.

See also

goalEquivalentManifold(), toleranceNoiseCovariation()

Examples

set.seed(1)
# redundant task: total = a1 + a2 held constant; variance mostly along the UCM
a1 <- rnorm(200, 30, 4); a2 <- 60 - a1 + rnorm(200, 0, 0.5)
uncontrolledManifold(cbind(a1, a2), jacobian = matrix(c(1, 1), nrow = 1))$delta_v
#> [1] 1.981731