Figure 1.

Conceptual idea of using shared synergies in dynamical systems. (A) A synergy is constructed by a superposition of parametrized Gaussians. The parameters are the amplitude a_{m, n}, the mean μ_{m, n} and the bandwidth h_{m, n}. In the example two Gaussians (n = 1..2) are used to model the first m = 1 synergy. (B) For each task only the activation β_m of a synergy is learned. Time-varying synergies additionally implement a time-shift Δs_m. The key concept is that multiple tasks share the same parametrized synergies shown in (A), which represent task related common knowledge. (C) For each task the non-linear function f(s) is given by the weighted sum of the (time-shifted) synergies. Shown is a normalized version of f(s) to illustrate the effects of the superposition also at the end of the movement, which would usually converge toward zero. (D) Finally, the non-linear function f(s) is used to modulate a dynamical system. The unperturbed system with f(s) = 0 is denoted by the dashed line which is attracted by the goal state that is indicated by the large dot.