Optimality and Modularity in Human Movement: From Optimal Control to Muscle Synergies

Abstract

In this chapter, we review recent work related to the optimal and modular control hypotheses for human movement. Optimal control theory is often thought to imply that the brain continuously computes global optima for each motor task it faces. Modular control theory typically assumes that the brain explicitly stores genuine synergies in specific neural circuits whose combined recruitment yields task-effective motor inputs to muscles. Put this way, these two influential motor control theories are pushed to extreme positions. A more nuanced view, framed within Marr’s tri-level taxonomy of a computational theory of movement neuroscience, is discussed here. We argue that optimal control is best viewed as helping to understand “why” certain movements are preferred over others but does not say much about how the brain would practically trigger optimal strategies. We also argue that dimensionality reduction found in muscle activities may be a by-product of optimality and cannot be attributed to neurally hardwired synergies stricto sensu, in particular when the synergies are extracted from simple factorization algorithms applied to electromyographic data; their putative nature is indeed strongly dictated by the methodology itself. Hence, more modeling work is required to critically test the modularity hypothesis and assess its potential neural origins. We propose that an adequate mathematical formulation of hierarchical motor control could help to bridge the gap between optimality and modularity, thereby accounting for the most appealing aspects of the human motor controller that robotic controllers would like to mimic: rapidity, efficiency, and robustness.

Appendix

In this appendix, we provide a tutorial example illustrating the concepts of optimality and modularity that we discussed in the main text.

To this aim, we consider a simple controlled pendulum whose dynamic equation is

$$ \ddot{\theta }=-\theta +u $$

where $\theta$ is the angular position and u the input torque. This may represent a simplified human arm model in the gravity field (with normalized anthropometry and small angle assumption).

We define an optimal control problem as follows: find the controller that drives the system from a given state $x_{0}=(\theta _{0},\dot{\theta }_{0})$ to a final state $x_{f}=(\theta _{f},\dot{\theta }_{f})$ in time T and minimizes the cost function

$$ C(u)=\int \limits _{0}^{T}[u^{2}+\theta ^{2}+\dot{\theta }^{2}]dt. $$

This is a linear-quadratic (LQ) problem of the form $\dot{x}=Ax+Bu$ (linear dynamics) and $C(u)=\int _{0}^{T}u^{T}Ru+x^{T}Qxdt$ (quadratic optimality criterion) where the matrices are identified as follows:

$$ A=\left( \begin{array}{cc} 0 &{} 1\\ -1 &{} 0 \end{array}\right) ,\,B=\left( \begin{array}{c} 0\\ 1 \end{array}\right) ,\,R=1,\,Q=I. $$

It can be shown that the optimal control can be written formally as (see [51] for mathematical proofs):

$$\begin{aligned} u(t)=-B^{T}P_{+}\text {e}^{A_{+}t}p-B^{T}P_{-}\text {e}^{A_{-}(t-T)}q \end{aligned}$$

(5)

where $P_{+}$ and $P_{-}$ are the maximal and minimal solutions of the associated Riccati equation $PA+A^{T}P-PBB^{T}P+Q=0$. The matrices $A_{+}$ and $A_{-}$ are defined as $A_{+}=A-BB^{T}P_{+}$ and $A_{-}=A-BB^{T}P_{-}$, and p and q are some vectors depending on the initial/final states and on movement duration T. Importantly, the matrices $P_{\pm }$ and $A_{\pm }$ just depend on the optimal control problem specification, i.e. the matrices $A,B,R,\,\text {and}\,Q$.

The optimal control u(t) can thus be written as a function of the eigenvalues of $A_{\pm }.$ Simple computations show that the 4 eigenvalues are of the form $\pm \alpha \pm \text {i}\beta$ with $\alpha =\sqrt{(2\sqrt{2}-1)}/2$ and $\beta =\sqrt{(2\sqrt{2}+1)}/2.$

Therefore, the optimal control can be rewritten as the following linear combination:

$$ u(t)=a_{1}\text {e}^{-\alpha t}\cos (\beta t)+a_{2}\text {e}^{\alpha t}\cos (\beta t)+a_{3}\text {e}^{-\alpha t}\sin (\beta t)+a_{4}\text {e}^{\alpha t}\sin (\beta t), $$

where the coefficients $(a_{i})_{1\le i\le 4}$ have to be adjusted depending on the constraints $x(0)=(\theta _{0},\dot{\theta }_{0})\,\text {and}\,x(T)=(\theta _{f},\dot{\theta }_{f})$.

In summary, this simple example illustrates that for this system all optimal motor commands can be decomposed as follows:

$$ u(t)=\sum _{i=1}^{4}a_{i}v_{i}(t) $$

where the functions $v_{i}(t)$ are time-varying primitives that are invariants of the problem and can thus be stored once for all (modularity). In contrast, the activation coefficients $a_{i}$ must be set for each single movement in order to start/end in the adequate states and times. Note the similarity between the present equations and Eq. 3 in the main text. Storing invariant building blocks $v_{i}$ and adjusting activation coefficients $a_{i}$ in order to produce controllers that allow task achievement in an optimal fashion are the core concepts discussed in the present Chapter. It is worth noting that the same conclusion could actually be drawn for any well-defined LQ problem given the general form of Eq. 5.