Vigour of self-paced reaching movement: cost of time and individual traits

People usually move at a self-selected pace in everyday life. Yet, the principles underlying the formation of human movement vigour remain unclear, particularly in view of intriguing inter-individual variability. It has been hypothesized that how the brain values time may be the cornerstone of such differences, beyond biomechanics. Here, we focused on the vigour of self-paced reaching movement and assessed the stability of vigour via repeated measurements within participants. We used an optimal control methodology to identify a cost of time (CoT) function underlying each participant’s vigour, considering a model of the biomechanical cost of movement. We then tested the extent to which anthropometric or psychological traits, namely boredom proneness and impulsivity, could account for a significant part of inter-individual variance in vigour and CoT parameters. Our findings show that the vigour of reaching is largely idiosyncratic and tend to corroborate a relation between the relative steepness of the identified CoT and boredom proneness, a psychological trait relevant to one’s relationship with time in decision-making.


Identification procedure of the CoT value g(T )
To calculate the value g(T ), for a given time T , we consider the fixed-time optimal control problem (in time T ) associated with the free-time problem formulated in the main text (which includes the time cost g). We recall that the dynamical system under consideration is linear as follows: x = Ax + Bu where x = (θ,θ,θ) is the system state and u the control variable.
The goal is to find the optimal control u(·) and the associated trajectory x(·) joining an initial state x(0) = x 0 to a final state x(T ) = x f that minimize the following integral cost (called physical effort): To solve this problem, we use Pontryagin Maximum Principle [1] which gives necessary optimality conditions, and define the Hamiltonian as follows: where p is the co-state (or adjoint) vector.
We can compute the optimal control that minimizes the Hamiltonian with respect to u and get: A star denotes a quantity related to the optimal solution. The co-state equation is given by: Let us define the matrix: , we get the hamiltonian systeṁ ξ = Hξ whose solution writes as follows: We can partition the matrix Φ(t) in blocks as follows: From this block matrix and ξ(t) = Φ(t)ξ 0 , we conclude in particular that: Therefore, if Φ −1 12 (T ) exists (which is the case if the system is fully controllable), we get: Since the Hamiltonian is constant along the optimal trajectory we obtain its value as follows: Finally, the infinitesimal CoT at time T is (see [2,3]): Note that only one matrix exponential, i.e. exp(HT ), needs to be computed in order to obtain the value g(T ). If we repeat this procedure for different durations T we can get different values g(T ) and, therefore, we can infer the shape of g(·) on some time interval, and eventually integrate it to recover the genuine cost of time G(T ) − G(0) =´T 0 g(t)dt.

Link between CoT parameters and traits for asymmetric sigmoids
In the main text, we fitted the CoT to symmetrical sigmoid functions. Here we fitted the CoT to asymmetric sigmoids of the form G(T ) = α − α/(1 + ( T δ ) β ) 0.1 . Indeed, visual inspection indicated that the inferred time costs may be asymmetrical logistic functions.
Here are the regression analyses (Fig. 1). A B Figure 1: Regression analyses between the main CoT parameters and individual traits. A. Relationships between moment of inertia, boredom proneness, impulsivity, and log α. B. Relationships between moment of inertia, boredom proneness, impulsivity, and β.

Link between CoT parameters and traits for the minimum torque change cost
Here we tested a different measure of the biomechanical cost of movement. We considered a cost based on the minimum torque change model. We fitted time costs to standard sigmoids. Goodness of fit was R 2 = 0.88 ± 0.04.
The results of the regression analyses are presented in Figure 2.