Electronic Supplementary Material 1

1. The Rayleigh-Duffing model

1.1 Background. The only existing dynamical model to account for Fitts’ law was developed by(Mottet & Bootsma, 1999), who proposed the so-called Rayleigh-Duffing model (incorporating Rayleigh damping and Duffing stiffness; see section 1.2 for details). Briefly, these authors analyzed the movement patterns produced in a reciprocal aiming task under 20 different combinations of D and W giving rise to 18 differentID’s, ranging from 3.0 to 6.9. Patterns were identified using different graphical representation techniques, including (normalized) Hooke’s portraits (the space spanning position xversus acceleration) allowing (in particular, locally) assessment of the stiffness function (Guiard, 1993). In the Hooke’s space, a purely linear (harmonic) oscillator appears as a straight line.Deviations thereof reflect the influence of nonlinearities in the underlying dynamics. The data clearly showed that participants moved in an almost pure harmonic fashion at low ID’s (evidenced by a straight line in the Hooke’s portrait) and that nonlinearities gradually came to the fore as ID increased (as evidenced by the straight line becoming graduallyN-shaped and asymmetric around the origin). The RD-model was able to reproduce these key aspects through an appropriate parameterization of the linear an non-linear damping and stiffness terms.

Fitts’ reciprocal aiming task has thus been characterized as revealing gradual changes in the kinematic patterning of movement as a function of the gradual change in task difficulty (ID). (This feature can also be gleaned from our data; see Figure 2 in the main text.). The work of Mottet and Bootsma (1999) further suggests that these gradual changes in movement patterns (underlying Fitts’ law) are due to a gradual change in the model parameters. Here we show that a gradual parameterization of the model leads to a so-called homoclinic bifurcation (cf. Strogatz, 1994; Jordan & Smith, 1999). In the present case, this implies a bifurcation in which a limit cycle coalesces with two saddle points (one on each side of the origin). In other words, the phase space is subject to a topological reorganization. From a motor control perspective, this implies the utilization of distinct control mechanisms utilizing different timing mechanism (Huys, Studenka, Rheaume, Zelaznik, Jirsa, 2008).

1.2 Analysis. We rewrite the model equation as

(eq. 1)

The fixed points are located at and (see also (6)). The latter fixed points are real if the signs of c10 and c30 are the same. As c10 can be associated with 2 ( being the system’s eigenfrequency), we assume it to be positive. The Jacobian of equation 1 equals

,

which has as eigenvalues. The origin will either be a node or a spiral, depending on the specific values of c10 and c01, but will always be unstable. For , the eigenvalues equal . For c100 and c30 > 0(i.e., fixed points are real) 1 is always positive and 2 is always negative, i.e., the fixed points are saddles (see also Zaal, Bootsma, van Wieringen, 1999).

We estimate the amplitude of the Rayleight-Duffing limit cycle using the harmonic balance approximation (cf. Jordan & Smith, 1999). The approximate solution of eq. 1 is x= Acos(t), where A and  represent movement amplitude and frequency, respectively. Substitution of this solution in eq.1, neglecting the resulting sin(3t) and cos(3t), and matching sine and cosine terms leads to and . Concentrating on amplitude A only, after substituting for 2 we find. Solving for A2 leads to the solution

(eq. 2),

which exists if is satisfied. In other words, when sufficiently far away from the bifurcation point the limit cycle amplitude obeys eq. 2. Furthermore, scales ~, that is, the amplitude A of the limit cycle grows faster than the distance of the fixed point to the origin, x*, as increases.

References

Guiard, Y.1993On Fitts' and Hooke's laws Simple harmonic movement in upper-limb cyclical aimingActa Psychol,82, 139-159.

Huys, R., Studenka, B. E., Rheaume, N. L., Zelaznik, H. N. & Jirsa, V. K. 2008 Distinct timing mechanisms produce discrete and continuous movements. PLoS Comput Biol, 4, e1000061.

Jordan, D.W. & Smith, P. 1999 Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems. Oxford University Press; 3rd Revised edition.

Mottet, D. & Bootsma, R. J. 1999 The dynamics of goal directed rhythmical aiming. Biol Cybern, 80, 235-245.

Strogatz, S. H. 1994 Nonlinear dynamics and chaos. With applications to physics, biology, chemistry, and engineering. CambridgeMassachusetts: Perseus.

Zaal, F. T. J. M., Bootsma, R. J. & van Wieringen, P. C. W. 1999Dynamics of reaching for stationary and moving objects: Data and model. J Exp Psychol Hum Percept Perform, 25,149-161.

1