Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.

Figure 1

Qualitative functional dependencies of variances ${\sigma }_p^2$ on time delays $\tau$ for first-order dynamical models of the form (5).

Figure 2

Stationary probability densities of the system (2, 3) represented by various values of parameter $b$ while the other parameters are fixed. Symbols represent the numerical simulation and the solid lines represent the analytical results given by Eq. (8). The parameters are fixed by $a=2$, $k=0.8$, $\tau =0.2$, $Q=1$, and $m=1$. Using $b=8,2,1$, the left panels show the stationary probability densities which, for the $b$ values used, correspond to $b>a$ (diamonds), $b=a$ (circles), and $a^2>b^2>a^2-(4∕km)$ (stars), respectively. Using $b=1,0.5,0$, the right panels show the stationary probability densities which, for the $b$ values used, correspond to $a^2>b^2>a^2-(4k∕m)$ (stars), $b^2<a^2-(4k∕m)$ (triangles), and $b=0$ (squares), respectively.

Figure 3

Stability diagram in the $a\text{−}\tau$ plane for the case $b>a$. Panel (a): the parameters $Q=m=k=1$ and $b=1.6$ are fixed, whereas the parameter $a$ is varied from 0 to $b$. Three double Hopf bifurcation points can be found at approximately (1.30, 8.53), (1.46, 5.98), and (1.57, 12.39). Panel (b): a detail of panel (a) for $a∊[1.5,1.6]$. Panel (c): stability diagram revealing (right hand side) boundary points given by $a\to b-0$ and ${\tau }_c=\pi (1+2j)∕{\omega }_0$. Parameters: $Q=m=b=1$ and $k=10$. Panel (d): a detail of panel (c).

Figure 4

Variances for the case $b>a$ versus delay. The diamonds show simulation results and the lines show analytical results given by Eqs. (17, 31). The parameters are $a=0.1$ and $b=k=m=Q=1$, which implies that the critical delay is about 1.04.

Figure 5

Variances for the case $b=a$ versus delay. The diamonds show simulation results and the lines show analytical results given by Eqs. (C8, C9). The parameters are $a=b=Q=m=1$ and $k=10$, which yields a critical delay of about 0.99.

Figure 6

Variances at large delays for the underdamped case with $b<a$ and $b^2>a^2-(4k∕m)$. The diamonds show the simulation results and the lines show the analytical results given by Eqs. (17, 43, 45). Parameters: $a=2$ and $b=k=m=Q=1$.

Figure 7

Asymptotic behavior of the variances for the overdamped case with $b<a$ and $b^2<a^2+(4k∕m)$. The diamonds show the simulation results and the lines show the analytical results given by Eqs. (17, 41, 42). Parameters: $a=1.2$, $b=m=Q=1$, and $k=0.1$.

Figure 8

(a) Variance of $p$ and (b) variance of $x$ for $b>a$ in the case of a re-entrant bifurcation. There are two stable domains in which the variance is finite: $[0,{\tau }_{u,0})$ and $({\tau }_{s,0},{\tau }_{u,1})$. For $\tau ⩾{\tau }_{u,1}$ stationary distributions do not exist. Vertical bars correspond to ${\tau }_{u,0}$, ${\tau }_{s,0}$, and ${\tau }_{u,1}$. The parameters: $a=1.5$, $b=1.6$, and $Q=m=k=1$. (c) Variance of $p$ and (d) variance of $x$ for $b=a$. Singularities occur at regular intervals. Parameters: $a=b=Q=m=1$ and $k=10$.

Figure 9

(a) Variance of $p$, (b) variance of $x$, (c) energy $U\left(\tau \right)$, and (d) entropy $S\left(\tau \right)$ with respect to time delay for the underdamped case with $b<a$ and $b^2>a^2-(4k∕m)$. Parameters: $a=1.01$, $b=Q=m=1$, and $k=10$.

Figure 10

(a) Variance of $p$, (b) variance of $x$, (c) energy $U\left(\tau \right)$, and (d) entropy $S\left(\tau \right)$ with respect to time delay for the overdamped case with $b<a$ and $b^2<a^2-(4k∕m)$. Parameters: $a=2$, $b=Q=m=1$, and $k=0.74$.

Figure 11

(a) Variance of $p$, (b) variance of $x$, (c) energy $U\left(\tau \right)$, and (d) entropy $S\left(\tau \right)$ with respect to time delay for the overdamped case $b<a$, $b^2<a^2-(4k∕m)$. The parameters: $Q=1$, $m=1,k=0.1$, $a=1.2$, and $b=0.0001$.

Figure 12

Reinterpretation of experimental results found in two previous studies on tracking movements under time-delayed visual feedback: a study by Tass et al. [15] (a) and a study by Langenberg et al. [16] (b). Here, $a^{*}=\sqrt{a^2-4k∕m}$. See text for details.

Figure 13

Qualitative functional dependencies of variances ${\sigma }_p^2$ on time delays $\tau$ for second-order dynamical models of the form (2, 3) [solid line: $b>a$; dashed line: $b<a$ with $b^2>a^2-(4k∕m)$].