Velocity control in Parkinson’s disease: a quantitative analysis of isochrony in scribbling movements

Abstract

An experiment was conducted to contrast the motor performance of three groups (N = 20) of participants: (1) patients with confirmed Parkinson Disease (PD) diagnose; (2) age-matched controls; (3) young adults. The task consisted of scribbling freely for 10 s within circular frames of different sizes. Comparison among groups focused on the relation between the figural elements of the trace (overall size and trace length) and the velocity of the drawing movements. Results were analysed within the framework of previous work on normal individuals showing that instantaneous velocity of drawing movements depends jointly on trace curvature (Two-thirds Power Law) and trace extent (Isochrony principle). The motor behaviour of PD patients exhibited all classical symptoms of the disease (reduced average velocity, reduced fluency, micrographia). At a coarse level of analysis both isochrony and the dependence of velocity on curvature, which are supposed to reflect cortical mechanisms, were spared in PD patients. Instead, significant differences with respects to the control groups emerged from an in-depth analysis of the velocity control suggesting that patients did not scale average velocity as effectively as controls. We factored out velocity control by distinguishing the influence of the broad context in which movement is planned—i.e. the size of the limiting frames—from the influence of the local context—i.e. the linear extent of the unit of motor action being executed. The balance between the two factors was found to be distinctively different in PD patients and controls. This difference is discussed in the light of current theorizing on the role of cortical and sub-cortical mechanisms in the aetiology of PD. We argue that the results are congruent with the notion that cortical mechanisms are responsible for generating a parametric template of the desired movement and the BG specify the actual spatio-temporal parameters through a multiplicative gain factor acting on both size and velocity.

Appendices

Appendix 1

We define the parameters selected for the analysis of the movements and detail the data processing for their estimation.

Scaling and smoothing

Traces were aligned by computing the centre of gravity of the samples and shifting all samples so that the new centre was at the origin of the coordinate system. Most of the parameters involve the computation of time derivatives. Because we needed explicit expressions for derivatives (see later), we adopted an interpolation method based on harmonic analysis. The coordinates x = x(t) and y = y(t) of the movement were decomposed in Fourier series:

$$ \begin{aligned} x(t) = & \frac{1}{2}a_{{x_{0} }} + \sum\limits_{k = 1}^{\infty } {\left( {a_{xk} \cos \left( {k\omega_{0} t} \right) + b_{xk} \sin \left( {k\omega_{0} t} \right)} \right)} \\ y(t) = & \frac{1}{2}a_{{y_{0} }} + \sum\limits_{k = 1}^{\infty } {\left( {a_{yk} \cos \left( {k\omega_{0} t} \right) + b_{yk} \sin \left( {k\omega_{0} t} \right)} \right)} \\ \end{aligned} $$

Preliminary tests showed that retaining the first 50 terms of the series yields an excellent approximation to the traces and is also effective for eliminating uncorrelated noise from the data. All further processing was applied to the truncated series. First and second time derivatives were computed analytically.

Computing the characteristic parameters of the trace

The following parameters were computed from each trace:

• (a) Tangential velocity [V(t)]:

  $$ V(t) = \sqrt {\left( {\frac{{\rm d}x}{{\rm d}t}} \right)^{2} + \left( {\frac{{\rm d}y}{{\rm d}t}} \right)^{2} } $$

  The average velocity of the trace V ₀ is related to the total trace length (L) by the equation $V_{0} = {L \mathord{\left/ {\vphantom {L T}} \right. \kern-\nulldelimiterspace} T}.$

• (b) Total length (L):

  $$ L = \int\limits_{0}^{T} {V(t){\text{d}}t} = \int\limits_{0}^{T} {\sqrt {\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^{2} + \left( {\frac{{{\text{d}}y}}{{{\text{d}}t}}} \right)^{2} } } {\text{d}}t $$

  This parameter was computed by integrating numerically the tangential velocity.

• (c) Number of inflections [N _I]. An inflection in a 2D trajectory is a point were the curvature of the trajectory changes sign. Inflections were located by identifying the sample index k such that the quantity

  $$ \frac{{{\text{d}}x(t_{k} )}}{{{\text{d}}t}}\frac{{d^{2} y(t_{k} )}}{{{\text{d}}t^{2} }} - \frac{{d^{2} x(t_{k} )}}{{{\text{d}}t^{2} }}\frac{{{\text{d}}y(t_{k} )}}{{{\text{d}}t}} $$

  changes sign between k and k + 1.

• (d) Length of trajectory segments (L _seg). Segments were defined as portions of the trajectory bounded by two successive points of inflection. Thus, in a trace there were N _s = N _I − 1 segments. When trajectory was almost straight, several inflections occurred in close succession.

• (e) Time average of the one-third power of the radius $\left[ {R_{m}^{{*{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} } \right].$ The function R(t) that describes how the radius of curvature of the trajectory changes in time is

  $$ R(t) = \frac{{V(t)^{3} }}{{\left| {\frac{{{\text{d}}x}}{{{\text{d}}t}}\frac{{d^{2} y}}{{{\text{d}}t^{2} }} - \frac{{\text {d}}y}{{{\text{d}}t}}\left. {\frac{{d^{2} x}}{{{\text{d}}t^{2} }}} \right|} \right.}} $$

  The radius of curvature increases in the proximity of a point of inflection where it becomes infinite. The most recent formulation of Two-thirds Power Law circumvents this difficulty by expressing the velocity as a power function of the normalized radius R ^*(t) defined as

  $$ R^{*} (t) = \frac{R(t)}{1 + \alpha R(t)} $$

  As the movement approaches an inflection, R ^* stays finite and tends to the limit 1/α. The required time average $\langle R^{{*{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} \rangle$ was computed by numerical evaluation of the integral

  $$ \langle R^{{*{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} \rangle = \frac{1}{T}\int\limits_{0}^{T} {R^{*} (t)^{{\frac{1}{3}}} {\text{d}}t} $$

• (f) Average curvature [C _m]. Curvature is the inverse of the radius of curvature. It would be inappropriate to compute the average curvature using directly the Fourier approximation of the trajectory. Because the instantaneous velocity decreases with curvature as prescribed by the Two-thirds Power Law, and because the sampling rate is constant, the sample density around the high-curvatures portions of the trajectory is much higher than the density within low-curvature portions. Thus, numerical integration of the inverse of the radius R(t) would severely overestimate the average geometric curvature. Instead, we adopted the following resampling strategy. Let us consider a doubly differentiable time function φ = φ(t) such that φ(0) = 0 and φ(T) = T, where T is the total duration of the movement. The parametric equations [x = x(φ(t)), y = y(φ(t))] describe the same trajectory C as the original ones [x = x(t), y = y(t)]. However, the kinematics of the movement depends on the function φ, and the correspondent transformed velocity is in general different from the velocity of the actual movement:

  $$ V_{\varphi } (t) = \frac{{{\text{d}}\varphi }}{{{\text{d}}t}}\left[ {\left( {\frac{{{\text{d}}x}}{{{\text{d}}\varphi }}} \right)^{2} + \left( {\frac{{{\text{d}}y}}{{{\text{d}}\varphi }}} \right)^{2} } \right]^{{\frac{1}{2}}} $$

  The expression above can be rewritten as a separable nonlinear differential equation:

  $$ {\text{d}}\varphi = V_{\varphi } (t)\left[ {\left( {\frac{{{\text{d}}x}}{{{\text{d}}\varphi }}} \right)^{2} + \left( {\frac{{{\text{d}}y}}{{{\text{d}}\varphi }}} \right)^{2} } \right]^{{\frac{1}{2}}} {\text{d}}t $$

  Under mild continuity conditions, for any choice of the transformed velocity function V _φ(t), solving the equation above yields the unique function φ that is compatible with this choice. We imposed the condition that the tangential velocity is constant and equal to the average velocity of the actual movement (V _φ(t) = L/V _m) and computed the solution φ(t) with a fourth-order Runge–Kutta algorithm with the boundary conditions φ(0) = 0 and φ(T) = T. By inserting the solution φ(t) back into the parametric equations, the original movement was resampled so that successive data points were spaced by a constant fraction of the total length rather then by a constant time interval (the total number of samples was kept equal to 2,000 as in the original trace). Finally, the average geometric curvature was calculated as the mean over all samples of the inverse of the radius. Note that this strategy was possible only because the Fourier series affords an analytical approximation to the traces.

• (g) Average gain factor [K _m]. According to the Two-Thirds Power Law, the multiplicative parameter K (gain factor) is approximately constant over successive units of motor action. We computed the average gain factor for the entire trace by numerical estimation of the integral

  $$ K_{m} = \frac{1}{T}\int\limits_{0}^{T} {\left| {\frac{{{\text{d}}x(t)}}{{{\text{d}}t}}\frac{{d^{2} y(t)}}{{{\text{d}}t^{2} }} - \frac{{{\text{d}}y(t)}}{{{\text{d}}t}}\frac{{d^{2} x(t)}}{{{\text{d}}t^{2} }}} \right|^{{\frac{1}{3}}} {\text{d}}t} $$

  In the same manner, we computed the average gain factors K _seg for each segment within the complete trace.

Appendix 2

We specify the method for testing the statistical significance of parameters in the equation relating frame area A and segment length L_seg to segment gain K_seg. The three equations to be combined are (relevant stochastic variables in boldface)

$$ \begin{aligned} &{\text{K}}_{\text{seg}}/{\text{K}}_{\text{av}} = {\mathbf{c}}_{0} \left( {{\text{L}}_{\text{seg}} /{\text{L}}_{\text{av}} } \right)^{\varvec{\xi}}\\ & {\text{K}}_{\text{av}} = {\mathbf{c}}_{1} {\text{A}}^{\varvec{\theta}}\\ & {\text{L}}_{\text{av}} = {\mathbf{c}}_{2} {\text{A}}^{\varvec{\psi}} \end{aligned} $$

As expected, c₀ was almost indistinguishable from 1. Therefore

$$ {{K}}_{\text{seg}} = {\mathbf{c_1}} {\mathbf{c_2}}^{ -\varvec{\xi}} {{A}}^{(\varvec{\psi} - \varvec{\theta \xi })} {\text{L}}_{\text{seg}} = {\mathbf{C}}{{A}}^{\varvec{\varphi}} {\text{L}}_{\text{seg}} $$

where C = c₁c ^-ξ₂ and φ = θ − ψξ. The variances σ ²_e of the exponents ξ, θ and ψ are estimated directly from the regression equations

$$ \log \left( {K_{\text{av}} } \right) = \log \left( {{\mathbf{c}}_{1} } \right) + {\varvec\theta} \log \left( A \right);\quad \log \left( {L_{\text{av}} } \right) = \log \left( {{\mathbf{c}}_{2} } \right) + {\varvec\psi} \log \left( A \right) $$

through the formula (Kendall and Stuart 1968, p. 395):

$$ \sigma_{e}^{2} = \left( {\sigma_{y}^{2} /\sigma_{x}^{2} } \right)\left( {1 - \rho^{2} } \right)/\left( {N-2} \right) $$

where ρ is Fisher’s correlation coefficient and N is the sample size. The averages of the regression coefficients a₁ = log(c₁) and a₂ = log(c₂) are μ_a1 = a₁ and μ_a1 = a₁, respectively. Their variances σ ²_a are estimated by

$$ \sigma_{\text{a}}^{2} = \sigma_{\text{e}}^{2} \left( {\sigma_{\text{x}}^{2} \left[ {N - 1} \right]/N + \mu_{\text{x}}^{2} } \right) $$

If two stochastic variables are related by a monotonic function y = g(x) the pdf of y is given by f_y(y) = f_x(g⁻¹(y))/dg(g⁻¹(y))/dx (Papoulis 1965, p. 126). We assume that both a₁ and a₂ have a Gaussian pdf. Thus, because c₁ = exp(a₁) and c₂ = exp(a₂) the pdf of c₁ and c₂ have the common expression

$$\frac{\exp \left( { - \frac{{\left( {\log (c) - \mu_{c} } \right)^{2} }}{{2\sigma_{c}^{2} }}} \right)}{{{c\sqrt {2\pi \sigma_{c}^{2} } }}} $$

By computing first and second moments of this distribution, average and variance of c₁ and c₂ can expressed in terms of known quantities

$$ \begin{aligned} {{\upmu}}_{\text{c}} = & \exp \left( {{{\upmu}}_{\text{a}} + {{\upsigma}}_{\text{a}}^{2} /2} \right) \\ {{\upsigma}}_{c}^{2} = & \exp \left( {{{\upmu}}_{\text{a}} + 2{{\upsigma}}_{a}^{2} } \right) - \exp \left( {{{\upmu}}_{\text{a}} + {{\upsigma}}_{a}^{2} } \right) \\ \end{aligned} $$

Finally, the pdf of C = c₁c ^−ξ₂ and φ = θ − ψξ cannot be computed in closed form. However, the pdf of all stochastic variables appearing in the expressions of C and φ are known. Thus, the variances σ ²_C and σ ²_φ and the corresponding 99% confidence intervals were finally estimated with a Montecarlo procedure (n = 30,000).