Stroke survivors exhibit stronger lower extremity synergies in more challenging walking conditions

Abstract

The aim of this study was to examine how kinematic synergies are utilised as compensatory movements to stabilise foot positions under different walking task constraints in people with stroke. Ten (Males = 6, Females = 4) hemiplegic chronic stroke survivors volunteered to participate in this study, recruited from a rehabilitation centre. They completed a consent form and participated in treadmill walking tasks; flat, uphill, and crossing over a moving obstacle. The uncontrolled manifold method was used to quantify kinematic synergies in the paretic and non-paretic legs during their swing phase. The results of this study showed the strength of synergies was significantly greater in the obstacle task than in the uphill walking tasks at mid and terminal swing phases. In conclusion, the results suggest that walking in the challenging situations caused people with stroke to control step stability with greater compensation between lower extremity joints. Participants adapted to the increased challenge by increasing the amount of ‘good variability’, which could be a strategy to reduce the risks of falling.

Appendix: Details of the UCM method adapted from Krishnan et al (2013)

Creating a model for elemental variables

The UCM method that is used in this study has 10 elemental variables and 1 performance variable (mediolateral trajectory of swing foot). A geometric model was created to relate elemental variables to the performance variable. This comprised a five-segment and 10 DoFs model (Fig. 1). The segments were stance leg, pelvis, swing-leg thigh and swing-leg shank with lengths L₁, L₂, L₃, and L₄ respectively. For the paretic swing leg in gait cycle, 5 DoFs are in the frontal plane, Ɵ₁ = stance-leg shank_non-paretic, Ɵ₂ = stance-leg thigh_non-paretic, Ɵ₃ = pelvis, Ɵ₄ swing-leg thigh_paretic and Ɵ₅ = swing-leg shank_paretic, and 4 DoFs are out of the frontal plane; α₁ = stance-leg shank_non-paretic, α₂ = stance-leg thigh_non-paretic, α₃ = pelvis, α₄ = swing-leg thigh_paretic and s₅ = swing-leg shank_paretic. The same model is used when for the non-paretic swing leg.

Segmental configuration

An ankle joint trajectory (AJT) and functions of ten DoFs ($\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }$) from the segmental model was created by a custom-written code in Matlab 2018a (Mathworks, Natick, MA) according to the UCM procedure in Krishnan et al. (2013). Prior to UCM analysis, all segmental configuration and ankle joint trajectory (AJT) data were normalised for swing phases (0–100%):

$${\text{AJT}} = - L_{1} \cos \alpha_{1} \sin \theta_{1} - L_{2} \cos \alpha_{2} \sin \theta_{2} + L_{3} \cos \alpha_{3} \cos \theta_{3} + L_{4} \cos \alpha_{4} \sin \theta_{4} + L_{5} \cos \alpha_{5} \sin \theta_{5}$$

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta } = \left[ {\theta_{1} \theta_{2} \theta_{3} \theta_{4} \theta_{5} \alpha_{1} \alpha_{2} \alpha_{3} \alpha_{4} \alpha_{5} } \right].$$

(1)

Calculation of V _UCM and V _ORT

As geometric models describing the position of an end effector are generally non-linear, uncontrolled manifolds are often curved. Therefore, the first step in calculating $V_{\text{UCM}}$ and $V_{\text{ORT}}$ is to perform a linearization around a reference configuration. A linear approximation of the model can be obtained by calculating the Jacobian matrix with respect to a reference configuration—the mean segment configuration across trials (Scholz and Schoner 1999). Deviations of segment vectors, for a particular trial, from the mean segment configuration can then be projected onto the null space ($\varepsilon$) of this Jacobian matrix:

$$J = \frac{\partial D}{\partial \varTheta } = \left[ \begin{aligned} - & L_{1} \cos \alpha_{1} \cos \theta_{1} , - L_{2} \cos \alpha_{2} \cos \theta_{2} , - L_{3} \cos \alpha_{3} \sin \theta_{3} ,L_{4} \cos \alpha_{4} \cos \theta_{4} ,L_{5} \cos \alpha_{5} \cos \theta_{5} ,\; \\ L_{1} \sin \alpha_{1} \sin \theta_{1} ,L_{2} \sin \alpha_{2} \sin \theta_{2} , - L_{3} \sin \alpha_{3} \cos \theta_{3} , - L_{4} \sin \alpha_{4} \sin \theta_{4} - L_{5} \sin \alpha_{5} \sin \theta_{5} \\ \end{aligned} \right]$$

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }_{\parallel } = \mathop \sum \limits_{i = 1}^{n - d} \left( {\varepsilon_{i} \cdot \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }^{0} } \right)} \right)\varepsilon_{i} s,$$

(2)

where $n$ is the number of degrees of freedom of the model, $d$ is the number of dimensions of the performance variable, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }$ is a vector of segment angles for a particular trial and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }^{0}$ is a vector of segment angles in the reference configuration. For this work, n = 10 and d = 1. A component of the deviation of $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }$ from $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }^{0}$ that is perpendicular to the UCM can also be calculated as:

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }_{ \bot } = \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }^{0} } \right) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varTheta }_{\parallel } .$$

(3)

The variability per degree of freedom parallel ($V_{\text{UCM}}$) and perpendicular ($V_{\text{ORT}}$) to the UCM was then calculated using:

$$V_{\text{UCM}} = \left( {\frac{{\sum \varTheta_{\parallel }^{2} }}{{\left( {n - d} \right)N}}} \right),$$

(4)

and

$$V_{\text{ORT}} = \left( {\frac{{\sum \varTheta_{ \bot }^{2} }}{dN}} \right),$$

(5)

where N is the number of stride. V_UCM does not affect the variance in the mediolateral foot trajectory (Good variability), whereas V_ORT causes the variance in the foot trajectory (Bad variability).

Motor synergy index

If V_UCM > V_ORT, then lower limb synergy stabilise the foot trajectory in a gait cycle. A synergy index was calculated as:

$$\Delta V = \frac{{V_{\text{UCM}} - V_{\text{ORT}} }}{{V_{\text{ToT}} }},$$

(6)

where

$$V_{\text{ToT}} = \left( {\frac{1}{n}} \right)\left( {dV_{\text{ORT}} + \left( {n - d} \right)V_{\text{UCM}} } \right),$$

(7)

n = number of DoFs; d = number of performance variable.

The more positive ΔV, the stronger the motor synergy among segments. If V_ORT = 0, then ΔV = 10/9 and all variance lie within the UCM. If all variance lies in the orthogonal sub-space (V_UCM = 0), ΔV = − 10. For consistency among studies, ΔV was transformed to Fisher’s z-transformation index:

$$\Delta V_{Z} = \frac{1}{2}\log \left[ {\frac{10 + \Delta V}{{\frac{10}{9} - \Delta V}}} \right].$$

(8)