Walking is a fundamental human action that underpins daily activities throughout the lifespan. Examining the solutions for satisfying walking when neurological or neuromuscular impairments are present is key to understanding motor control and optimizing therapeutic strategies. Based on current understanding, the complexity and redundancy of human movement outdo our abilities to fully model its mechanics. Physical systems with time-evolving internal couplings, such as those between limb motions during gait, are commonplace in nature. However, the analytical characterisation of these motions, for example their governing equations, is only possible to determine through simplified linear models. Determining the extent to which one system drives the behavior of another through causality analysis may elegantly uncover the subtle, redundant interactions ubiquitous of biological systems that are likely to vary from stride to stride (Hausdorff et al., 1999) but are fundamental to maintaining the overall dynamic stability of the system (Orth et al., 2017; Hamacher et al., 2017; Ye et al., 2015; Wyatt et al., 2021; Strongman and Morrison, 2020).

The dynamical systems perspective contends that the timing and coordination of movements are emergent self-organized properties of the individual physical system and its interaction with the environment and the task (Newell, 1986; Kelso and Schöner, 1988). Complexity and stability are fundamental elements, characterizing a system’s dynamics with respect to attractor states (Newell, 1986; Kelso and Schöner, 1988; Kugler and Turvey, 2015; van Emmerik and van Wegen, 2000; Wade and Kazeck, 2018). Walking gait is considered a global attractor, for which complexity and stability can be estimated to capture the motor control solution (Buzzi et al., 2003; Raffalt et al., 2017; Stergiou and Decker, 2011). Based on the cyclic motion related to steps, walking can be described as a period limit cycle attractor (Buzzi et al., 2003). Complexity may be quantified by the dimensionality of a system, represented by the number of independent variables that are needed to describe its evolution (Buzzi et al., 2003; Lipsitz and Goldberger, 1992; Vicinanza et al., 2018). Specifically, a higher fractal dimension indicates a higher level of complexity, as more independent variables are needed to describe the system’s behavior. The fractal dimension can be quantified by the correlation dimension, which can be interpreted by the number of independent variables needed to describe the system’s evolution. In the study of human movement, the fractal dimension can be used to analyze and quantify the complexity of movement patterns. It can provide insights into the coordination and control of movement, as well as the adaptability and variability of human motor behavior. Meanwhile, local stability captures resilience to small perturbations, such as those naturally produced by the system during steady-state gait (Dingwell et al., 2000; Dingwell et al., 2001). Largest finite-time Lyapunov Exponent (LLE) has been used to evaluate stability in gait during steady-state walking, showing evidence that for joint angle variables, as gait speed increases local dynamic stability decreases (Dingwell et al., 2000; Dingwell and Cusumano, 2000; England and Granata, 2007), increases (Manor et al., 2008), or that an inverted U-shaped relationship exists where stability is highest at preferred stride frequency and lower at slower or faster stride frequencies (Raffalt et al., 2017; Russell and Haworth, 2014). In relation to intrinsic constraints, local dynamic stability is shown to be lower in fall-prone older adults compared to healthy older adults and young adults, even though walking speeds were slower and step lengths were smaller (Granata and Lockhart, 2008; Lockhart and Liu, 2008; Amirpourabasi et al., 2022), and decreased with age (Wade and Kazeck, 2018). The combination of complexity and stability has been theoretically linked to efficiency and adaptability in movement and is a promising area of future research (Buzzi et al., 2003; Raffalt et al., 2017; Stergiou and Decker, 2011). Therefore, in this work, we introduced an index (the Complexity and Instability Index, CII) which combines both complexity and stability as a comprehensive measure of the nonlinear dynamics characteristic related to functional biological systems that are relevant to gait stability and resilience.

Contractures can be defined by a limited extensibility or increased stiffness of the soft tissues surrounding the joints resulting in a reduced passive range of motion (ROM) (Attias et al., 2016; Fergusson et al., 2007). In order to better understand the impact of contracture on gait, experimental studies have simulated contractures on the ROM of joints of healthy participants using restrictive exoskeletons (Armand and Attias, 2019; Attias et al., 2023; Attias et al., 2019; Matjačić et al., 2006; Goodman et al., 2004; Houx et al., 2012; Harato et al., 2008; Whitehead et al., 2007; Attias et al., 2017; Attias et al., 2019). Gait adaptations have been demonstrated in Armand and Attias (Armand and Attias, 2019; Attias et al., 2023) with the aim to provide understanding of the mechanisms underpinning characteristics of pathological gait, as well as the biomechanical solutions for walking in their presence. The main results can be visualized on a web-application (pcgs.unige.ch).

This work seeks to further current understanding by exploring nonlinear causal interactions between limb motions during gait, underpinning the search for explanations or interpretations of complexities outside of mechanical relationships which might be inherently related to skill level or health status in human motor control. Therefore, the aim of this paper was to explore the causal drive and nonlinear dynamics characteristics of the legs during walking when contractures were bi- and uni- laterally emulated by a passive exoskeleton to the gastrocnemius. It is hypothesized that the unconstrained limb will predominantly drive the constrained limb, and that in constrained limbs complexity will increase and stability will decrease.

The aim of this paper was to explore the causal drive and nonlinear dynamic characteristics of the legs during walking when contractures were bi- and uni- laterally emulated by a passive exoskeleton to the gastrocnemius. During normal walking, there is a high mutual drive from one leg to the other. The CII is low, where the complexity of each leg motion is low and the stability is high (Figure 2 and Table 1). In line with the first part of the hypothesis, in bilateral exoskeleton condition, the causal drive of each leg to the other was reduced, and the CII for both legs was significantly increased. Contrary to the hypothesis, in the unilateral exoskeleton condition, the mechanically constrained leg strongly drives the unconstrained leg, and CII was significantly higher for the constrained leg compared to normal walking.

Fundamental mechanical links exist in inter-limb relations during walking, as well as subtle, redundant interactions ubiquitous of biological systems that are likely to vary from stride to stride (Hausdorff et al., 1999) but are essential to maintaining the overall dynamic stability of the system. The analysis of causality in inter-limb motion transcends information gained from pure mechanics, mean phase relations or simply unpacking the dynamics of individual limbs or segments signals. Rather, in biological systems, identifying the nature of nonlinear causal interactions between limb motions outside of mechanical relationships is likely inherently related to health status in motor control, and underpins the search for explanations or interpretations of complexities. When applied in this special case, we gain understanding of the solution of our coupled leg-leg system in satisfying walking gait in constrained conditions that reflect common pathological states, capturing how the causal relations and nonlinear dynamics are altered with the exoskeleton. Complexity and stability in actions are the hallmarks of dynamics, where complexity is represented by the number of independent variables that are needed to describe the system’s evolution in time (Vicinanza et al., 2018) and local stability as a measure of the strength of an attractor state (Dingwell et al., 2001; Dingwell and Cusumano, 2000). By considering the effects of induced contractures on the complexity and stability of leg actions during walking, we can have a window into understanding the effect of the contractures on the dynamics of this system.

During normal walking (no exoskeleton condition), there is a high causal drive of the action of each leg to the other, where each leg has a low CII (Figure 3 and Table 1). The effect of the bilateral exoskeleton is a reduction in strength of the causal drive of each leg to the other, and an increase in CII. These results strongly suggest that our biological system in its normal state uses redundancy to support causal interactions, which reduces complexity and increases stability in our leg dynamics during walking. In a broader sense, there might be initial evidence that the biological system in its normal redundant state uses causal interactions to reduce complexity and increase stability in our leg dynamics during walking.

The goal (natural) state of the systems complexity and stability demonstrates evidence of basic dynamical principles applied to human motor control. It is observed in tasks such as finger wagging, arm and leg swinging, and even during interpersonal coordination, that strict natural, simple and stable coupling relationships exist (Haken et al., 1985; Kelso et al., 1986). Causal aspects calculated here demonstrate motor control includes strong causal links for limbs for which CII remains low (Figure 3 and Table 1), and these characteristics are supported by redundancy. It seems then that the role of redundancy is to allow adaptability above being able to satisfy the overall biomechanical problem; and to allow the system to interact optimally. Redundancy allows the system to be influenced and driven by an externally or internally constrained subsystem in a degenerate manner. This is an important theoretical and applied point because it captures the notions of flexibility within biology being key to success, and highlights that in the instance of injury or disability the most important characteristics of functional motion might be captured by these nonlinear and causal variables, as well as the biomechanical aspects typically studied.

Causal drive between legs in the bilateral exoskeleton condition remains mutually low while the CII increases for both legs compared to normal walking (Figure 3 and Table 1). However, there is an asymmetrical increase where the left leg has higher CII than the right leg. Since both legs are constrained equally, this finding might reflect natural asymmetry associated with leggedness which is emphasized in bilaterally constrained conditions. Some evidence of natural asymmetry increasing with walking speed exists (Plotnik et al., 2013), however since most studies of gait symmetry have examined normal walking versus natural or imposed bilateral asymmetry (Kwek and Williams, 2021; Shorter et al., 2008) these findings require further investigation.

In comparison, when walking with the unilateral exoskeleton, the mechanically constrained leg (in this case the left leg) strongly drives the unconstrained leg (right leg), while the drive of the unconstrained leg to the constrained leg is limited (Figure 3). This nontrivial finding suggests when one leg is constrained, the system exploits redundancy in unconstrained parts of the system to be used in a compensatory rather than a driving capacity. In this scenario, we observe that exploiting redundancy facilitates an alternative technique that can be used to compensate for the constrained leg. From a theoretical standpoint, the motor control solution is then defined in light of the constraints imposed, and adaptability and exploiting redundancy are key to supporting functionality. Based on the large effect sizes reported, we provide strong evidence of this phenomenon in the current context, where constraints are imposed on healthy individuals. This finding is significant in a practical sense since optimal walking gait may be driven by the most constrained leg, and while less resembling ‘healthy’ walking kinematics, may provide a more optimal motor control solution. Indeed, some patients with unilateral paretic limb have greater deviation from normal, or greater gait impairment, on the non-paretic limb, which is hypothesized to be due to compensation strategies employed to accommodate for reduced function in the paretic limb (Devetak et al., 2016; Jarvis et al., 2022). The nonlinear aspects of leg motion or inter-leg interaction are not well studied, but could provide interesting insights into the underpinnings of the strategy.

Furthermore, it may be conjectured that the more severe the constraint, the stronger the unidirectional causal drive. While the causal drive from the constrained leg to the unconstrained leg is strong, we have evidence of a non-optimal solution since the CII for both legs remains high; which could be interpreted as the price for the compensatory nature of the actions (Figure 2). CII provided further evidence of an increase in complexity and a decrease in stability being associated with a non-optimal state of the human system during movement (Hausdorff et al., 1999; Buzzi et al., 2003; Stergiou and Decker, 2011; Vicinanza et al., 2018; Dingwell et al., 2001). Specifically, during normal walking, CII is low where the complexity of each leg motion is low, and the stability is high (Table 1). Theoretically, this pattern then allows both efficiency and adaptability and is in line with the findings of Raffalt et al. (2017), who showed minimum values of CD and LLE of kinematic variables when walking at preferred speeds. With bilateral exoskeleton conditions, a significantly higher CII for both legs was recorded, while when walking with the unilateral exoskeleton conditions, CII was significantly higher for the constrained leg compared to normal walking. This suggests that the redundancy available is used to stabilize the system components but that the compensatory actions evidenced by the causality analysis leave the system with higher CII characteristics.

Detecting causal interactions in complex biological systems is essential for understanding how these systems function and will be a promising area to understand human motor control. However, it is unknown whether a longer period of adaptation would change walking behavior in the exoskeleton conditions.

This study has several limitations, data were collected for a number of trials overground, rather than continuously. There is also a limited number of participants in the study.

Providing a new perspective on the theoretical notion of flexibility within biology, this work provides evidence that our biological system uses redundancy to support causal interactions, reducing complexity and increasing stability in our leg dynamics during walking. Normal walking was characterized by a high mutual drive of each leg to the other, where CII was lowest for both legs (complexity of each leg motion estimated with CD was low, and the stability estimated with LLE was high). The effect of the bilateral emulated contractures was a reduced drive of each leg to the other and an increased CII for both legs. With unilateral emulated contracture, the mechanically constrained leg strongly drives the unconstrained leg, and CII was significantly higher for the constrained leg compared to normal walking. From a dynamics system perspective, it is seen here that a constrained limb drives the free parts of the system, a nontrivial finding, suggesting that the role of redundancy is to allow adaptability above being able to satisfy the overall biomechanical problem; to allow the system to interact optimally.

From an applied perspective, in the instance of imposed constraint (injury or disability), the most important characteristics of functional motor control and the resulting motions might be captured by these nonlinear and causal variables, as well as the biomechanical aspects typically studied. Through understanding of nonlinear causal interactions between limb motions, we can drive the search for explanations or interpretations of complexities outside of mechanical relationships which might be inherently related to skill level, functionality or health status in human motor control.