## 1 Introduction
PCA reduces the number of variables while retaining essential information about the motor behavior of individual movement components, referred to as \"principal movements\" (PM\({}_{k}\)), where k denotes the order of the movement components [12,13]. In locomotion analysis, each principal movement (PM) can be visualized as a distinct movement synergy representing phases of a gait cycle, such as swing or stance, which work in tandem to accomplish the task [1,2,14]. This PCA-based approach also offers valuable insights into each PM's position and acceleration, reflecting the neuromuscular control associated with different movement strategies, including the forces exerted by the system [11,12] and related myoelectric activities [7]. Previous research has demonstrated the utility of PCA in identifying principal movement patterns [1,15,16] and assessing the effects of perturbations on running stability [14]. In motor control, variability refers to variations in the amplitude of a time series, quantified using measures like standard deviation [15]. Understanding the impact of treadmill gradients on neuromuscular control at the level of individual running movement components could inform the development of training programs and rehabilitation protocols tailored to specific goals. Temporal variations in gait are analyzed using non-linear methods, such as the largest Lyapunov exponent (LyE), which measures how quickly trajectories diverge [17, 18, 19, 20]. LyE reflects the \"predictability\" of a time series and is commonly interpreted as a measure of dynamic or local stability, indicating the stability of the system across step cycles [17; 18; 19; 20]. LyE has been used to evaluate the ability to compensate for small internal or external perturbations and maintain functional locomotion [17; 18; 19; 20]. In this context, stability refers to a system's (e.g., the movement system) capacity to preserve its original state when confronted with internal (e.g., neuromuscular) or external (e.g., environmental) disturbances [19]. Stability parameters provide insight into the variability in motor task performance and quantify the system's ability to correct dynamic errors [21]. LyE, when applied to the positions of individual principal movements (PMs), reflects the neuromuscular system's ability to control and adapt to minor disruptions in movement patterns [1; 14]. In running, local dynamic stability indicates the neuromuscular system's capability to handle these small perturbations [17; 18; 19; 20]. Reduced stability has been linked to an increased risk of overuse injuries, such as bone stress injuries, which occur when repetitive loads exceed the bone's capacity for recovery [22; 23]. The current study builds on these methodologies to investigate gradient-specific neuromuscular adaptations during running. In summary, this study aimed to explore how variations in treadmill gradient--both incline and decline--affect the local dynamic stability of individual running movement components/synergies, reflecting neuromuscular control during locomotion. It was hypothesized that changes in gradients would modulate the neuromuscular control of running movement components, as incline and decline treadmill gradients introduce altered biomechanical aspects to running [6]. It was also expected that the gradient effects on running stability would particularly influence the weight-bearing support phase, which is crucial for balance and stability [14]. The findings provide valuable insights into how treadmill gradients affect movement stability, with implications for optimizing training and rehabilitation, especially for patients recovering from lower-limb injuries.