## 1 Introduction  
Ankle sprains are prevalent injuries in sports, particularly impacting high-intensity lower-limb movements such as jumping and drop landings, which are common in basketball, soccer, and volleyball. The incidence of such injuries is alarmingly high, with estimates suggesting a risk as high as 50% [1, 2]. Inadequately treated initial sprains may progress to chronic conditions, which are characterized by persistent pain and recurrent edema, potentially diminishing postural control and proprioception [3, 4]. Such deterioration may culminate in the development of functional ankle instability (FAI) [5, 6], a condition that significantly degrades athletic performance and diminishes the quality of life for affected individuals [7; 8]. Furthermore, the central nervous system's adaptive response to ankle injuries is known to modify the postural control strategies, thereby increasing the risk of subsequent injuries for individuals with FAI during complex movements [9]. However, despite the high incidence of ankle sprains leading to FAI, there is a certain deficiency in our understanding of how the cognitive load affects the biomechanical responses during landing tasks in these individuals. This deficiency limits the development of targeted interventions to mitigate the risk of recurrent injuries in FAI patients. In the context of motor control, the cognitive dimension is critical, especially for individuals with FAI. The ability to maintain postural control and motor stability during the dynamic and unpredictable nature of sports largely relies on the cognitive resources of the brain, which are tasked with coordinating attention, memory, executive functions, and perceptual processing [10]. The dual-task paradigm, a widely recognized method in cognitive psychology and motor control research, posits that engaging in a secondary cognitive task while performing a primary motor task can lead to decrements in performance on one or both tasks due to the limited cognitive capacity [11]. Cognitive tasks, such as the n-back task, require the coordination of focused attention, memory, executive functions, and perceptual processing. This engagement recruits key brain regions including the prefrontal cortex, hippocampus, and parietal lobe, which are integral to motor control [12; 13]. Individuals with FAI may increasingly rely on cognitive information to compensate for the diminished perceptual feedback, highlighting the importance of cognitive performance in their overall athletic capabilities [14]. Landing movements are among the most common causes of ankle injuries and represent a complex biomechanical model [1]. Analyzing the differences in the ability to maintain postural control and motor stability during these high-impact movements between individuals with FAI and healthy populations is crucial in preventing subsequent injuries in individuals with FAI. The interplay between the cognitive load and motor control in FAI is particularly pronounced during sports movements, such as landings, where the dual-task condition may divert attention and adversely affect the postural control stability and movement precision [15]. Despite previous in-depth studies on single-leg drop landing movements in individuals with FAI, these studies often fail to fully account for the cognitive engagement present in actual sports activities [16; 17]. Elevated cognitive demands can necessitate strategic adjustments in movement execution, compelling athletes with FAI to expedite the reallocation of resources for postural stability and cognitive information processing [18]. However, there is a dearth of research on how individuals with FAI perform during drop landings under the influence of cognitive tasks. The two-back task is favored over the one-back task due to its requirement for higher-order cognitive resources, whereas the two-back task, in comparison to the three-back task, offers a more manageable balance between challenge and difficulty [19].