Novel stability measures (Gain Margin |GM|, and Phase Margin PM, from control theory) were used in this study to quantify the stability degree in patients with Parkinson’s disease, as well as evaluating the effects of balance training in PD. The pattern of improvements during a 6-week balance-training program, in terms of Gain and Phase Margin, was investigated in PD patients. |GM| and PM were calculated using the low-cost posturographic data and a computational subject-specific postural control model. Findings showed that stability safety margin (i.e. |GM| and PM) were smaller in patients compared to healthy control subjects (HCs). Patients, unlike HCs, significantly reduced PM on foam due to considerable time delay. Stability margin, as well as all clinical outcomes, improved in patients after balance training. Improvement in |GM| was characterized by an early improvement at week 4, followed by plateau during the next two ending weeks of training. In contrast, PM improved relatively late at week 6 in a rather continuous-progression form.
Reduced Gain Margin (|GM|) in PD, as we observed in Study1, is in-line with previous studies which reported lower margin of stability for PD patients compared to HCs in different tasks (e.g. perturbed quiet stance [19, 30], or perturbed gait ). However, considering the different tasks and techniques of quantification in those studies, any inference should be drawn cautiously. Although most of these studies employed the spatial term of margin of stability (i.e. the difference between peak COM and Peak COP), Patton et al.  showed that torque safety margin is highly correlated with spatial safety margin calculated from COP response characteristics. |GM| is substantially influenced by the value of controller gain parameters in a system, which in turn, is mainly associated with the muscular strength (control effort) of human body system; suggesting that the reduced |GM| in PD may be due to the weakened muscle strength [42, 43]. Recently, we showed that most of control parameters (such as KP, the pivotal ruling gain parameter) was lower in PD compared to HCs . Nevertheless, excessive amplification of gain parameters results in resonant instability . In other words, KP should remain between a lower and an upper bound to guarantee stability of the inverted-pendulum system. This implies that, in fact, CNS tunes all control gain parameters in a way to adjust the margin of stability; that is, |GM| is truly expressing the margin of stability rather than muscular strength (KP). That is, |GM| carried different implication of stability from KP (the other measure for quantification of stability degree). Furthermore, results showed that HCs and patients increased |GM| when closing eyes; supporting the impression that CNS adopted higher level of safety margin in more threatening and challenging tasks. Extended safety margin in EC tasks was seen in young  and healthy elderly subjects [46, 47], as well as PD’s  and multiple-sclerosis patients . Group by vision interaction in our study, however, revealed that PD patients did not adjust (augment) |GM| as much as HCs did in EC tasks; reiterating the contribution of reduced strength factor in patients, as evidenced by low KP in EC tasks in our recent study .
Unlike |GM|, PM was almost similar in PD patients and HCs, although with drastic drop for PD on foam. It seemed plausible that all subjects – patients or HCs – performed all tasks stably with moderately similar PM, nevertheless having different |GM|. Stability in a delayed-inverted-pendulum system relies largely on the adequacy of time-delay, which should not violate a critical value [49, 50]. Time delay, in part, has remarkable contribution to the amount of PM. Therefore, in stable performance, PM almost remains in a specific range, unless time delay varies significantly. Our findings showed that patients, contrary to HCs, exhibited drastic drop in PM while standing on foam (group × surface interaction), indicating patients’ deficit in preserving the level of PM on foam to that on rigid surface. Decline in PM for patients on foam may be a consequence of significant delayed-response (longer time delay) that patients had on foam , which brought patients to the verge of instability (smaller PM). Wright et al.  observed reduced margin of stability on compliant surface for group of elderly healthy women; yet some studies reported increased margin of stability for stepping onto and walking on foam for young adults . Different tasks, or definition for margin of stability may give rise to different possible results, which further highlights the importance of applying fundamental concepts for quantification of stability (or margin of stability) in future studies.
Tracking of stability margin measures (|GM|, PM) in multiple time points, during a course of balance-training program in Study2, disclosed their pattern of improvement; thereby enlightening how balance training can affect stability in PD. In addition, findings suggest that the improvement during training programs can be captured by |GM| and PM, as meaningful and sensitive measures for assessment of stability in PD. The pattern of improvement in |GM| was characterized by an early improvement at week 4, followed by a plateau during the next two closing weeks. In our recent study , we found similar early-improvement with plateaued behavior for stability-related measures such as KP; which we concluded, based on evidences from other studies [34, 53, 54], as the limited capacity for the development of strength in PD. Although PD patients demonstrated ability to learn, limited learning capacity in PD was numerously noted in other studies [10, 55]. Corcos et al  noted that 2-year progressive resistance exercise (PRE) in PD, at best, can lead to such plateaued behavior (stagnation at a specific level) for elbow flexion torque; in comparison to non-progressive exercise, which progressed during first months of training, and regressed back in the rest of training. Secondly, some studies showed early strength gain during first weeks of training in healthy elderlies  or PD patients [54, 57]. These findings indicate that the capability of patients for retaining or improving adequate margin of stability, besides issuing properly scaled motor commands, highly pertain to the capacity of regaining strength. Therefore, |GM|, which is per se, a reflection of properly tuned gain parameters, improved initially by training, but was later subject to plateau after week 4. Possibly, attaining greater values of |GM|, maybe as to the level of more skilled individuals, demands further focused training, particularly in PD. The other stability margin measure, PM, showed rather a continuous-progression form that improved late at the end of training at week 6. Our recent study  showed that time delay in PD improved with similar late and continuous-progression form, specifically in FO task (the very task in which PM was improved). Probably, improvement in PM highly demands correct timing of control commands in postural control. Findings of Study2 emphasized that the problem of poor stability safety margin in PD is amendable via balance training, with specific attention to the needed dosage on each intervals of training program.
Few studies considered the changes in stability, in terms of margin of stability, during a training program [7, 34, 58]. These studies reported improvements in the margin of stability throughout repeating trials, for young , healthy elderly , and PD patients [7, 34], although with different test protocols, or definition for margin of stability; still with intriguing results. Given that the spatial gain margin had high correlation to torque safety margin – which is relevant to gain parameters and therefore |GM| – facilitates reasoning from these studies. Peterson et al  realized that margin of stability (measured as the difference between Xcom and first stepping footfall in response to perturbation) improved in HCs continuously throughout trails; whereas, PD patients improved margin of stability primarily in the first blocks of trials and then plateaued out. Patton et al.  studied the improvement in relative stability while learning a dynamic task (pulling a handle) in ten young subjects. They found that both spatial (the distance-to-boundary for COP to either heel or toe) and temporal (time-to-boundary for COP, using 1st order predictive extrapolation – based on COP velocity–) safety margin increased with practices; however, progress in spatial margin was more significant than temporal one. Interestingly, they observed that spatial margin (for trials with different pulling forces) finally converged into a roughly similar specific value by 5 days of practices; suggesting that, in normal performances or due to biomechanical constraints, only a specific level of spatial margin is achievable. Furthermore, this study revealed that improvement in temporal margin is hardly achieved by practices, which is in favor of our finding on late and continuous-progression of PM.
This study had limitations. First, because of the novelty in quantification method of stability in this study (first-ever usage of GM and PM for quantification of stability in PD), lack of similar studies and evidences limited more in-depth interpretation of findings, particularly results related to the learning dynamics during a course of training program. Previously, spatial (distance-to-boundary) and temporal (time-to-boundary) margin of stability were proposed  and employed [7, 29, 58, 59] in studies; mostly based on characteristics of COP signal in response to different tasks. The prevailing spatial and temporal definition of margin of stability is simply an external manifestation of the genuine underlying control safety margin that CNS adopts (e.g. GM and PM, which are the characteristics of supraspinal control commands). Future studies are needed to disclose the relationship between intrinsic features of safety margin (e.g. GM, PM) and existing stability measures. Furthermore, this study was limited to static posturography. Basically, applying such fundamental concepts of theoretical stability in dynamic tasks (e.g. in perturbed quiet stance, perturbed gait) provides more comprehensive explanation of patients’ balance performance. Moreover, future studies can encompass complex models (e.g. double-inverted-pendulum), as well as applying more general stability theorems (e.g. Lyapunov stability criterion  for non-linear systems). Finally yet importantly, future studies carrying longer training programs with follow-up inspection, accompanied by more time points of assessment, or enjoying diversity in training regimen, are highly recommended.