Nearest points and delta convex functions in Banach spaces

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.Comment: To appear in Bull. Aust. Math. So

Is Freezing of Gait in Parkinson's Disease a Result of Multiple Gait Impairments? Implications for Treatment

Several gait impairments have been associated with freezing of gait (FOG) in patients with Parkinson's disease (PD). These include deteriorations in rhythm control, gait symmetry, bilateral coordination of gait, dynamic postural control and step scaling. We suggest that these seemingly independent gait features may have mutual interactions which, during certain circumstances, jointly drive the predisposed locomotion system into a FOG episode. This new theoretical framework is illustrated by the evaluation of the potential relationships between the so-called “sequence effect”, that is, impairments in step scaling, and gait asymmetry just prior to FOG. We further discuss what factors influence gait control to maintain functional gait. “Triggers”, for example, such as attention shifts or trajectory transitions, may precede FOG. We propose distinct categories of interventions and describe examples of existing work that support this idea: (a) interventions which aim to maintain a good level of locomotion control especially with respect to aspects related to FOG; (b) those that aim at avoiding FOG “triggers”; and (c) those that merely aim to escape from FOG once it occurs. The proposed theoretical framework sets the stage for testable hypotheses regarding the mechanisms that lead to FOG and may also lead to new treatment ideas

Objective Assessment of Fall Risk in Parkinson's Disease Using a Body-Fixed Sensor Worn for 3 Days

Background: Patients with Parkinson's disease (PD) suffer from a high fall risk. Previous approaches for evaluating fall risk are based on self-report or testing at a given time point and may, therefore, be insufficient to optimally capture fall risk. We tested, for the first time, whether metrics derived from 3 day continuous recordings are associated with fall risk in PD. Methods and Materials 107 patients (Hoehn & Yahr Stage: 2.6±0.7) wore a small, body-fixed sensor (3D accelerometer) on lower back for 3 days. Walking quantity (e.g., steps per 3-days) and quality (e.g., frequency-derived measures of gait variability) were determined. Subjects were classified as fallers or non-fallers based on fall history. Subjects were also followed for one year to evaluate predictors of the transition from non-faller to faller. Results: The 3 day acceleration derived measures were significantly different in fallers and non-fallers and were significantly correlated with previously validated measures of fall risk. Walking quantity was similar in the two groups. In contrast, the fallers walked with higher step-to-step variability, e.g., anterior-posterior width of the dominant frequency was larger (p = 0.012) in the fallers (0.78±0.17 Hz) compared to the non-fallers (0.71±0.07 Hz). Among subjects who reported no falls in the year prior to testing, sensor-derived measures predicted the time to first fall (p = 0.0034), whereas many traditional measures did not. Cox regression analysis showed that anterior-posterior width was significantly (p = 0.0039) associated with time to fall during the follow-up period, even after adjusting for traditional measures. Conclusions/Significance: These findings indicate that a body-fixed sensor worn continuously can evaluate fall risk in PD. This sensor-based approach was able to identify transition from non-faller to faller, whereas many traditional metrics were not successful. This approach may facilitate earlier detection of fall risk and may in the future, help reduce high costs associated with falls

Automated detection of near falls: algorithm development and preliminary results

<p>Abstract</p> <p>Background</p> <p>Falls are a major source of morbidity and mortality among older adults. Unfortunately, self-report is, to a large degree, the gold-standard method for characterizing and quantifying fall frequency. A number of studies have demonstrated that near falls predict falls and that near falls may occur more frequently than falls. These studies suggest that near falls might be an appropriate fall risk measure. However, to date, such investigations have also relied on self-report. The purpose of the present study was to develop a method for automatic detection of near falls, potentially a sensitive, objectivemarker of fall risk and to demonstrate the ability to detect near falls using this approach.</p> <p>Findings</p> <p>15 healthy subjects wore a tri-axial accelerometer on the pelvis as they walked on a treadmill under different conditions. Near falls were induced by placing obstacles on the treadmill and were defined using observational analysis. Acceleration-derived parameters were examined as potential indicators of near falls, alone and in various combinations. 21 near falls were observed and compared to 668 "non-near falls" segments, consisting of normal and abnormal (but not near falls) gait. The best single method was based on the maximum peak-to-peak vertical acceleration derivative, with detection rates better than 85% sensitivity and specificity.</p> <p>Conclusions</p> <p>These findings suggest that tri-axial accelerometers may be used to successfully distinguish near falls from other gait patterns observed in the gait laboratory and may have the potential for improving the objective evaluation of fall risk, perhaps both in the lab and in at home-settings.</p

Microbiota changes associated with ADNP deficiencies: rapid indicators for NAP (CP201) treatment of the ADNP syndrome and beyond

Activity-dependent neuroprotective protein (ADNP) and its protein snippet NAP (drug candidate CP201) regulate synapse formation and cognitive as well as behavioral functions, in part, through microtubule interaction. Given potential interactions between the microbiome and brain function, we now investigated the potential effects of the ADNP-deficient genotype, mimicking the ADNP syndrome on microbiota composition in the Adnp+/- mouse model. We have discovered a surprising robust sexually dichotomized Adnp genotype effect and correction by NAP (CP201) as follows. Most of the commensal bacterial microbiota tested were affected by the Adnp genotype and corrected by NAP treatment in a male sex-dependent manner. The following list includes all the bacterial groups tested-labeled in bold are male Adnp-genotype increased and corrected (decreased) by NAP. (1) Eubacteriaceae (EubV3), (2) Enterobacteriaceae (Entero), (3) Enterococcus genus (gEncocc), (4) Lactobacillus group (Lacto), (5) Bifidobacterium genus (BIF), (6) Bacteroides/Prevotella species (Bac), (7) Clostridium coccoides group (Coer), (8) Clostridium leptum group (Cluster IV, sgClep), and (9) Mouse intestinal Bacteroides (MIB). No similarities were found between males and females regarding sex- and genotype-dependent microbiota distributions. Furthermore, a female Adnp+/- genotype associated decrease (contrasting male increase) was observed in the Lactobacillus group (Lacto). Significant correlations were discovered between specific bacterial group loads and open-field behavior as well as social recognition behaviors. In summary, we discovered ADNP deficiency associated changes in commensal gut microbiota compositions, a sex-dependent biomarker for the ADNP syndrome and beyond. Strikingly, we discovered rapidly detected NAP (CP201) treatment-dependent biomarkers within the gut microbiota

Effect of gait speed on gait rhythmicity in Parkinson's disease: variability of stride time and swing time respond differently

BACKGROUND: The ability to maintain a steady gait rhythm is impaired in patients with Parkinson's disease (PD). This aspect of locomotor dyscontrol, which likely reflects impaired automaticity in PD, can be quantified by measuring the stride-to-stride variability of gait timing. Previous work has shown an increase in both the variability of the stride time and swing time in PD, but the origins of these changes are not fully understood. Patients with PD also generally walk with a reduced gait speed, a potential confounder of the observed changes in variability. The purpose of the present study was to examine the relationship between walking speed and gait variability. METHODS: Stride time variability and swing time variability were measured in 36 patients with PD (Hoehn and Yahr stage 2–2.5) and 30 healthy controls who walked on a treadmill at four different speeds: 1) Comfortable walking speed (CWS), 2) 80% of CWS 3) 90% of CWS, and 4) 110% of CWS. In addition, we studied the effects of walking slowly on level ground, both with and without a walker. RESULTS: Consistent with previous findings, increased variability of stride time and swing time was observed in the patients with PD in CWS, compared to controls. In both groups, there was a small but significant association between treadmill gait speed and stride time variability such that higher speeds were associated with lower (better) values of stride time variability (p = 0.0002). In contrast, swing time variability did not change in response to changes in gait speed. Similar results were observed with walking on level ground. CONCLUSION: The present results demonstrate that swing time variability is independent of gait speed, at least over the range studied, and therefore, that it may be used as a speed-independent marker of rhythmicity and gait steadiness. Since walking speed did not affect stride time variability and swing time variability in the same way, it appears that these two aspects of gait rhythmicity are not entirely controlled by the same mechanisms. The present findings also suggest that the increased gait variability in PD is disease-related, and not simply a consequence of bradykinesia

White Matter Hyperintensities in Parkinson’s Disease: Do They Explain the Disparity between the Postural Instability Gait Difficulty and Tremor Dominant Subtypes?

Background: Brain white matter hyperintensities (WMHs) commonly observed on brain imaging of older adults are associated with balance and gait impairment and have also been linked to cognitive deficits. Parkinson’s disease (PD) is traditionally sub-classified into the postural instability gait difficulty (PIGD) sub-type, and the tremor dominant (TD) sub-type. Considering the known association between WMHs and axial symptoms like gait disturbances and postural instability, one can hypothesize that WMHs might contribute to the disparate clinical sub-types of patients with PD. Methods: 110 patients with PD underwent a clinical evaluation and a 3T MRI exam. Based on the Unified Parkinson Disease Rating Scale, the patients were classified into motor sub-types, i.e., TD or PIGD, and scores reflecting PIGD and TD symptoms were computed. We compared white matter burden using three previously validated methods: one using a semi-quantitative visual rating scale in specific brain regions and two automated methods. Results: Overall, MRI data were obtained in 104 patients. The mean WMHs scores and the percent of subjects with lesions in specific brain regions were similar in the two subtypes, p = 0.678. The PIGD and the TD scores did not differ even when comparing patients with a relatively high burden of WMHs to patients with a relatively low burden. Across most of the brain regions, mild to moderate correlations between WMHs and age were found (r = 0.23 to 0.41; p<0.021). Conversely, no significant correlations were found between WMHs and the PIGD score or disease duration. In addition, depressive symptoms and cerebro-vascular risk factors were similar among the two subtypes. Conclusions: In contrast to what has been reported previously among older adults, the present study could not demonstrate any association between WMHs and the PIGD or TD motor sub-types in patients with PD