Movement Timing and Invariance Arise from Several Geometries

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations

Glycosaminoglycan and Proteoglycan Biotherapeutics in Articular Cartilage Protection and Repair Strategies: Novel Approaches to Visco?supplementation in Orthobiologics

The aim of this study is to review developments in glycosaminoglycan and proteoglycan research relevant to cartilage repair biology and in particular the treatment of osteoarthritis (OA). Glycosaminoglycans decorate a diverse range of extracellular matrix and cell associated proteoglycans conveying structural organization and physico‐chemical properties to tissues. They play key roles mediating cellular interactions with bioactive growth factors, cytokines, and morphogenetic proteins, and structural fibrillar collagens, cell interactive and extracellular matrix proteoglycans, and glycoproteins which define tissue function. Proteoglycan degradation detrimentally affects tissue functional properties. Therapeutic strategies have been developed to counter these degenerative changes. Neo‐proteoglycans prepared from chondroitin sulfate or hyaluronan and hyaluronan or collagen‐binding peptides emulate the interactive, water imbibing, weight bearing, and surface lubricative properties of native proteoglycans. Many neo‐proteoglycans outperform native proteoglycans in terms of water imbibition, matrix stabilization, and resistance to proteolytic degradation. The biospecificity of recombinant proteoglycans however, provides precise attachment to native target molecules. Visco‐supplements augmented with growth factors/therapeutic cells, hyaluronan, and lubricin (orthobiologicals) have the capacity to lubricate and protect cartilage, control inflammation, and promote cartilage repair and regeneration of early cartilage lesions and may represent a more effective therapeutic approach to the treatment of mild to moderate OA and deserve further study