Affine differential geometry analysis of human arm movements

Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law” which connects path curvature with velocity, and “local isochrony” which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production

Voluntary Associations and Net Fertility During the Swedish Demographic Transition

This study investigates the role of changing social relations for fertility decline during the European fertility transition. The growth of voluntary associations at the end of the nineteenth century entailed a radical shift in the landscape of social relations in Sweden. By combining micro-census data from 1890 to 1900 with local-level membership data for three voluntary association groups, this article assesses the effect of parish-level voluntary association size on net fertility in Sweden using mixed-effects Poisson regression models. The results show that the adoption of fertility limitation during the transition period was associated with the creation and diffusion of the idea of respectability within large social network organisations, an idea that has previously been shown to be connected to fertility limitation. Furthermore, by applying a social network perspective, the results show that the strength of the effect was dependent on the structure of the social networks in terms of size, density, and homogeneity. Voluntary association size had the strongest effect for the free churches, which created dense heterogeneous networks through systems of social control, while the size of the temperance association showed no effect on fertility because the connections between nodes were sparse