Wishart laws and variance function on homogeneous cones

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

Beyond cones: an improved model of whisker bending based on measured mechanics and tapering

The sense of touch is represented by neural activity patterns evoked by mechanosensory input forces. The rodent whisker system is exceptional for studying the neurophysiology of touch in part because these forces can be precisely computed from video of whisker deformation. We evaluate the accuracy of a standard model of whisker bending, which assumes quasi-static dynamics and a linearly tapered conical profile, using controlled whisker deflections. We find significant discrepancies between model and experiment: real whiskers bend more than predicted upon contact at locations in the middle of the whisker and less at distal locations. Thus whiskers behave as if their stiffness near the base and near the tip is larger than expected for a homogeneous cone. We assess whether contact direction, friction, inhomogeneous elasticity, whisker orientation, or nonconical shape could explain these deviations. We show that a thin-middle taper of mouse whisker shape accounts for the majority of this behavior. This taper is conserved across rows and columns of the whisker array. The taper has a large effect on the touch-evoked forces and the ease with which whiskers slip past objects, which are key drivers of neural activity in tactile object localization and identification. This holds for orientations with intrinsic whisker curvature pointed toward, away from, or down from objects, validating two-dimensional models of simple whisker-object interactions. The precision of computational models relating sensory input forces to neural activity patterns can be quantitatively enhanced by taking thin-middle taper into account with a simple corrective function that we provide.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

In this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].