Critical firing levels (CFLs) of single motor units (MU?.) in the long head of the human biceps brachii muscle were determined in combinations of two isometric tasks flexion of the elbow, supination of the lower arm, and exorotation of the humerus, as well as the corresponding antagonistic tasks. The MU activity was recorded by 25-pm bipolar wire electrodes. Four main patterns of MU recruitment, related to the recording location in the muscle, were found: (i) MUs active only when flexing the elbow were located mostly laterally. (ii) MUs active only when supinating were all located medially. (iii) MUs whose CFL depended on a linear combination of flexion and supination forces were all located medially. Some of these MUs could not be recruited during pronation. (iv) Nonlinearly behaving MUs, located centrally. The relative weights of tlexion and supination input were constant for all units, whose CFL depended on a linear sum of flexion and supination forces, as well as for the nonlinearly behaving units. Supination and exorotation showed equivalent CFL changes when they were combined with the flexion task. Extension did not change the CFL for supination-or exorotation tasks. No clear difference was found between the ratios of the peak twitch forces in flexion and supination direction for laterally and medially located small muscle areas or single MUs. A simple model of the motoneuron pool organization is proposed to explain our findings.Abbreviations: CFL-critical firing level, MU-motor unit, BLH-human biceps brachii long head, MVC-maximum voluntary contraction, S-supination, E-exorotation, F-flexion.

Abstract. Left-invariant PDE-evolutions on the roto-translation group SE(2) (and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to SE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDEevolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

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