When we grasp and manipulate an object, populations of tactile nerve fibers become activated and convey information about the shape, size, and texture of the object and its motion across the skin. The response properties of tactile fibers have been extensively characterized in single-unit recordings, yielding important insights into how individual fibers encode tactile information. A recurring finding in this extensive body of work is that stimulus information is distributed over many fibers. However, our understanding of population-level representations remains primitive. To fill this gap, we have developed a model to simulate the responses of all tactile fibers innervating the glabrous skin of the hand to any spatiotemporal stimulus applied to the skin. The model first reconstructs the stresses experienced by mechanoreceptors when the skin is deformed and then simulates the spiking response that would be produced in the nerve fiber innervating that receptor. By simulating skin deformations across the palmar surface of the hand and tiling it with receptors at their known densities, we reconstruct the responses of entire populations of nerve fibers. We show that the simulated responses closely match their measured counterparts, down to the precise timing of the evoked spikes, across a wide variety of experimental conditions sampled from the literature. We then conduct three virtual experiments to illustrate how the simulation can provide powerful insights into population coding in touch. Finally, we discuss how the model provides a means to establish naturalistic artificial touch in bionic hands.mechanoreceptor | tactile afferent | somatosensory periphery | skin mechanics | computational model T he human hand is endowed with thousands of mechanoreceptors of different types distributed across the skin, each innervated by one or more large myelinated nerve fibers (1). These fibers convey detailed information about contact events and provide us with an exquisite sensitivity to the form and surface properties of grasped objects (2, 3). During object manipulation and tactile exploration, the glabrous skin of the hand undergoes complex spatiotemporal mechanical deformations, which in turn, drive very precise spiking responses in individual afferents. Coarse object features, such as edges and corners, are reflected in spatial patterns of activation in slowly adapting type I (SA1) and rapidly adapting (RA) fibers, which are densely packed in the fingertip (3, 4). At the same time, interactions with objects and surfaces elicit high-frequency, low-amplitude surface waves that propagate across the skin of the finger and palm and excite vibration-sensitive Pacinian (PC) afferents all over the hand (5-8).Recording the activity of tactile nerve fibers in monkeys or humans is technically difficult, is slow, and generally yields responses from a single unit at a time (9, 10). Although such recordings have yielded powerful insights into the neural basis of touch, they provide a limited window into the information that the hand c...

We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight onehalf that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay-Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.

As Mathieu moonshine is a special case of umbral moonshine, Thompson moonshine (in half-integral weight) is a special case of a family of similar relationships between finite groups and vector-valued modular forms of a certain kind. We call this penumbral moonshine. We

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