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Although the orientation of an arm in space or the static view of an object may be represented by a population of neurons in complex ways, how these variables change with movement often follows simple linear rules, reflecting the underlying geometric constraints in the physical world. A theoretical analysis is presented for how such constraints affect the average firing rates of sensory and motor neurons during natural movements with low degrees of freedom, such as a limb movement and rigid object motion. When applied to nonrigid reaching arm movements, the linear theory accounts for cosine directional tuning with linear speed modulation, predicts a curl-free spatial distribution of preferred directions, and also explains why the instantaneous motion of the hand can be recovered from the neural population activity. For three-dimensional motion of a rigid object, the theory predicts that, to a first approximation, the response of a sensory neuron should have a preferred translational direction and a preferred rotation axis in space, both with cosine tuning functions modulated multiplicatively by speed and angular speed, respectively. Some known tuning properties of motion-sensitive neurons follow as special cases. Acceleration tuning and nonlinear speed modulation are considered in an extension of the linear theory. This general approach provides a principled method to derive mechanism-insensitive neuronal properties by exploiting the inherently low dimensionality of natural movements.

A Theory of Geometric Constraints on Neural Activity for Natural Three-Dimensional Movement