In this letter, I have extended the generality of previous results [Georgopoulos et al. 1988,Mussa-Ivaldi 1988] to show that cosine tuning curves will be found for large classes of arbitrary response functions if they are analyzed according to the statistical techniques in [Schwartz et al. 1988], and that the existence of a population vector as found in [Georgopoulos et al. 1988] is determined by very general necessary and sufficient conditions which depend only on the distribution of preferred directions rather than on any intrinsically coded coordinate system. The concept that a distributed representation codes a particular coordinate system may not be well-defined, since certain types of representations can be considered ``coordinate-free''. These considerations imply that experiments of the type described may yield population vectors which predict many different 3-dimensional coordinates (such as Cartesian, polar, muscle lengths, or joint angles).
It is important to understand that the considerations presented here in no way reduce the importance of the results reported in [Schwartz et al. 1988,Georgopoulos et al. 1988,Kettner et al. 1988,Caminiti et al. 1990] and elsewhere. The fact that results R2, R4, and R5 are direct consequences of R1 and R3 only serves to underscore the significance of these two results. They show that large populations of motor cortical cells respond to hand motion in a predictable way, and that the preferred directions are approximately uniformly distributed with respect to a Cartesian representation of extrapersonal space. No additional conclusions can be drawn from the population vector, since its existence is a mathematical consequence of these two facts. However, if the distribution of preferred directions is nonuniform with respect to other coordinate systems or if the distribution can be modified through experience, then this would provide significant information about cortical representations. In addition, if cosine tuning can be verified by explicitly fitting cell tuning curves to a linear regression model, then further studies may discover constraints which explain why more than 486 linear cells are needed to code for only 3 linearly independent components of hand direction.