Introduction

Recent results suggest that the representation of arm movement in motor cortex involves the simultaneous activity of many cells, and that the pattern of activation over the group of cells specifies the motion which occurs [Caminiti et al. 1990,Kalaska and Crammond 1992, for review,]. These results have led many researchers to ask whether movement variables are coded internally in terms of a particular coordinate system such as a Cartesian or polar representation of extrapersonal space, a representation of muscle lengths around relevant joints, or some other set of coordinates. In the following, I distinguish between a coded variable (such as hand position) and the coordinates used to represent that variable (such as Cartesian coordinates). I will summarize certain experiments which demonstrate that hand movement direction is represented within motor cortex, but I claim that these experiments cannot be used to determine the coordinate system in which movements are coded.

I discuss a set of experiments that investigated the relationship between cell firing rates during free arm movements in awake monkeys, and the direction in which the hand was moved to a target in space [Georgopoulos et al. 1988,Kettner et al. 1988,Schwartz et al. 1988]. These experiments led to the following results:

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R1:

The firing rate of 89.1% (486/568) of the cells tested in motor cortex varied consistently with the direction of hand motion within a limited region of space, and many cells would be simultaneously active for any given direction.  

R2:

A statistically significant component of the variance of the firing rate of 83.6% of the cells could be accounted for by a broadly-tuned function of the form

 

where is the firing rate of cell i for hand motion in the direction of a unit vector M, is the direction of motion in which the cell has maximal response, is the angle between the direction of hand motion and the cell's preferred direction, and and determine the average firing rate and modulation depth, respectively. (Here and in the following, capital letters indicate vector quantities.)  

R3:

The preferred directions are approximately uniformly distributed with respect to directions in the workspace.  

R4:

The hand direction vector M can be approximated in Cartesian coordinates by a population vector P computed from a linear combination of the cell firing rates.  

R5:

The coefficients of this linear combination are given by unit vectors along the preferred direction for each cell, so that

 

where the have been normalized to account for resting firing rate and response amplitude, and both the movement direction M and the preferred direction vectors are given in Cartesian coordinates with respect to the external workspace.  

I will show that results R2, R4, and R5 are direct consequences of results R1, R3, and the experimental design. This in no way reduces the importance of these experiments, but rather emphasizes the fact that results R1 and R3 contain the most significant information. Although their importance was recognized in [Georgopoulos et al. 1988], the fact that they imply the other results was not. Previous investigations have studied the conditions under which results R4 and R5 hold, and it has been shown that the population vector predicts the direction of hand motion if the tuning curve is symmetric and the distribution of preferred directions is uniform [Georgopoulos et al. 1988] or has no second harmonic components [Mussa-Ivaldi 1988]. I derive a necessary and sufficient condition which is even broader, since it requires only that the three components of the preferred directions be uncorrelated with each other over the population. Before doing this, I will first show that the cosine tuning curves found in [Schwartz et al. 1988] may be an artifact of the analytic techniques used.