Lecture 2:

SPATIAL REPRESENTATIONS AND TRANSFORMATIONS: A PRIMER

In order to talk about spatial transformations, we need a clear language. This topic has been reviewed in detail elsewhere [], so here will just define the most necessary terms.

Positions and movements are normally represented by scientists as vectors, defined in some coordinate frame. The latter incorporates two concepts: A /reference frame/ is some other rigid body, useful for describing the relative location or orientation of the body we want to represent. Normally, in our terrestrial environment one choses a frame that is either more stable or familiar. In the case of motor control, where muscles and tendons generally cause one segment to move relative to another, one generally chooses the more stable of the two (relative to the earth) as the frame. So for example, the head is a good frame for the eye, and the torso is a good frame for the head. This is not quite the same as a set of /coordinate axes/, which are a really a special set of vectors (each of one unit length, and often mutually orthogonal for convenience) chosen to describe the components of any other arbitrary vector. One last point: for locations and orientations, one needs a /reference position/, which corresponds to the zero vector in the associated coordinate system. Other positions are measured relative to this point.

To take a familiar example, the 2-D location of any point on earth (the frame) can be described as a set of angles (usually in degrees) latitude and longitude (the coordinate system) relative to the intersection of the equator and the prime meridian (the reference position).

Once one is clear on coordinate system being used, one can use this to define the components of a vector that may represent the amplitude and direction of some kinematic variable (i.e., related to position and motion). In terms of position, any unrestrained rigid body has six degrees of freedom, where three dimensions are required to describe its /location/ in space, and three are required to describe its /orientation/. Likewise, changes in location are called /translation/, whereas changes in orientation are called /rotation/. The distinction between location/translation and rotation/orientation is important because these two types of motion have very different mathematical properties. Vectors that represent translations can be added commutatively (in any order) to get the correct final result, whereas rotations are non-commutative: the order of operations gives different results, which also means one must account for initial orientation []. The latter point was first raised for systems neuroscientists in the context of 3-D eye control, but as we shall see here, it has broad, almost pernicious implications vision and motor control.

These formalizations are generally defined rigorously in physics or linear algebra, but they tend to be used loosely, and very often incorrectly, in neuroscience. Common examples include using ‘coordinate systems’ and ‘reference frames’ interchangeably (e.g. ‘head coordinates’), confusing reference frames with reference positions (they are not the same thing), and confusing the thing being represented with the coordinates used to represent it (e.g., poorly defined terms such as ‘hand movement coordinates’). Perhaps none of us in this field are without sin, but we shall try to use these terms correctly in the following review and amend some common misconceptions.

Finally, when we talk here about /transformations/, we refers to some change, either an operation within some coordinate system that generates a new representation from one or more inputs, or a transformation of the same representation into another coordinate system.

These terms are inexpendible tools if we wish to describe and model the spatial aspects of sensorimotor behavior. However, the notion that they are explicitly used in brain function is much more controversial []. We shall deal with some of these controversies as we proceed through the review. But just to be clear on one point; when we say here that the brain represents something, we simply mean that experimenters have established some useful correspondence between some event within the brain and an externally measurable variable, not that the brain is necessarily trying to represent something in the sense of algebra or imagery.