In his article "The Problem of the Interrelation of Co-ordination and Lateralization," Bernstein (1935) drew attention to invariance in the shape of drawings made under very different conditions, such as using different effectors or different combinations of muscles and joints. He called these invariances "topological" and contrasted them with other "metric" aspects of movement, such as size and location in space. He then speculated on the brain representation of motor topology, as follows:

There is the deeply seated inherent indifference of the motor control centre to the scale and position of the movement effected. ... it is clear that each of the variations of a movement (for example, drawing a circle large or small...) demands a quite different muscular formula; and even more than this, involves a completely different set of muscles in the action. The almost equal facility and accuracy with which all these variations can be performed is evidence for the fact that they are ultimately determined by one and the same higher directional engram in relation to which dimensions and position play a secondary role." (Bernstein, 1935; see Whiting, 1984, p. 109)

This insight of Bernstein's, from 45 years ago, into the "directional engram" is remarkable. In a way, our work during the past 20 years on the neural coding of motor direction can be seen as addressing precisely this issue (i.e., the extraction of directional information from the impulse activity of single cells and cell populations in cortical areas). The discovery of directional tuning provided the key link between neural activity and the direction of movement and made possible the neural construction of a motor trajectory in space (Georgopoulos et al., 1988; Schwartz, 1994). This "neural trajectory" proved to be an accurate and isomorphic representation of the actual motor trajectory. Remarkably, this was also predicted by Bernstein (1935), who state that "the higher engram ... is extremely geometrical, representing a very abstract motor image of space" (Whiting, 1984, p. 109). Indeed, space is pervasive in figure drawing. Unlike relatively pure temporal functions, such as tapping, figure drawing cannot be conceived apart from the spatial relations connecting the elements of the figure. Therefore, the geometric aspects of the shape are of fundamental importance for its drawing.In this chapter, I review the results of recent studies that have dealt with issues at the heart of Bernstein's concerns: motor topology and the neural representation of direction, size, and location of movement in space. Specifically, I discuss the results of behavioral studies of drawing geometrical figures and then turn to the issue of representation of topological features and how these can be invariantly extracted from neuronal populations irrespective of attributes of size, location, and the muscles effecting the motor trajectory. Finally, I discuss applications of such neurally inspired operations to artificial neural networks trained to draw figures.