Transformations of neural representations are functions of variables that are represented by neural populations. Transformations are determined using an alternately weighted linear decoding (i.e., the transformational decoding asopposed to the representational decoding).

Quantitatively, we assume the same encoding as described in principle 1 and define the decoding:

$$\hat{f}(\mathbf{x}(t)) = \sum_i a_i (\mathbf{x}(t)) \phi_i^f$$

This decoding is the similar to that in principle 1, except the decoders are determined such that a function, $f$, of the original input signal is estimated.

The comments in principle 1 about representational decoders apply equally to transformational decoders. This should be no surprise given our prior discussion (in section 1.3; this and all subsequent section references are to the book) in which we noted that defining a transformation is just like defining a representation (although with different decoders). However, we did not previously emphasize the kinds of transformations that can be supported with linear decoding.

It has often been argued that nonlinear transformations are by far the most common kind of transformations found in neurobiological systems (see, e.g., Freeman 1987). This should not be surprising to engineers, as most real-world control problems require complex, nonlinear control analyses; a good contemporary example being the remote manipulator system on the international space station. This should be even less of a surprise to neuroscientists who study the subtle behavior of natural systems. As Pouget and Sejnowski (1997) note, even a relatively simple task, such as determining the head-centered coordinates of a target given retinal coordinates, requires nonlinear computation when considered fully (i.e., including the geometry of rotation in three dimensions). Thus, it is essential that we be able to account for nonlinear as well as linear transformations. In section 6.3 we discuss how to characterize nonlinear transformations in general. We provide a neurobiological example of a nonlinear transformation (determining the cross product) that allows us to account for a number of experimental results (see section 6.5). Thus we show that assumptions about the linearity of decoding do not limit the possible functions that can be supported by neurobiological systems.

This result will not be surprising to researchers familiar with current computational neuroscience. It has long been known that linear decoding of nonlinear basis functions can be used to approximate nonlinear functions (see section 7.4). Nevertheless, our analysis sheds new light on standard approaches. Specifically, we: 1) show how observations about neural systems can determine which nonlinear functions can be well-approximated by those systems (section 7.3); 2) apply these results to large-scale, fully spiking networks (section 6.5); and 3) integrate these results with a characterization of neural dynamics and representation (section 8.1.3).