Perceptual Coherence : Hearing and Seeing

Characteristics of Auditory and Visual Scenes 135
nervous system is organized. These outputs, when combined, would optimally
reproduce the original image.
Hyvärinen and Hoyer (2001) followed this strategy to derive the receptive
spatial fields and spatial arrangement of the complex cells found in the
primary visual cortex V1. Simple cells will increase their firing rate above
baseline if a white band falls on the excitation region and a black band falls
on the inhibition region but reduce their firing rate below baseline for the reverse
arrangement. Complex cells, in contrast, will increase their firing rate
for either arrangement; they are phase insensitive. Intuitively, the simple
cells feed forward to the complex cells such that either an increase or decrease
in firing rate of simple cells increases the firing rate of the complex
cell. Hyvärinen and Hoyer (2001) modeled this process by assuming that (1)
the outputs of the simple cells (the difference between the firing rate and the
baseline-firing rate) are squared so that all changes in the light pattern due to
movements of edges increase the firing rate, and (2) the outputs from nearby
simple cells (a 5 × 5 grid of cells) converge on every complex cell. They
then use independent component analysis to maximize the sparseness or independence
of the complex cell responses. To maximize the sparseness of
the complex cells means that the simple cells that converge on a complex
cell should have as similar a filter response as possible. The authors argued
that if the responses of the convergent simple cells were independent, then
the response of the complex cell would be more normal and less peaky (an
outcome predicted from the central limit theorem).
The results can be considered in terms of the simple cells and in terms
of how the activations of the simple cells create the complex cells. Each
retinal region is represented by a set of adjacent overlapping simple cells.
The receptive fields of the simple cells in figure 3.11 differ in retinal location
and orientation of the brightness contrast, and respond to a narrow
range of spatial frequencies. The topographic map of each set of simple
cells shows that the cells are smoothly arranged spatially in terms of orientation
and position. The majority of the receptive fields are simply scaled
versions of another field. Thus the receptive fields possess differing spatial
and temporal scales that create the multiresolution characteristics of perceptual
systems described in chapter 1. The simple cells that are pooled to
create complex cells have similar receptive fields.
The results of Hyvärinen and Hoyer (2001) provide a model to explain
the retinal topographic organization of the visual cortex. Starting with a
simple feed-forward model in which simple cells converge on complex
cells and invoking a criterion of maximal independence creates the spatial
arrangement of receptive fields found in all species.
Körding, Käyser, Einhouser, and König (2004) followed a different route
to derive the properties of complex cells. They argued that nearly all visual
objects and higher-level variables change slowly or not at all over time, even