Computational models can play a fundamental role in understanding the development and function of complex systems such as the cortex. With the introduction of massively parallel computers in the last five years, it has become possible to simulate large numbers of neural units and their connections. At the same time, neurobiological techniques for mapping the response properties and connectivity of neurons have become sophisticated enough to constrain and validate such models. This provides a timely opportunity to test the inhibition theory of the TAE through large-scale computational experiments. By iteratively making and testing hypotheses about which cortical features are responsible for which behaviors, one can extend and revise a computational model so that it can predict cortical behavior in detail. At each step of the way, the model itself can provide predictions for verification or refutation in human and animal subjects. The overall goal is to represent the essential computational and organizational principles of the cortex in as simple a model as possible.
To test the inhibition theory, a model of the orientation detectors is needed. Several computational models have shown how receptive fields (such as selectivity to different orientations) and their global organization in the cortical network can develop through Hebbian self-organization of afferent synapses (Erwin et al., 1995; Goodhill, 1993; Kohonen, 1982; Miller et al., 1989; Miller, 1994; Obermayer et al., 1990; von der Malsburg, 1973). These models have not taken the lateral interactions between cells into account, or have assumed that they are preset and fixed and have a regular profile. Thus they are unsuitable for testing the inhibitory theory of tilt aftereffects, since it depends upon modifiable connections between specific orientation detectors.
Of the models that have been suitable for examining tilt aftereffects, none have successfully accounted for both direct and indirect effects in a biologically realistic manner. The virtual axis/lateral inhibition theory of Wenderoth et al. (1989) has been implemented in a very simple computational model by Spivey-Knowlton (1993). He found that the model could produce an S-shaped curve for the angular function of the tilt illusion; the curve was a reasonably good match to human data. Presumably a similar model could be used for the tilt aftereffect, but this has not yet been implemented. However, since the virtual axis theory was explicitly developed as an explanation for the indirect tilt illusion data, demonstrating it in a computational model only serves to verify that the theory is internally consistent, rather than supporting or explaining it. The model to be described in this thesis, in contrast, was developed primarily as a model of self-organization of cortical structures, and exhibits tilt aftereffects only as an emergent phenomenon.
Another mathematical model for neural development by Dong (1994, 1996) has also been shown to exhibit the tilt illusion and tilt aftereffect. The model is a formal expression of the information processing principle of decorrelation, that is, the reduction of redundancy in an image or other sensory data. All natural images are redundant to some extent, and the nervous system appears to make wide use of this fact (Barlow, 1990). The amount of redundancy may be expressed in terms of correlations: two pieces of information that are highly correlated with each other could in theory be represented as a single entity, and thus they contain redundant information. A network or algorithm that compacts the image by removing such redundancies is said to decorrelate. After full decorrelation, an input image would consist of purely white noise, where every bit of information is independent of each of the others. Dong shows that a network that satisfies a simple approximation to full decorrelation will exhibit both direct and indirect tilt aftereffects for Gaussian-shaped inputs. The model predicts an S-shaped curve that is a reasonably good approximation to the average human TAE data from Campbell and Maffei (1971). Dong's theory may be seen as a mathematical expression of some of the principles also operating in the RF-LISSOM model; it is complementary to the more detailed column-level description of actual neural behavior presented in later chapters of this thesis.
Finally, Wilson and Humanski (1993) proposed a linear differential equation model of a cortical gain control mechanism that exhibits direct tilt aftereffects. Their model is a proposed circuit for achieving contrast independence via a divisive inhibitory gain control. The model had been shown to predict contrast sensitivity before and after adaptation, and with the same parameters it was found to exhibit direct tilt effects that have an angular function somewhat similar to the measured direct effects for humans. The same type of contrast gain control is exhibited by RF-LISSOM (Sirosh, 1995), so one can consider their model to represent one part of the cortical processes modeled by RF-LISSOM. However, they do not discuss indirect effects, and it is difficult to see how those would occur in their model. Thus this model, like the others, does not fully account for both direct and indirect tilt aftereffects in terms of actual known cortical structure.