The results presented above suggest that the same self-organizing principles that result in sparse coding and reduce redundant activation in the visual cortex may also be operating over short time intervals in the adult, with quantifiable psychological consequences such as the TAE. This finding demonstrates a potentially important computational link between development, structure, and function.
Even though the RF-LISSOM model was not originally developed as an explanation for the tilt aftereffect, it exhibits tilt aftereffects that have nearly all of the features of those measured in humans. The effect of varying angular separation between the test and adaptation lines is similar to human data, the time course is approximately logarithmic, and the TAE is localized to the retinal location that experienced the stimulus. With minor extensions, the model should account for other features of the TAE as well, such as higher variance at oblique orientations, frequency localization, movement direction specificity, and ocular transfer (Bednar, 1997).
One difference, however, shows up in prolonged adaptation: the TAE in humans eventually saturates near 4-5° (Greenlee and Magnussen, 1987; Magnussen and Johnsen, 1986; Wolfe and O'Connell, 1986). The TAE in the RF-LISSOM model does not saturate quite as quickly (figure 5), and it also appears to saturate for different reasons. In the model, saturation occurs only after the inhibitory weights have been strengthened so much that the cortical response to the adaptation line is entirely suppressed. However, in humans, the adaptation line is still easily detectable even after saturation (Magnussen and Johnsen, 1986). Apparently, the human visual system has additional constraints on how much adaptation can occur in the short term, and those constraints are currently not part of the model.
Another feature of the human TAE that does not directly follow from the model is the gradual recovery of accurate perception even in complete darkness (Greenlee and Magnussen, 1987; Magnussen and Johnsen, 1986). With the purely Hebbian learning used in the model, only active units are adapted, and thus no weight changes will occur for blank inputs. One possible explanation for both saturation and dark recovery is that the inhibitory weights modified during tilt adaptation are a set of small, temporary weights adding to or multiplying more permanent connections. Changes to these weights would be limited in magnitude, and the changes would gradually decay in the the absence of visual stimulation. Such a mechanism was proposed by von der Malsburg (1987) as an explanation of visual object segmentation, and was implemented in the RF-LISSOM model of segmentation by Choe and Miikkulainen (1998) and Miikkulainen et al. (1997). A variety of physical substrates for temporary modifications in efficacy have been found (reviewed in Zucker, 1989). Such effects could also be included in the TAE model, which would allow it to replicate the saturation and dark recovery aspects of the human TAE.
The main contribution of the RF-LISSOM model of the TAE is its novel explanation of the indirect aftereffect. Proponents of the lateral inhibitory theory of the TAE and tilt illusion have generally ignored indirect effects, or postulated that they occur only at higher cortical levels (Wenderoth and Johnstone, 1988; van der Zwan and Wenderoth, 1995), partly because it has not been clear how they could arise through inhibition in V1. However, recent theoretical models have suggested that indirect effects could occur as early as V1 for simultaneous stimuli (Mundel et al., 1997), and RF-LISSOM demonstrates that a quite simple, local mechanism in V1 is also sufficient to produce indirect aftereffects: If the total synaptic strength at each neuron is limited, bolstering the lateral inhibitory connections between active neurons eventually weakens their inactive inhibitory connections, causing the indirect aftereffect.
Such weight normalization is computationally necessary: weights governed by a pure Hebbian rule would otherwise increase indefinitely, or would have to reach a fixed maximum strength (Miller and MacKay, 1994; Rochester et al., 1956; von der Malsburg, 1973), and neither of these outcomes is biologically plausible. Furthermore, very recent experimental results demonstrate that several processes of normalization are actually occurring in visual cortex neurons near the time scales at which the indirect effect is seen (Turrigiano et al., 1994, 1998; Turrigiano, 1998). Such regulation changes the efficacy of each synapse while keeping their relative weights constant (Turrigiano et al., 1998), as in the RF-LISSOM model. The current RF-LISSOM results predict that similar whole-cell regulation of total synaptic strength underlies the indirect tilt aftereffect in V1.
The RF-LISSOM results may also help explain why there is relatively large inter-subject variability in the shape and magnitude of the angular function of the indirect effect (Mitchell and Muir, 1976). Although the overall fit in figure 4 is quite good, there is a small discrepancy at very large angles. This may be an interesting artifact of the visual environment of modern humans. As humans orient themselves with respect to manmade objects such as books, pictures, etc., they will often see angles near 90°, whereas the model was trained with only single lines. If the model were trained on more realistic visual inputs that included corners, crosses, and approximately orthogonal angles, it would develop more connections between neurons with orthogonal preferences. These connections would have an inhibitory effect, and would decrease through normalization during adaptation. As a result, the model would show an increased indirect effect near ±90°, matching the human data. The prediction is, therefore, that the indirect effect differences between subjects arise from differences in their long-term visual experience, due to factors such as attention and different environments.
Through mechanisms similar to those causing the TAE, the RF-LISSOM model should also be able to explain tilt illusions between two stimuli presented simultaneously. Such an explanation was originally proposed by Carpenter and Blakemore (1973), and the principles have been demonstrated recently in an abstract model of orientation (Mundel et al., 1997). Testing this hypothesis in RF-LISSOM for spatially-separated stimuli would require a much larger inhibitory connection radius than used for these experiments; such simulations are not yet practical because of computational constraints. Bayesian analysis may offer a way to extract the response to two simultaneously-presented overlapping lines (Zemel et al., 1998), which could allow the tilt illusion to be measured in the current model. The tilt illusion can also be tested with existing computing hardware by first combining the current orientation map simulation with an ocular dominance simulation (such as that of Sirosh and Miikkulainen, 1997). Then perceived orientations can be computed for inputs in different eyes; if the tilt illusion is occurring then the perceived orientation for an input in one eye will change when a different orientation is presented to the other eye (as found in humans, Carpenter and Blakemore, 1973).
In addition, many similar phenomena such as aftereffects of curvature, motion, spatial frequency, size, position, and color have been documented in humans (Barlow, 1990). Since specific detectors for most of these features have been found in the cortex, RF-LISSOM should be able to account for them by the same process of decorrelation mediated by self-organizing lateral connections.