5.2 Angular function of the tilt aftereffect

The TAE testing procedure described in the previous section was used to determine the amount of aftereffect for each difference between adaptation and testing orientations in the model. Figure 5.2 plots the aftereffects after adaptation for 90 iterations of the RF-LISSOM algorithm. For comparison, figure 5.2 also shows the most detailed data available for the TAE in human foveal vision (Mitchell and Muir, 1976).

**Figure 5.2:** **Tilt aftereffect versus retinal angle.**
The open circles represent the average tilt aftereffect for a single human subject (DEM) from Mitchell and Muir (1976) over ten trials. For each angle in each trial, the subject adapted for three minutes on a sinusoidal grating of a given angle, then was tested for the effect on a horizontal grating. Error bars indicate ±1 standard error of measurement. The subject shown had the most complete data of the four in the study. All four showed very similar effects in the range ±40°; the indirect TAE for the larger angles varied widely in the range ±2.5°. The graph is roughly anti-symmetric around 0°, so the TAE is essentially the same in both directions relative to the adaptation line. For comparison, the heavy line shows the average magnitude of the tilt aftereffect in the RF-LISSOM model over nine trials, as described in section 5.1. Error bars indicate ±1 standard error of measurement. The network adapted to a vertical adaptation line at a particular position for 90 iterations, then the TAE was measured for test lines oriented at each angle. Positive values of aftereffect denote a counterclockwise change in the perceived orientation of the test line. The duration of adaptation was chosen so that the magnitude of the TAE matches the human data; as discussed in section 5.5 the shape of the curve is nearly constant, but the magnitude increases with adaptation. Learning rates were $\alpha_A=\alpha_E=\alpha_I=0.00005$ . The result from the model closely resembles the curve for humans at all angles, showing both direct and indirect tilt aftereffects.
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The results from the RF-LISSOM simulation are strikingly similar to the psychophysical results. For the range 5° to 40°, all subjects in the human study (including the one shown) exhibited angle repulsion effects nearly identical to those found in the RF-LISSOM model. The magnitude of the TAE increases very rapidly to a maximum angle repulsion at approximately 10°, falling off somewhat more gradually to zero as the angular separation increases. Section 5.4.2 examines the mechanisms responsible for these direct effects in RF-LISSOM.

The results for larger angular separations (from 45° to 85°) show a greater inter-subject variability in the psychophysical literature, but those found for the RF-LISSOM model are well within the range seen for human subjects. The indirect effects for the subject shown were typical for that study, although some subjects showed effects up to 2.5°. Section 5.4.3 examines the mechanisms responsible for these indirect effects in the RF-LISSOM model, and section 6.2.4 of the discussion proposes explanations for the variety of effects seen in human subjects.

The TAE seen in figure 5.2 must result from changes in the connection strengths between neurons, since there is no other component of the model which changes as adaptation progresses. In particular, there is no lasting change in the neuron's inherent excitability or sustained activation level. Thus there is nothing that could correspond to the concept of whole-cell neural fatigue.

Three sets of weights adapt synergetically: the afferent weights, the lateral excitatory weights, and the lateral inhibitory weights. The lateral inhibitory theory would predict that the inhibitory weights are primarily responsible for the TAE magnitude at each angle. To determine whether this is the case, the contribution of each of the weight types was evaluated independently of the others (figure 5.3).

**Figure 5.3:** **Components of the TAE due to each weight type.**
The heavy line shows the magnitude of the TAE for a single trial from the average shown in figure 5.2. This trial was at the center of the retina, and is typical of the effect seen at the other eight locations. The other curves in this figure show the contribution of adapting each weight type separately; other than the learning rates the parameters for each were identical. The thin dashed line represents the contribution from the afferent weights ( $\alpha_A=0.00005$ ; $\alpha_E=\alpha_I=0$ ). Adaptation of the afferent weights contributes in a direction opposite to that of the overall TAE curve. The dotted line represents the contribution from the lateral excitatory weights ( $\alpha_E=0.00005$ ; $\alpha_A=\alpha_I=0$ ). Note that the x-axis is not shown because it would have covered up this line. Adapting the lateral excitatory weights results in a curve with a similar shape as for afferent weights, but it is so small in magnitude that it is insignificant. The thick dashed line represents the contribution from the inhibitory weights ( $\alpha_I=0.00005$ ; $\alpha_A=\alpha_E=0$ ). The adaptation of the lateral inhibitory weights clearly determines the shape of the overall curve, though it is somewhat reduced in magnitude by the afferent contribution.
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While the excitatory connections are adapting, the inhibitory connections adapt as well. Each inhibitory connection adapts with the same learning rate as the excitatory connections ( $\alpha_I=\alpha_A=\alpha_E=0.00005$ ), but there are many more inhibitory connections than excitatory connections. The combined strength of all the small inhibitory changes outweighs the excitatory changes, and results in a curve with a sign opposite that of the components from the excitatory weights. If excitatory learning is turned off altogether, the magnitude of the TAE increases slightly on average, but the shape of the curve does not change significantly (figure 5.3). Similar results should be obtained if there are not as many inhibitory connections as used here, but the ones present change more rapidly than afferent connections; this would represent an alternative interpretation of the results.

Thus the inhibitory connections are clearly responsible for both the direct and indirect tilt aftereffects observed in the RF-LISSOM model. The following sections will examine exactly how changing the inhibitory weight strength produces the effects seen. To make the analysis clear and unambiguous, the simplest case that shows a realistic TAE was chosen. The discussion will center upon a single trial, using a Gaussian at the exact center of the retina, with only the inhibitory weights adapting. The TAE curve for that case is shown in figure 5.3. The curve is still quite similar to the average result when all weights were adapting at the same rate, as well as to the human data (figure 5.2). The analysis will show that Hebbian modification of the lateral inhibitory connection weights, followed by normalization of the total connection strength, systematically alters the response in a way that results in the tilt aftereffect.