3 Experiments

The model consisted of an array of 192 × 192 neurons, and a retina of 36 × 36 ganglion cells. The center of the anatomical receptive field of each neuron was placed at the location in the central 24 × 24 portion of the retina corresponding to the location of the neuron in the cortex, so that every neuron would have a complete set of afferent connections. The RF had a circular shape, consisting of random-strength connections to all ganglion cells within 6 units from the RF center. The cortex was self-organized for 20,000 iterations on oriented Gaussian inputs with major and minor axes of half-width $\sigma=7.5$and 1.5 , respectively. The training took 2.5 hours on 16 processors of a Cray T3E supercomputer at the Texas Advanced Computing Center. The model requires 1.5 gigabytes of physical memory to represent the 400 million connections in this small section of the cortex.

3.1 Orientation map organization

In the self-organization process, the neurons developed oriented receptive fields organized into orientation columns very similar to those observed in the primary visual cortex (figure 3b).

 

Figure 3:  Orientation map activation.

The strongest lateral connections of highly-tuned cells link areas of similar orientation preference, and avoid neurons with the orthogonal orientation preference. Furthermore, the connection patterns of highly oriented neurons are typically elongated along the direction in the map that corresponds to the neuron's preferred stimulus orientation (as subsequently found experimentally by Bosking et al. 1997.) This organization reflects the activity correlations caused by the elongated Gaussian input pattern: such a stimulus activates primarily those neurons that are tuned to the same orientation as the stimulus, and located along its length (Sirosh et al., 1996). Since the long-range lateral connections are inhibitory, the net result is decorrelation : redundant activation is removed, resulting in a sparse representation of the novel features of each input (Barlow, 1990; Field, 1994; Sirosh et al., 1996). As a side effect, illusions and aftereffects may sometimes occur, as will be shown below.

3.2 Aftereffect simulations

In psychophysical measurements of the TAE, a fixed stimulus is presented at a particular location on the retina. To simulate these conditions in the RF-LISSOM model, the position and angle of the inputs were fixed to a single value for a number of iterations. To permit more detailed analysis of behavior at short time scales, the learning rates were reduced from those used during self-organization to $\alpha_A=\alpha_E=\alpha_I=$0.000005. All other parameters remained as in self-organization.

To compare with the psychophysical experiments, it is necessary to determine what orientation the model ``perceives'' for any given input. Precisely how neural responses are interpreted for perception remains quite controversial (see Parker and Newsome 1998 for review), but results in primates suggest that behavioral performance approaches the statistical optimum given the measured properties of cortical neurons (Geisler and Albrecht, 1997). Accordingly, we extracted the perceived orientation using a vector sum procedure, which has been shown to be optimal under conditions present in the model (Snippe, 1996). For this procedure, each active neuron was represented by a vector whose magnitude corresponded to the activation level and whose direction corresponded to the orientation preference of the neuron. Perceived orientation was then measured as a vector sum over all neurons that responded to the input. In the model, the perceived orientation was found to match the absolute orientation of the input pattern to within a few degrees (Bednar, 1997).

To determine the tilt aftereffect in the model, the perceived orientation was first computed for inputs of all orientations. The model then adapted to a fixed input for 90 iterations, and the perceived orientation was again computed for all inputs. The difference between the initial perceived angle and the one perceived after adaptation was taken as the magnitude of the TAE. Figure 4 plots these differences for the different angles. For comparison, figure 4 also shows the most detailed data available for the TAE in human foveal vision (Mitchell and Muir, 1976).

 

Figure 4:  Tilt aftereffect at different angles.

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 exhibited angle repulsion effects nearly identical to those found in the RF-LISSOM model; the data was most complete for the subject shown. The magnitude of this direct TAE increases very rapidly to a maximum angle repulsion at 8-10°, falling off somewhat more gradually to zero as the angular separation increases. The simulations with larger angular separations (from 40° to 85°) show a smaller angle attraction, i.e. the indirect effect. Although there is a greater inter-subject variability in the psychophysical literature for the indirect effect than the direct effect, those found for the RF-LISSOM model are well within the range seen for human subjects.

In parallel with the angular changes in the TAE, its peak magnitude in humans increases logarithmically with adaptation time (Gibson and Radner, 1937), eventually saturating at a level that depends upon the experimental protocol used (Greenlee and Magnussen, 1987; Magnussen and Johnsen, 1986). As the number of adaptation iterations is increased in the model, the magnitude of the TAE also increases logarithmically (figure 5). (The single curve that best matched the magnitude of the human data was shown in figure 4, but the ones for different amounts of adaptation all had the same basic shape.) The time course of the TAE in the RF-LISSOM model is qualitatively similar to the human data, but it does not completely saturate over the adaptation amounts tested so far.

 

Figure 5:  Direct TAE over time.

3.3 How does the TAE arise in the model?

The TAE seen in figure 4 must result from changes in the connection strengths between neurons, since no other component of the model changes as adaptation progresses. Simulations performed with only one type of weight adapting (either afferent, lateral excitatory, or lateral inhibitory) show that the inhibitory weight changes determine the shape of the curve for all angles (Bednar, 1997).

In what way do the changing inhibitory connections cause these effects? We will demonstrate this process in a simulation where only inhibitory weights adapted, the adaptation period was longer, and a higher learning rate was used, all in order to exaggerate the effect and make its causes more clearly visible. First of all, the connections change in a way that leaves the perceived orientation of the adaptation line unchanged. Figure 6a

 

Figure 6:  Explanation of the TAE.

shows the initial response (I), the settled response (S), and the settled response after adaptation (A) for a vertical input (0°, marked with a vertical line.) The response after adaptation is more diffuse because the most active neurons have become inhibited (equation 3). Throughout adaptation, the distribution of active orientation detectors is centered around approximately the same angle, and a constant angle is perceived.

The initial and settled responses to a test line with a slightly different orientation (e.g. 10°, marked with a dotted line in figure 6b ) are again centered around that orientation (6b I and S). However, comparing the histograms of settled activity before and after adaptation (6b S and A), it is clear that fewer neurons close to 0° respond after adaptation, but an increased number of those representing distant angles (over 10°) do. This is because during adaptation, inhibition was strengthened primarily between neurons close to the 0° adaptation angle, and not between those that prefer larger orientations.

Such adaptation increases the ability of the map to detect small differences from the adaptation line. Before adaptation, the settled histograms for 0° and 10° are fairly similar, with averages differing by only 9.2° in this simulation (6a S and 6b S). After adaptation, the histograms are very different, resulting in a 23° difference in perceived orientation (compare 6a A with 6b A). This adaptation is manifested at the psychological level as a shift of the perceived orientation away from the adaptation line, that is, the direct TAE.

Meanwhile, the response to a very different test line (e.g. 50°, the dotted line in figure 6c ) becomes broader and stronger after adaptation (compare 6c S and A). Adaptation occurred only in activated neurons, so neurons with orientation preferences greater than 50° are unchanged. However, those with preferences less than 50° actually now respond more strongly (6c S and A). The reason is that during adaptation, the inhibitory connections of these neurons with each other became stronger. Because of normalization (equation 3), their connections to other neurons, i.e. those representing distant angles such as 50°, became weaker. As a result, the 50° line now inhibits them less than before adaptation. Thus they are more active, and the perceived orientation shifts towards 0°, causing the indirect tilt aftereffect.

The indirect effect is therefore true to its name, caused indirectly by the strengthening of inhibitory connections during adaptation. This explanation of the indirect effect is novel, and emerges automatically from the RF-LISSOM model. The model thus shows computationally how both the direct and indirect effects can be caused by the same activity-dependent adaptation process, and that it is the same process that drives the development of the map.

Footnotes

...respectively.

The initial lateral excitation radius was 19 and was gradually decreased to 1 . The lateral inhibitory radius of each neuron was 47 , and inhibitory connections whose strength was below 0.00005 were pruned away at 20,000 iterations. The lateral inhibitory connections were initialized to a Gaussian profile with $\sigma=100$, and the lateral excitatory connections to a Gaussian with $\sigma=15$, with no connections outside the nominal circular radius. The lateral excitation $\gamma_e$ and inhibition strength $\gamma_i$ were both 0.9 . The learning rate $\alpha_{\rm A}$ was gradually decreased from 0.007 to 0.0015 , $\alpha_{\rm E}$ from 0.002 to 0.001 and $\alpha_{\rm I}$ was a constant 0.00025 . The lower and upper thresholds of the sigmoid were increased from 0.1 to 0.24 and from 0.65 to 0.88 , respectively. The number of iterations for which the lateral connections were allowed to settle at each training iteration was initially 9 , and was increased to 13 over the course of training. These parameter settings were used by Sirosh et al. to model development of the orientation map, and were not tuned or tweaked for the tilt aftereffect simulations. Small variations produce roughly equivalent results (Sirosh, 1995).