Ординатура / Офтальмология / Английские материалы / Computational Maps in the Visual Cortex_Miikkulainen_2005

182 8 HLISSOM: A Hierarchical Model

Fig. 8.3. Effect of afferent normalization on V1 neuron tuning. The differences in V1 population activities shown in Figure 8.2 are due to changes in how individual neurons respond at different contrasts. These plots show orientation tuning curves of the neuron at the center of the cortex, which prefers stimuli oriented at 60◦. Each curve shows the peak settled responses of this neuron to sine gratings whose orientations are indicated in the x-axis and contrast specified in the legend at right. In each case, the sine grating phase was used that resulted in the largest response. (a) Without afferent normalization, the neuron becomes less selective for orientation as contrast increases. Given enough contrast (above 50%), the neuron responds at full strength to inputs of all orientations, and thus no longer provides information about the input orientation. (b) With normalization, the tuning curve is the same over a wide range of contrasts, allowing the neuron to respond only to inputs that match its orientation preference. The curves are similar at 20% contrast (solid line), but the neuron now responds selectively to other contrasts as well. Afferent normalization is therefore crucial for preserving orientation selectivity over a wide range of contrasts.

to demonstrate which essential features of the input are responsible for orientation, ocular dominance, and direction selectivity, each simulation used only a single type of input. In contrast, with HLISSOM the goal is to understand the how internally generated inputs and environmental inputs together influence self-organization, and thus HLISSOM simulations include the input activity patterns thought to occur at each developmental age.

Internally generated activity has been observed in several locations in the developing visual system (Section 2.3). Given the current evidence, retinal waves are the most likely source for prenatal self-organization of V1 orientation maps (as will be discussed in Section 16.2.1). One such pattern is reproduced in Figure 8.4a, showing activity in retinal cells of the ferret before photoreceptors have developed. Although their precise origin still needs to be determined, retinal waves activate the ON and OFF neurons differently (Section 16.2.1; Myhr, Lukasiewicz, and Wong 2001). Such patterns can be modeled as “noisy disks”, i.e. large active areas (modeling ON channel activation) and large inactive areas (modeling OFF channel activation) with oriented edges in a noisy background (Figure 8.4b). These patterns will be used to

184 8 HLISSOM: A Hierarchical Model

dark outlines of the two eyes and the nose and mouth area, could explain why human newborns are drawn to facelike visual inputs. Like retinal waves, PGO waves are not the only possible source for such patterns, but they are the most likely cause for prenatal self-organization of higher levels (Section 16.2.2).

Postnatal training, on the other hand, is based on natural visual inputs in the retina, including photographic images of natural objects, landscapes, and faces (Figure 8.4d–f ). Each of these datasets has a slightly different distribution of orientations: Whereas objects have more horizontal and vertical edges than other orientations, landscapes are predominantly horizontal and faces mostly vertical. Compared with the prenatal PGO patterns, the face images include strong outlines and more detailed internal features. While it is difficult to obtain a realistic set of such patterns that would match the experience of an infant (as will be discussed in Section 17.3.3), it is possible to demonstrate what effects the variations of such patterns might have. In several experiments in Chapters 9 and 10, variations of these patterns as well as the prenatal ones will be tested to evaluate how strongly the developmental process depends on specific input features.

The ON and OFF sheets of the LGN in HLISSOM are identical to those in LISSOM, with DoG receptive fields (Equation 4.1) that filter out large, uniformly bright or dark areas, leaving only edges and lines (through Equation 4.3). The cortical sheets, i.e. V1 and the FSA, are similar to V1 in LISSOM. Each consists of initially unselective, laterally connected units that become selective through learning. V1 receives input from the ON and OFF cells of the LGN, while the FSA receives input from the laterally settled response of V1. The mapping from V1 to the FSA is constructed just like the mapping from the LGN to V1 in Figure A.1(a), so that no FSA neuron has a receptive field that is cropped by the border of V1.

HLISSOM simulations with large natural images generally start with an initial normalization strength γn of zero, because neurons are initially unselective. As neurons become selective over the course of training, γn is gradually increased. To prevent the net responses from decreasing, the scaling factor γA is set manually to compensate for each change to γn. The goal is to ensure that the afferent response ζ will continue to have values in the full range [0..1] for typical input patterns, regardless of the γn value. At the same time, γn ensures that the cortex responds to all areas of the input, not just the areas with the highest contrast.

After the afferent normalization, the initial cortical response is calculated from the afferent response using a sigmoid activation function (Equations 4.4–4.6). Activation then settles due to the lateral connections (Equation 4.7), and each weight is updated as in LISSOM. Through this process, HLISSOM develops realistic ordered maps, receptive fields, and lateral connections.

8.4 Effect of Input Sequence and Initial Organization

Before analyzing how the different training sequences affect self-organization in HLISSOM, it is necessary to verify that the resulting organization is indeed primarily determined by the inputs, and not by the initial random state of the network. This

hypothesis is experimentally verified in this section using HLISSOM, but it applies to all LISSOM models and also to other similar self-organizing models. There are two types of variability between runs with different random numbers: the random order, location, and position of the individual input patterns at each iteration, and the random initial values of the connection weights. Each one can be independently varied while the other is kept constant, and the resulting differences can be observed.

A series of orientation map simulations similar to those in Section 5.3 were run in this way (Figure 8.5; Appendix C.1). The results demonstrate that the map shape does not depend on the random initial values of the weights, as long as the initial weights are drawn from the same random distribution. This observation is consistent with those on the SOM model (Cottrell, de Bodt, and Verleysen 2001), confirming that self-organization is less sensitive to initial conditions than e.g. backpropagation learning (Kolen and Pollack 1990). Instead, the self-organized orientation map pattern in HLISSOM depends crucially on the stream of inputs. Two different streams lead to different orientation maps, even if the streams are drawn from the same distribution. The overall properties of the maps (such as the distance between orientation patches, the number of pinwheels, etc.) are very similar, but different input streams lead to different arrangements of patches and pinwheels.

HLISSOM is insensitive to initial weights for three reasons, all of which are common properties of most incremental Hebbian models. First, because input patterns vary smoothly, the receptive fields are relatively large, and early in self-organization the afferent weights are uniformly random, the initial scalar product responses between the input and weight vectors (Equation 4.5) are similar regardless of the specific weight values. Second, because these responses settle through lateral excitation (Equation 4.7), the final activity levels are even more similar. Third, with a high enough learning rate, the initial weight values are soon overwritten by the Hebbian learning based on the final responses (Equation 4.8). The net result is that as long as the initial weights are generated from the same distribution, their precise values do not significantly affect map organization. Similar invariance to the initial weights should be found in other Hebbian models that compute the scalar product of the input and a weight vector, particularly if they include lateral excitation and use a high learning rate in the beginning of self-organization.

In animals, maps are also similar between members of the same species, but they differ in the specific arrangements of orientation patches and pinwheels (Blasdel 1992b). The HLISSOM model predicts that the specific orientation map pattern in the adult animal depends primarily on the order and type of activity seen by the cortex in early development, and not on the details of the initial connectivity. This result also means that it is very important to study how different input streams affect the self-organization process, as will be done in the next two chapters.

8.5 Conclusion

The HLISSOM model includes the retina and a brainstem pattern generator, the LGN (both ON and OFF channels), V1, and a higher level face-selective region. It can be

190 9 Understanding Low-Level Development: Orientation Maps

This result allows asking the next question: How are such internal inputs combined with external ones during development? None of the map models discussed in Section 5.2 has yet demonstrated how the orientation map can smoothly integrate information from these two sources. A number of the models have simulated spontaneously generated activity (e.g. Burger and Lang 1999; Linsker 1986a,b,c; Mayer, Herrmann, and Geisel 2001; Miller 1994; Piepenbrock, Ritter, and Obermayer 1996), and a few models have shown self-organization based on natural images (as reviewed in Section 5.2). Yet, to our knowledge, the only orientation map model to be tested on a prenatal phase with spontaneous patterns followed by a postnatal phase is the Burger and Lang (1999) model. They found that if a map organized based on uniformly random noise was subsequently trained on natural images (actually, patches from a single natural image), the initial structure was soon overwritten. As was discussed in Section 8.1, this is a curious result because animal maps instead maintain the same overall structure during postnatal development.

Section 9.3 will demonstrate how the HLISSOM prenatal map is smoothly refined in postnatal training with natural images. The prenatal map organization is not very different from that of a na¨ıve network, i.e. an initially random map trained only with natural images. Therefore, postnatal training with natural images will only locally adjust the map, not replace it with something else. In this way HLISSOM will show how internal inputs and natural images can interact to construct realistic orientation maps, an important finding that has not been explained by previous models.

After demonstrating that prenatal and postnatal training together can account for the experimental data, the question is: Why are there two phases? There is indirect evidence that the visual cortex keeps getting trained with internally generated inputs at least for several weeks after birth (Crair et al. 1998). Would it be possible to obtain the refined adult orientation map in such continued training with internal inputs? Conversely, is the prenatal phase necessary, or would an adult map form just as well through postnatal training with natural images only?

Further simulations in Section 9.4 demonstrate that accurate adult maps can be obtained with internally generated patterns alone, and with natural images alone. However, there are good reasons why both phases exist: Prenatal training is an advantage because it allows the animal to have a functional visual system already at birth, and its further development will be more robust. Postnatal adaptation, on the other hand, allows it to form an accurate representation of the environment that it actually encounters during its life. Therefore, both phases serve a distinctly different role in constructing the visual system. How these two processes could continue interacting throughout the animal’s life, balancing the need to adapt to the environment and the need to maintain stable visual abilities, is in important further research question, discussed in Section 17.2.4.

9.2 Prenatal Development

In this section, HLISSOM is trained with internally generated patterns to develop a rudimentary orientation map similar to e.g. those of newborn kittens. Maps trained