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Self-organization

The extent of receptive fields and the degree of initial order in undeveloped cortical networks have not been characterized in great detail. There are three possibilities:

  1. Cortical neurons receive afferents from local but overlapping receptive fields on the retina, and the receptive field centers are topographically ordered (i.e. the centers are randomly scattered around their ordered positions within a small enough radius of so that the centers of neighboring neurons do not cross).

  2. The receptive fields are local and overlapping, but their centers are scattered within a larger radius, causing centers of neighboring neurons to sometimes cross.

  3. The receptive fields are extremely large, so that all cortical neurons receive inputs from almost the entire retina. gif

The plausibility of these hypotheses can be tested computationally using networks with (1) topographically ordered RF centers, (2) more widely scattered nontopographic RF centers, and (3) networks where all neurons are connected to all receptors. In each case, all synaptic weights are initially random.

The three networks were simulated with Gaussian spots of ``light'' on the retina as input. At each presentation, the activation at the receptor was given by:

 

where n is the number of spots, specifies the width of the Gaussian, and the spot centers (,): , were chosen randomly.

When the networks were trained with single light spots (n=1), similar afferent and lateral connection structures developed in all three cases. With more realistic input consisting of multiple light spots (n>1), the first and the third network self-organized just as robustly. Figures 2-4 illustrate the self-organization of the first network. The initial rough pattern of afferent weights of each neuron evolved into a hill-shaped profile (figure 2). The afferent weight profiles of different neurons peaked over different parts of the retina, and their center of gravities (calculated in retinal coordinates) formed a topographical map (figure 3).

  
Figure 2: Self-organization of the afferent input weights. The afferent weights of five neurons (located at the center and at the four corners of the network) are superimposed on the retinal surface in this figure. The retina had receptors, and the receptive field radius was chosen to be 8. Therefore, neurons could have anywhere from to afferents depending on their distance from the network boundary. ( a) The anatomical RF centers were topographically ordered, and the weights were initialized randomly. There are four concentrated areas of weights slightly displaced from the corners, and one larger one in the middle. At the corners, the profiles are taller because the normalization divides the total afferent weight among a smaller numer of connections. ( b) As the self-organization progresses, the weights organize into smooth hill-shaped profiles. In this simulation, each input consisted of 3 randomly-located Gaussian spots with . The lateral interaction strengths were , with total lateral excitation = total inhibition = . The learning rates were , and the upper and lower thresholds of the sigmoid were 0.65 and 0.1 respectively. Only the parameters and the sigmoid's upper threshold were somewhat sensitive.

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Figure 3: Self-organization of scattered receptive fields into a topographic map. The center of gravity of the afferent weight vector of each neuron in the network is projected onto the receptive surface (represented by the square). Each center of gravity point is connected to those of the four immediately neighboring neurons by a line. The resulting dark grid depicts the topographical organization of the map. In (a), the anatomical RF centers were topographically ordered, but because the afferent weights were initially random, the center of gravities are locally scattered. As the self-organization progresses, the map unfolds and the weight vectors spread out to form a regular topographic map of the receptive surface, such as shown in ( b). The self-organization is robust with single- and multi-spot inputs.

With multi-spot inputs, the second network failed to develop global order. It is interesting to analyze why. With single light spots, the afferent weights of an active neuron always change towards a single, local input pattern. Eventually these weights become concentrated around a local area in the RF such that the global distribution of the centers best approximates the input distribution. However, when multiple spots occur in the receptive field at the same time, the weights change towards several different locations. Since the spots are uniformly distributed, these changes cancel over time, and the weights end up concentrated somewhere around the center of the anatomical RF. If the RF centers are already ordered, this process will simply refine the already existing topographic map; if the anatomical RF centers cross, the weight centers will stay crossed. On the other hand, in the third case (of extremely large receptive fields), all neurons see essentially the same input patterns, and develop into a map of the input space as in the standard self-organizing map. The model therefore predicts that the initial afferent connections to the undeveloped cortex must either cover almost the entire retina, or the receptive field centers must be initially ordered.

The lateral connections evolve together with the afferents. By the normalized Hebbian rule (equation 3), the lateral connection weights of each neuron are distributed according to how well the neuron's activity correlates with the activities of the other neurons. As the afferent receptive fields organize into a uniform map (figure 3), these correlations fall off with distance approximately like a Gaussian, with strong correlations to near neighbors and weaker correlations to more distant neurons. The lateral excitatory and inhibitory connections acquire the Gaussian shape, and the combined lateral excitation and inhibition becomes an approximate difference of Gaussians (or a ``Mexican hat''; figure 4).

  
Figure 4: Self-organization of the lateral interaction. The lateral interaction profile for a neuron at position in the network is plotted. The excitation and inhibition weights are initially randomly distributed within radii 3 and 18. The combined interaction is the sum of the excitatory and inhibitory weights and illustrates the total effect of the lateral connections. The sums of excitation and inhibition were chosen to be equal, but because there are fewer excitatory connections, the interaction has the shape of a rough plateau with a central peak ( a). During self-organization, smooth patterns of excitatory and inhibitory weights evolve, resulting in a smooth ``Mexican hat'' shaped lateral interaction profile ( b).

The activity patterns in cortical networks are not uniformly random as in the above experiments-the visual cortex, for example, has a rich structure of orientation and ocular dominance columns which restricts the type of activity patterns that can occur. As a first step to studying how lateral connections evolve in the visual cortex, the development of ocular dominance was simulated. A second retina was added to the model and afferent connections were set up exactly as for the first retina (with local receptive fields and topographically ordered RF centers). Multiple Gaussian spots were presented in each eye as input, with no between-eye correlations (as in strabismus). Because of cooperation and competition between inputs from the two eyes, groups of neurons developed strong afferent connections to one eye or the other, resulting in patterns of ocular dominance in the network (cf. [19,11]).

As figure 5 illustrates, the final lateral connection patterns closely follow the ocular dominance organization. As neurons become better tuned to one eye or the other, activity correlations between regions tuned to the same eye become stronger, and correlations between opposite eye areas weaker. As a result, monocular neurons develop strong lateral connections to regions with the same eye preference, and weak connections to regions of opposite eye preference. Most neurons are monocular, but a few binocular neurons occur at the boundaries of ocular dominance regions. Because they are equally tuned to the two eyes, the binocular neurons have activity correlations with both ocular dominance regions, and their lateral connection weights are distributed more or less equally between them. The lateral connections thus represent the simple Gestalt rule that under strabismus, the significant visual correlations are between monocular regions.

  
Figure 5: Patterned long-range lateral connections in an ocular dominance model. Each neuron is colored with a color value ( black orange white) that represents continuously changing eye preference from exclusive left through binocular to exclusive right. Small blue dots indicate the strongest lateral input connections to the neuron marked with a big blue dot. ( a) The lateral connections of a left monocular neuron predominantly link areas of the same ocular dominance. ( b) The lateral connections of a binocular neuron come from both eye regions.

The lateral connection patterns shown above closely match observations in the primary visual cortex. When between-eye correlations are abolished in kittens by surgically induced strabismus, long-range lateral connections primarily link areas of the same ocular dominance [9]. However, binocular neurons, located between ocular dominance columns, retained connections to both eye regions. The receptive field model confirms that such patterned lateral connections develop based on correlated neuronal activity, and demonstrates that they can self-organize simultaneously with ocular dominance columns. The model also predicts that the long-range connections have an inhibitory function.



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Next: Conclusion Up: Topographic Receptive Fields and Previous: The Receptive-Field LISSOM