Cortical Maps

Nerve cells (neurons) have different response properties, which have been studied in many cortical areas. Thus, in primary visual cortex (V1) each neuron is characterized by preferred position of stimulus in the visual space, preferred stimulus orientation, spatial and temporal frequencies, etc [1]. These response properties allow representing each neuron as a single point in a multi-dimensional space of features (position, orientation, etc). For the case of V1, this feature space has about 10 dimensions. In order to maximize the individual's chances of survival the neuronal population would sample the ~10 dimensional feature space uniformly at the maximum density. This would happen in an 'unconstrained' brain. In reality, however, the neural system faces an uneasy task to sample the feature space in the presence of various biological constraints. Thus, in primary visual cortex, only a three-dimensional subspace of the 10 dimensional feature space is actually sampled [2]. The geometry of this subspace is not fully understood. To understand the nature of cortical computation it is essential therefore to study how biological constraints shape the coordinated strategy of the visual system.

Perhaps a simpler example of the non-uniform sampling of the feature space can be found in mammalian retina. It is clear from everyone's experience that acuity of vision is maximal at positions corresponding to the center of the eye. For this reason, to decipher an object in the periphery we have to move our eyes and aim the eye center at the object. Thus, the two-dimensional visual space is represented densely only in the small fraction of the whole retina (area centralis). This non-uniformity is controlled by limits on the total number of retinal ganglion cells, imposed by the flexibility of the optic nerve [3,4].

Another example of the interplay between function and constraints are patterns and various other anatomical features found in visual cortical areas. Such patterns refer to arrangement of neurons relative to each other based on their response properties. In an 'unconstrained' brain the neurons would not be pressured into any definite layout and would form mixed arrangements, controlled by random factors. We argue that the patterns observed in visual cortical areas can be explained in great detail on the basis of various limitations imposed evolutionary.

Figure 1: The pattern of preferred orientation and ocular dominance combined on the same picture, which is obtained in a computer simulation minimizing the total length of connections. The preferred orientation is coded by color. The left-eye dominated areas are shown by a darker color.

Figure 2: Connections established by two neurons (stars) in the binary mixture of cells dominated by left and right eyes (squares and dots). Each neurons makes more connections to the neurons of the same ocular dominance, than to the others. This leads to the segregation of mixture and formation of ocular dominance pattern. Therefore, wiring economy is the first and the only principle thus far, which assigns any functional significance to the ocular dominance pattern.

It has been suggested that wiring economy constraint is an important principle of cortical design [5, 6]. Since connections (axons and dendrites) take up a significant fraction (about 60%) of the cortical volume, limitations on the brain size require keeping the connection length as short as possible. It is likely that for a given neuronal circuit evolution has found an optimal neuronal arrangement. Therefore, the wiring economy principle may be a link connecting the wiring rules and the neuronal arrangement.

As mentioned above neurons in the visual areas respond best to edges in their receptive field [7]. Edge orientation that evokes most vigorous response determines the orientation preference of a neuron. Preferred orientation of the neurons changes smoothly along the cortical surface with occasional singularities, such as pinwheels (vortices) and fractures (strings). The reason for existence of these singularities has remained elusive for the past 30 years. We showed that under reasonable assumptions singularities are necessary to minimize the connection length (Figure 1). The reduction of the cortical volume due to the formation of singularities can sometimes reach 30% [8]. The connection rules leading to pinwheels and fractures are consistent with recently measured in ferret primary and cat secondary visual areas [9].

Neurons in primary visual cortex are either right or left eye dominated. The binary mixture of right and left eye dominated neurons is segregated resulting in a system of alternating monocular regions, known as ocular dominance pattern. Mitchison previously suggested [6] that the emergence of ocular dominance patterns is a result of wiring economy. To make a quantitative assessment of this hypothesis we studied the phase diagram of the ocular dominance patterns. We obtained a phase transition from the unidirectional "zebra skin" pattern of ocular dominance to a two-dimensional "leopard skin" pattern, driven by the decreasing relative fraction of neurons dominated by one of the eyes (ipsilateral). The relative fraction at which the transition occurs according to our calculations is 38%. This prediction agrees with the experimental observations in macaque and Cebus monkeys [10].

These findings provide support for wiring economy principle. At the same time, the abundance of existing experimental data on cortical maps demands more quantitative work in this direction. One can apply the principle to cytochrome-oxidase blobs, for which the connection matrix is already studied [11]. It is also important to understand the coupling between different maps in primary visual cortex. Why for instance the pinwheels tend to be at the center of the blobs? Why the gradient of orientation tends to be parallel to the interface between ocular dominance patterns? Why high gradient of orientation tends to be coupled to the high gradient of visuotopic coordinates [12]? Directional maps in area MT reveal many unusual properties absent in the lower visual areas, such as interchanging columns of opposite direction preference but similar orientation of the axis of motion [13]. Understanding these structures will help us understand the design principles of the brain.

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