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Introduction to Neural NetworksU. Minn. Psy 5038Self-organizing Adaptive MapsInitializationIn[151]:=Off[SetDelayed::write]Off[General::spell1]Cortical mapsWork in monkey, and human brain, shows that the cortex is characterized by numerous distinct areas. It has been esti-mated that there are more than 30 visual areas alone in the macaque cortex. The earlier areas typically show a spatial topographic representation of visual space--nearby regions of visual space map to nearby regions of cortex. The so-called retinotopic map of the primary visual area (V1) is the clearest example of this (cf. Engel et al., 1994). Other visual areas of the brain also show geometrical organization (Wandell et al. 2005). We earlier saw that more abstract features, such as orientation, show spatial organization where similar orientations map to nearby spatial locations. Other areas of the brain show spatial organization of "non-spatial features". For example, the auditory cortex has tonotopic maps in which the spatial order of cell responses corresponds to pitch or acoustic frequency (Talavage et al. (2004) describe neuroimaging results in humans). The somatosensory cortex also shows a spatial organization (the "cortical homunculus").In regions of the cortex with no obvious maps, it is quite possible that other kinds of maps wait to be discovered. Tanaka and colleagues (Tanaka, 1996; 2003) have shown that region TE of the monkey inferotemporal cortex has columns with cells that have similar visual shape preferences. Along the surface of the cortex, receptive field properties may correspond to other kinds of variation, such as rotation in depth of a face, over limited extents (on the order of 1 mm or so).The widespread use of spatial organization in cortex suggests the possibility of a general constraint underlying the develop-ment of neural receptive field organization. We know more about primary visual cortex than any other area, so let's take a closer look at what it does.Quantitative modeling of the retinotopic map to V1We've learned that primary cortex is spatially organized so that nearby image points map to nearby cortical points. Can we say more about the metrical structure of this mapping? As one moves from an image point above the foveal/fixation point (i.e. starting at a point a fixed distance along the vertical meridian) along an arc (say counter-clockwise), the correspond-ing point on V1 moves up in a roughly straight line from the lower bank (towards the lingual gyrus) of calcarine the to the midline and then up on the upper bank (towards the cuneus). In other words, retinal rings map (approximately) to vertical cortical lines. If one moves from the fovea along a "spoke" to the periphery, the corresponding point on V1 moves from near the pole (most posterior point) of the occipital cortex toward interior and anterior region of V1. In other words, retinal spokes map (approximately) to horizontal lines. The change from image coordinates to cortical coordinates has been modeled as a log polar or complex log map (Schwarz, 1977). For a demo, see smallRetinaCortexMap.nb or this demo. These topographic properties are used to distinguish the boundaries between visual areas such as V1 and V2.Let's treat the cortex as a 2D sheet. The topographical map says how to map retinal positions to cortical positions: i.e. take 2D inputs to 2D outputs. But we know that cortex represents more than positional features. Cells show selectivity for the degree of ocularity, orientation, motion,...This suggests that a functional role for the spatial organization of cortex is to map N-D inputs (in feature space) to 2-D outputs (topographic cortical space), where N>2.Dimension reduction framework for understanding cortical mapsPrimary visual cortex does not simply have the job of representing nearby retinal points at nearby cortical locations. Much physiological research has shown that V1 brings together information from the two eyes, along similar orientations, as well as location. Together with anatomical studies, it is now commonly accepted that in many species, including humans, neurons with similar orientation preferences and various degrees of relative input from the two eyes are organized into "hypercolumns" (See Figure below, and earlier Lecture). (A major puzzle, however, is the observation that not all species have ocular dominance columns, and the function of such columns is not understood (Horton and Adams, 2005). ) Hyper-columns preserve spatial contiguity and smoothness of the placement of neurons selective for features of the input. This observation suggests that a general principle may account for the organization and development of cortical maps: Neighboring points in feature or parameter space (e.g. orientation, ocular dominance, as well as retinal position) should map to nearby points on the 2D cortical sheet. (See: Durbin & Mitchison, 1990)The underlying assumption is that most operations performed in the cortex are local, thus the related input for these computations should be physically near the computing units. For example, one task of vision is to go beyond the mere detection of contour segments, but to link contours that are likely to belong together to form a global object outline. Thus it would make sense to have the cells that signal similar orientations to be near. Visual information from the two eyes is close in the world, but separated by a great distance anatomically in the left and right eyes. This information needs to be brought physically together to process the two images binocularly, for example, to compute stereoscopic depth. Further, operations that occur frequently, that need to combine many sources of information, and that need to be done quickly could be done more efficiently if the brain could avoid having too many long connections. But there seems to be a problem: How to map a high dimensional feature vector to a 2-dimensional surface?2 Lect_21_AdaptMaps.nbLect_21_AdaptMaps.nb 3Minimum wiring length constraint‡NematodeA number of people have sought a simple organizational principle that would predict the spatial layout of neurons. One such principle is that the layout of nervous system components minimizes total connection cost. Christopher Cherniak, a philosopher at the University of Maryland calculated the total wiring length for the ~40,000,000 (11!) possible layouts of the 11 hypothetically "moveable" ganglia (connecting 302 neurons) in the nematode worm C.


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