Copyright © 1999 Massachusetts Institute of Technology
The term computational anatomy was introduced in Schwartz 1980, which suggested that the observables of functional anatomy, such as columnar and topographic structure, be made the basis of the state variables for perceptual computations (rather than, as is universally assumed, some combination of single neuronal response properties). The goal of computational neuroanatomy is to construct a supraneuronal continuum, or "field theory," approach to neural structure and function, to the "particulate" approach associated with the properties of single neurons. In particular, the details of spatial neuroanatomy of the nervous system are viewed as computational, rather than as mere "packaging" (see Schwartz 1994 for a comprehensive review of the structural and functional correlates of neuroanatomy).
Global Map Structure
It is widely accepted that the cortical magnification factor in primates is approximately inverse linear, at least for the central twenty degrees of field (e.g., Tootell et al. 1985; van Essen, Newsome, and Maunsell 1984; Dow, Vautin, and Bauer 1985), and preliminary results from functional MAGNETIC RESONANCE IMAGING (fMRI) suggest roughly the same for humans. The simple mathematical argument showing there is only one complex analytic two-dimensional map function that has this property, namely, the complex logarithm (Schwartz 1977, 1994), suggested an experiment. Determine the point correspondence data from a 2DG (2-deoxyglucose) experiment, accurately measure the flattened cortical surface, and check the validity of the Reimann mapping theorem prediction of primary visual cortex (V1) topography, or, equivalently, of the hypothesis that global topography in V1 is a generalized conformal map. The results of this experiment (Schwartz, Munsif, and Albright 1989; Schwartz 1994) confirmed that cortical topography is in strong agreement with the conformal mapping hypothesis, up to an error estimated to be roughly 20 percent.
Local Map Structure
A large number of NEURAL NETWORK models have been constructed to address the generation of ocular dominance and orientation columns in cat and monkey VISUAL CORTEX. Surprisingly, the common element in all these models, often not explicitly stated, is the use of a spatial filter applied to spatial white noise (Rojer and Schwartz 1990a). Thus ocular dominance columns are the result of band-pass filtering a scalar white noise variable (ocularity). Orientation columns, as originally pointed out by Rojer and Schwartz (1990a), could be understood as the result of applying a band-pass filter to vector noise, that is, to a vector quantity whose magnitude represented strength of tuning, and whose argument represented orientation. One recent result of this analysis is that the zero-crossings of the cortical orientation map were predicted, on topological grounds, to provide a coordinate system in which left- and right-handed orientation vortices should alternate in handedness (i.e., clockwise or counterclockwise orientation change). This prediction was tested with optical recording data on primate visual cortex orientation maps and found to be in perfect agreement with the data (Tal and Schwartz 1997).
Unified Global and Local Map Structure
A joint map structure to express the global conformal topographic structure, and, at the same time, the local orientation column and ocular dominance column structure of primate V1, was introduced by Landau and Schwartz (1994), making use of a new construct in computational geometry called the "protocolumn."
Global Map Function
One obvious functional advantage of using strongly space-variant (e.g., foveal) architecture in vision is data compression. It has been estimated that a constant-resolution version of visual cortex, were it to retain the full human visual field and maximum human visual resolution, would require roughly 104 as many cells as our actual cortex (and would weigh, by inference, roughly 15,000 pounds; Rojer and Schwartz 1990b). The problem of viewing a wide-angle work space at high resolution would seem to be best performed with space-variant visual architectures, an important theme in MACHINE VISION (Schwartz, Greve, and Bonmassar 1995). The complex logarithmic mapping has special properties with respect to size and rotation invariance. For a given fixation point, changing the size or rotating a stimulus causes its cortical representation to shift, but to otherwise remain invariant (Schwartz 1977). This symmetry property provides an excellent example of computational neuroanatomy: simply by virtue of the spatial properties of cortical topography, size and rotation symmetries may be converted into the simpler symmetry of shift. One obvious problem with this idea is that it only works for a given fixation direction. As the eye scans an image, translation invariance is badly broken. Recently, a computational solution to this problem has been found, by generalizing the Fourier transform to complex logarithmic coordinate systems, resulting in a new form of spatial transform, called the "exponential chirp transform" (Bonmassar and Schwartz forthcoming.) The exponential chirp transform, unlike earlier attempts to incorporate Fourier analysis in the context of human vision (e.g., Cavanagh 1978), provides size, rotation, and shift invariance properties, while retaining the fundamental space-variant structure of the visual field.
Local Map Function
The ocular dominance column presents a binocular view of the visual world in the form of thin "stripes," alternating between left- and right-eye representations. One question that immediately arises is how this aspect of cortical anatomy functionally relates to binocular stereopsis. Yeshurun and Schwartz (1989) constructed a computational stereo algorithm based on the assumption that the ocular dominance column structure is a direct representation, as an anatomical pattern, of the stereo percept. It was shown that the power spectrum of the log power spectrum (also known as the "cepstrum") of the interlaced cortical "image" provided a simple and direct measure of stereo disparity of objects in the visual scene. This idea has been subsequently used in a successful machine vision algorithm for stereo vision (Ballard, Becker, and Brown 1988), and provides another excellent illustration of computational neuroanatomy.
The regular local spatial map of orientation response in cat and monkey, originally described by Hubel and Wiesel (1974), suggested the hypothesis that a local analysis of shape, in terms of periodic changes in orientation of a stimulus outline, might provide a basis for shape analysis (Schwartz 1984). A parametric set of shape descriptors, based on shapes whose boundary curvature varied sinusoidally, was used as a probe for the response properties of neurons in infero-temporal cortex, which is one of the final targets for V1, and which is widely believed to be an important site for shape recognition. This work found that a subset of the infero-temporal neurons examined were tuned to stimuli with sinusoidal curvature variation (so-called Fourier descriptors), and that these responses showed a significant amount of size, rotation, and shift invariance (Schwartz et al. 1983).
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Copyright © 1999 Massachusetts Institute of Technology