Eric L. Schwartz

Computational anatomy and functional architecture of striate cortex: A spatial mapping approach to perceptual coding

Abstract The spatial inhomogeneity of the retino-striate system is summarized by the vector cortical magnification factor. The logarithm of retinal eccentricity provides a good fit to the integrated cortical magnification factor. Under the assumption that the cortical map is analytic (conformal), this implies that a complex logarithmic function of retinal coordinates describes the two-dimensional structure of the cortical representation of a visual stimulus. This hypothesis is in good agreement with the measured global structure of rhesus, squirrel, and owl monkey retino-striate mappings, as well as that of the upper visual field of the cat. The geometric structure of the local hypercolumnar unit of striate cortex may also be characterized in terms of the complex logarithmic mapping: thus, the retino-cortical system may be thought of as a concatenated complex logarithmic mapping. A simple developmental mechanism is capable of constructing a map of this form, and the general mathematical properties of conformal mappings allow some insight into the nature of the minimal coding requirements which must be specified to encode a neural map. Complex logarithmic mapping yields a cortical “Gestalt” which is pseudo-invariant to size, rotation, and projection scaling: these symmetries, for a given fixation point, result in a linear shift of an invariant cortical pattern. The term computational anatomy refers to the possibility that the anatomical structure of the retinotopic mapping may simplify certain aspects of perceptual coding. Similar uses of the complex logarithmic mapping, in computer pattern recognition, are cited in support of this concept. Furthermore, it is shown that columnar structure, together with topographic mapping, may also provide a direct computational function. If two topographic mappings are appropriately interlaced, by columns, then the difference mapping of the two independent inputs is encoded within a spatial frequency channel determined by the period of the columns. A quantitative model of the human visual cortex is developed and used to portray the detailed structure of certain visual stimuli, as they would appear at the level of the striate cortex. The local and global geometric structure of the striate map suggests a simple explanation for several visual illusions. Thus, it is demonstrated that the geometric structure of visual stimuli, at the level of striate cortex, may be of significance to perception. Finally, the concept of computational anatomy is discussed in relation to other contemporary notions of perceptual coding. It is argued that both single cell feature extraction models and Fourier analysis models of visual coding are inconsistent with the known properties of the visual system, and moreover, have never been adequately formulated in computational terms. The approach of the present paper is to suggest that the basic data structure of perceptual coding consists of two-dimensional laminar mapping, and that successive stages of remapping, along with columnar architecture, may provide important computational functions.

Spatial mapping in the primate sensory projection: Analytic structure and relevance to perception

The retinotopic mapping of the visual field to the surface of the striate cortex is characterized as a longarithmic conformal mapping. This summarizes in a concise way the observed curve of cortical magnification, the linear scaling of receptive field size with eccentricity, and the mapping of global visual field landmarks. It is shown that if this global structure is reiterated at the local level, then the sequence regularity of the simple cells of area 17 may be accounted for as well. Recently published data on the secondary visual area, the medial visual area, and the inferior pulvinar of the owl monkey suggests that same global logarithmic structure holds for these areas as well. The available data on the structure of the somatotopic mapping (areaS-1) supports a similar analysis. The possible relevance of the analytical form of the cortical receptotopic maps to perception is examined and a brief discussion of the developmental implications of these findings is presented.

Isoperimetric graph partitioning for image segmentation

Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations of spectral methods, but with improved speed and stability.

Perceptual framing and cortical alpha rhythm

Abstract Apparent motion was used as a probe to test the hypothesis that perceptual framing is correlated to the phase of the alpha rhythm. Stimuli were presented phasically with respect to the occipital and parietal alpha cycle, and subjects were asked to judge whether the stimuli appeared simultaneous or sequential. The probability of perceived simultaneity was maximal for the positive phase of the occipital alpha cycle. Visual evoked potentials recorded during stimulus presentation were significantly different, in the late components, for the cases of perceived simultaneity or sequential motion. A brief review of previous experimental and theoretical studies of the relationship of perceptual framing to alpha rhythm is presented.

Effects of mindful-attention and compassion meditation training on amygdala response to emotional stimuli in an ordinary, non-meditative state

The amygdala has been repeatedly implicated in emotional processing of both positive and negative-valence stimuli. Previous studies suggest that the amygdala response to emotional stimuli is lower when the subject is in a meditative state of mindful-attention, both in beginner meditators after an 8-week meditation intervention and in expert meditators. However, the longitudinal effects of meditation training on amygdala responses have not been reported when participants are in an ordinary, non-meditative state. In this study, we investigated how 8 weeks of training in meditation affects amygdala responses to emotional stimuli in subjects when in a non-meditative state. Healthy adults with no prior meditation experience took part in 8 weeks of either Mindful Attention Training (MAT), Cognitively-Based Compassion Training (CBCT; a program based on Tibetan Buddhist compassion meditation practices), or an active control intervention. Before and after the intervention, participants underwent an fMRI experiment during which they were presented images with positive, negative, and neutral emotional valences from the IAPS database while remaining in an ordinary, non-meditative state. Using a region-of-interest analysis, we found a longitudinal decrease in right amygdala activation in the Mindful Attention group in response to positive images, and in response to images of all valences overall. In the CBCT group, we found a trend increase in right amygdala response to negative images, which was significantly correlated with a decrease in depression score. No effects or trends were observed in the control group. This finding suggests that the effects of meditation training on emotional processing might transfer to non-meditative states. This is consistent with the hypothesis that meditation training may induce learning that is not stimulus- or task-specific, but process-specific, and thereby may result in enduring changes in mental function.

A numerical solution to the generalized mapmaker's problem: flattening nonconvex polyhedral surfaces

Methods are described to unfold and flatten the curved, convoluted surfaces of the brain in order to study the functional architectures and neural maps embedded in them. In order to do this, it is necessary to solve the general mapmakers problem for representing curved surfaces by planar models. This algorithm has applications in areas other than computer-aided neuroanatomy, such as robotics motion planning and geophysics. The algorithm maximizes the goodness of fit distances in these surfaces to distances in a planar configuration of points. It is illustrated with a flattening of monkey visual cortex, which is an extremely complex folded surface. Distance errors in the range of several percent are found, with isolated regions of larger error, for the class of cortical surfaces studied so far. >

Accurate prediction of V1 location from cortical folds in a surface coordinate system

Previous studies demonstrated substantial variability of the location of primary visual cortex (V1) in stereotaxic coordinates when linear volume-based registration is used to match volumetric image intensities [Amunts, K., Malikovic, A., Mohlberg, H., Schormann, T., and Zilles, K. (2000). Brodmanns areas 17 and 18 brought into stereotaxic space-where and how variable? Neuroimage, 11(1):66-84]. However, other qualitative reports of V1 location [Smith, G. (1904). The morphology of the occipital region of the cerebral hemisphere in man and the apes. Anatomischer Anzeiger, 24:436-451; Stensaas, S.S., Eddington, D.K., and Dobelle, W.H. (1974). The topography and variability of the primary visual cortex in man. J Neurosurg, 40(6):747-755; Rademacher, J., Caviness, V.S., Steinmetz, H., and Galaburda, A.M. (1993). Topographical variation of the human primary cortices: implications for neuroimaging, brain mapping, and neurobiology. Cereb Cortex, 3(4):313-329] suggested a consistent relationship between V1 and the surrounding cortical folds. Here, the relationship between folds and the location of V1 is quantified using surface-based analysis to generate a probabilistic atlas of human V1. High-resolution (about 200 microm) magnetic resonance imaging (MRI) at 7 T of ex vivo human cerebral hemispheres allowed identification of the full area via the stria of Gennari: a myeloarchitectonic feature specific to V1. Separate, whole-brain scans were acquired using MRI at 1.5 T to allow segmentation and mesh reconstruction of the cortical gray matter. For each individual, V1 was manually identified in the high-resolution volume and projected onto the cortical surface. Surface-based intersubject registration [Fischl, B., Sereno, M.I., Tootell, R.B., and Dale, A.M. (1999b). High-resolution intersubject averaging and a coordinate system for the cortical surface. Hum Brain Mapp, 8(4):272-84] was performed to align the primary cortical folds of individual hemispheres to those of a reference template representing the average folding pattern. An atlas of V1 location was constructed by computing the probability of V1 inclusion for each cortical location in the template space. This probabilistic atlas of V1 exhibits low prediction error compared to previous V1 probabilistic atlases built in volumetric coordinates. The increased predictability observed under surface-based registration suggests that the location of V1 is more accurately predicted by the cortical folds than by the shape of the brain embedded in the volume of the skull. In addition, the high quality of this atlas provides direct evidence that surface-based intersubject registration methods are superior to volume-based methods at superimposing functional areas of cortex and therefore are better suited to support multisubject averaging for functional imaging experiments targeting the cerebral cortex.

Space Variant Image Processing

This paper describes a graph-based approach to image processing, intended for use with images obtained from sensors having space variant sampling grids. The connectivity graph (CG) is presented as a fundamental framework for posing image operations in any kind of space variant sensor. Partially motivated by the observation that human vision is strongly space variant, a number of research groups have been experimenting with space variant sensors. Such systems cover wide solid angles yet maintain high acuity in their central regions. Implementation of space variant systems pose at least two outstanding problems. First, such a system must be active, in order to utilize its high acuity region; second, there are significant image processing problems introduced by the non-uniform pixel size, shape and connectivity. Familiar image processing operations such as connected components, convolution, template matching, and even image translation, take on new and different forms when defined on space variant images. The present paper provides a general method for space variant image processing, based on a connectivity graph which represents the neighbor-relations in an arbitrarily structured sensor. We illustrate this approach with the following applications: (1) Connected components is reduced to its graph theoretic counterpart. We illustrate this on a logmap sensor, which possesses a difficult topology due to the branch cut associated with the complex logarithm function. (2) We show how to write local image operators in the connectivity graph that are independent of the sensor geometry. (3) We relate the connectivity graph to pyramids over irregular tessalations, and implement a local binarization operator in a 2-level pyramid. (4) Finally, we expand the connectivity graph into a structure we call a transformation graph, which represents the effects of geometric transformations in space variant image sensors. Using the transformation graph, we define an efficient algorithm for matching in the logmap images and solve the template matching problem for space variant images. Because of the very small number of pixels typical of logarithmic structured space variant arrays, the connectivity graph approach to image processing is suitable for real-time implementation, and provides a generic solution to a wide range of image processing applications with space variant sensors.

Design considerations for a space-variant visual sensor with complex-logarithmic geometry

A space-variant sensor design based on the conformal mapping of the half disk, w=log (z+a), with real a>0, which characterizes the anatomical structure of the primate and human visual systems is discussed. There are three relevant parameters: the circumferential index kappa which is defined as the number of pixels around the periphery of the sensor, the visual field radius R (of the half-disk to be mapped), and the map parameter a, which displaces the logarithms singularity at the origin out of the domain of the mapping. It is shown that the log sensor requires O( kappa /sup 2/log (R/a)) pixels. An analysis is presented which makes it possible to compare directly the space complexity of different sensor designs in the complex logarithmic family. In particular, rough estimates can be obtained of the parameters necessary to duplicate the field width/resolution performance of the human visual system.<<ETX>>

Cepstral filtering on a columnar image architecture: a fast algorithm for binocular stereo segmentation

Many primate visual cortex architectures have a prominent feature responsible for the mixing of left and right eye visual data: ocular dominance columns represent thin (about 5-10 minutes of arc) strips of alternating left and right eye input to the brain. It is shown that such an architecture, when operated upon with a cepstral filter, provides a strong cue for binocular stereopsis. Specifically, the vector of binocular disparity may be easily identified in the output of the (columnar based) cepstral filter. This algorithm is illustrated with application to a random dot stereogram and to natural images. The authors suggest that this provides a fast algorithm for stereo segmentation, in a machine vision context. In a biological context, it may provide a computational rationale for the existence of columnar systems with regard to both ocular mixing and other visual modalities that have a columnar architecture. >