A model for the formation of orientation columns

A mathematical model is proposed to describe the formation of orientation columns in mammalian visual cortex. The model is similar in concept to that proposed for ocular dominance column formation (Swindale 1980), the essential difference being that orientation is a vector rather than a scalar variable. It is assumed that initially orientation selectivity is weak and randomly distributed, and that selectivity develops in such a way that the orientation preferences of neurons less than about 200 microns apart tend to change in a similar direction, whereas the preferences of cells further apart tend to develop in opposite directions. No hypotheses are made about the anatomical or physiological basis of these interactions, and it is not necessary to assume that they are the result of environmental stimulation, as with existing models for the development of orientation selectivity (see, for example, von der Malsburg, 1973). The model reproduces the experimental data on orientation columns: roughly linear sequences of orientation change are produced, and these alternate unpredictably between clockwise and anticlockwise directions of change. Continuous sequences may span several 180 degrees cycles of rotation. The sequences are generally smooth, but abrupt discontinuities of up to 90 degrees also occur. The iso-orientation domains for large orientation ranges (60-90 degrees) are periodically spaced branching stripes that resemble those demonstrated in animals by the 2-deoxyglucose technique. The domains for narrower orientation ranges are periodically spaced but are more irregular in shape, though sometimes thin and elongated. The model makes a number of predictions that can be tested experimentally. Of particular interest are the discontinuities in the orientation sequences: these should be distributed with a spacing roughly equal to, or half, that of the iso-orientation domains. Each should be surrounded by one or two complete sets of iso-orientation domains, and each may be associated with regions where cells are not orientation selective. These regions may be more extensive in younger animals, when the columns are at an intermediate stage of formation, and less numerous where the columns run parallel and unbranched over large areas.     

A model for the coordinated development of columnar systems in primate striate cortex

The existence of patchy regions in primate striate cortex in which orientation selectivity is reduced, and which lie in the centers of ocular dominance stripes is well established (Hubel and Livingstone 1981). Analysis of functional maps obtained with voltage sensitive dyes (Blasdel and Salama 1986) has suggested that regions where the spatial rate of change of orientation preference is high, tend to be aligned either along the centers of ocular dominance stripes, or to intersect stripe borders at right angles. In this paper I present results from a developmental model which show that a tendency for orientation selectivity to develop more slowly in the centers of ocular dominance stripes would lead to the observed relationships between the layout of ocular dominance and the map of orientation gradient. This occurs despite the fact that there is no direct connection between the measures of preferred orientation (from which the gradient map is derived) and orientation selectivity (which is independent of preferred orientation). I also show that in both the monkey and the model, orientation singularities have an irregular distribution, but tend to be concentrated in the centers of the ocular dominance stripes. The average density of singularities is about 3/ ², where is the period of the orientation columns. The results are based on an elaboration of previous models (Swindale 1980, 1982) which show how, given initially disordered starting conditions, lateral interactions that are short-range excitatory and long-range inhibitory can lead to the development of patterns of orientation or ocular dominance that resemble those found in monkey striate cortex. To explain the coordinated development of the two kinds of column, it is proposed that there is an additional tendency in development for the rate of increase in orientation selectivity to be reduced in the centers of emerging ocular dominance stripes. This might come about if a single factor modulates plasticity in each cell, or column of cells. Thus plasticity may be turned off first in regions in the centers of ocular dominance stripes where relatively extreme and therefore stable ocular dominance values are achieved early in development. Consequently it will be hard for cells in these columns to modify other properties such as orientation preference or selectivity.     

Coverage and the design of striate cortex

Hubel and Wiesel (1977) suggested that ocular dominance and orientation columns in the macaque monkey striate cortex might be bands of uniform width that intersected orthogonally. They pointed out that if this were the case, there would be an equal allocation of cells of different orientation preference to each eye and to each point in visual space. However, orientation and ocular dominance columns have a more complex structural organization than is implied by this model: for example, iso-orientation domains do not intersect ocular dominance stripes at right angles and the two columnar systems have different periodicities. This raises the question as to how well the striate cortex manages to allocate equal numbers of neurons of different orientation preference to each eye and to each region of visual space, a factor referred to here as coverage. This paper defines a measure of uniformity of coverage, c, and investigates its dependence on several different parameters of columnar organisation. Calculations were done first using a simplified one-dimensional model of orientation and ocular dominance columns and were then repeated using more realistic two-dimensional models, generated with the algorithms described in the preceding paper (Swindale 1991). Factors investigated include the relative periodicities of the two columnar systems, the size of the cortical point image, the width of orientation tuning curves, whether columns are spatially anisotropic or not, and the role of the structural relationships between columns described by Blasdel and Salama (1986). The results demonstrate that coverage is most uniform when orientation hypercolumns are about half the size of ocular dominance hypercolumns. Coverage is most uneven when the hypercolumns are the same size, unless they are related in the way described by Blasdel and Salama, in which case coverage gets only slightly worse as the size ratio (ori/od) increases above 0.5. The minimum diameter of cortical point image that ensures reasonably uniform coverage is about twice the size of an ocular dominance hypercolumn i.e. about 1.5–2.0 mm.     

Statistics and geometry of orientation selectivity in primary visual cortex

Orientation maps are a prominent feature of the primary visual cortex of higher mammals. In macaques and cats, for example, preferred orientations of neurons are organized in a specific pattern, where cells with similar selectivity are clustered in iso-orientation domains. However, the map is not always continuous, and there are pinwheel-like singularities around which all orientations are arranged in an orderly fashion. Although subject of intense investigation for half a century now, it is still not entirely clear how these maps emerge and what function they might serve. Here, we suggest a new model of orientation selectivity that combines the geometry and statistics of clustered thalamocortical afferents to explain the emergence of orientation maps. We show that the model can generate spatial patterns of orientation selectivity closely resembling the maps found in cats or monkeys. Without any additional assumptions, we further show that the pattern of ocular dominance columns is inherently connected to the spatial pattern of orientation.     

Bi-sensory, striped representations: comparative insights from owl and platypus.

Bi-sensory striped arrays are described in owl and platypus that share some similarities with the other variant of bi-sensory striped array found in primate and carnivore striate cortex: ocular dominance columns. Like ocular dominance columns, the owl and platypus striped systems each involve two different topographic arrays that are cut into parallel stripes, and interdigitated, so that higher-order neurons can integrate across both arrays. Unlike ocular dominance stripes, which have a separate array for each eye, the striped array in the middle third of the owl tectum has a separate array for each cerebral hemisphere. Binocular neurons send outputs from both hemispheres to the striped array where they are segregated into parallel stripes according to hemisphere of origin. In platypus primary somatosensory cortex (S1), the two arrays of interdigitated stripes are derived from separate sensory systems in the bill, 40,000 electroreceptors and 60,000 mechanoreceptors. The stripes in platypus S1 cortex produce bimodal electrosensory-mechanosensory neurons with specificity for the time-of-arrival difference between the two systems. This "thunder-and-lightning" system would allow the platypus to estimate the distance of the prey using time disparities generated at the bill between the earlier electrical wave and the later mechanical wave caused by the motion of benthic prey. The functional significance of parallel, striped arrays is not clear, even for the highly-studied ocular dominance system, but a general strategy is proposed here that is based on the detection of temporal disparities between the two arrays that can be used to estimate distance.     

Differences in orientation and receptive field position between supra- and infragranular cells of cat striate cortex and their possible functional implications

On the postlateral gyrus of the cat striate cortex the cells' preferred orientation and the location of their receptive fields was measured as a function of cortical depth in penetrations as parallel as possible to the radiating fibres. In most penetrations the majority of infragranular cells showed orientation preferences 45 degrees-90 degrees different from the preferred orientations of supragranular cells. In addition, aggregate receptive fields from the same eye of supra- and infragranular cells were spatially shifted against each other. Using different columnar models these results are discussed in terms of spatial contrast enhancement for two parallel mechanisms in upper and lower layers, determined for pattern discrimination and movement detection.     

The radial bias: a different slant on visual orientation sensitivity in human and nonhuman primates

It is generally assumed that sensitivity to different stimulus orientations is mapped in a globally equivalent fashion across primate visual cortex, at a spatial scale larger than that of orientation columns. However, some evidence predicts instead that radial orientations should produce higher activity than other orientations, throughout visual cortex. Here, this radial orientation bias was robustly confirmed using (1) human psychophysics, plus fMRI in (2) humans and (3) behaving monkeys. In visual cortex, fMRI activity was at least 20% higher in the retinotopic representations of polar angle which corresponded to the radial stimulus orientations (relative to tangential). In a global demonstration of this, we activated complementary retinotopic quadrants of visual cortex by simply changing stimulus orientation, without changing stimulus location in the visual field. This evidence reveals a neural link between orientation sensitivity and the cortical retinotopy, which have previously been considered independent.     

Cortical templates for the self-organization of orientation-specific d- and l-hypercolumns in monkeys and cats

Blasdel and Salama's sensory maps of orientation-selective edge detectors in the monkey striate cortex can be reduced to an idealized scheme in which orientation hypercolumns of the d- and l-type occur in alternating sequence (Fig. 1). This scheme resolves the apparent contradiction between linear and circular arrangements of successive edge directions in earlier accounts. The actual configuration of hypercolumns is in register with two possible templates for the self-organization of orientation selectivity: the isometric cytochrome oxidase blobs of the colour system, and the anisometric slabs of the ocular dominance system. The centers of the hypercolumns coincide with the blobs. Simulation of cortical self-organization shows this co-incidence even in the absence of template-specific interactions. However, blobs and slabs are symmetrical to these centers, and therefore no templates for the asymmetrical distribution of preferred orientation in the hypercolumns. The present simulation derives the pre-natal formation of an initial scheme from a hypothetical gradient of nervous activity. Post-natal formation, or maturation, of this scheme is achieved by visual experience. Simulation of corresponding interactions between simultaneously activated neurons illustrates both the gain in orientation selectivity (Figs. 2 and 3), and the optimization of farfield diversity and nearfield conformity (Figs. 4 and 5). The results are compatible with the actual distribution of blob-centered d- and l-hypercolumns, iso-orientation modules and orientation fractures in the monkey. A surprisingly similar distribution of blobless d- and l-hypercolumns is expected in the absence of the colour system. Applied to the apparently blobless cortex of the cat, the scheme explains the modulation of deoxyglucose uptake along the iso-orientation bands in a report of Löwel, Freeman, and Singer.     

Topography and ocular dominance: a model exploring positive correlations

The map from eye to brain in vertebrates is topographic, i.e. neighbouring points in the eye map to neighbouring points in the brain. In addition, when two eyes innervate the same target structure, the two sets of fibres segregate to form ocular dominance stripes. Experimental evidence from the frog and goldfish suggests that these two phenomena may be subserved by the same mechanisms. We present a computational model that addresses the formation of both topography and ocular dominance. The model is based on a form of competitive learning with subtractive enforcement of a weight normalization rule. Inputs to the model are distributed patterns of activity presented simultaneously in both eyes. An important aspect of this model is that ocular dominance segregation can occur when the two eyes are positively correlated, whereas previous models have tended to assume zero or negative correlations between the eyes. This allows investigation of the dependence of the pattern of stripes on the degree of correlation between the eyes: we find that increasing correlation leads to narrower stripes. Experiments are suggested to test this prediction.     

How can squint change the spacing of ocular dominance columns?

The pattern of ocular dominance columns in primary visual cortex of mammals such as cats and macaque monkeys arises during development by the activity-dependent refinement of thalamocortical connections. Manipulating visual experience in kittens by the induction of squint leads to the emergence of ocular dominance columns with a larger size and larger column-to-column spacing than in normally raised animals. The mechanism underlying this phenomenon is presently unknown. Theory suggests that experience cannot influence the spacing of columns if the development proceeds through purely Hebbian mechanisms. Here we study a developmental model in which Hebbian mechanisms are complemented by activity-dependent regulation of the total strength of afferent synapses converging onto a cortical neurone. We show that this model implies an influence of visual experience on the spacing of ocular dominance columns and provides a conceptually simple explanation for the emergence of larger sized columns in squinting animals. Assuming that during development cortical neurones become active in local groups, which we call co-activated cortical domains (CCDs), ocular dominance segregation is controlled by the size of these groups: (1) Size and spacing of ocular dominance columns are proportional to the size sigma of CCDs. (2) There is a critical size sigma* of CCDs such that ocular dominance columns form if sigmasigma*. This critical size of CCDs is determined by the correlation functions of activity patterns in the two eyes and specifies the influence of experience on ocular dominance segregation. We show that sigma* is larger with squint than with normal visual experience. Since experimental evidence indicates that the size of CCDs decreases during development, ocular dominance columns are predicted to form earlier and with a larger spacing in squinters compared to normal animals.     

Human ocular dominance columns as revealed by high-field functional magnetic resonance imaging.

We mapped ocular dominance columns (ODCs) in normal human subjects using high-field (4 T) functional magnetic resonance imaging (fMRI) with a segmented echo planar imaging technique and an in-plane resolution of 0.47 x 0.47 mm(2). The differential responses to left or right eye stimulation could be reliably resolved in anatomically well-defined sections of V1. The orientation and width ( approximately 1 mm) of mapped ODC stripes conformed to those previously revealed in postmortem brains stained with cytochrome oxidase. In addition, we showed that mapped ODC patterns could be largely reproduced in different experiments conducted within the same experimental session or over different sessions. Our results demonstrate that high-field fMRI can be used for studying the functions of human brains at columnar spatial resolution.     

Recovery from monocular deprivation in the monkey. I. Reversal of physiological effects in the visual cortex

This is a study of the effects of monocular deprivation, reverse suturing (opening the deprived eye with closure of the other) and reopening of the deprived eye alone (without closing the other) on the physiological organization of the primary visual cortex in monkeys (Erythrocebus patas). All animals were initially monocularly deprived by suture of the lids of the right eye from soon after birth until about 4 weeks of age (24-29 days). In a monocularly deprived animal, recordings were taken from area 17 at 24 days. Already most neurons recorded outside layer IVc, were strongly or completely dominated by functional input from the left eye. The Non-oriented cells of layer IVc, where the bulk of the afferent input terminates, were also mainly dominated by the left eye. Although segregation of input from the two eyes was not complete, large areas of layer IVc were already monocularly dominated by the left eye. Four animals were reverse-sutured at about 4 weeks and recorded 3, 6, 15 and 126 days later. In each animal the pattern of ocular dominance was fairly similar within and outside layer IVc. Even with only 3 days of forced usage of the initially deprived right eye, about half of all cells recorded had become dominated by it, and the process of "recapture' of cortical cells by the initially deprived eye was apparently complete within 15 days. In layer IVc, the recovery took the form of an expansion of zones dominated by the deprived eye, as if the originally shrunken stripes of afferent termination had become enlarged. Binocularly driven neurons were rare at all stages, in all layers, but when present and orientation-selective, they had similar preferred orientations in the two eyes. Likewise the "columnar' sequences of preferred orientation continued without obvious disruption on shifting from regions dominated by one eye to those dominated by the other. Simply reopening the deprived eye at about 4 weeks, for 15 to 96 days caused no detectable change in the overall ocular dominance of cortical cells and, on average, no expansion of right-eye dominance columns in layer IVc. Therefore the recovery seen after reverse suturing depends not just on the restoration of normal activity to axons carrying information from the right eye, but on the establishment of a competitive advantage, through the right eye being made more active than the left.     

Neural networks with constrained inputs as models for pattern formation in primate visual cortex

We present a neural network model for the formation of ocular dominance stripes on primate visual cortex and examine the generic phase behavior and dynamics of the model. The dynamical equation of ocular dominance development can be identified with a class of Langevin equations with a nonconserved order parameter. We first set up and examine an Ising model with long-range interactions in an external field, which is equivalent to the model described by the Langevin equation. We use both mean-field theory and Monte-Carlo simulations to study the equilibrium phase diagram of this equivalent Ising model. The phase diagram comprises three phases: a striped phase, a hexagonal bubble phase, and a uniform paramagnetic phase. We then examine the dynamics of the striped phase by solving the Langevin equation both numerically and by singular perturbation theory. Finally, we compare the results of the model with physiological data. The typical striped structure of the ocular dominance columns corresponds to the zero-field configurations of the model. Monocular deprivation can be simulated by allowing the system to evolve in the absence of an external field at early times and then continuing the simulation in the presence of an external field. The physical and physiological applications of our model are discussed in the conclusion.     

Topographical maps of orientation specificity

A spatially congruent framework for orientation encoding in the primate striate visual cortex is proposed and discussed. This framework, which is based on the foot-of-normal representation of straight lines, not only provides a reasonable explanation for the centric organization of the orientation specificity in the primate striate visual cortex but also accounts for a series of experimentally verified intriguing phenomena such as the lack of orientation specificity around the centres of the orientation modules (i.e. the singularities), the increased neural activity at these same places, and the relatively uniform distribution of the singularities along the ocular dominance columns. The proposed framework can also explain and predict the possible existence of centric modules in other cortical regions containing topographical maps of two-dimensional sensory spaces (e.g. pre-striate and somatic sensory cortex). A simple one-layer neural model of the basic centric module in the framework is presented, and simulation results are discussed.     

Ocular dominance stripe formation by regenerated isogenic double temporal retina in Xenopus laevis

In lower vertebrates such as frogs and fish, long ocular dominance stripes with anterior-posterior (A-P) orientation can be produced by causing both eyes to innervate one optic tectum during the course of development. Similar experiments on adult animals usually produce patches rather than stripes. During development, new retinal fibers from the nasal retina segregate into appropriate stripes at the growing edge of the posterior (P) tectum while new temporal fibers segregate at the non-growing anterior (A) tectal edge. Fiber segregation into long A-P oriented stripes might depend upon a template produced by new nasal fibers initiating stripe orientation in the vicinity of new tectal cells; new nasal fibers would orient to the nascent (posterior) edge of the template while temporal fibers would orient to the anterior (non-growing) end of the template. To test the dependence of stripe formation on the matching of nascent retinal cells with nascent tectal cells, we compared stripe orientation in animals with isogenic double nasal innervation and isogenic double temporal innervation of the tectum. In double nasal innervation, the oldest retinal cells innervate the anterior tectum; new fibers from the entire retinal periphery always innervate the newest tectal cells at the posterior tectum. Stripes are oriented A-P, consistent with a maturation front model. In contrast, the oldest retinal cells innervate the newest (posterior) tectal cells in double temporal innervation of the tectum; the growing retinal periphery innervates the non-growing anterior tectum. Stripes are also oriented A-P, indicating that the production of long stripes does not depend upon maturation front matching of nascent retinal fibers and nascent tectal cells.     

A reaction-diffusion model to capture disparity selectivity in primary visual cortex

Decades of experimental studies are available on disparity selective cells in visual cortex of macaque and cat. Recently, local disparity map for iso-orientation sites for near-vertical edge preference is reported in area 18 of cat visual cortex. No experiment is yet reported on complete disparity map in V1. Disparity map for layer IV in V1 can provide insight into how disparity selective complex cell receptive field is organized from simple cell subunits. Though substantial amounts of experimental data on disparity selective cells is available, no model on receptive field development of such cells or disparity map development exists in literature. We model disparity selectivity in layer IV of cat V1 using a reaction-diffusion two-eye paradigm. In this model, the wiring between LGN and cortical layer IV is determined by resource an LGN cell has for supporting connections to cortical cells and competition for target space in layer IV. While competing for target space, the same type of LGN cells, irrespective of whether it belongs to left-eye-specific or right-eye-specific LGN layer, cooperate with each other while trying to push off the other type. Our model captures realistic 2D disparity selective simple cell receptive fields, their response properties and disparity map along with orientation and ocular dominance maps. There is lack of correlation between ocular dominance and disparity selectivity at the cell population level. At the map level, disparity selectivity topography is not random but weakly clustered for similar preferred disparities. This is similar to the experimental result reported for macaque. The details of weakly clustered disparity selectivity map in V1 indicate two types of complex cell receptive field organization.     

Self-organizing maps for visual feature representation based on natural binocular stimuli

We model the stimulus-induced development of the topography of the primary visual cortex. The analysis uses a self-organizing Kohonen model based on high-dimensional coding. It allows us to obtain an arbitrary number of feature maps by defining different operators. Using natural binocular stimuli, we concentrate on discussing the orientation, ocular dominance, and disparity maps. We obtain orientation and ocular dominance maps that agree with essential aspects of biological findings. In contrast to orientation and ocular dominance, not much is known about the cortical representation of disparity. As a result of numerical simulations, we predict substructures of orientation and ocular dominance maps that correspond to disparity maps. In regions of constant orientation, we find a wide range of horizontal disparities to be represented. This points to geometrical relations between orientation, ocular dominance, and disparity maps that might be tested in experiments. Received: 9 July 1998 / Accepted in revised form: 2 June 1999     

Phase transition theory for abnormal ocular dominance column formation

A thermodynamic theory has previously been introduced for explaining the formation of ocular dominance columns in the visual cortex. This paper extends the theory to account for the variation in patterns of ocular dominance columns as a phase transition phenomenon. For this purpose, an "Order parameter" is calculated by Monte Carlo simulation. On a phase diagram representing a two-dimensional parameter space, the conditions in which abnormal ocular dominance columns arise are visualized, and several visual deprivation experiments are successfully explained.