A Computational Model for the Development of Multiple Maps in Primary Visual Cortex

A unifying framework for functional organization in early and higher ventral visual cortex

A key feature of cortical systems is functional organization: the arrangement of functionally distinct neurons in characteristic spatial patterns. However, the principles underlying the emergence of functional organization in the cortex are poorly understood. Here, we develop the topographic deep artificial neural network (TDANN), the first model to predict several aspects of the functional organization of multiple cortical areas in the primate visual system. We analyze the factors driving the TDANN's success and find that it balances two objectives: learning a task-general sensory representation and maximizing the spatial smoothness of responses according to a metric that scales with cortical surface area. In turn, the representations learned by the TDANN are more brain-like than in spatially unconstrained models. Finally, we provide evidence that the TDANN's functional organization balances performance with between-area connection length. Our results offer a unified principle for understanding the functional organization of the primate ventral visual system.

Mutual consistency of multiple visual feature maps constrains combined map topology

The primary visual cortex (V1) is the first cortical area that processes visual information relayed from the thalamus. The topologies permitted in joint ocular dominance (OD), orientation preference (OP), and direction preference (DP) maps in V1 are considered, with the aim of finding a maximally symmetric periodic case that can serve as the basis for perturbations toward natural realizations. It is shown that mutual consistency of the maps selects just two possible such lattice structures, and that one of these is much closer to experiment than the other. This comprises a hexagonal lattice of alternating positive and negative OP singularities, with each unit cell or hypercolumn containing four such singularities, each of which radiates three DP discontinuities that follow OP contours and end at OP singularities of opposite sign. Other DP discontinuities emanate at 90 degrees to the midpoints of the ones that link OP singularities, and cross OP contours perpendicularly. These features explain experimentally observed relationships between DP discontinuities and OP contours, including sudden approximately 90-degree changes of direction in the former.

A Unifying Principle for the Functional Organization of Visual Cortex

A key feature of many cortical systems is functional organization: the arrangement of neurons with specific functional properties in characteristic spatial patterns across the cortical surface. However, the principles underlying the emergence and utility of functional organization are poorly understood. Here we develop the Topographic Deep Artificial Neural Network (TDANN), the first unified model to accurately predict the functional organization of multiple cortical areas in the primate visual system. We analyze the key factors responsible for the TDANN's success and find that it strikes a balance between two specific objectives: achieving a task-general sensory representation that is self-supervised, and maximizing the smoothness of responses across the cortical sheet according to a metric that scales relative to cortical surface area. In turn, the representations learned by the TDANN are lower dimensional and more brain-like than those in models that lack a spatial smoothness constraint. Finally, we provide evidence that the TDANN's functional organization balances performance with inter-area connection length, and use the resulting models for a proof-of-principle optimization of cortical prosthetic design. Our results thus offer a unified principle for understanding functional organization and a novel view of the functional role of the visual system in particular.

Orientation pinwheels in primary visual cortex of a highly visual marsupial

Primary visual cortices in many mammalian species exhibit modular and periodic orientation preference maps arranged in pinwheel-like layouts. The role of inherited traits as opposed to environmental influences in determining this organization remains unclear. Here, we characterize the cortical organization of an Australian marsupial, revealing pinwheel organization resembling that of eutherian carnivores and primates but distinctly different from the simpler salt-and-pepper arrangement of eutherian rodents and rabbits. The divergence of marsupials from eutherians 160 million years ago and the later emergence of rodents and rabbits suggest that the salt-and-pepper structure is not the primitive ancestral form. Rather, the genetic code that enables complex pinwheel formation is likely widespread, perhaps extending back to the common therian ancestors of modern mammals.

A theory of cortical map formation in the visual brain

The cerebral cortex receives multiple afferents from the thalamus that segregate by stimulus modality forming cortical maps for each sense. In vision, the primary visual cortex maps the multiple dimensions of the visual stimulus in patterns that vary across species for reasons unknown. Here we introduce a general theory of cortical map formation, which proposes that map diversity emerges from species variations in the thalamic afferent density sampling sensory space. In the theory, increasing afferent sampling density enlarges the cortical domains representing the same visual point, allowing the segregation of afferents and cortical targets by multiple stimulus dimensions. We illustrate the theory with an afferent-density model that accurately replicates the maps of different species through afferent segregation followed by thalamocortical convergence pruned by visual experience. Because thalamocortical pathways use similar mechanisms for axon segregation and pruning, the theory may extend to other sensory areas of the mammalian brain.

Najafian et al. introduce a developmental theory of map formation in the cerebral cortex. The theory proposes that increases in the density of thalamic afferents sampling sensory space make cortical maps to segregate more stimulus dimensions.

The Phenomenology and Neurobiology of Visual Distortions and Hallucinations in Schizophrenia: An Update

Schizophrenia is characterized by visual distortions in ~60% of cases, and visual hallucinations (VH) in ~25-50% of cases, depending on the sample. These symptoms have received relatively little attention in the literature, perhaps due to the higher rate of auditory vs. visual hallucinations in psychotic disorders, which is the reverse of what is found in other neuropsychiatric conditions. Given the clinical significance of these perceptual disturbances, our aim is to help address this gap by updating and expanding upon prior reviews. Specifically, we: (1) present findings on the nature and frequency of VH and distortions in schizophrenia; (2) review proposed syndromes of VH in neuro-ophthalmology and neuropsychiatry, and discuss the extent to which these characterize VH in schizophrenia; (3) review potential cortical mechanisms of VH in schizophrenia; (4) review retinal changes that could contribute to VH in schizophrenia; (5) discuss relationships between findings from laboratory measures of visual processing and VH in schizophrenia; and (6) integrate findings across biological and psychological levels to propose an updated model of VH mechanisms, including how their content is determined, and how they may reflect vulnerabilities in the maintenance of a sense of self. In particular, we emphasize the potential role of alterations at multiple points in the visual pathway, including the retina, the roles of multiple neurotransmitters, and the role of a combination of disinhibited default mode network activity and enhanced state-related apical/contextual drive in determining the onset and content of VH. In short, our goal is to cast a fresh light on the under-studied symptoms of VH and visual distortions in schizophrenia for the purposes of informing future work on mechanisms and the development of targeted therapeutic interventions.

Development and Reorganization of Orientation Representation in the Cat Visual Cortex: Experience-Dependent Synaptic Rewiring in Early Life

To date, numerous mathematical models have been proposed on the basis of some types of Hebbian synaptic learning to account for the activity-dependent development of orientation maps as well as neuronal orientation selectivity. These models successfully reproduced orientation map-like spatial patterns. Nevertheless, we still have questions: (1) How does synaptic rewiring occur in the visual cortex during the formation of orderly orientation maps in early life? (2) How does visual experience contribute to the maturation of orientation selectivity of visual cortical neurons and reorganize orientation maps? (3) How does the sensitive period for orientation plasticity end? In this study, we performed animal experiments and mathematical modeling to understand the mechanisms underlying synaptic rewiring for experience-dependent formation and reorganization of orientation maps. At first, we visualized orientation maps from the intrinsic signal optical imaging in area 17 of kittens reared under single-orientation exposure through cylindrical-lens-fitted goggles. The experiments revealed that the degree of expansion of cortical domains representing the experienced orientation depends on the age at which the single-orientation exposure starts. As a result, we obtained the sensitive period profile for orientation plasticity. Next, we refined our previously proposed mathematical model for the activity-dependent self-organization of thalamo-cortical inputs on the assumption that rewiring is caused by the competitive interactions among transient synaptic contacts on the same dendritic spine. Although various kinds of molecules have been reported to be involved in such interactions, we attempt to build a mathematical model to describe synaptic rewiring focusing on brain-derived neurotrophic factor (BDNF) and its related molecules. Performing computer simulations based on the refined model, we successfully reproduced orientation maps reorganized in kittens reared under single-orientation exposure as well as normal visual experience. We also reproduced the experimentally obtained sensitive period profile for orientation plasticity. The excellent agreement between experimental observations and theoretical reproductions suggests that the BDNF-induced competitive interaction among synaptic contacts from different axons on the same spine is an important factor for the experience-dependent formation and reorganization of orientation selectivity and orientation maps.

Columnar connectome: toward a mathematics of brain function

Understanding brain networks is important for many fields, including neuroscience, psychology, medicine, and artificial intelligence. To address this fundamental need, there are multiple ongoing connectome projects in the United States, Europe, and Asia producing brain connection maps with resolutions at macro- and microscales. However, still lacking is a mesoscale connectome. This viewpoint (1) explains the need for a mesoscale connectome in the primate brain (the columnar connectome), (2) presents a new method for acquiring such data rapidly on a large scale, and (3) proposes how one might use such data to achieve a mathematics of brain function.

Estimating Cortical Feature Maps with Dependent Gaussian Processes

A striking example of brain organisation is the stereotyped arrangement of cell preferences in the visual cortex for edges of particular orientations in the visual image. These "orientation preference maps" appear to have remarkably consistent statistical properties across many species. However fine scale analysis of these properties requires the accurate reconstruction of maps from imaging data which is highly noisy. A new approach for solving this reconstruction problem is to use Bayesian Gaussian process methods, which produce more accurate results than classical techniques. However, so far this work has not considered the fact that maps for several other features of visual input coexist with the orientation preference map and that these maps have mutually dependent spatial arrangements. Here we extend the Gaussian process framework to the multiple output case, so that we can consider multiple maps simultaneously. We demonstrate that this improves reconstruction of multiple maps compared to both classical techniques and the single output approach, can encode the empirically observed relationships, and is easily extendible. This provides the first principled approach for studying the spatial relationships between feature maps in visual cortex.

The temporal-spatial dynamics of feature maps during monocular deprivation revealed by chronic imaging and self-organization model simulation

Experiments on the adult visual cortex of cats, ferrets and monkeys have revealed organized spatial relationships between multiple feature maps which can also be reproduced by the Kohonen and elastic net self-organization models. However, attempts to apply these models to simulate the temporal kinetics of monocular deprivation (MD) during the critical period, and their effects on the spatial arrangement of feature maps, have led to conflicting results. In this study, we performed MD and chronic imaging in the ferret visual cortex during the critical period of ocular dominance (OD) plasticity. We also used the Kohonen model to simulate the effects of MD on OD and orientation map development. Both the experiments and simulations demonstrated two general parameter-insensitive findings. Specifically, our first finding demonstrated that the OD index shift resulting from MD, and its subsequent recovery during binocular vision (BV), were both nonlinear, with a significantly stronger shift occurring during the initial period. Meanwhile, spatial reorganization of feature maps led to globally unchanged but locally shifted map patterns. In detail, we found that the periodicity of OD and orientation maps remained unchanged during, and after, deprivation. Relationships between OD and orientation maps remained similar but were significantly weakened due to OD border shifts. These results indicate that orthogonal gradient relationships between maps may be preset and are only mildly modifiable during the critical period. The Kohonen model was able to reproduce these experimental results, hence its role is further extended to the description of cortical feature map dynamics during development.

Sensory experience modifies feature map relationships in visual cortex

The extent to which brain structure is influenced by sensory input during development is a critical but controversial question. A paradigmatic system for studying this is the mammalian visual cortex. Maps of orientation preference (OP) and ocular dominance (OD) in the primary visual cortex of ferrets, cats and monkeys can be individually changed by altered visual input. However, the spatial relationship between OP and OD maps has appeared immutable. Using a computational model we predicted that biasing the visual input to orthogonal orientation in the two eyes should cause a shift of OP pinwheels towards the border of OD columns. We then confirmed this prediction by rearing cats wearing orthogonally oriented cylindrical lenses over each eye. Thus, the spatial relationship between OP and OD maps can be modified by visual experience, revealing a previously unknown degree of brain plasticity in response to sensory input.

DOI:http://dx.doi.org/10.7554/eLife.13911.001

The structure of the brain results from a combination of nature (genes) and nurture (environment). The brain’s ability to adapt to changes in the environment is known as plasticity, and the young brain is especially plastic. An animal’s sensory experiences in early life help to determine how its brain will process sensory input as an adult. One of the best sensory systems in which to study this process is the visual system.

Within the visual system, some brain cells respond only to input from the left eye and others only to input from the right eye. Cells that respond to input from the same eye are arranged to form columns. Within each column, some cells respond only to lines with a particular orientation. Cells with different preferred orientations are grouped together in patterns that resemble pinwheels. The relative positions of the pinwheels and eye-specific columns within the brain tissue belonging to the visual system have so far been robust to changes in visual experience during development, suggesting that they are determined by an animal’s genes.

However, Cloherty, Hughes et al. have now tested the unexpected predictions of a computer model. The model suggested that rearing animals so that they saw mostly vertical lines through one eye, and mostly horizontal lines through the other, would cause a form of plasticity that had never been observed before. Specifically, it would change the relative positions of the pinwheels and eye-specific columns within the visual parts of the brain. This prediction turned out to be correct. Young cats that wore special lenses – which slightly distorted what they saw but did not obviously affect their behavior – showed the predicted changes in brain structure.

The results confirm that this aspect of brain structure is partly determined by nurture, as opposed to being entirely specified by nature. A key future challenge is to identify the chemical signaling that enables sensory input to have these effects on brain structure. It might then be possible to use drugs to restore normal brain activity in cases where abnormal sensory input has altered the brain, for example in the condition known as amblyopia (or “lazy eye”).

DOI:http://dx.doi.org/10.7554/eLife.13911.002

Stripe-rearing changes multiple aspects of the structure of primary visual cortex

An important example of brain plasticity is the change in the structure of the orientation map in mammalian primary visual cortex in response to a visual environment consisting of stripes of one orientation. In principle there are many different ways in which the structure of a normal map could change to accommodate increased preference for one orientation. However, until now these changes have been characterised only by the relative sizes of the areas of primary visual cortex representing different orientations. Here we extend to the stripe-reared case a recently proposed Bayesian method for reconstructing orientation maps from intrinsic signal optical imaging data. We first formulated a suitable prior for the stripe-reared case, and developed an efficient method for maximising the marginal likelihood of the model in order to determine the optimal parameters. We then applied this to a set of orientation maps from normal and stripe-reared cats. This analysis revealed that several parameters of overall map structure, specifically the difference between wavelength, scaling and mean of the two vector components of maps, changed in response to stripe-rearing, which together give a more nuanced assessment of the effect of rearing condition on map structure than previous measures. Overall this work expands our understanding of the effects of the environment on brain structure.

Overlapping structures in sensory-motor mappings

This paper examines a biologically-inspired representation technique designed for the support of sensory-motor learning in developmental robotics. An interesting feature of the many topographic neural sheets in the brain is that closely packed receptive fields must overlap in order to fully cover a spatial region. This raises interesting scientific questions with engineering implications: e.g. is overlap detrimental? does it have any benefits? This paper examines the effects and properties of overlap between elements arranged in arrays or maps. In particular we investigate how overlap affects the representation and transmission of spatial location information on and between topographic maps. Through a series of experiments we determine the conditions under which overlap offers advantages and identify useful ranges of overlap for building mappings in cognitive robotic systems. Our motivation is to understand the phenomena of overlap in order to provide guidance for application in sensory-motor learning robots.

On the dynamics of cortical development: synchrony and synaptic self-organization

We describe a model for cortical development that resolves long-standing difficulties of earlier models. It is proposed that, during embryonic development, synchronous firing of neurons and their competition for limited metabolic resources leads to selection of an array of neurons with ultra-small-world characteristics. Consequently, in the visual cortex, macrocolumns linked by superficial patchy connections emerge in anatomically realistic patterns, with an ante-natal arrangement which projects signals from the surrounding cortex onto each macrocolumn in a form analogous to the projection of a Euclidean plane onto a Möbius strip. This configuration reproduces typical cortical response maps, and simulations of signal flow explain cortical responses to moving lines as functions of stimulus velocity, length, and orientation. With the introduction of direct visual inputs, under the operation of Hebbian learning, development of mature selective response “tuning” to stimuli of given orientation, spatial frequency, and temporal frequency would then take place, overwriting the earlier ante-natal configuration. The model is provisionally extended to hierarchical interactions of the visual cortex with higher centers, and a general principle for cortical processing of spatio-temporal images is sketched.

Coordinated optimization of visual cortical maps (I) symmetry-based analysis

In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps.

Coordinated optimization of visual cortical maps (II) numerical studies

In the juvenile brain, the synaptic architecture of the visual cortex remains in a state of flux for months after the natural onset of vision and the initial emergence of feature selectivity in visual cortical neurons. It is an attractive hypothesis that visual cortical architecture is shaped during this extended period of juvenile plasticity by the coordinated optimization of multiple visual cortical maps such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we introduced a class of analytically tractable coordinated optimization models and solved representative examples, in which a spatially complex organization of the OP map is induced by interactions between the maps. We found that these solutions near symmetry breaking threshold predict a highly ordered map layout. Here we examine the time course of the convergence towards attractor states and optima of these models. In particular, we determine the timescales on which map optimization takes place and how these timescales can be compared to those of visual cortical development and plasticity. We also assess whether our models exhibit biologically more realistic, spatially irregular solutions at a finite distance from threshold, when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. We show that, although maps typically undergo substantial rearrangement, no other solutions than pinwheel crystals and stripes dominate in the emerging layouts. Pinwheel crystallization takes place on a rather short timescale and can also occur for detuned wavelengths of different maps. Our numerical results thus support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the architecture of the visual cortex. We discuss several alternative scenarios that may improve the agreement between model solutions and biological observations.

Neurons in the visual cortex of carnivores, primates and their close relatives form spatial representations or maps of multiple stimulus features. In part (I) of this study we theoretically predicted maps that are optima of a variety of optimization principles. When analyzing the joint optimization of two interacting maps we showed that for different optimization principles the resulting optima show a stereotyped, spatially perfectly periodic layout. Experimental maps, however, are much more irregular. In particular, in case of orientation columns it was found that different species show apparently species invariant statistics of point defects, so-called pinwheels. In this paper, we numerically investigate whether the spatial features of the stereotyped optima described in part (I) are expressed on biologically relevant timescales and whether other, spatially irregular, long-living states emerge that better reproduce the experimentally observed statistical properties of orientation maps. Moreover, we explore whether the coordinated optimization of more than two maps can lead to spatially irregular optima.

Link between orientation and retinotopic maps in primary visual cortex

Maps representing the preference of neurons for the location and orientation of a stimulus on the visual field are a hallmark of primary visual cortex. It is not yet known how these maps develop and what function they play in visual processing. One hypothesis postulates that orientation maps are initially seeded by the spatial interference of ON- and OFF-center retinal receptive field mosaics. Here we show that such a mechanism predicts a link between the layout of orientation preferences around singularities of different signs and the cardinal axes of the retinotopic map. Moreover, we confirm the predicted relationship holds in tree shrew primary visual cortex. These findings provide additional support for the notion that spatially structured input from the retina may provide a blueprint for the early development of cortical maps and receptive fields. More broadly, it raises the possibility that spatially structured input from the periphery may shape the organization of primary sensory cortex of other modalities as well.

Online learning and stimulus-driven responses of neurons in visual cortex

In understanding how visual scene is processed in visual cortex, it has been an intriguing problem for theoretical and experimental neuroscientists to examine the relationship between visual stimuli and the induced responses of visual cortex. In particular, it is less explored whether and how the collective responses of visual neurons are patterned to reflect the geometrical regularities. In this paper, through a computation model and statistical analysis, we show that the orientation preference maps induced from correlated visual stimuli exhibit geometrical regularities similar as observed in natural images.

Coverage, continuity, and visual cortical architecture

BACKGROUND

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far.

RESULTS

We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all the previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise.

CONCLUSIONS

Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.

Computational modeling of neuronal map development: insights into disease

The study of the formation of neuronal maps in the brain has greatly increased our understanding of how the brain develops and, in some cases, regenerates. Computational modeling of neuronal map development has been invaluable in integrating complex biological phenomena and synthesizing them into quantitative and predictive frameworks. These models allow us to investigate how neuronal map development is perturbed under conditions of altered development, disease and regeneration. In this article, we use examples of activity-dependent and activity-independent models of retinotopic map formation to illustrate how they can aid our understanding of developmental and acquired disease processes. We note that fully extending these models to specific clinically relevant problems is a largely unexplored domain and suggest future work in this direction. We argue that this type of modeling will be necessary in furthering our understanding of the pathophysiology of neurological diseases and in developing treatments for them. Furthermore, we discuss how the nature of computational and theoretical approaches uniquely places them to bridge the gap between the bench and the clinic.

Theory for the alignment of cortical feature maps during development

We present a developmental model of ocular dominance column formation that takes into account the existence of an array of intrinsically specified cytochrome oxidase blobs. We assume that there is some molecular substrate for the blobs early in development, which generates a spatially periodic modulation of experience-dependent plasticity. We determine the effects of such a modulation on a competitive Hebbian mechanism for the modification of the feedforward afferents from the left and right eyes. We show how alternating left and right eye dominated columns can develop, in which the blobs are aligned with the centers of the ocular dominance columns and receive a greater density of feedforward connections, thus becoming defined extrinsically. More generally, our results suggest that the presence of periodically distributed anatomical markers early in development could provide a mechanism for the alignment of cortical feature maps.

Mapping of contextual modulation in the population response of primary visual cortex

We review the evidence of long-range contextual modulation in V1. Populations of neurons in V1 are activated by a wide variety of stimuli outside of their classical receptive fields (RF), well beyond their surround region. These effects generally involve extra-RF features with an orientation component. The population mapping of orientation preferences to the upper layers of V1 is well understood, as far as the classical RF properties are concerned, and involves organization into pinwheel-like structures. We introduce a novel hypothesis regarding the organization of V1's contextual response. We show that RF and extra-RF orientation preferences are mapped in related ways. Orientation pinwheels are the foci of both types of features. The mapping of contextual features onto the orientation pinwheel has a form that recapitulates the organization of the visual field: an iso-orientation patch within the pinwheel also responds to extra-RF stimuli of the same orientation. We hypothesize that the same form of mapping applies to other stimulus properties that are mapped out in V1, such as colour and contrast selectivity. A specific consequence is that fovea-like properties will be mapped in a systematic way to orientation pinwheels. We review the evidence that cytochrome oxidase blobs comprise the foci of this contextual remapping for colour and low contrasts. Neurodynamics and motion in the visual field are argued to play an important role in the shaping and maintenance of this type of mapping in V1.

Modeling magnification and anisotropy in the primate foveal confluence

A basic organizational principle of the primate visual system is that it maps the visual environment repeatedly and retinotopically onto cortex. Simple algebraic models can be used to describe the projection from visual space to cortical space not only for V1, but also for the complex of areas V1, V2 and V3. Typically a conformal (angle-preserving) projection ensuring local isotropy is regarded as ideal and primate visual cortex is often regarded as an approximation of this ideal. However, empirical data show systematic deviations from this ideal that are especially relevant in the foveal projection. The aims of this study were to map the nature of anisotropy predicted by existing models, to investigate the optimization targets faced by different types of retino-cortical maps, and finally to propose a novel map that better models empirical data than other candidates. The retino-cortical map can be optimized towards a space-conserving homogenous representation or a quasi-conformal mapping. The latter would require a significantly enlarged representation of specific parts of the cortical maps. In particular it would require significant enlargement of parafoveal V2 and V3 which is not supported by empirical data. Further, the recently published principal layout of the foveal singularity cannot be explained by existing models. We suggest a new model that accurately describes foveal data, minimizing cortical surface area in the periphery but suggesting that local isotropy dominates the most foveal part at the expense of additional cortical surface. The foveal confluence is an important example of the detailed trade-offs between the compromises required for the mapping of environmental space to a complex of neighboring cortical areas. Our models demonstrate that the organization follows clear morphogenetic principles that are essential for our understanding of foveal vision in daily life.

Theoretical models of neural circuit development

Proper wiring up of the nervous system is critical to the development of organisms capable of complex and adaptable behaviors. Besides the many experimental advances in determining the cellular and molecular machinery that carries out this remarkable task precisely and robustly, theoretical approaches have also proven to be useful tools in analyzing this machinery. A quantitative understanding of these processes can allow us to make predictions, test hypotheses, and appraise established concepts in a new light. Three areas that have been fruitful in this regard are axon guidance, retinotectal mapping, and activity-dependent development. This chapter reviews some of the contributions made by mathematical modeling in these areas, illustrated by important examples of models in each section. For axon guidance, we discuss models of how growth cones respond to their environment, and how this environment can place constraints on growth cone behavior. Retinotectal mapping looks at computational models for how topography can be generated in populations of neurons based on molecular gradients and other mechanisms such as competition. In activity-dependent development, we discuss theoretical approaches largely based on Hebbian synaptic plasticity rules, and how they can generate maps in the visual cortex very similar to those seen in vivo. We show how theoretical approaches have substantially contributed to the advancement of developmental neuroscience, and discuss future directions for mathematical modeling in the field.

Natural scene statistics and the structure of orientation maps in the visual cortex

Visual activity after eye-opening influences feature map structure in primary visual cortex (V1). For instance, rearing cats in an environment of stripes of one orientation yields an over-representation of that orientation in V1. However, whether such changes also affect the higher-order statistics of orientation maps is unknown. A statistical bias of orientation maps in normally raised animals is that the probability of the angular difference in orientation preference between each pair of points in the cortex depends on the angle of the line joining those points relative to a fixed but arbitrary set of axes. Natural images show an analogous statistical bias; however, whether this drives the development of comparable structure in V1 is unknown. We examined these statistics for normal, stripe-reared and dark-reared cats, and found that the biases present were not consistently related to those present in the input, or to genetic relationships. We compared these results with two computational models of orientation map development, an analytical model and a Hebbian model. The analytical model failed to reproduce the experimentally observed statistics. In the Hebbian model, while orientation difference statistics could be strongly driven by the input, statistics similar to those seen in experimental maps arose only when symmetry breaking was allowed to occur spontaneously. These results suggest that these statistical biases of orientation maps arise primarily spontaneously, rather than being governed by either input statistics or genetic mechanisms.

Innate visual learning through spontaneous activity patterns

Patterns of spontaneous activity in the developing retina, LGN, and cortex are necessary for the proper development of visual cortex. With these patterns intact, the primary visual cortices of many newborn animals develop properties similar to those of the adult cortex but without the training benefit of visual experience. Previous models have demonstrated how V1 responses can be initialized through mechanisms specific to development and prior to visual experience, such as using axonal guidance cues or relying on simple, pairwise correlations on spontaneous activity with additional developmental constraints. We argue that these spontaneous patterns may be better understood as part of an "innate learning" strategy, which learns similarly on activity both before and during visual experience. With an abstraction of spontaneous activity models, we show how the visual system may be able to bootstrap an efficient code for its natural environment prior to external visual experience, and we continue the same refinement strategy upon natural experience. The patterns are generated through simple, local interactions and contain the same relevant statistical properties of retinal waves and hypothesized waves in the LGN and V1. An efficient encoding of these patterns resembles a sparse coding of natural images by producing neurons with localized, oriented, bandpass structure-the same code found in early visual cortical cells. We address the relevance of higher-order statistical properties of spontaneous activity, how this relates to a system that may adapt similarly on activity prior to and during natural experience, and how these concepts ultimately relate to an efficient coding of our natural world.

Establishment of a scaffold for orientation maps in primary visual cortex of higher mammals

In higher mammals, environmentally driven patterns of neural activity do not play a role in the establishment of orientation specificity and maps. It has been proposed that specific long-range interactions provide the scaffold for developing orientation maps. Our model aims at explaining how such a scaffold could develop in the first place. Broad spontaneous activity waves and locally evoked spatially periodic response pattern are used. The model is discussed in relation to biological evidence, and experiments to test the model are proposed. We show that reliable orientation specificity cannot be a result of haphazard cortical wiring, as has been proposed.

Visual cortex: more wiggle room for the brain

Experiments in which one eye of a ferret is removed at birth show subtle effects on the development of visual cortex maps that are in agreement with those predicted by theory.

A model for the thick, thin and pale stripe organization of primate V2

Models based on the idea of dimension reduction have been successful in describing the patterns of ocular dominance, spatial frequency and orientation preference found in primate V1. It is shown here that this approach can be extended to describe the organization of thick, thin and pale cytochrome oxidase stripes of primate V2 given an appropriately constructed stimulus space which includes a 3-valued variable which co-varies with color, orientation and disparity. The model successfully describes several aspects of V2 organization, including the fact that there are two pale stripes for each thick and thin stripe and the strong tendency for stripes to run perpendicular to the V1 border. In addition it predicts the presence of reversals in the direction of mapping of retinal eccentricity which should be more common in the pale stripes than elsewhere.

Alteration of visual input results in a coordinated reorganization of multiple visual cortex maps

In the adult visual cortex, multiple feature maps exist and have characteristic spatial relationships with one another. The relationships can be reproduced by "dimension-reduction" computational models, suggesting that the principles of continuity and coverage may underlie cortical map organization. However, the mechanisms responsible for establishing these relationships are unknown. We explored whether removing one feature map during development causes a coordinated reorganization of the remaining maps or whether the remaining maps are unaffected. We removed the ocular dominance map by monocular enucleation in newborn ferrets, so that single eye stimulation drove the cortex in a more spatially uniform manner in adult monocular animals compared with normal animals. Maps of orientation, spatial frequency, and retinotopy formed in monocular ferrets, but their structures and spatial relationships differed from those in normal ferrets. The wavelength of the orientation map increased, so that the average orientation gradient across the cortex decreased. The decrease in the orientation gradient in monocular animals was most prominent in the high gradient regions of the spatial frequency map, indicating a coordinated reorganization between these two maps. In monocular animals, the orthogonal relationship between the orientation and spatial frequency maps was preserved, and the orthogonal relationship between the orientation and retinotopic maps became more pronounced. These results were consistent with detailed predictions of a dimension-reduction model of cortical organization. Thus, the number of feature maps in a cortical area influences the relationships between them, and inputs to the cortex have a significant role in generating these relationships.

The effect of angioscotomas on map structure in primary visual cortex

When blood vessels occlude the photoreceptor layer in the retina, they cast shadows onto the photoreceptors, creating angioscotomas (regions of the visual field to which that eye is blind). Remarkably, Adams and Horton (2002) have recently shown that it is sometimes possible to observe representations of these angioscotomas anatomically in the primary visual cortices of squirrel monkeys. However, there is substantial variability in the degree and form of these representations. The source of this variability is difficult to determine experimentally, because experimental studies are unavoidably limited by small sample size. In addition, experimental studies cannot compare the map structure that would develop with and without an angioscotoma. Here, we investigate these phenomena computationally using feature-mapping models of visual cortical development, which are not subject to the same limitations. These models suggest that the primary source of variability in angioscotoma representation is the precise timing of the onset of visual experience relative to the time course of ocular dominance column segregation. Furthermore, the models predict that angioscotomas could compete for control of local column layout with other influences such as cortical shape but that they have a small effect on the structure of orientation preference maps.

The development of retinotectal maps: a review of models based on molecular gradients

Information about the world is often represented in the brain in the form of topographic maps. A paradigm example is the topographic representation of the visual world in the optic tectum/superior colliculus. This map initially forms during neural development using activity-independent molecular cues, most notably some type of chemospecific matching between molecular gradients in the retina and corresponding gradients in the tectum/superior colliculus. Exactly how this process might work has been studied both experimentally and theoretically for several decades. This review discusses the experimental data briefly, and then in more detail the theoretical models proposed. The principal conclusions are that (1) theoretical models have helped clarify several important ideas in the field, (2) earlier models were often more sophisticated than more recent models, and (3) substantial revisions to current modelling approaches are probably required to account for more than isolated subsets of the experimental data.