Coverage, continuity, and visual cortical architecture

Reconstructing Group Wavelet Transform From Feature Maps With a Reproducing Kernel Iteration

In this article, we consider the problem of reconstructing an image that is downsampled in the space of its SE(2) wavelet transform, which is motivated by classical models of simple cell receptive fields and feature preference maps in the primary visual cortex. We prove that, whenever the problem is solvable, the reconstruction can be obtained by an elementary project and replace iterative scheme based on the reproducing kernel arising from the group structure, and show numerical results on real images.

Orientation Preference Maps in Microcebus murinus Reveal Size-Invariant Design Principles in Primate Visual Cortex

Orientation preference maps (OPMs) are a prominent feature of primary visual cortex (V1) organization in many primates and carnivores. In rodents, neurons are not organized in OPMs but are instead interspersed in a “salt and pepper” fashion, although clusters of orientation-selective neurons have been reported. Does this fundamental difference reflect the existence of a lower size limit for orientation columns (OCs) below which they cannot be scaled down with decreasing V1 size? To address this question, we examined V1 of one of the smallest living primates, the 60-g prosimian mouse lemur (Microcebus murinus). Using chronic intrinsic signal imaging, we found that mouse lemur V1 contains robust OCs, which are arranged in a pinwheel-like fashion. OC size in mouse lemurs was found to be only marginally smaller compared to the macaque, suggesting that these circuit elements are nearly incompressible. The spatial arrangement of pinwheels is well described by a common mathematical design of primate V1 circuit organization. In order to accommodate OPMs, we found that the mouse lemur V1 covers one-fifth of the cortical surface, which is one of the largest V1-to-cortex ratios found in primates. These results indicate that the primate-type visual cortical circuit organization is constrained by a size limitation and raises the possibility that its emergence might have evolved by disruptive innovation rather than gradual change.

Orientation preference maps are a hallmark of V1 organization in all primates studied thus far, yet they are absent in rodents. It is uncertain whether these structures scale with body or brain size. Using intrinsic signal imaging, Ho et al. reveal the presence of such maps in the V1 of the world’s smallest primate, the mouse lemur (Microcebus murinus).

Neural field model to reconcile structure with function in primary visual cortex

Voltage-sensitive dye imaging experiments in primary visual cortex (V1) have shown that local, oriented visual stimuli elicit stable orientation-selective activation within the stimulus retinotopic footprint. The cortical activation dynamically extends far beyond the retinotopic footprint, but the peripheral spread stays non-selective-a surprising finding given a number of anatomo-functional studies showing the orientation specificity of long-range connections. Here we use a computational model to investigate this apparent discrepancy by studying the expected population response using known published anatomical constraints. The dynamics of input-driven localized states were simulated in a planar neural field model with multiple sub-populations encoding orientation. The realistic connectivity profile has parameters controlling the clustering of long-range connections and their orientation bias. We found substantial overlap between the anatomically relevant parameter range and a steep decay in orientation selective activation that is consistent with the imaging experiments. In this way our study reconciles the reported orientation bias of long-range connections with the functional expression of orientation selective neural activity. Our results demonstrate this sharp decay is contingent on three factors, that long-range connections are sufficiently diffuse, that the orientation bias of these connections is in an intermediate range (consistent with anatomy) and that excitation is sufficiently balanced by inhibition. Conversely, our modelling results predict that, for reduced inhibition strength, spurious orientation selective activation could be generated through long-range lateral connections. Furthermore, if the orientation bias of lateral connections is very strong, or if inhibition is particularly weak, the network operates close to an instability leading to unbounded cortical activation.

Universal transition from unstructured to structured neural maps

Neurons sharing similar features are often selectively connected with a higher probability and should be located in close vicinity to save wiring. Selective connectivity has, therefore, been proposed to be the cause for spatial organization in cortical maps. Interestingly, orientation preference (OP) maps in the visual cortex are found in carnivores, ungulates, and primates but are not found in rodents, indicating fundamental differences in selective connectivity that seem unexpected for closely related species. Here, we investigate this finding by using multidimensional scaling to predict the locations of neurons based on minimizing wiring costs for any given connectivity. Our model shows a transition from an unstructured salt-and-pepper organization to a pinwheel arrangement when increasing the number of neurons, even without changing the selectivity of the connections. Increasing neuronal numbers also leads to the emergence of layers, retinotopy, or ocular dominance columns for the selective connectivity corresponding to each arrangement. We further show that neuron numbers impact overall interconnectivity as the primary reason for the appearance of neural maps, which we link to a known phase transition in an Ising-like model from statistical mechanics. Finally, we curated biological data from the literature to show that neural maps appear as the number of neurons in visual cortex increases over a wide range of mammalian species. Our results provide a simple explanation for the existence of salt-and-pepper arrangements in rodents and pinwheel arrangements in the visual cortex of primates, carnivores, and ungulates without assuming differences in the general visual cortex architecture and connectivity.

Parsimony, Exhaustivity and Balanced Detection in Neocortex

The layout of sensory brain areas is thought to subtend perception. The principles shaping these architectures and their role in information processing are still poorly understood. We investigate mathematically and computationally the representation of orientation and spatial frequency in cat primary visual cortex. We prove that two natural principles, local exhaustivity and parsimony of representation, would constrain the orientation and spatial frequency maps to display a very specific pinwheel-dipole singularity. This is particularly interesting since recent experimental evidences show a dipolar structures of the spatial frequency map co-localized with pinwheels in cat. These structures have important properties on information processing capabilities. In particular, we show using a computational model of visual information processing that this architecture allows a trade-off in the local detection of orientation and spatial frequency, but this property occurs for spatial frequency selectivity sharper than reported in the literature. We validated this sharpening on high-resolution optical imaging experimental data. These results shed new light on the principles at play in the emergence of functional architecture of cortical maps, as well as their potential role in processing information.

Random Wiring, Ganglion Cell Mosaics, and the Functional Architecture of the Visual Cortex

The architecture of iso-orientation domains in the primary visual cortex (V1) of placental carnivores and primates apparently follows species invariant quantitative laws. Dynamical optimization models assuming that neurons coordinate their stimulus preferences throughout cortical circuits linking millions of cells specifically predict these invariants. This might indicate that V1's intrinsic connectome and its functional architecture adhere to a single optimization principle with high precision and robustness. To validate this hypothesis, it is critical to closely examine the quantitative predictions of alternative candidate theories. Random feedforward wiring within the retino-cortical pathway represents a conceptually appealing alternative to dynamical circuit optimization because random dimension-expanding projections are believed to generically exhibit computationally favorable properties for stimulus representations. Here, we ask whether the quantitative invariants of V1 architecture can be explained as a generic emergent property of random wiring. We generalize and examine the stochastic wiring model proposed by Ringach and coworkers, in which iso-orientation domains in the visual cortex arise through random feedforward connections between semi-regular mosaics of retinal ganglion cells (RGCs) and visual cortical neurons. We derive closed-form expressions for cortical receptive fields and domain layouts predicted by the model for perfectly hexagonal RGC mosaics. Including spatial disorder in the RGC positions considerably changes the domain layout properties as a function of disorder parameters such as position scatter and its correlations across the retina. However, independent of parameter choice, we find that the model predictions substantially deviate from the layout laws of iso-orientation domains observed experimentally. Considering random wiring with the currently most realistic model of RGC mosaic layouts, a pairwise interacting point process, the predicted layouts remain distinct from experimental observations and resemble Gaussian random fields. We conclude that V1 layout invariants are specific quantitative signatures of visual cortical optimization, which cannot be explained by generic random feedforward-wiring models.

What, if anything, are topological maps for?

What, if anything, is the functional significance of spatial patterning in cortical feature maps? We ask this question of four major theories of cortical map formation: self-organizing maps, wiring optimization, place coding, and reaction-diffusion. We argue that (i) self-organizing maps yield spatial patterning only as a by-product of efficient mechanisms for developing environmentally appropriate distributions of feature preferences, (ii) wiring optimization assumes rather than explains a map-like organization, (iii) place-coding mechanisms can at best explain only a subset of maps in functional terms, and (iv) reaction-diffusion models suggest two factors in the evolution of maps, the first based on efficient development of feature distributions, and the second based on generating feature-specific long-range recurrent cortical circuitry. None of these explanations for the existence of topological maps requires spatial patterning in maps to be useful. Thus despite these useful frameworks for understanding how maps form and how they are wired, the possibility that patterns are merely epiphenomena in the evolution of mammalian neocortex cannot be rejected. The article is intended as a nontechnical introduction to the assumptions and predictions of these four important classes of models, along with other possible functional explanations for maps.

Multimap formation in visual cortex

An extrastriate visual area such as V2 or V4 contains neurons selective for a multitude of complex shapes, all sharing a common topographic organization. Simultaneously developing multiple interdigitated maps--hereafter a "multimap"--is challenging in that neurons must compete to generate a diversity of response types locally, while cooperating with their dispersed same-type neighbors to achieve uniform visual field coverage for their response type at all orientations, scales, etc. Previously proposed map development schemes have relied on smooth spatial interaction functions to establish both topography and columnar organization, but by locally homogenizing cells' response properties, local smoothing mechanisms effectively rule out multimap formation. We found in computer simulations that the key requirements for multimap development are that neurons are enabled for plasticity only within highly active regions of cortex designated "learning eligibility regions" (LERs), but within an LER, each cell's learning rate is determined only by its activity level with no dependence on location. We show that a hybrid developmental rule that combines spatial and activity-dependent learning criteria in this way successfully produces multimaps when the input stream contains multiple distinct feature types, or in the degenerate case of a single feature type, produces a V1-like map with "salt-and-pepper" structure. Our results support the hypothesis that cortical maps containing a fine mixture of different response types, whether in monkey extrastriate cortex, mouse V1 or elsewhere in the cortex, rather than signaling a breakdown of map formation mechanisms at the fine scale, are a product of a generic cortical developmental scheme designed to map cells with a diversity of response properties across a shared topographic space.

How uncertainty bounds the shape index of simple cells

We propose a theoretical motivation to quantify actual physiological features, such as the shape index distributions measured by Jones and Palmer in cats and by Ringach in macaque monkeys. We will adopt the uncertainty principle associated to the task of detection of position and orientation as the main tool to provide quantitative bounds on the family of simple cells concretely implemented in primary visual cortex.Mathematics Subject Classification (2000)2010: 62P10, 43A32, 81R15.

Can retinal ganglion cell dipoles seed iso-orientation domains in the visual cortex?

It has been argued that the emergence of roughly periodic orientation preference maps (OPMs) in the primary visual cortex (V1) of carnivores and primates can be explained by a so-called statistical connectivity model. This model assumes that input to V1 neurons is dominated by feed-forward projections originating from a small set of retinal ganglion cells (RGCs). The typical spacing between adjacent cortical orientation columns preferring the same orientation then arises via Moiré-Interference between hexagonal ON/OFF RGC mosaics. While this Moiré-Interference critically depends on long-range hexagonal order within the RGC mosaics, a recent statistical analysis of RGC receptive field positions found no evidence for such long-range positional order. Hexagonal order may be only one of several ways to obtain spatially repetitive OPMs in the statistical connectivity model. Here, we investigate a more general requirement on the spatial structure of RGC mosaics that can seed the emergence of spatially repetitive cortical OPMs, namely that angular correlations between so-called RGC dipoles exhibit a spatial structure similar to that of OPM autocorrelation functions. Both in cat beta cell mosaics as well as primate parasol receptive field mosaics we find that RGC dipole angles are spatially uncorrelated. To help assess the level of these correlations, we introduce a novel point process that generates mosaics with realistic nearest neighbor statistics and a tunable degree of spatial correlations of dipole angles. Using this process, we show that given the size of available data sets, the presence of even weak angular correlations in the data is very unlikely. We conclude that the layout of ON/OFF ganglion cell mosaics lacks the spatial structure necessary to seed iso-orientation domains in the primary visual cortex.

Mechanisms for stable, robust, and adaptive development of orientation maps in the primary visual cortex

Development of orientation maps in ferret and cat primary visual cortex (V1) has been shown to be stable, in that the earliest measurable maps are similar in form to the eventual adult map, robust, in that similar maps develop in both dark rearing and in a variety of normal visual environments, and yet adaptive, in that the final map pattern reflects the statistics of the specific visual environment. How can these three properties be reconciled? Using mechanistic models of the development of neural connectivity in V1, we show for the first time that realistic stable, robust, and adaptive map development can be achieved by including two low-level mechanisms originally motivated from single-neuron results. Specifically, contrast-gain control in the retinal ganglion cells and the lateral geniculate nucleus reduces variation in the presynaptic drive due to differences in input patterns, while homeostatic plasticity of V1 neuron excitability reduces the postsynaptic variability in firing rates. Together these two mechanisms, thought to be applicable across sensory systems in general, lead to biological maps that develop stably and robustly, yet adapt to the visual environment. The modeling results suggest that topographic map stability is a natural outcome of low-level processes of adaptation and normalization. The resulting model is more realistic, simpler, and far more robust, and is thus a good starting point for future studies of cortical map development.

Neural maps versus salt-and-pepper organization in visual cortex

Theoretical neuroscientists have long been intrigued by the spatial patterns of neuronal selectivities observed in the visual cortices of many mammals, including primates. While theoretical studies have contributed significantly to our understanding of how the brain learns to see, recent experimental discoveries of the spatial irregularity of visual response properties in the rodent visual cortex have prompted new questions about the origin and functional significance of cortical maps. Characterizing the marked differences of cortical design principles among species and comparing them may provide us with a deeper understanding of primate and non-primate vision.