Induction of epithelial-mesenchymal transition (EMT) in breast cancer cells is calcium signal dependent

Signals from the tumor microenvironment trigger cancer cells to adopt an invasive phenotype through epithelial-mesenchymal transition (EMT). Relatively little is known regarding key signal transduction pathways that serve as cytosolic bridges between cell surface receptors and nuclear transcription factors to induce EMT. A better understanding of these early EMT events may identify potential targets for the control of metastasis. One rapid intracellular signaling pathway that has not yet been explored during EMT induction is calcium. Here we show that stimuli used to induce EMT produce a transient increase in cytosolic calcium levels in human breast cancer cells. Attenuation of the calcium signal by intracellular calcium chelation significantly reduced epidermal growth factor (EGF)- and hypoxia-induced EMT. Intracellular calcium chelation also inhibited EGF-induced activation of signal transducer and activator of transcription 3 (STAT3), while preserving other signal transduction pathways such as Akt and extracellular signal-regulated kinase 1/2 (ERK1/2) phosphorylation. To identify calcium-permeable channels that may regulate EMT induction in breast cancer cells, we performed a targeted siRNA-based screen. We found that transient receptor potential-melastatin-like 7 (TRPM7) channel expression regulated EGF-induced STAT3 phosphorylation and expression of the EMT marker vimentin. Although intracellular calcium chelation almost completely blocked the induction of many EMT markers, including vimentin, Twist and N-cadherin, the effect of TRPM7 silencing was specific for vimentin protein expression and STAT3 phosphorylation. These results indicate that TRPM7 is a partial regulator of EMT in breast cancer cells, and that other calcium-permeable ion channels are also involved in calcium-dependent EMT induction. In summary, this work establishes an important role for the intracellular calcium signal in the induction of EMT in human breast cancer cells. Manipulation of calcium-signaling pathways controlling EMT induction in cancer cells may therefore be an important therapeutic strategy for preventing metastases.

Sparse coding can predict primary visual cortex receptive field changes induced by abnormal visual input

Receptive fields acquired through unsupervised learning of sparse representations of natural scenes have similar properties to primary visual cortex (V1) simple cell receptive fields. However, what drives in vivo development of receptive fields remains controversial. The strongest evidence for the importance of sensory experience in visual development comes from receptive field changes in animals reared with abnormal visual input. However, most sparse coding accounts have considered only normal visual input and the development of monocular receptive fields. Here, we applied three sparse coding models to binocular receptive field development across six abnormal rearing conditions. In every condition, the changes in receptive field properties previously observed experimentally were matched to a similar and highly faithful degree by all the models, suggesting that early sensory development can indeed be understood in terms of an impetus towards sparsity. As previously predicted in the literature, we found that asymmetries in inter-ocular correlation across orientations lead to orientation-specific binocular receptive fields. Finally we used our models to design a novel stimulus that, if present during rearing, is predicted by the sparsity principle to lead robustly to radically abnormal receptive fields.

The responses of neurons in the primary visual cortex (V1), a region of the brain involved in encoding visual input, are modified by the visual experience of the animal during development. For example, most neurons in animals reared viewing stripes of a particular orientation only respond to the orientation that the animal experienced. The responses of V1 cells in normal animals are similar to responses that simple optimisation algorithms can learn when trained on images. However, whether the similarity between these algorithms and V1 responses is merely coincidental has been unclear. Here, we used the results of a number of experiments where animals were reared with modified visual experience to test the explanatory power of three related optimisation algorithms. We did this by filtering the images for the algorithms in ways that mimicked the visual experience of the animals. This allowed us to show that the changes in V1 responses in experiment were consistent with the algorithms. This is evidence that the precepts of the algorithms, notably sparsity, can be used to understand the development of V1 responses. Further, we used our model to propose a novel rearing condition which we expect to have a dramatic effect on development.

Optimality and saturation in axonal chemotaxis

Chemotaxis (detecting and following chemical gradients) plays a crucial role in the function of many biological systems. In particular, gradient following by neuronal growth cones is important for the correct wiring of the nervous system. There is increasing interest in the constraints that determine how small chemotacting devices respond to gradients, but little quantitative information is available in this regard for neuronal growth cones. Mortimer et al. ( 2009 ) and Mortimer, Dayan, Burrage, and Goodhill ( 2011 ) proposed a Bayesian ideal observer model that predicts chemotactic performance for shallow gradients. Here we investigated two important aspects of this model. First, we found by numerical simulation that although the analytical predictions of the model assume shallow gradients, these predictions remain remarkably robust to large deviations in gradient steepness. Second, we found experimentally that the chemotactic response increased linearly with gradient steepness for very shallow gradients as predicted by the model; however, the response saturated for steeper gradients. This saturation could be reproduced in simulations of a growth rate modulation response mechanism. Together these results illuminate the domain of validity of the Bayesian model and give further insight into the biological mechanisms of axonal chemotaxis.

Calcium and cAMP levels interact to determine attraction versus repulsion in axon guidance

Correct guidance of axons to their targets depends on an intricate network of signaling molecules in the growth cone. Calcium and cAMP are two key regulators of whether axons are attracted or repelled by molecular gradients, but how these molecules interact to determine guidance responses remains unclear. Here, we constructed a mathematical model for the relevant signaling network, which explained a large range of previous biological data and made predictions for when axons will be attracted or repelled. We then confirmed these predictions experimentally, in particular showing that while small increases in cAMP levels promote attraction large increases do not, and that under some circumstances reducing cAMP levels promotes attraction. Together, these results show that a relatively simple mathematical model can quantitatively predict guidance decisions across a wide range of conditions, and that calcium and cAMP levels play a more complex role in these decisions than previously determined.

Sparse coding on the spot: spontaneous retinal waves suffice for orientation selectivity

Ohshiro, Hussain, and Weliky (2011) recently showed that ferrets reared with exposure to flickering spot stimuli, in the absence of oriented visual experience, develop oriented receptive fields. They interpreted this as refutation of efficient coding models, which require oriented input in order to develop oriented receptive fields. Here we show that these data are compatible with the efficient coding hypothesis if the influence of spontaneous retinal waves is considered. We demonstrate that independent component analysis learns predominantly oriented receptive fields when trained on a mixture of spot stimuli and spontaneous retinal waves. Further, we show that the efficient coding hypothesis provides a compelling explanation for the contrast between the lack of receptive field changes seen in animals reared with spot stimuli and the significant cortical reorganisation observed in stripe-reared animals.

The Combinatorics of Neurite Self-Avoidance

During neural development in Drosophila, the ability of neurite branches to recognize whether they are from the same or different neurons depends crucially on the molecule Dscam1. In particular, this recognition depends on the stochastic acquisition of a unique combination of Dscam1 isoforms out of a large set of possible isoforms. To properly interpret these findings, it is crucial to understand the combinatorics involved, which has previously been attempted only using stochastic simulations for some specific parameter combinations. Here we present closed-form solutions for the general case. These reveal the relationships among the key variables and how these constrain possible biological scenarios.

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.

Cyclic nucleotide-dependent switching of mammalian axon guidance depends on gradient steepness

Correct wiring of the nervous system during development requires axons to respond appropriately to gradients of attractive and repulsive guidance cues. However, the steepness and concentration of these gradients vary in vivo, for instance, with distance from the target. Understanding how these changing conditions affect the navigation strategies used by developing axons is important for understanding how they are guided over long distances. Previous work has shown that cyclic nucleotide levels determine whether axons are attracted or repelled by steep gradients of the same guidance cue, but it is unknown whether this is also true for shallow gradients. We therefore investigated the guidance responses of rat superior cervical ganglion (SCG) axons in both steep and shallow gradients of nerve growth factor (NGF). In steep gradients we found that cyclic nucleotide-dependent switching occurred, consistent with previous reports. Surprisingly however, we found that in shallow NGF gradients, cyclic nucleotide-dependent switching did not occur. These results suggest that there may be substantial differences in the way axons respond to gradient-based guidance cues depending on where they are within the gradient.

A simple model can unify a broad range of phenomena in retinotectal map development

A paradigm model system for studying the development of patterned connections in the nervous system is the topographic map formed by retinal axons in the optic tectum/superior colliculus. Starting in the 1970s, a series of computational models have been proposed to explain map development in both normal conditions, and perturbed conditions where the retina and/or tectum/superior colliculus are altered. This stands in contrast to more recent models that have often been simpler than older ones, and tend to address more limited data sets, but include more recent genetic manipulations. The original exploration of many of the early models was one-dimensional and limited by the computational resources of the time. This leaves open the ability of these early models to explain both map development in two dimensions, and the genetic manipulation data that have only appeared more recently. In this article, we show that a two-dimensional and updated version of the XBAM model (eXtended Branch Arrow Model), first proposed in 1982, reproduces a range of surgical map manipulations not yet demonstrated by more modern models. A systematic exploration of the parameter space of this model in two dimensions also reveals richer behavior than that apparent from the original one-dimensional versions. Furthermore, we show that including a specific type of axon-axon interaction can account for the map collapse recently observed when particular receptor levels are genetically manipulated in a subset of retinal ganglion cells. Together these results demonstrate that balancing multiple influences on map development seems to be necessary to explain many biological phenomena in retinotectal map formation, and suggest important constraints on the underlying biological variables.

Statistical structure of lateral connections in the primary visual cortex

BACKGROUND

The statistical structure of the visual world offers many useful clues for understanding how biological visual systems may understand natural scenes. One particularly important early process in visual object recognition is that of grouping together edges which belong to the same contour. The layout of edges in natural scenes have strong statistical structure. One such statistical property is that edges tend to lie on a common circle, and this 'co-circularity' can predict human performance at contour grouping. We therefore tested the hypothesis that long-range excitatory lateral connections in the primary visual cortex, which are believed to be involved in contour grouping, display a similar co-circular structure.

RESULTS

By analyzing data from tree shrews, where information on both lateral connectivity and the overall structure of the orientation map was available, we found a surprising diversity in the relevant statistical structure of the connections. In particular, the extent to which co-circularity was displayed varied significantly.

CONCLUSIONS

Overall, these data suggest the intriguing possibility that V1 may contain both co-circular and anti-cocircular connections.

The response of dorsal root ganglion axons to nerve growth factor gradients depends on spinal level

Directed sensory axon regeneration has the potential to promote functional recovery after peripheral nerve injury. Using a novel guidance assay to generate precisely controllable nerve growth factor gradients, we show for the first time that the guidance and outgrowth response of rat dorsal root ganglion neurons to identical nerve growth factor gradients depends on the rostrocaudal origin of the dorsal root ganglion explant. These findings have implications for the study of peripheral nerve regeneration in response to exogenous neurotrophins such as nerve growth factor, and provide new insight into the clinical potential of nerve growth factor in the treatment of nerve injury.

Bayes-optimal chemotaxis

Chemotaxis plays a crucial role in many biological processes, including nervous system development. However, fundamental physical constraints limit the ability of a small sensing device such as a cell or growth cone to detect an external chemical gradient. One of these is the stochastic nature of receptor binding, leading to a constantly fluctuating binding pattern across the cell's array of receptors. This is analogous to the uncertainty in sensory information often encountered by the brain at the systems level. Here we derive analytically the Bayes-optimal strategy for combining information from a spatial array of receptors in both one and two dimensions to determine gradient direction. We also show how information from more than one receptor species can be optimally integrated, derive the gradient shapes that are optimal for guiding cells or growth cones over the longest possible distances, and illustrate that polarized cell behavior might arise as an adaptation to slowly varying environments. Together our results provide closed-form predictions for variations in chemotactic performance over a wide range of gradient conditions.

A Boolean model of the gene regulatory network underlying Mammalian cortical area development

The cerebral cortex is divided into many functionally distinct areas. The emergence of these areas during neural development is dependent on the expression patterns of several genes. Along the anterior-posterior axis, gradients of Fgf8, Emx2, Pax6, Coup-tfi, and Sp8 play a particularly strong role in specifying areal identity. However, our understanding of the regulatory interactions between these genes that lead to their confinement to particular spatial patterns is currently qualitative and incomplete. We therefore used a computational model of the interactions between these five genes to determine which interactions, and combinations of interactions, occur in networks that reproduce the anterior-posterior expression patterns observed experimentally. The model treats expression levels as Boolean, reflecting the qualitative nature of the expression data currently available. We simulated gene expression patterns created by all possible networks containing the five genes of interest. We found that only of these networks were able to reproduce the experimentally observed expression patterns. These networks all lacked certain interactions and combinations of interactions including auto-regulation and inductive loops. Many higher order combinations of interactions also never appeared in networks that satisfied our criteria for good performance. While there was remarkable diversity in the structure of the networks that perform well, an analysis of the probability of each interaction gave an indication of which interactions are most likely to be present in the gene network regulating cortical area development. We found that in general, repressive interactions are much more likely than inductive ones, but that mutually repressive loops are not critical for correct network functioning. Overall, our model illuminates the design principles of the gene network regulating cortical area development, and makes novel predictions that can be tested experimentally.

Understanding the development of the brain is an important challenge. Progress on this problem will give insight into how the brain works and what can go wrong to cause developmental disorders like autism and learning disability. This paper examines the development of the outer part of the mammalian brain, the cerebral cortex. This part of the brain contains different areas with specialised functions. Over the past decade, several genes have been identified that play a major role in the development of cortical areas. During development, these genes are expressed in different patterns across the surface of the cortex. Experiments have shown that these genes interact with each other so that they each regulate how much other genes in the group are expressed. However, the experimental data are consistent with many different regulatory networks. In this study, we use a computational model to systematically screen many possible networks. This allows us to predict which regulatory interactions between these genes are important for the patterns of gene expression in the cortex to develop correctly.

Assays for eukaryotic cell chemotaxis

Chemotaxis is essential for many biological processes. Much of our understanding of the mechanisms underlying chemotaxis is based on a variety of in vitro assays. We review these assays, dividing them into groups depending on the process used to generate the gradient. We describe how each method works, its strengths and limitations, and provide some information about the kinds of cells that have been studied with each assay.

Quantitative studies of neuronal chemotaxis in 3D

During development a variety of cell types are guided by molecular concentration gradients to form tissues and organ systems. In the nervous system, the migration and neuronal pathfinding that occurs during development is organized and driven by "guidance cues." Some of these cues are substrate bound or nondiffusible, while many are diffusible and form gradients within the developing embryo to guide neurons and neurites to their appropriate destination. There have been many approaches used to discover and characterize the multitude of guidance cues, their cognate receptors, and how these cues and receptors are regulated to achieve the highly detailed connections found in the nervous system. Here we present a method for creating precisely controlled gradients of molecular factors within a three-dimensional culture environment. The method is based on a non contact mediated delivery of biomolecules to the surface of a collagen gel. The factors are printed in a pattern on the top of a gel containing the tissue or cell type of interest embedded in the gel. The formation of the gradient is dependent upon the diffusion of the printed molecule in the gel. The concentration of the factor within the gel becomes independent of depth rapidly, and the gradient becomes smooth on a similar time scale. The gradients formed can remain relatively stable for a day or more. Moreover, the steepness and molar concentration of tropic or trophic factors within the gradient can be controlled.

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.

Generating controlled molecular gradients in 3D gels

A new method for producing molecular gradients of arbitrary shape in thin three dimensional gels is described. Patterns are produced on the surface of the gel by printing with a micropump that dispenses small droplets of solution at controlled rates. The molecules in the solution rapidly diffuse into the gel and create a smooth concentration profile that is independent of depth. The pattern is relatively stable for long times, and its evolution can be accurately described by finite element modeling of the diffusion equation. As a demonstration of the method, direct measurements of protein gradients are performed by quantitative fluorescence microscopy. A complementary technique for measuring diffusion coefficients is also presented. This rapid, flexible, contactless approach to gradient generation is ideally suited for cell culture experiments to investigate the role of gradients of diffusible substances in processes such as chemotaxis, morphogenesis, and pattern formation, as well as for high-throughput screening of system responses to a wide range of chemical concentrations.

Adaptation is not required to explain the long-term response of axons to molecular gradients

It has been suggested that growth cones navigating through the developing nervous system might display adaptation, so that their response to gradient signals is conserved over wide variations in ligand concentration. Recently however, a new chemotaxis assay that allows the effect of gradient parameters on axonal trajectories to be finely varied has revealed a decline in gradient sensitivity on either side of an optimal concentration. We show that this behavior can be quantitatively reproduced with a computational model of axonal chemotaxis that does not employ explicit adaptation. Two crucial components of this model required to reproduce the observed sensitivity are spatial and temporal averaging. These can be interpreted as corresponding, respectively, to the spatial spread of signaling effects downstream from receptor binding, and to the finite time over which these signaling effects decay. For spatial averaging, the model predicts that an effective range of roughly one-third of the extent of the growth cone is optimal for detecting small gradient signals. For temporal decay, a timescale of about 3 minutes is required for the model to reproduce the experimentally observed sensitivity.

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.

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

Primary visual cortex contains multiple maps of features of the visual scene, including visual field position, orientation, direction, ocular dominance and spatial frequency. The complex relationships between these maps provide clues to the strategies the cortex uses for representing and processing information. Here we simulate the combined development of all these map systems using a computational model, the elastic net. We show that this model robustly produces combined maps of these four variables that bear a close resemblance to experimental maps. In addition we show that the experimentally observed effects of monocular deprivation and single-orientation rearing can be reproduced in this model, and we make some testable predictions. These results provide strong support for the hypothesis that cortical representations attempt to optimize a trade-off between coverage and continuity.

Influence of Lateral Connections on the Structure of Cortical Maps

Maps of ocular dominance and orientation in primary visual cortex have a highly characteristic structure. The factors that determine this structure are still largely unknown. In particular, it is unclear how short-range excitatory and inhibitory connections between nearby neurons influence structure both within and between maps. Using a generalized version of a well-known computational model of visual cortical map development, we show that the number of excitatory and inhibitory oscillations in this interaction function critically influences map structure. Specifically, we demonstrate that functions that oscillate more than once do not produce maps closely resembling those seen biologically. This strongly suggests that local lateral connections in visual cortex oscillate only once and have the form of a Mexican hat.

Predicting Axonal Response to Molecular Gradients with a Computational Model of Filopodial Dynamics

Axons are often guided to their targets in the developing nervous system by attractive or repulsive molecular concentration gradients. We propose a computational model for gradient sensing and directed movement of the growth cone mediated by filopodia. We show that relatively simple mechanisms are sufficient to generate realistic trajectories for both the short-term response of axons to steep gradients and the long-term response of axons to shallow gradients. The model makes testable predictions for axonal response to attractive and repulsive gradients of different concentrations and steepness, the size of the intracellular amplification of the gradient signal, and the differences in intracellular signaling required for repulsive versus attractive turning.

A new chemotaxis assay shows the extreme sensitivity of axons to molecular gradients

Axonal chemotaxis is believed to be important in wiring up the developing and regenerating nervous system, but little is known about how axons actually respond to molecular gradients. We report a new quantitative assay that allows the long-term response of axons to gradients of known and controllable shape to be examined in a three-dimensional gel. Using this assay, we show that axons may be nature's most-sensitive gradient detectors, but this sensitivity exists only within a narrow range of ligand concentrations. This assay should also be applicable to other biological processes that are controlled by molecular gradients, such as cell migration and morphogenesis.

A theoretical model of axon guidance by the Robo code

After crossing the midline, different populations of commissural axons in Drosophila target specific longitudinal pathways at different distances from the midline. It has recently been shown that this choice of lateral position is governed by the particular combination of Robo receptors expressed by these axons, presumably in response to a gradient of Slit released by the midline. Here we propose a simple theoretical model of this combinatorial coding scheme. The principal results of the model are that purely quantitative rather than qualitative differences between the different Robo receptors are sufficient to account for the effects observed following removal or ectopic expression of specific Robo receptors, and that the steepness of the Slit gradient in vivo must exceed a certain minimum for the results observed experimentally to be consistent.

Are visual cortex maps optimized for coverage?

The elegant regularity of maps of variables such as ocular dominance, orientation, and spatial frequency in primary visual cortex has prompted many people to suggest that their structure could be explained by an optimization principle. Up to now, the standard way to test this hypothesis has been to generate artificial maps by optimizing a hypothesized objective function and then to compare these artificial maps with real maps using a variety of quantitative criteria. If the artificial maps are similar to the real maps, this provides some evidence that the real cortex may be optimizing a similar function to the one hypothesized. Recently, a more direct method has been proposed for testing whether real maps represent local optima of an objective function (Swindale, Shoham, Grinvald, Bonhoeffer, & Hübener, 2000). In this approach, the value of the hypothesized function is calculated for a real map, and then the real map is perturbed in certain ways and the function recalculated. If each of these perturbations leads to a worsening of the function, it is tempting to conclude that the real map is quite likely to represent a local optimum of that function. In this article, we argue that such perturbation results provide only weak evidence in favor of the optimization hypothesis.

Auto-SOM: recursive parameter estimation for guidance of self-organizing feature maps

An important technique for exploratory data analysis is to form a mapping from the high-dimensional data space to a low-dimensional representation space such that neighborhoods are preserved. A popular method for achieving this is Kohonen's self-organizing map (SOM) algorithm. However, in its original form, this requires the user to choose the values of several parameters heuristically to achieve good performance. Here we present the Auto-SOM, an algorithm that estimates the learning parameters during the training of SOMs automatically. The application of Auto-SOM provides the facility to avoid neighborhood violations up to a user-defined degree in either mapping direction. Auto-SOM consists of a Kalman filter implementation of the SOM coupled with a recursive parameter estimation method. The Kalman filter trains the neurons' weights with estimated learning coefficients so as to minimize the variance of the estimation error. The recursive parameter estimation method estimates the width of the neighborhood function by minimizing the prediction error variance of the Kalman filter. In addition, the "topographic function" is incorporated to measure neighborhood violations and prevent the map's converging to configurations with neighborhood violations. It is demonstrated that neighborhoods can be preserved in both mapping directions as desired for dimension-reducing applications. The development of neighborhood-preserving maps and their convergence behavior is demonstrated by three examples accounting for the basic applications of self-organizing feature maps.

Analysis of the elastic net model applied to the formation of ocular dominance and orientation columns

The development and structure of orientation (OR) and ocular dominance (OD) maps in the primary visual cortex of cats and monkeys can be modelled using the elastic net algorithm, which attempts to find an 'optimal' cortical representation of the input features. Here we analyse this behaviour in terms of parameters of the feature space. We derive expressions for the OR periodicity, and the first bifurcation point as a function of the annealing parameter using the methods of Durbin et al (Durbin R, Szeliski R and Yuille A 1989 Neural Computation 1 348-58). We also investigate the effect of the relative order of OR and OD development on overall map structure. This analysis suggests that developmental order can be predicted from the final OR and OD periodicities. In conjunction with experimentally measured values for these periodicities, the model predicts that (i) in normal macaques OD develops first, (ii) in normal cats OR develops first and (iii) in strabismic cats OD develops first.

Retinotectal maps: molecules, models and misplaced data

The mechanisms underlying the formation of topographic maps in the retinotectal system have long been debated. Recently, members of the Eph and ephrin receptor-ligand family have been found to provide a molecular substrate for one type of mechanism, that of chemospecific gradient matching, as proposed by Sperry. However, experiments over several decades have demonstrated that there is more to map formation than gradient matching. This article briefly reviews the old and new findings, argues that these two types of data must be properly integrated in order to understand map formation fully, and suggests some experimental and theoretical ways to begin this process.

Theoretical analysis of gradient detection by growth cones

Gradients of diffusible and substrate-bound molecules play an important role in guiding axons to appropriate targets in the developing nervous system. Although some of the molecules involved have recently been identified, little is known about the physical mechanisms by which growth cones sense gradients. This article applies the seminal Berg and Purcell (1977) model of gradient sensing to this problem. The model provides estimates for the statistical fluctuations in the measurement of concentration by a small sensing device. By assuming that gradient detection consists of the comparison of concentrations at two spatially or temporally separated points, the model therefore provides an estimate for the steepness of gradient that can be detected as a function of physiological parameters. The model makes the following specific predictions. (a) It is more likely that growth cones use a spatial rather than temporal sensing strategy. (b) Growth cone sensitivity increases with the concentration of ligand, the speed of ligand diffusion, the size of the growth cone, and the time over which it averages the gradient signal. (c) The minimum detectable gradient steepness for growth cones is roughly in the range 1-10%. (d) This value varies depending on whether a bound or freely diffusing ligand is being sensed, and on whether the sensing occurs in three or two dimensions. The model also makes predictions concerning the role of filopodia in gradient detection.

The influence of neural activity and intracortical connectivity on the periodicity of ocular dominance stripes

Several factors may interact to determine the periodicity of ocular dominance stripes in cat and monkey visual cortex. Previous theoretical work has suggested roles for the width of cortical interactions and the strength of between-eye correlations. Here, a model based on an explicit optimization is presented that allows a thorough characterization of how these and other parameters of the afferent input could affect ocular dominance stripe periodicity. The principle conclusions are that increasing the width of within-eye correlations leads to wider columns, and, surprisingly, that increasing the width of cortical interactions can sometimes lead to narrower columns.

Mathematical guidance for axons

Axon guidance by gradients plays an important role in wiring up the developing nervous system. Growth cones seem to sense a concentration difference across their spatial extent, and convert this into a signal to move up or down a gradient. In this article, a simple mathematical framework is developed to understand when and where gradient detection can occur as a function of gradient shape. This framework is applied to two examples:the guidance of axons by target-derived diffusible factors in vivo and in collagen gels, and guidance by substrate-bound gradients of optimal shape, as might be relevant in the retinotectal system.Two distinct spatial limits on guidance emerge: I mm for a target-derived diffusible gradient, and I cm for a substrate-bound gradient.

Axon guidance: stretching gradients to the limit

Neuronal growth cones, the sensory-motile structures at the tips of developing axons, navigate to their targets over distances that can be many times greater than their diameter. They may accomplish this impressive task by following spatial gradients of axon guidance molecules in their environment (Bonhoeffer & Gierer, 1984; Tessier-Lavigne & Placzek, 1991; Baier & Bonhoeffer, 1994). We calculate the optimal shape of a gradient and the distance over which it can be detected by a growth cone for two competing mechanistic models of axon guidance. The results are surprisingly simple: Regardless of the mechanism, the maximum distance is about 1 cm. Since gradients and growth cones have coevolved, we suggest that the shape of the gradient in situ will predict the mechanism of gradient detection. In addition, we show that the experimentally determined dissociation constants for receptor-ligand complexes implicated in axon guidance are about optimal with respect to maximizing guidance distance. The relevance of these results to the retinotectal system is discussed.

A Unifying Objective Function for Topographic Mappings

Many different algorithms and objective functions for topographic mappings have been proposed. We show that several of these approaches can be seen as particular cases of a more general objective function. Consideration of a very simple mapping problem reveals large differences in the form of the map that each particular case favors. These differences have important consequences for the practical application of topographic mapping methods.

Diffusion in axon guidance

Axon guidance by target-derived diffusible factors plays an important role in the development of the nervous system. This paper considers the constraints imposed on this process by the mathematics of diffusion. A point source continuously producing a factor into an infinite three-dimensional volume is considered as a model for both the in vivo and in vitro situation. Basic constraints for effective guidance are assumed to be that the concentration falls between certain maximum and minimum limits, and that the percentage change in concentration across the width of the growth cone exceeds a certain minimum value. The evolution of the shape of the gradient over time is analysed. Using biologically reasonable parameter values, it is shown that the maximum range over which growth cone guidance by a diffusible factor is possible for large times (several days) after the start of the production of the factor is 500-1000 microm. This maximum distance is independent of the diffusion constant of the diffusing molecule, applies to both chemoattractants and chemorepellents, and agrees with experimental data. At earlier times, however, the constraints may be satisfied for distances up to several millimetres. The time it takes for this maximum guidance distance to fall to the asymptotic value depends on the diffusion constant. This time is a few hours for a small molecule but as much as a few days for a large molecule. The model therefore predicts that guidance over distances larger than 1000 microm is possible if the start of production of the factor is carefully matched to the time when guidance is required.

Influences on the global structure of cortical maps

Cortical maps often contain global spatial structure: however, theoretical accounts for their development have generally concentrated on reproducing only local structure. We show that the elastic net model of cortical map formation can closely approximate the global structure of the ocular dominance column map observed in macaque primary visual cortex. A key component is the assumption of spatially non-uniform and anisotropic correlations in the retina. This work shows how genetic and epigenetic effects could combine to establish characteristic global structure in cortical maps.

Theory meets experiment: correlated neural activity helps determine ocular dominance column periodicity

The development of ocular dominance columns in primary visual cortex has attracted much interest from both experimentalists and theoreticians. One key parameter of these columns is their periodicity - it is thus important to understand how this is determined. Novel experimental work demonstrates that the periodicity is influenced by the temporal patterning of afferent activity, as predicted by recent theoretical work.

An evaluation of the use of multidimensional scaling for understanding brain connectivity

A large amount of data is now available about the pattern of connections between brain regions. Computational methods are increasingly relevant for uncovering structure in such datasets. There has been recent interest in the use of non-metric multidimensional scaling (NMDS) for such analysis. NMDS produces a spatial representation of the 'dissimilarities' between a number of entities. Normally, it is applied to data matrices containing a large number of levels of dissimilarity, whereas for brain connectivity data there is a very small number. We address the suitability of NMDS for this case. Systematic numerical studies are presented to evaluate the ability of this method to reconstruct known geometrical configurations from dissimilarity data processing few levels. In this case there is a strong bias for NMDS to produce annular configurations, whether or not such structure exists in the original data. For the case of a connectivity dataset derived from the primate cortical visual system, we demonstrate that great caution is needed in interpreting the resulting configuration. Application of an independent method that we developed also strongly suggests that the visual system NMDS configuration is affected by an annular bias. We question the strength of support that an NMDS analysis of the visual system data provides for the two streams view of visual processing.

Elastic Net Model of Ocular Dominance: Overall Stripe Pattern and Monocular Deprivation

The elastic net (Durbin and Willshaw 1987) can account for the development of both topography and ocular dominance in the mapping from the lateral geniculate nucleus to primary visual cortex (Goodhill and Willshaw 1990). Here it is further shown for this model that (1) the overall pattern of stripes produced is strongly influenced by the shape of the cortex: in particular, stripes with a global order similar to that seen biologically can be produced under appropriate conditions, and (2) the observed changes in stripe width associated with monocular deprivation are reproduced in the model.