Dating behavior of the retinal ganglion cell.
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Animals↗
Biomedical subjects
G J Goodhill
Publications and source records attributed to G J Goodhill.
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.
Algorithms↗
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.
Axons↗
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.
Animals↗
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.
Animals↗
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.
Animals↗
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.
Animals↗
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.
Animals↗
Stimulating issues in cortical map development.
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Brain Mapping↗
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.
Animals↗
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.
Animals↗
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.
Animals↗
Scaling and brain connectivity.
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Animals↗
Topography and ocular dominance: a model exploring positive correlations.
The map from eye to brain in vertebrates is topographic, i.e., neighbouring points in the eye map to neighbouring points in the brain. In addition, when two eyes innervate the same target structure, the two sets of fibres segregate to form ocular dominance stripes. Experimental evidence from the frog and goldfish suggests that these two phenomena may be subserved by the same mechanisms. We present a computational model that addresses the formation of both topography and ocular dominance. The model is based on a form of competitive learning with subtractive enforcement of a weight normalization rule. Inputs to the model are distributed patterns of activity presented simultaneously in both eyes. An important aspect of this model is that ocular dominance segregation can occur when the two eyes are positively correlated, whereas previous models have tended to assume zero or negative correlations between the eyes. This allows investigation of the dependence of the pattern of stripes on the degree of correlation between the eyes: we find that increasing correlation leads to narrower stripes. Experiments are suggested to test this prediction.
Animals↗