Understanding brain function requires disentangling the high-dimensional activity of populations of neurons. Calcium imaging is an increasingly popular technique for monitoring such neural activity, but computational tools for interpreting extracted calcium signals are lacking. While there has been a substantial development of factor analysis-type methods for neural spike train analysis, similar methods targeted at calcium imaging data are only beginning to emerge. Here we develop a flexible modeling framework that identifies low-dimensional latent factors in calcium imaging data with distinct additive and multiplicative modulatory effects. Our model includes spike-and-slab sparse priors that regularize additive factor activity and gaussian process priors that constrain multiplicative effects to vary only gradually, allowing for the identification of smooth and interpretable changes in multiplicative gain. These factors are estimated from the data using a variational expectation-maximization algorithm that requires a differentiable reparameterization of both continuous and discrete latent variables. After demonstrating our method on simulated data, we apply it to experimental data from the zebrafish optic tectum, uncovering low-dimensional fluctuations in multiplicative excitability that govern trial-to-trial variation in evoked responses.

The pattern of neural activity evoked by a stimulus can be substantially affected by ongoing spontaneous activity. Separating these two types of activity is particularly important for calcium imaging data given the slow temporal dynamics of calcium indicators. Here we present a statistical model that decouples stimulus-driven activity from low dimensional spontaneous activity in this case. The model identifies hidden factors giving rise to spontaneous activity while jointly estimating stimulus tuning properties that account for the confounding effects that these factors introduce. By applying our model to data from zebrafish optic tectum and mouse visual cortex, we obtain quantitative measurements of the extent that neurons in each case are driven by evoked activity, spontaneous activity, and their interaction. By not averaging away potentially important information encoded in spontaneous activity, this broadly applicable model brings new insight into population-level neural activity within single trials.

The development of a functional nervous system requires tight control of neurite growth and guidance by extracellular chemical cues. Neurite growth is astonishingly sensitive to shallow concentration gradients, but a widely observed feature of both growth and guidance regulation, with important consequences for development and regeneration, is that both are only elicited over the same relatively narrow range of concentrations. Here we show that all these phenomena can be explained within one theoretical framework. We first test long-standing explanations for the suppression of the trophic effects of nerve growth factor at high concentrations, and find they are contradicted by experiment. Instead we propose a new hypothesis involving inhibitory signalling among the cell bodies, and then extend this hypothesis to show how both growth and guidance can be understood in terms of a common underlying signalling mechanism. This new model for the first time unifies several key features of neurite growth regulation, quantitatively explains many aspects of experimental data, and makes new predictions about unknown details of developmental signalling.

Spontaneous activity is a fundamental characteristic of the developing nervous system. Intriguingly, it often takes the form of multiple structured assemblies of neurons. Such assemblies can form even in the absence of afferent input, for instance in the zebrafish optic tectum after bilateral enucleation early in life. While the development of neural assemblies based on structured afferent input has been theoretically well-studied, it is less clear how they could arise in systems without afferent input. Here we show that a recurrent network of binary threshold neurons with initially random weights can form neural assemblies based on a simple Hebbian learning rule. Over development the network becomes increasingly modular while being driven by initially unstructured spontaneous activity, leading to the emergence of neural assemblies. Surprisingly, the set of neurons making up each assembly then continues to evolve, despite the number of assemblies remaining roughly constant. In the mature network assembly activity builds over several timesteps before the activation of the full assembly, as recently observed in calcium-imaging experiments. Our results show that Hebbian learning is sufficient to explain the emergence of highly structured patterns of neural activity in the absence of structured input.

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.

Branching is an important mechanism by which axons navigate to their targets during neural development. For instance, in the developing zebrafish retinotectal system, selective branching plays a critical role during both initial pathfinding and subsequent arborisation once the target zone has been reached. Here we show how quantitative methods can help extract new information from time-lapse imaging about the nature of the underlying branch dynamics. First, we introduce Dynamic Time Warping to this domain as a method for automatically matching branches between frames, replacing the effort required for manual matching. Second, we model branch dynamics as a birth-death process, i.e. a special case of a continuous-time Markov process. This reveals that the birth rate for branches from zebrafish retinotectal axons, as they navigate across the tectum, increased over time. We observed no significant change in the death rate for branches over this time period. However, blocking neuronal activity with TTX slightly increased the death rate, without a detectable change in the birth rate. Third, we show how the extraction of these rates allows computational simulations of branch dynamics whose statistics closely match the data. Together these results reveal new aspects of the biology of retinotectal pathfinding, and introduce computational techniques which are applicable to the study of axon branching more generally.

The colorful representation of orientation preference maps in primary visual cortex has become iconic. However, the standard representation is misleading because it uses a color mapping to indicate orientations based on the HSV (hue, saturation, value) color space, for which important perceptual features such as brightness, and not just hue, vary among orientations. This means that some orientations stand out more than others, conveying a distorted visual impression. This is particularly problematic for visualizing subtle biases caused by slight overrepresentation of some orientations due to, for example, stripe rearing. We show that displaying orientation maps with a color mapping based on a slightly modified version of the HCL (hue, chroma, lightness) color space, so that primarily only hue varies between orientations, leads to a more balanced visual impression. This makes it easier to perceive the true structure of this seminal example of functional brain architecture.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Abstract We examine the dynamics of development of cortical maps of spatial position, ocular dominance and orientation preference in the elastic net model. Analysis and simulation results are presented for the order in which each of these maps first emerges as a function of the parameters of the feature space.

This paper applies a simple analysis of the statistical noise inherent in sensing concentrations of diffusible chemical factors to the problem of guidance of developing axons. We show that growth cones may be able to detect chemical changes of 0.5% across their width, and that guidance by a gradient is limited to distances below about 1 cm, even for a gradient of ideal shape.

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.

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.

The effect of different kinds of weight normalization on the outcome of a simple competitive learning rule is analyzed. It is shown that there are important differences in the representation formed depending on whether the constraint is enforced by dividing each weight by the same amount (“divisive enforcement”) or subtracting a fixed amount from each weight (“subtractive enforcement”). For the divisive cases weight vectors spread out over the space so as to evenly represent “typical” inputs, whereas for the subtractive cases the weight vectors tend to the axes of the space, so as to represent “extreme” inputs. The consequences of these differences are examined.

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.