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 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.

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.

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.

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 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.

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.