Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows

We present Sequential Neural Likelihood (SNL), a new method for Bayesian inference in simulator models, where the likelihood is intractable but simulating data from the model is possible. SNL trains an autoregressive flow on simulated data in order to learn a model of the likelihood in the region of high posterior density. A sequential training procedure guides simulations and reduces simulation cost by orders of magnitude. We show that SNL is more robust, more accurate and requires less tuning than related neural-based methods, and we discuss diagnostics for assessing calibration, convergence and goodness-of-fit.Comment: Accepted for publication at AISTATS 201

Analysis of local and global topographic order in mouse retinocollicular maps

We introduce the Lattice Method for the quantitative assessment of the topographic order within the pattern of connections between two structures. We apply this method to published visuocollicular mapping data obtained by Fourier-based intrinsic imaging of mouse colliculus. We find that, in maps from wild types and β2 knock-outs, at least 150 points on the colliculus are represented in the visual field in the correct relative order. In maps from animals with knock-out of the three ephrinA ligands (TKO), thought to specify the rostrocaudal axis of the map, the projection on the colliculus of each small circular area of visual field is elongated approximately rostrocaudally. Of these projections, 9% are made up of two distinct regions lying along the direction of ingrowth of retinal fibers. These are similar to the ectopic projections found in other ephrinA knock-out data. Coexisting with the ectopic projections, each TKO map contains a submap where neighbor–neighbor relations are preserved, which is ordered along both rostrocaudal and mediolateral axes, in the orientation found in wild-type maps. The submaps vary in size with order well above chance level, which can approach the order in wild-type maps. Knock-out of both β2 and two of the three ephrinAs yields maps with some order. The ordered TKO maps cannot be produced by correlated neural activity acting alone, as this mechanism is unable to specify map orientation. These results invite reassessment of the role of molecular signaling, particularly that of ephrinAs, in the formation of ordered nerve connections

Quantitative assessment of computational models for retinotopic map formation.

Molecular and activity-based cues acting together are thought to guide retinal axons to their terminal sites in vertebrate optic tectum or superior colliculus (SC) to form an ordered map of connections. The details of mechanisms involved, and the degree to which they might interact, are still not well understood. We have developed a framework within which existing computational models can be assessed in an unbiased and quantitative manner against a set of experimental data curated from the mouse retinocollicular system. Our framework facilitates comparison between models, testing new models against known phenotypes and simulating new phenotypes in existing models. We have used this framework to assess four representative models that combine Eph/ephrin gradients and/or activity-based mechanisms and competition. Two of the models were updated from their original form to fit into our framework. The models were tested against five different phenotypes: wild type, Isl2-EphA3(ki/ki), Isl2-EphA3(ki/+), ephrin-A2,A3,A5 triple knock-out (TKO), and Math5(-/-) (Atoh7). Two models successfully reproduced the extent of the Math5(-/-) anteromedial projection, but only one of those could account for the collapse point in Isl2-EphA3(ki/+). The models needed a weak anteroposterior gradient in the SC to reproduce the residual order in the ephrin-A2,A3,A5 TKO phenotype, suggesting either an incomplete knock-out or the presence of another guidance molecule. Our article demonstrates the importance of testing retinotopic models against as full a range of phenotypes as possible, and we have made available MATLAB software, we wrote to facilitate this process.Contract grant sponsors: Wellcome Trust; contract grant numbers: 083205, Engineering and Physical Sciences Research Council (EPSRC) (CSC).This is the published version. It's also available from Wiley at http://onlinelibrary.wiley.com/doi/10.1002/dneu.22241/abstract

Estimating the location and size of retinal injections from orthogonal images of an intact retina.

BACKGROUND: To study the mapping from the retina to the brain, typically a small region of the retina is injected with a dye, which then propagates to the retina's target structures. To determine the location of the injection, usually the retina is dissected out of the eye, flattened and then imaged, causing tears and stretching of the retina. The location of the injection is then estimated from the image of the flattened retina. Here we propose a new method that avoids dissection of the retina. RESULTS: We have developed IntactEye, a software package that uses two orthogonal images of the intact retina to locate focal injections of a dye. The two images are taken while the retina is still inside the eye. This bypasses the dissection step, avoiding unnecessary damage to the retina, and speeds up data acquisition. By using the native spherical coordinates of the eye, we avoid distortions caused by interpreting a curved structure in a flat coordinate system. Our method compares well to the projection method and to the Retistruct package, which both use the flattened retina as a starting point. We have tested the method also on synthetic data, where the injection location is known. Our method has been designed for analysing mouse retinas, where there are no visible landmarks for discerning retinal orientation, but can also be applied to retinas from other species. CONCLUSIONS: IntactEye allows the user to precisely specify the location and size of a retinal injection from two orthogonal images taken of the eye. We are solving the abstract problem of locating a point on a spherical object from two orthogonal images, which might have applications outside the field of neuroscience.SJE and MR gratefully acknowledge the support of the University of Strasbourg Institute for Advanced Study (USIAS). SJE and JJJH were supported by the Wellcome Trust (grant number 083205). The authors wish to thank Ellese Cotterill for analysing synthetic data for verification of accuracy

Optimal learning rules for familiarity detection

It has been suggested that the mammalian memory system has both familiarity and recollection components. Recently, a high-capacity network to store familiarity has been proposed. Here we derive analytically the optimal learning rule for such a familiarity memory using a signalto- noise ratio analysis. We find that in the limit of large networks the covariance rule, known to be the optimal local, linear learning rule for pattern association, is also the optimal learning rule for familiarity discrimination. The capacity is independent of the sparseness of the patterns, as long as the patterns have a fixed number of bits set. The corresponding information capacity is 0.057 bits per synapse, less than typically found for associative networks

On the Importance of Countergradients for the Development of Retinotopy: Insights from a Generalised Gierer Model

During the development of the topographic map from vertebrate retina to superior colliculus (SC), EphA receptors are expressed in a gradient along the nasotemporal retinal axis. Their ligands, ephrin-As, are expressed in a gradient along the rostrocaudal axis of the SC. Countergradients of ephrin-As in the retina and EphAs in the SC are also expressed. Disruption of any of these gradients leads to mapping errors. Gierer's (1981) model, which uses well-matched pairs of gradients and countergradients to establish the mapping, can account for the formation of wild type maps, but not the double maps found in EphA knock-in experiments. I show that these maps can be explained by models, such as Gierer's (1983), which have gradients and no countergradients, together with a powerful compensatory mechanism that helps to distribute connections evenly over the target region. However, this type of model cannot explain mapping errors found when the countergradients are knocked out partially. I examine the relative importance of countergradients as against compensatory mechanisms by generalising Gierer's (1983) model so that the strength of compensation is adjustable. Either matching gradients and countergradients alone or poorly matching gradients and countergradients together with a strong compensatory mechanism are sufficient to establish an ordered mapping. With a weaker compensatory mechanism, gradients without countergradients lead to a poorer map, but the addition of countergradients improves the mapping. This model produces the double maps in simulated EphA knock-in experiments and a map consistent with the Math5 knock-out phenotype. Simulations of a set of phenotypes from the literature substantiate the finding that countergradients and compensation can be traded off against each other to give similar maps. I conclude that a successful model of retinotopy should contain countergradients and some form of compensation mechanism, but not in the strong form put forward by Gierer

Standard Anatomical and Visual Space for the Mouse Retina: Computational Reconstruction and Transformation of Flattened Retinae with the Retistruct Package

The concept of topographic mapping is central to the understanding of the visual system at many levels, from the developmental to the computational. It is important to be able to relate different coordinate systems, e.g. maps of the visual field and maps of the retina. Retinal maps are frequently based on flat-mount preparations. These use dissection and relaxing cuts to render the quasi-spherical retina into a 2D preparation. The variable nature of relaxing cuts and associated tears limits quantitative cross-animal comparisons. We present an algorithm, "Retistruct," that reconstructs retinal flat-mounts by mapping them into a standard, spherical retinal space. This is achieved by: stitching the marked-up cuts of the flat-mount outline; dividing the stitched outline into a mesh whose vertices then are mapped onto a curtailed sphere; and finally moving the vertices so as to minimise a physically-inspired deformation energy function. Our validation studies indicate that the algorithm can estimate the position of a point on the intact adult retina to within 8° of arc (3.6% of nasotemporal axis). The coordinates in reconstructed retinae can be transformed to visuotopic coordinates. Retistruct is used to investigate the organisation of the adult mouse visual system. We orient the retina relative to the nictitating membrane and compare this to eye muscle insertions. To align the retinotopic and visuotopic coordinate systems in the mouse, we utilised the geometry of binocular vision. In standard retinal space, the composite decussation line for the uncrossed retinal projection is located 64° away from the retinal pole. Projecting anatomically defined uncrossed retinal projections into visual space gives binocular congruence if the optical axis of the mouse eye is oriented at 64° azimuth and 22° elevation, in concordance with previous results. Moreover, using these coordinates, the dorsoventral boundary for S-opsin expressing cones closely matches the horizontal meridian