Understanding of neuronal circuitry at cellular resolution within the brain has relied on neuron tracing methods which involve careful observation and interpretation by experienced neuroscientists. With recent developments in imaging and digitization, this approach is no longer feasible with the large scale (terabyte to petabyte range) images. Machine learning based techniques, using deep networks, provide an efficient alternative to the problem. However, these methods rely on very large volumes of annotated images for training and have error rates that are too high for scientific data analysis, and thus requires a significant volume of human-in-the-loop proofreading. Here we introduce a hybrid architecture combining prior structure in the form of topological data analysis methods, based on discrete Morse theory, with the best-in-class deep-net architectures for the neuronal connectivity analysis. We show significant performance gains using our hybrid architecture on detection of topological structure (e.g. connectivity of neuronal processes and local intensity maxima on axons corresponding to synaptic swellings) with precision/recall close to 90% compared with human observers. We have adapted our architecture to a high performance pipeline capable of semantic segmentation of light microscopic whole-brain image data into a hierarchy of neuronal compartments. We expect that the hybrid architecture incorporating discrete Morse techniques into deep nets will generalize to other data domains. Author ContributionsThe idea of using topological priors in the pipeline was conceptualized by Y.W. and P.M. Algorithmic design and development was performed by S. B. and L. M.. Proof reading assistance and neuroanatomical expertise for neuroanatomical ground truth data was provided by J. J. and K. M.. Data preparation, including quality control and acquisition were performed by B. H., J. J. and K. M. under the supervision of J. H. and P. M.. The ALBU baseline was tested by D. W.. Evaluation of the algorithm was conducted by S. B., D.

Understanding of neuronal circuitry at cellular resolution within the brain has relied on tract tracing methods which involve careful observation and interpretation by experienced neuroscientists. With recent developments in imaging and digitization, this approach is no longer feasible with the large scale (terabyte to petabyte range) images. Machine learning based techniques, using deep networks, provide an efficient alternative to the problem. However, these methods rely on very large volumes of annotated images for training and have error rates that are too high for scientific data analysis, and thus requires a significant volume of human-in-the-loop proofreading. Here we introduce a hybrid architecture combining prior structure in the form of topological data analysis methods, based on discrete Morse theory, with the best-in-class deep-net architectures for the neuronal connectivity analysis. We show significant performance gains using our hybrid architecture on detection of topological structure (e.g. connectivity of neuronal processes and local intensity maxima on axons corresponding to synaptic swellings) with precision/recall close to 90% compared with human observers. We have adapted our architecture to a high performance pipeline capable of semantic segmentation of light microscopic whole-brain image data into a hierarchy of neuronal compartments. We expect that the hybrid architecture incorporating discrete Morse techniques into deep nets will generalize to other data domains.

Neuroscientific data analysis has traditionally relied on linear algebra and stochastic process theory. However, the tree-like shapes of neurons cannot be described easily as points in a vector space (the subtraction of two neuronal shapes is not a meaningful operation), and methods from computational topology are better suited to their analysis. Here we introduce methods from Discrete Morse (DM) Theory to extract the tree-skeletons of individual neurons from volumetric brain image data, and to summarize collections of neurons labelled by tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons using a consensus tree-shape that provides a richer information summary than the traditional regional 'connectivity matrix' approach. The conceptually elegant DM approach lacks handtuned parameters and captures global properties of the data as opposed to previous approaches which are inherently local. For individual skeletonization of sparsely labelled neurons we obtain substantial performance gains over state-of-the-art non-topological methods (over 10% improvements in precision and faster proofreading). The consensus-tree summary of tracer injections incorporates the regional connectivity matrix information, but in addition captures the collective collateral branching patterns of the set of neurons connected to the injection site, and provides a bridge between single-neuron morphology and tracer-injection data. SummaryNeuroscientific data analysis has traditionally involved methods for statistical signal and image processing, drawing on linear algebra and stochastic process theory. However, digitized neuroanatomical datasets containing labelled neurons, either individually or in groups labelled by tracer injections, do not fit into this classical framework. The tree-like shapes of neurons cannot be adequately described as points in a vector space (e.g. the subtraction of two neuronal shapes is not a meaningful operation). There is therefore a need for new approaches, which has become more urgent given the growth in whole-brain datasets with sparsely labelled neurons or tracer injections.Methods from computational topology and geometry are naturally suited to the analysis of neuronal shapes. In this paper we introduce methods from Discrete Morse Theory to extract tree-skeletons of individual neurons from volumetric brain image data, and to summarize collections of neurons labelled by anterograde tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons using a consensus tree-shape. This consensus tree provides a richer information summary than the regional or voxel-based "connectivity matrix" approach that has previously been used in the literature.The algorithmic procedure for single neuron skeletonization includes an initial neurite-detection step to extract a density field from the raw volumetric image data, followed by Dis-crete Morse theoretic skeleton extraction from the density field us...

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicts the formation factor and effective permeability from micro-CT images. Fast Fourier Transform (FFT) solvers are established to compute both the formation factor and effective permeability, while the topology and geometry of the pore space are represented by a persistence-based Morse graph. Together, they constitute the database for training, validating, and testing the neural networks. While the graph and Euclidean convolutional approaches both employ neural networks to generate low-dimensional latent space to represent the features of the microstructures for forward predictions, the SE(3) equivariant neural network is found to generate more accurate predictions, especially when the training data are limited. Numerical experiments have also shown that the new SE(3) approach leads to predictions that fulfill the material frame indifference whereas the predictions from classical convolutional neural networks (CNNs) may suffer from spurious dependence on the coordinate system of the training data. Comparisons among predictions inferred from training the CNN and those from graph convolutional neural networks with and without the equivariant constraint indicate that the equivariant graph neural network seems to perform better than the CNN and GNN without enforcing equivariant constraints.

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