École Polytechnique Fédérale de Lausanne (EPFL)

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

State-of-the art experimental data promises exquisite insight into the spatial heterogeneity in tissue samples. However, the high level of detail in such data is contrasted with a lack of methods that allow an analysis that fully exploits the available spatial information. Persistent Homology (PH) has been very successfully applied to many biological datasets, but it is typically limited to the analysis of single species data. In the first part of my talk, I will highlight two novel techniques in relational PH that we develop to encode spatial heterogeneity of multi species data. Our approaches are based on Dowker complexes and Witness complexes. We apply the methods to synthetic images generated by an agent-based model of tumour-immune cell interactions. We demonstrate that relational PH features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. I will present an extension to our pipeline which combines graph neural networks (GNN) with local relational PH and significantly enhances the performance of the GNN on the synthetic data. In the second part of the talk, I will showcase a noise-robust extension of Reani and Bobrowski’s cycle registration algorithm  (2023) to reconstruct 3D brain atlases of Drosophila flies from a sequence of μ-CT images.

Donaldson-Thomas theory is classically defined for moduli spaces of sheaves over a Calabi-Yau threefold. Thanks to recent foundational work of Cao-Leung, Borisov-Joyce and Oh-Thomas, DT theory has been extended to Calabi-Yau 4-folds. We discuss how, in this context, one can define natural K-theoretic refinements of Donaldson-Thomas invariants (counting sheaves on Hilbert schemes) and Pandharipande-Thomas invariants (counting sheaves on moduli spaces of stable pairs) and how — conjecturally — they are related. Finally, we introduce an extension of DT invariants to Calabi-Yau 4-orbifolds, and propose a McKay-type correspondence, which we expect to be suitably interpreted as a wall-crossing phenomenon. Joint work (in progress) with Yalong Cao and Martijn Kool.

It has been known for some time that the contact sets between
self-avoiding idealised tubes (i.e. with exactly circular, normal
cross-sections) in various highly compact, tight structures comprise
double lines of contact. I will re-visit those results for two canonical
examples, namely the orthogonal clasp and the ideal trefoil knot. I will
then show experimental and 3D FEM simulation data for deformable elastic
tubes (obtained within the group of Pedro Reis at the EPFL) which
reveals that the ideal contact set lines bound (in a non-rigorous sense)
the actual contact patches that arise in reality.

[1] The shapes of physical trefoil knots, P. Johanns, P. Grandgeorge, C.
Baek, T.G. Sano, J.H. Maddocks, P.M. Reis, Extreme Mechanics Letters 43
(2021), p. 101172, DOI:10.1016/j.eml.2021.101172
[2]  Mechanics of two filaments in tight orthogonal contact, P.
Grandgeorge, C. Baek, H. Singh, P. Johanns, T.G. Sano, A. Flynn, J.H.
Maddocks, and P.M. Reis, Proceedings of the National Academy of Sciences
of the United States of America 118, no. 15 (2021), p. e2021684118
DOI:10.1073/pnas.2021684118

In this talk, we will present a new class of invariants of multi-parameter persistence modules : \emph{projected barcodes}. Relying on Grothendieck's six operations for sheaves, projected barcodes are defined as derived pushforwards of persistence modules onto $\R$ (which can be seen as sheaves on a vector space in a precise sense). We will prove that the well-known fibered barcode is a particular instance of projected barcodes. Moreover, our construction is able to distinguish persistence modules that have the same fibered barcodes but are not isomorphic. We will present a systematic study of the stability of projected barcodes. Given F a subset of the 1-Lipschitz functions, this leads us to define a new class of well-behaved distances between persistence modules, the  F-Integral Sheaf Metrics (F-ISM), as the supremum over p in F of the bottleneck distance of the projected barcodes by p of two persistence modules. 

In the case where M is the collection in all degrees of the sublevel-sets persistence modules of a function f : X -> R^n, we prove that the projected barcode of M by a linear map p : R^n \to R is nothing but the collection of sublevel-sets barcodes of the post-composition of f by p. In particular, it can be computed using already existing softwares, without having to compute entirely M. We also provide an explicit formula for the gradient with respect to p of the bottleneck distance between projected barcodes, allowing to use a gradient ascent scheme of approximation for the linear ISM. This is joint work with François Petit.

Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD),  that assigns a so-called “barcode" to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines  reliable clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the relatively small number of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and  connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks.  We have also successfully applied these classification and synthesis techniques to microglia and astrocytes, two other types of cells that populate the brain.

In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties.

This talk is based on work in collaborations led by Lida Kanari at the Blue Brain Project.

The well known node2vec algorithm has been used to explore network structures and represent the nodes of a graph in a vector space in a way that reflects the structure of the graph. Random walks in node2vec have been used to study the local structure through pairwise interactions. Our motivation for this project comes from a desire to understand higher-order relationships by a similar approach. To this end, we propose an extension of node2vec to a method for representing the k-simplices of a simplicial complex into Euclidean space. 

In this talk I outline a way to do this by performing random walks on simplicial complexes, which have a greater variety of adjacency relations to take into account than in the case of graphs. The walks on simplices are then used to obtain a representation of the simplices. We will show cases in which this method can uncover the roles of higher order simplices in a network and help understand structures in graphs that cannot be seen by using just the random walks on the nodes. 

Computer-based simulation of partial differential equations (PDEs) involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions. In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used for the simulation of the mechanical behaviour of the designed object. This workflow is often repeated many times as part of a shape optimisation loop. Within this loop, the use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-) meshing and re-interpolation techniques that often require human intervention and result in inaccurate solutions and lack of robustness of the whole process. In my talk, I will present the mathematical counterpart of this problem, I will discuss the mismatch in the mathematical representations of geometries and PDEs unknowns and introduce a promising framework where geometric objects and PDEs unknowns are represented in a compatible way. Within this framework, the challenges to be addressed in order to construct robust PDE solvers are many and I will discuss some of them. Mathematical results will besupported by numerical validation.

The TMD algorithm (Kanari et al. 2018) computes the barcode of a neuron (tree) with respect to the radial or path distance from the soma (root). We are interested in the inverse problem: how to understand the space of trees that are represented by the same barcode. Our tool to study this spaces is the stochastic TNS algorithm (Kanari et al. 2020) which generates trees from a given barcode in a biologically meaningful way. 

I will present some theoretical results on the space of trees that have the same barcode, as well as the effect of adding noise to the barcode. In addition, I will provide a more combinatorial perspective on the space of barcodes, expressed in terms of the symmetric group. I will illustrate these results with experiments based on the TNS.

This is joint work with L. Kanari and K. Hess.