Publications
41 results found
Christgau AM, Arnaudon A, Sommer S, 2022, Moment Evolution Equations and Moment Matching for Stochastic Image EPDiff, JOURNAL OF MATHEMATICAL IMAGING AND VISION, ISSN: 0924-9907
Sapienza R, Barahona M, Saxena D, et al., 2022, Sensitivity and spectral control of network lasers, Nature Communications, Vol: 13, Pages: 1-7, ISSN: 2041-1723
Recently, random lasing in complex networks has shown efficient lasing over more than 50 localised modes, promoted by multiple scattering over the underlying graph. If controlled, these network lasers can lead to fast-switching multifunctional light sources with synthesised spectrum. Here, we observe both in experiment and theory high sensitivity of the network laser spectrum to the spatial shape of the pump profile, with some modes for example increasing in intensity by 280% when switching off 7% of the pump beam. We solve the nonlinear equations within the steady state ab-initio laser theory (SALT) approximation over a graph and we show selective lasing of around 90% of the strongest intensity modes, effectively programming the spectrum of the lasing networks. In our experiments with polymer networks, this high sensitivity enables control of the lasing spectrum through non-uniform pump patterns. We propose the underlying complexity of the network modes as the key element behind efficient spectral control opening the way for the development of optical devices with wide impact for on-chip photonics for communication, sensing, and computation.
Reimann MW, Bolaños-Puchet S, Courcol J-D, et al., 2022, Modeling and Simulation of Neocortical Micro- and Mesocircuitry. Part I: Anatomy
<jats:title>Abstract</jats:title><jats:p>The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, interregional connectivity. We present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, integrating whole-brain scale data while providing cellular and subcellular specificity. The model consists of 4.2 million morphologically detailed neurons, placed in a digital brain atlas. They are connected by 14.2 billion synapses, comprising local, long-range and extrinsic connectivity. We delineated the limits of determining connectivity from anatomy, finding that it reproduces the targeting of PV+ and VIP+ interneurons only with explicitly added specificity, but the one of Sst+ neurons even without. Globally, connectivity was characterized by local clusters tied together through hub neurons in layer 5, demonstrating how local and interegional connectivity are complicit, inseparable networks. A 211,712 neuron subvolume of the model has been made freely and openly available to the community.</jats:p>
Arnaudon A, Peach R, Petri G, et al., 2022, Connecting Hodge and Sakaguchi-Kuramoto: a mathematical framework for coupled oscillators on simplicial complexes, Publisher: ArXiv
We formulate a general Kuramoto model on weighted simplicial complexes wherephases oscillators are supported on simplices of any order $k$. Crucially, weintroduce linear and non-linear frustration terms that are independent of theorientation of the $k+1$ simplices, providing a natural generalization of theSakaguchi-Kuramoto model. In turn, this provides a generalized formulation ofthe Kuramoto higher-order parameter as a potential function to write thedynamics as a gradient flow. With a selection of simplicial complexes ofincreasingly complex structure, we study the properties of the dynamics of thesimplicial Sakaguchi-Kuramoto model with oscillators on edges to highlight thecomplexity of dynamical behaviors emerging from even simple simplicialcomplexes. We place ourselves in the case where the vector of internalfrequencies of the edge oscillators lies in the kernel of the Hodge Laplacian,or vanishing linear frustration, and, using the Hodge decomposition of thesolution, we understand how the nonlinear frustration couples the dynamics inorthogonal subspaces. We discover various dynamical phenomena, such as thepartial loss of synchronization in subspaces aligned with the Hodge subspacesand the emergence of simplicial phase re-locking in regimes of highfrustration.
Peach R, Arnaudon A, Barahona M, 2022, Relative, local and global dimension in complex networks, Nature Communications, Vol: 13, ISSN: 2041-1723
Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. To take into account locality, finiteness and discreteness, dynamical processes can be used to probe the space geometry and define its dimension. Here we show that each point in space can be assigned a relative dimension with respect to the source of a diffusive process, a concept that provides a scale-dependent definition for local and global dimension also applicable to networks. To showcase its application to physical systems, we demonstrate that the local dimension of structural protein graphs correlates with structural flexibility, and the relative dimension with respect to the active site uncovers regions involved in allosteric communication. In simple models of epidemics on networks, the relative dimension is predictive of the spreading capability of nodes, and identifies scales at which the graph structure is predictive of infectivity. We further apply our dimension measures to neuronal networks, economic trade, social networks, ocean flows, and to the comparison of random graphs.
Kanari L, Dictus H, Chalimourda A, et al., 2022, Computational synthesis of cortical dendritic morphologies, CELL REPORTS, Vol: 39, ISSN: 2211-1247
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- Citations: 1
Arnaudon A, van der Meulen F, Schauer M, et al., 2022, Diffusion Bridges for Stochastic Hamiltonian Systems and Shape Evolutions\ast, SIAM JOURNAL ON IMAGING SCIENCES, Vol: 15, Pages: 293-323, ISSN: 1936-4954
Zisis E, Keller D, Kanari L, et al., 2021, Digital Reconstruction of the Neuro-Glia-Vascular Architecture, CEREBRAL CORTEX, Vol: 31, Pages: 5686-5703, ISSN: 1047-3211
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- Citations: 9
Gosztolai A, Arnaudon A, 2021, Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature., Nat Commun, Vol: 12
Describing networks geometrically through low-dimensional latent metric spaces has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, latent space embeddings are limited to specific classes of networks because incompatible metric spaces generally result in information loss. Here, we study arbitrary networks geometrically by defining a dynamic edge curvature measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps are robust to large fluctuations in node degrees, encoding communities until the phase transition of detectability, where spectral and node-clustering methods fail. Using this insight, we derive geometric modularity to find multiscale communities based on deviations from constant network curvature in generative and real-world networks, significantly outperforming most previous methods. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.
Peach R, Arnaudon A, Schmidt J, et al., 2021, HCGA: Highly comparative graph analysis for network phenotyping, Patterns, Vol: 2, ISSN: 2666-3899
Networks are widely used as mathematical models of complex systems across many scientific disciplines. Decades of work have produced a vast corpus of research characterizing the topological, combinatorial, statistical, and spectral properties of graphs. Each graph property can be thought of as a feature that captures important (and sometimes overlapping) characteristics of a network. In this paper, we introduce HCGA, a framework for highly comparative analysis of graph datasets that computes several thousands of graph features from any given network. HCGA also offers a suite of statistical learning and data analysis tools for automated identification and selection of important and interpretable features underpinning the characterization of graph datasets. We show that HCGA outperforms other methodologies on supervised classification tasks on benchmark datasets while retaining the interpretability of network features. We exemplify HCGA by predicting the charge transfer in organic semiconductors and clustering a dataset of neuronal morphology images.
Gosztolai A, Arnaudon A, 2021, Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature, Publisher: arXiv
Defining the geometry of networks is typically associated with embedding inlow-dimensional spaces such as manifolds. This approach has helped designefficient learning algorithms, unveil network symmetries and study dynamicalnetwork processes. However, the choice of embedding space is network-specific,and incompatible spaces can result in information loss. Here, we define adynamic edge curvature for the study of arbitrary networks measuring thedeformation between pairs of evolving dynamical network processes on differenttimescales. We show that the curvature distribution exhibits gaps atcharacteristic timescales indicating bottleneck-edges that limit informationspreading. Importantly, curvature gaps robustly encode communities until thephase transition of detectability, where spectral clustering methods fail. Weuse this insight to derive geometric modularity optimisation and demonstrate iton the European power grid and the C. elegans homeobox gene regulatory networkfinding previously unidentified communities on multiple scales. Our worksuggests using network geometry for studying and controlling the structure ofand information spreading on networks.
Zisis E, Keller D, Kanari L, et al., 2021, Architecture of the Neuro-Glia-Vascular System
<jats:title>Abstract</jats:title><jats:p>Astrocytes connect the vasculature to neurons and mediate the supply of nutrients and biochemicals. They also remove metabolites from the neurons and extracellular environment. They are involved in a growing number of physiological and pathophysiological processes. Understanding the biophysical, physiological, and molecular interactions in this neuro-glia-vascular ensemble (NGV) and how they support brain function is severely restricted by the lack of detailed cytoarchitecture. To address this problem, we used data from multiple sources to create a data-driven digital reconstruction of the NGV at micrometer anatomical resolution. We reconstructed 0.2 mm<jats:sup>3</jats:sup> of rat somatosensory cortical tissue with approximately 16000 morphologically detailed neurons, its microvasculature, and approximately 2500 morphologically detailed protoplasmic astrocytes. The consistency of the reconstruction with a wide array of experimental measurements allows novel predictions of the numbers and locations of astrocytes and astrocytic processes that support different types of neurons. This allows anatomical reconstruction of the spatial microdomains of astrocytes and their overlapping regions. The number and locations of end-feet connecting each astrocyte to the vasculature can be determined as well as the extent to which they cover the microvasculature. The structural analysis of the NGV circuit showed that astrocytic shape and numbers are constrained by vasculature’s spatial occupancy and their functional role to form NGV connections. The digital reconstruction of the NGV is a resource that will enable a better understanding of the anatomical principles and geometric constraints which govern how astrocytes support brain function.</jats:p><jats:sec><jats:title>Table of contents</jats:title><jats:sec><jats:title>Main points</jats:title><jats:list list-ty
Arnaudon A, Holm D, Sommer S, 2021, Stochastic Shape Analysis, Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, Publisher: Springer International Publishing, Pages: 1-24, ISBN: 9783030030094
Palacios J, Zisis E, Coste B, et al., 2020, BlueBrain/NeuroM: New function extract_dataframe that extract morphometrics as a dataframe
Neuronal Morphology Analysis Tool
Peach R, Arnaudon A, Barahona M, 2020, Graph centrality is a question of scale, Physical Review Research, Vol: 2, ISSN: 2643-1564
Classic measures of graph centrality capture distinct aspects of node importance, from the local (e.g., degree) to the global (e.g., closeness). Here we exploit the connection between diffusion and geometry to introduce a multiscale centrality measure. A node is defined to be central if it breaks the metricity of the diffusion as a consequence of the effective boundaries and inhomogeneities in the graph. Our measure is naturally multiscale, as it is computed relative to graph neighbourhoods within the varying time horizon of the diffusion. We find that the centrality of nodes can differ widely at different scales. In particular, our measure correlates with degree (i.e., hubs) at small scales and with closeness (i.e., bridges) at large scales, and also reveals the existence of multi-centric structures in complex networks. By examining centrality across scales, our measure thus provides an evaluation of node importance relative to local and global processes on the network.
Peach RL, Arnaudon A, Barahona M, 2020, Semi-supervised classification on graphs using explicit diffusion dynamics, Foundations of Data Science, Vol: 2, Pages: 19-33, ISSN: 2639-8001
Classification tasks based on feature vectors can be significantly improved by including within deep learning a graph that summarises pairwise relationships between the samples. Intuitively, the graph acts as a conduit to channel and bias the inference of class labels. Here, we study classification methods that consider the graph as the originator of an explicit graph diffusion. We show that appending graph diffusion to feature-based learning as a posteriori refinement achieves state-of-the-art classification accuracy. This method, which we call Graph Diffusion Reclassification (GDR), uses overshooting events of a diffusive graph dynamics to reclassify individual nodes. The method uses intrinsic measures of node influence, which are distinct for each node, and allows the evaluation of the relationship and importance of features and graph for classification. We also present diff-GCN, a simple extension of Graph Convolutional Neural Network (GCN) architectures that leverages explicit diffusion dynamics, and allows the natural use of directed graphs. To showcase our methods, we use benchmark datasets of documents with associated citation data.
Kuhnel L, Sommer S, Arnaudon A, 2019, Differential geometry and stochastic dynamics with deep learning numerics, Publisher: ELSEVIER SCIENCE INC
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- Citations: 6
Arnaudon A, Holm DD, Sommer S, 2019, Stochastic Metamorphosis with Template Uncertainties
In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex uids when the template properties are regarded as order parameters. The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.
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- Citations: 1
Arnaudon A, Barp A, Takao S, 2019, Irreversible Langevin MCMC on Lie Groups, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG
Bock A, Arnaudon A, Cotter C, 2019, Selective Metamorphosis for Growth Modelling with Applications to Landmarks, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG
Kühnel L, Arnaudon A, Fletcher T, et al., 2018, Stochastic Image Deformation in Frequency Domain and Parameter Estimation using Moment Evolutions
Modelling deformation of anatomical objects observed in medical images canhelp describe disease progression patterns and variations in anatomy acrosspopulations. We apply a stochastic generalisation of the Large DeformationDiffeomorphic Metric Mapping (LDDMM) framework to model differences in theevolution of anatomical objects detected in populations of image data. Thecomputational challenges that are prevalent even in the deterministic LDDMMsetting are handled by extending the FLASH LDDMM representation to thestochastic setting keeping a finite discretisation of the infinite dimensionalspace of image deformations. In this computationally efficient setting, weperform estimation to infer parameters for noise correlations and localvariability in datasets of images. Fundamental for the optimisation procedureis using the finite dimensional Fourier representation to derive approximationsof the evolution of moments for the stochastic warps. Particularly, the firstmoment allows us to infer deformation mean trajectories. The second momentencodes variation around the mean, and thus provides information on the noisecorrelation. We show on simulated datasets of 2D MR brain images that theestimation algorithm can successfully recover parameters of the stochasticmodel.
Saravanan M, Arnaudon A, 2018, Engineering solitons and breathers in a deformed ferromagnet: Effect of localised inhomogeneities, PHYSICS LETTERS A, Vol: 382, Pages: 2638-2644, ISSN: 0375-9601
Arnaudon A, Holm DD, Sommer S, 2018, A Geometric Framework for Stochastic Shape Analysis, Foundations of Computational Mathematics, ISSN: 1615-3375
We introduce a stochastic model of diffeomorphisms, whose action on a varietyof data types descends to stochastic evolution of shapes, images and landmarks.The stochasticity is introduced in the vector field which transports the datain the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework forshape analysis and image registration. The stochasticity thereby models errorsor uncertainties of the flow in following the prescribed deformation velocity.The approach is illustrated in the example of finite dimensional landmarkmanifolds, whose stochastic evolution is studied both via the Fokker-Planckequation and by numerical simulations. We derive two approaches for inferringparameters of the stochastic model from landmark configurations observed atdiscrete time points. The first of the two approaches matches moments of theFokker-Planck equation to sample moments of the data, while the second approachemploys an Expectation-Maximisation based algorithm using a Monte Carlo bridgesampling scheme to optimise the data likelihood. We derive and numerically testthe ability of the two approaches to infer the spatial correlation length ofthe underlying noise.
Arnaudon A, Holm D, Sommer S, 2018, String methods for stochastic image and shape matching, Journal of Mathematical Imaging and Vision, Vol: 60, Pages: 953-967, ISSN: 0924-9907
Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the large deformation diffeomorphic metric mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite-temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.
Arnaudon A, Takao S, 2018, Networks of coadjoint orbits: from geometric to statistical mechanics
A class of network models with symmetry group $G$ that evolve as aLie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising `triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.
Arnaudon A, 2018, Structure preserving noise and dissipation in the Toda lattice, Journal of Physics A: Mathematical and Theoretical, Vol: 51, ISSN: 1751-8113
In this paper, we use Flaschka's change of variables of the open Toda latticeand its interpretation in term of the group structure of the LU factorisationas a coadjoint motion on a certain dual of Lie algebra to implement a structurepreserving noise and dissipation. Both preserve the structure of coadjointorbit, that is the space of symmetric tri-diagonal matrices and arise as a newtype of multiplicative noise and nonlinear dissipation of the Toda lattice. Weinvestigate some of the properties of these deformations and in particular thecontinuum limit as a stochastic Burger equation with a nonlinear viscosity.This work is meant to be exploratory, and open more questions that we cananswer with simple mathematical tools and without numerical simulations.
Arnaudon A, Ganaba N, Holm DD, 2018, The stochastic energy-Casimir method, Comptes Rendus Mécanique, Vol: 346, Pages: 279-290, ISSN: 1631-0721
Akylzhanov R, Arnaudon A, 2018, Contractions of group representations via geometric quantisation
We propose a general framework to contract unitary dual of Lie groups viaholomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint orbits. This condition is checked explicitly for the contraction of ${\mathrm SU}_2$ into $\mathbb{H}$. The main tool is the geometric quantization. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.
Arnaudon A, De Castro AL, Holm DD, 2018, Noise and Dissipation on Coadjoint Orbits, Journal of Nonlinear Science, Vol: 28, Pages: 91-145, ISSN: 0938-8974
Arnaudon A, López MC, Holm DD, 2018, Un-reduction in field theory, Letters in Mathematical Physics, Vol: 108, Pages: 225-247, ISSN: 0377-9017
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