Lab news

Abstract

The brain's synaptic network, characterized by parallel connections and feedback loops, drives information flow between neurons through a large system with infinitely many degrees of freedom. This system is best modeled by the graph C∗-algebra of the underlying directed graph, the Toeplitz-Cuntz-Krieger algebra, which captures the diversity of potential information pathways. Coupled with the gauge action, this graph algebra defines an {\em algebraic quantum system}, and here we demonstrate that its thermodynamic properties provide a natural framework for describing the dynamic mappings of information flow within the network. Specifically, we show that the KMS states of this system yield global statistical measures of neuronal interactions, with computational illustrations based on the {\em C. elegans} synaptic network.

Abstract

A fundamental paradigm in neuroscience is that cognitive functions -- such as perception, learning, memory, and locomotion -- are shaped by the brain's structural organization. However, the theoretical principles explaining how this physical architecture governs its function remain elusive. Here, we propose an algebraic quantum mechanics (AQM) framework in which the functional states of a structural connectome emerge as thermal equilibrium states of an algebraic quantum system defined on the underlying directed multigraph. These equilibrium states, derived from the Kubo-Martin-Schwinger (KMS) states formalism, capture the contribution of each neuron to the overall information flow. We apply this framework to the connectome of the nematode {\em Caenorhabditis elegans}, providing a detailed description of the KMS states, exploring their functional implications, and predicting functional networks based on anatomical connectivity. Ultimately, our approach reveals functional circuits predicted by the topology of the connectome and illuminates on the mechanisms linking structure to function.

Description

Phase Aberration Correction: A Deep Learning-Based Aberration-to-Aberration Approach

One of the primary sources of suboptimal image quality in ultrasound imaging is phase aberration. It is caused by spatial changes in sound speed over a heterogeneous medium, which disturbs the transmitted waves and prevents coherent summation of echo signals. Obtaining non-aberrated ground truths in real-world scenarios can be extremely challenging, if not impossible. This challenge hinders the performance of deep learning-based techniques due to the domain shift between simulated and experimental data. In this talk, I will present one of our recent studies wherein we propose a deep learning-based method that does not require ground truth to correct the phase aberration problem and, as such, can be directly trained on real data. We trained a network wherein both the input and target output are randomly aberrated radio frequency (RF) data. Moreover, we demonstrated that a conventional loss function such as mean square error is inadequate for training such a network to achieve optimal performance. Instead, we proposed an adaptive mixed loss function that employs both B-mode and RF data.

Topology and spectral interconnectivities of higher-order multilayer networks

By E. Moutuou, O. B-K. Ali, & H. Benali

Abstract

Multilayer networks have permeated all areas of science as an abstraction for interdependent heterogeneous complex systems. However, describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures are only pairwise-based, a strong assumption likely leading to oversimplification. Most interdependent systems intrinsically involve higher-order intra- and inter-layer interactions. For instance, ecological systems involve interactions among groups within and in-between species, collaborations and citations link teams of coauthors to articles and vice versa, and interactions might exist among groups of friends from different social networks. Although higher-order interactions have been studied for monolayer systems through the language of simplicial complexes and hypergraphs, a systematic formalism incorporating them into the realm of multilayer systems is still lacking. Here, we introduce the concept of crossimplicial multicomplexes as a general formalism for modeling interdependent systems involving higher-order intra- and inter-layer connections. Subsequently, we introduce cross-homology and its spectral counterpart, the cross-Laplacian operators, to establish a rigorous mathematical framework for quantifying global and local intra- and inter-layer topological structures in such systems. Using synthetic and empirical datasets, we show that the spectra of the cross-Laplacians of a multilayer network detect different types of clusters in one layer that are controlled by hubs in another layer. We call such hubs spectral cross-hubs and define spectral persistence as a way to rank them, according to their emergence along the spectra. Our framework is broad and can especially be used to study structural and functional connectomes combining connectivities of different types and orders.

Workshop: Vascular and Metabolic Modeling of the Brain at Large Scale

While brain function originates in neuron, its energy is supported by the vascular system. With the recent development of anatomically detailed synthetic network models of the entire cerebral circulation, it is now possible to perform detailed simulations of physiology in realistic brains of small animals and humans on the computer. Modeling microcirculatory blood flow as a biphasic suspension of red-blood-cell and plasma, blood pressure, flow and hematocrit can be simulated in whole brains on the computer. Combining Poiseuille’s hemodynamic simulations with advection/diffusion equations describing oxygen diffusion, the human brain metabolism can now be modeled in-silico, by integrating tissue oxygen consumption and differential equations describing compartments associated with neural and glial brain cells. In parallel, energy metabolism regulation in the brain and its modeling has greatly progressed over the past few decades.

This workshop will thus be focused on digital human brains, and their use to predict critical metabolic functions namely blood flow, oxygen extraction and cellular metabolism across all length scales down to the level of individual capillaries and cells. Courses and conferences will be combined with research presentations and practical workshops.

Organizers: Frédéric Lesage, Polytechnique

Location: Center de Recherche Mathématique, Université de Montréal

Dates: October 12-20

Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks

By A. Rahimabadi, H. Benali

Abstract

In a variety of practical applications, there is a need to investigate diffusion or reaction–diffusion processes on complex structures, including brain networks, that can be modeled as weighted undirected and directed graphs. As an instance, the celebrated Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) reaction–diffusion equation is becoming increasingly popular for use in graph frameworks by substituting the standard graph Laplacian operator for the continuous one to study the progression of neurodegenerative diseases such as Alzheimer’s disease (AD). In this work, we establish existence, uniqueness, and boundedness of solutions for generalized Fisher–KPP reaction–diffusion equations on undirected and directed networks with fractional polynomial (FP) terms. This type of model has possible applications for modeling spreading of diseases within neuronal fibers whose porous structure may cause particles to diffuse anomalously. In the case of pure diffusion, convergence of solutions and stability of equilibria are also analyzed. Moreover, different families of positively invariant sets for the proposed equations are derived. Finally, we conclude by investigating nonlinear diffusion on a directed one-dimensional lattice.

Abstract

Multilayer networks have permeated all the sciences as a powerful mathematical abstraction for interdependent heterogenous systems such as multimodal brain connectomes, transportation, ecological systems, and scientific collaboration. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures and support their functions are only pairwise-based; a strong assumption likely leading to oversimplifications. Indeed, most interdependent systems intrinsically involve higher-order intra- and inter-layer interactions. For instance, ecological systems involve interactions among groups within and in-between species, collaborations and citations link teams of coauthors to articles and vice versa, interactions might exist among groups of friends from different social networks, etc. While higher-order interactions have been studied for monolayer systems through the language of simplicial complexes and hypergraphs, a broad and systematic formalism incorporating them into the realm of multilayer systems is still lacking. Here, we introduce the concept of crossimplicial multicomplexes as a general formalism for modelling interdependent systems involving higher-order intra- and inter-layer connections. Subsequently, we introduce cross-homology and its spectral counterpart, the cross-Laplacian operators, to establish a rigorous mathematical framework for quantifying global and local intra- and inter-layer topological structures in such systems. When applied to multilayer networks, these cross-Laplacians provide powerful methods for detecting clusters in one layer that are controlled by hubs in another layer. We call such hubs spectral cross-hubs and define spectral persistence as a way to rank them according to their emergence along the cross-Laplacian spectra.

Link

ArXiv: https://arxiv.org/abs/2305.05860

Abstract

In a variety of practical applications, there is a need to investigate diffusion or reaction-diffusion processes on complex structures, including brain networks, that can be modeled as weighted undirected and directed graphs. As an instance, the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) reaction-diffusion equation are becoming increasingly popular for use in graph frameworks by substituting the standard graph Laplacian operator for the continuous one to study the progression of neurodegenerative diseases such as tauopathies including Alzheimer’s disease (AD). However, due to the porous structure of neuronal fibers, the spreading of toxic species can be governed by an anomalous diffusion process rather than a normal one, and if this is the case, the standard graph Laplacian cannot adequately describe the dynamics of the spreading process. To capture such more complicated dynamics, we propose a diffusion equation with a nonlinear Laplacian operator and a generalization of the Fisher-KPP reaction-diffusion equation on undirected and directed networks using extensions of fractional polynomial (FP) functions. A complete analysis is also provided for the extended FP diffusion equation, including existence, uniqueness, and convergence of solutions, as well as stability of equilibria. Moreover, for the extended FP Fisher-KPP reaction-diffusion equation, we derive a family of positively invariant sets allowing us to establish existence, uniqueness, and boundedness of solutions. Finally, we conclude by investigating nonlinear diffusion on a directed one-dimensional lattice and then modeling tauopathy progression in the mouse brain to gain a deeper understanding of the potential applications of the proposed extended FP equations.