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Dense cell-by-cell systems of PDEs: approximation, spectral analysis, and preconditioning

Pietro Benedusi, Paola Ferrari, Marius Causemann, Stefano Serra-Capizzano

TL;DR

The work analyzes the cell-by-cell EMI framework for excitable tissues, deriving a large, structured linear system from Galerkin discretizations and performing a detailed spectral analysis using Toeplitz/GLT tools. It shows that EMI matrices decompose into a dominant block-diagonal (Laplacian-like) component plus low-rank couplings, enabling effective preconditioning via a block-diagonal or monolithic multilevel strategy. Numerical experiments across nervous-system-like, cardiac-like, and cortex tissue geometries demonstrate that a monolithic multilevel solver (AMG) provides robust performance across discretization and geometry, while block-diagonal preconditioners have limited robustness in dense, multilayer settings. The results support scalable simulations of large-scale tissue models (up to $N>10^5$ cells) and highlight open problems related to outliers, spectral transitions, and GLT-based symbol design with SMW corrections for more general discretizations.

Abstract

In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc boundary conditions, relevant to model portions of excitable tissues, composed of several cells. In particular, we study two geometrical settings: computational cardiology and neuroscience. The Galerkin approximations to the considered system of PDEs lead to large linear systems of algebraic equations, where the coefficient matrices depend on the number $N$ of cells and the fineness parameters. We give a structural and spectral analysis of the related matrix-sequences with $N$ fixed and with fineness parameters tending to zero. Based on the theoretical results, we propose preconditioners and specific multilevel solvers. Numerical experiments are presented and critically discussed, showing that a monolithic multilevel solver is efficient and robust with respect to all the problem and discretization parameters. In particular, we include numerical results increasing the number of cells $N$, both for idealized geometries (with $N$ exceeding $10^5$) and for realistic, densely populated 3D tissue reconstruction.

Dense cell-by-cell systems of PDEs: approximation, spectral analysis, and preconditioning

TL;DR

The work analyzes the cell-by-cell EMI framework for excitable tissues, deriving a large, structured linear system from Galerkin discretizations and performing a detailed spectral analysis using Toeplitz/GLT tools. It shows that EMI matrices decompose into a dominant block-diagonal (Laplacian-like) component plus low-rank couplings, enabling effective preconditioning via a block-diagonal or monolithic multilevel strategy. Numerical experiments across nervous-system-like, cardiac-like, and cortex tissue geometries demonstrate that a monolithic multilevel solver (AMG) provides robust performance across discretization and geometry, while block-diagonal preconditioners have limited robustness in dense, multilayer settings. The results support scalable simulations of large-scale tissue models (up to cells) and highlight open problems related to outliers, spectral transitions, and GLT-based symbol design with SMW corrections for more general discretizations.

Abstract

In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc boundary conditions, relevant to model portions of excitable tissues, composed of several cells. In particular, we study two geometrical settings: computational cardiology and neuroscience. The Galerkin approximations to the considered system of PDEs lead to large linear systems of algebraic equations, where the coefficient matrices depend on the number of cells and the fineness parameters. We give a structural and spectral analysis of the related matrix-sequences with fixed and with fineness parameters tending to zero. Based on the theoretical results, we propose preconditioners and specific multilevel solvers. Numerical experiments are presented and critically discussed, showing that a monolithic multilevel solver is efficient and robust with respect to all the problem and discretization parameters. In particular, we include numerical results increasing the number of cells , both for idealized geometries (with exceeding ) and for realistic, densely populated 3D tissue reconstruction.
Paper Structure (17 sections, 5 theorems, 61 equations, 7 figures, 7 tables)

This paper contains 17 sections, 5 theorems, 61 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The EMI model
  3. Weak formulation and discrete operators
  4. Nervous system setting
  5. Cardiac setting
  6. Spectral analysis
  7. Toeplitz structures, spectral symbol, GLT tools
  8. Symbol analysis
  9. Varying and adapting the symbol analysis
  10. Solution strategy and numerical experiments
  11. Problem and discretization settings
  12. Model A: idealized nervous system setting
  13. Model B: idealized myocytes
  14. Model C: mouse visual cortex tissue
  15. Implementation and solution strategies
  16. ...and 2 more sections

Key Result

Theorem 4.5

TilliNota \newlabelszego-herm0 Let $f\in L^1(Q^d,s_1\times s_2)$, then $\{T_{\nu}(f)\}_{{\nu}}\sim_\sigma(f,Q^d).$ If $s_1=s_2=s$ and if $f$ is a Hermitian matrix-valued function, then $\{T_{\nu}(f)\}_{{\nu}}\sim_\lambda(f,Q^d)$.

Figures (7)

  • Figure 1: Example of an EMI geometry with $N=5$ cells with $\Omega=\left(\bigcup_{i=0}^5\Omega_i \right)\cup\left(\bigcup_{i=1}^5\Gamma_i \right)$. Here, boundary conditions are enforced only for $\Omega_0$ and $\Omega_5$ and $\Gamma_{34}$ is an example of a common interface between two cells (e.g. a gap junction).
  • Figure 1: Model A: two dimensional geometry varying the number of cells $N$. The extra-cellular space $\Omega_0$ is colored in blue, while cells $\Omega_1,\Omega_2,\ldots,\Omega_N$ in red. No gap junction are present, i.e. $\Gamma_{ij}=varnothing$ for $i,j\neq 0$.
  • Figure 2: Examples of geometrical setting for the nervous system (left) and cardiac tissue (right) for $d=2$.
  • Figure 2: Model A. First row: overall initial conditions $v_{\mathrm{in}}(x,y)$ for $N=1$ (left) and $N=25$ (right). While $v_{\mathrm{in}}$ is imposed only for $(x,y)\in\Gamma$, we also show it in the whole domain $\Omega$. Second row: corresponding solutions for $N_h = 128$.
  • Figure 3: Model B. First row: 2D geometry with $N=4$ and $N=16$ cells; the extra-cellular space $\Omega_0$ is colored in blue, while cells $\Omega_1,\ldots,\Omega_N$ in red and membranes in white. Second row: corresponding solutions $u_0,\ldots,u_N$ for $N_h = 512$.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • Theorem 4.6
  • Remark 4.7
  • Theorem 4.8
  • Corollary 4.9
  • Corollary 4.10
  • ...and 2 more