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Weakly nonlinear analysis of a reaction-diffusion model for demyelinating lesions in Multiple Sclerosis

Romina Travaglini, Rossella Della Marca

TL;DR

This study examines how key parameters, including the squeezing probability of immune cells and the chemotactic response, impact pattern formation, and performs a Turing instability analysis and a weakly nonlinear analysis to investigate different spatial patterns that may emerge.

Abstract

Multiple Sclerosis is a chronic autoimmune disorder characterized by the degradation of the myelin sheath in the central nervous system, leading to neurological impairments. In this work, we analyze a reaction-diffusion model derived from kinetic theory to study the formation of demyelinating lesions. We perform a Turing instability analysis and a weakly nonlinear analysis to investigate different spatial patterns that may emerge. Our study examines how key parameters, including the squeezing probability of immune cells and the chemotactic response, impact pattern formation. Numerical simulations confirm the analytical results, revealing the emergence of distinct spatial structures.

Weakly nonlinear analysis of a reaction-diffusion model for demyelinating lesions in Multiple Sclerosis

TL;DR

This study examines how key parameters, including the squeezing probability of immune cells and the chemotactic response, impact pattern formation, and performs a Turing instability analysis and a weakly nonlinear analysis to investigate different spatial patterns that may emerge.

Abstract

Multiple Sclerosis is a chronic autoimmune disorder characterized by the degradation of the myelin sheath in the central nervous system, leading to neurological impairments. In this work, we analyze a reaction-diffusion model derived from kinetic theory to study the formation of demyelinating lesions. We perform a Turing instability analysis and a weakly nonlinear analysis to investigate different spatial patterns that may emerge. Our study examines how key parameters, including the squeezing probability of immune cells and the chemotactic response, impact pattern formation. Numerical simulations confirm the analytical results, revealing the emergence of distinct spatial structures.
Paper Structure (12 sections, 6 theorems, 93 equations, 5 figures)

This paper contains 12 sections, 6 theorems, 93 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Pattern formation analysis
  4. Turing instability
  5. Weakly nonlinear analysis
  6. Numerical simulations
  7. Conclusions
  8. Proof of results in Subsection \ref{['SubWeak']}
  9. Proof of Proposition \ref{['Prop2']}
  10. Proof of Proposition \ref{['Prop3']}
  11. Proof of Proposition \ref{['Prop4']}
  12. Proof of Proposition \ref{['Prop5']}

Key Result

Proposition 1

Let us suppose that the time derivatives of the two populations of self-antigen presenting cells $A(t,\mathbf{x})$ and immunosuppressive cells $S(t,\mathbf{x})$ are zero for all $\mathbf{x}\in\Gamma_{\mathbf{x}}$ and that the two populations are not constantly equal to zero. Then, the system eq:macA

Figures (5)

  • Figure 1: Values for $\mathcal{R}$ and $\delta$ satisfying condition \ref{['StabStri']} (region I) or condition \ref{['StabSqua']} (region II), taking values as in \ref{['Pars']} and fixing $\gamma=1$ in panel (a) and $\gamma=2$ in panel (b).
  • Figure 2: Long-time patterning of destroyed myelin portion ($E$), described by system \ref{['eq:rdsR']}, taking parameters $\gamma=1$, $\delta=0.6$ and $\mathcal{R}=3.2$, $\xi=13.31$.
  • Figure 3: Long-time patterning of destroyed myelin portion ($E$), described by system \ref{['eq:rdsR']}, taking parameters $\gamma=2$, $\delta=0.6$ and $\mathcal{R}=3.2$, $\xi=10.88$.
  • Figure 4: Long-time patterning of destroyed myelin portion ($E$), described by system \ref{['eq:rdsR']} , taking parameters $\gamma=2$, $\delta=0.6$ and $\mathcal{R}=3.2$, $\xi=14.5$.
  • Figure 5: Long-time patterning of destroyed myelin portion ($E$), described by system \ref{['eq:rdsR']} starting from a random perturbation of equilibrium, with parameters $\gamma=1$, $\delta=0.6$ and $\mathcal{R}=3.2$. Panel (a): $\xi=13.31$. Panel (b): $\xi=14.5$.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2