Immune network behavior—II. From oscillations to chaos and stationary states

Abstract

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

Immune network behavior—I. From stationary states to limit cycle oscillations

“…In this paper we describe the basic properties of the steady states and the dynamic behavior of the simplified models. In a second paper (De Boer et al, 1993), Part II, we will build on the results obtained here and study the detailed behavior of the full model. 5.…”

The evolution of mathematical immunology

“…If in the past, mathematical models in immunology were limited to a few groups, such as the Los Alamos T-10 group led by Perelson and his trainees (8,15,82,(131)(132)(133)(134) and…”

“…If in the past, mathematical models in immunology were limited to a few groups, such as the Los Alamos T‐10 group led by Perelson and his trainees (8, 15, 82, 131–134) and Nowak (23, 135–140), there are now current practices in all fields of immunology. Mathematical models that were once a minor branch based on theoretical considerations are now associated with most domains of molecular immunology and are often driven by precise experimental results.…”

Immune networks: multi-tasking capabilities at medium load

“…After a long period of dormancy since the pionering paper [1], we have in recent years seen a renewed interest in statistical mechanical models of the immune system [2,3,4,5,6,7,8,9,10]. These complement the standard approaches to immune system modelling, which are formulated in terms of dynamical systems [11,12,13,14]. However, to make further progress, we need quantitative tools that are able to handle the complexity of the immune system's intricate signalling patterns.…”