TY - JOUR

T1 - Immune network behavior-II. From oscillations to chaos and stationary states

AU - De Boer, Rob J.

AU - Perelson, Alan S.

AU - Kevrekidis, Ioannis G.

N1 - Funding Information:
Since the SL coordinate system has not revealed any new insights, it might have been better to present all figures in the original coordinate system. Thus, we could have computed the Poincar6 sections in the SL coordinate system, but represented their results in the original coordinate system. One of the most interesting new results found for the CABG model is that for high rates of complex removal, #, sustained oscillatory/chaotic behavior persists, even ifo-, the source of cells from the bone marrow, is zero. This may be particularly important for models in which instead of a continuous source of cells of established clones there is a stochastic source of novel clones (De Boer and Perelson, 1991). Stewart and Varela (1989, 1990) have also searched for conditions for which oscillatory/chaotic behavior could be sustained in the absence of a continuous bone marrow source. They suggested that a necessary condition is that each idiotype should recognize itself, i.e. should have a low affinity for binding itself. Although such antibodies have been described (Kang and K6hler, 1986), it seems much simpler to change to the parameter regime with higher rates of complex removal. This has the second advantage of the close correspondence between the model behavior (see Fig. 12) and the experimental data (Lundkvist et al., 1989; Varela et al., 1991). This work was performed under the auspices of the U.S. Department of Energy. It was supported in part by NIH Grants AI28433 and RR06555 (A.P.), by NSF Grant CTS89-57213 (I.K.), and a Packard Foundation Fellowship (I.K.). It was also supported by the Santa Fe Institute through their Theoretical Immunology Program and by the Los Alamos National Laboratory Center for Nonlinear Studies.

PY - 1993/7

Y1 - 1993/7

N2 - 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.

AB - 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.

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U2 - 10.1007/BF02460673

DO - 10.1007/BF02460673

M3 - Article

C2 - 8318930

AN - SCOPUS:0027348346

VL - 55

SP - 781

EP - 816

JO - The Bulletin of Mathematical Biophysics

JF - The Bulletin of Mathematical Biophysics

SN - 0092-8240

IS - 4

ER -