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Complex behaviours of AB model describing idiotypic network

L B Zhang¹, C Y Du, A S Qi

¹Physics Department, Beijing Normal University, China.

Bulletin of Mathematical Biology

|March 1, 1994

Summary

The AB model, a chemical immune system model, exhibits complex behaviors like periodic oscillations and chaos. Lyapounov exponent and fractal dimension analysis reveal parameter domains for these dynamics.

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Area of Science:

Immunology
Theoretical Biology
Chemical Kinetics

Background:

The idiotypic network is a complex regulatory system within the immune system.
De Boer et al. developed the AB model, a simplified chemical representation of the immune network.
Previous studies examined the AB model's steady states, periodic oscillations, and chaotic motions.

Purpose of the Study:

To intuitively describe the periodic motions and chaotic behaviors of the AB model.
To identify parameter domains that lead to periodic oscillations versus chaotic behaviors.
To characterize the attractors of the system using fractal dimension.

Main Methods:

Qualitative description of periodic and chaotic dynamics.
Lyapounov exponent calculation to delineate parameter regions.
Fractal dimension analysis to quantify attractor complexity.

Main Results:

The AB model demonstrates both periodic oscillations and chaotic behaviors.
Lyapounov exponent analysis successfully distinguishes parameter domains for each behavior.
Fractal dimension calculations provide insights into the 'strangeness' of the attractors.

Conclusions:

The AB model offers a framework for understanding immune network complexity.
Lyapounov exponents and fractal dimensions are crucial tools for analyzing such complex systems.
This study clarifies the conditions under which the AB model exhibits different dynamic regimes.

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