by Alan S. Perelson, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 U.S.A. and G\'erard Weisbuch.

The immune system is a complex system of cells and molecules distributed throughout our bodies that can provide us with a basic defense against pathogenic organisms. Like the nervous system, the immune system performs pattern recognition tasks, learns and retains a memory of the antigens that it has fought. The immune system contains more than 10 000 000 different clones of cells that communicate via cell-cell contact and the secretion of molecules. Performing complex tasks such as learning and memory involves cooperativity among large numbers of components of the immune system and hence there is interest in using the methods and the concepts of statistical mechanics. Furthermore, the immune response develops in time and the description of this time evolution is an interesting problem in dynamical systems. In this review, we provide a brief introduction to the biology of the immune system and summarize some of the uses of physical concepts and mathematical methods in understanding its operation.

One goal of modeling in immunology, which we shall focus on here, is to deduce macroscopic properties of the system from the properties and interactions among the elementary components. This goal is similar to the purposes of statistical mechanics. The interactions among the components of the immune system are extremely intricate and they are not fully understood. Further, unlike neurophysiology, the behavior of single cells has not been described quantitatively. There are no equivalents of the Hodgkin-Huxley equations in immunology. Yet the ``macroscopic behavior" of the immune system, as probed in a specific experiment, can be well characterized. The problem then arises of selecting a simple representation for the elementary interactions that would give rise to the organized behavior observed in the immune system. The adventure of statistical physics is full of equivalent endeavors, starting from the description of thermal properties of gases and solids based respectively on the assumption of independent particles composing a perfect gas and the coupling of harmonic oscillators, to the more recent description of neural nets. This kind of approach is especially suited to theoretical immunology because of our ignorance about the detailed mechanisms responsible for the observed behaviors of the immune system. To be more specific, we shall look for generic properties among models of the immune system. As in the case of phase transitions in condensed matter physics, we are interested in semi-quantitative laws, such as scaling laws, which only depend on the general features of the model, and not on its details.

Cells are already macroscopic systems far from equilibria from the point of view of thermodynamics, and there is little hope to start from a simple Hamiltonian as is often done in statistical mechanics. But basically, the simple system of differential equations described in section IV plays the role of an Ising Hamiltonian with respect to diluted magnetic systems or of the logistic equation for chaos and turbulence: although it represents a strong simplification of the interactions present in the system, it is expected to belong to the same class of universality as a "true" model of the immune system, and to exhibit the same generic properties. In the case of a dynamical system, the generic properties concern the attractors of the dynamics. Some of the questions that we shall address are: Are the attractors limits points, limit cycles or chaotic? What is their number? What are their basins of attraction? How do these properties relate to the parameters parameters of the differential system? How can one force transitions among the different dynamical regimes? If our hypotheses about the universality of a model are true, the generic properties, qualitative classification of the attractors, and scaling laws, should be evident in the phenomenology of the mammalian immune system.. The above items were discussed in a framework where the proliferation of lymphocytes is controlled by the network. What about maturation of lymphocytes?.

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