The present paper investigates conditions under which immunological memory can be maintained by stimulatory idiotypic network interactions. The paper was motivated by the work of (De Boer & Hogeweg, 1989b, Bull. math. Biol. 51, 381-408.) which claimed that idiotypic memory is not possible because of percolation within the network.Here we reinvestigate the issue of percolation using both the previous model and a simpler one (Weisbuch, 1990, J. theor. Biol. 143, 507-522.) that allows analytic analysis. We focus on network topologies in which each Abl is connected to several Ab2s, which in turn are connected to several Ab3s. It is demonstrated that, for a considerable range of parameters, both models account for the existence of localized memory-states in which only the Ab 1 and the Ab2 clones are activated and the clones of the Ab3 level remain virgin.The existence of localized memory-states seems to contradict the previous percolation result. This discrepancy will be shown to depend on the system dynamics. By simulation we explore the parameter regimes for which one finds percolation and those for which localized memory-states exists. We show that the conditions required for attaining the localized memory-state are considerably more stringent than those required for its existence and local stability. We conclude that both localized memory and percolation are possible in stimulatory idiotypic networks.

Models of the dynamical interactions important in generating immune reactivity have generally assumed that the immune system is a single well-stirred compartment. Here we explicitly take into account the compartmentalized nature of the immune system and show that qualitative conclusions, such as the stability of the immune steady state, depend on architectural details. We examine a simple model idiotypic network involving only two types of B cells and antibody molecules. We show, for model parameters used by De Boer et al. (1990, Chem. Eng. Sci. 45, 2375-2382), that the immune steady state is unstable in a one compartmental model but stable in a two compartment model that contains both a lymphoid organ, such as the spleen, and the circulatory system.

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