A Cayley Tree Immune Network Model with Antibody Dynamics
A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analyzed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters, percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvist et al. (1989), injection of $Ab_1$ antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations of $Ab_1$ and $Ab_2$ antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvist et al. but not accounted for previously.