  We consider self-tolerance and its failure -autoimmunity- in a minimal
mathematical model of the idiotypic network. A node in the network represents a
clone of B-lymphocytes and its antibodies of the same idiotype which is encoded
by a bitstring. The links between nodes represent possible interactions between
clones of almost complementary idiotype. A clone survives only if the number of
populated neighbored nodes is neither too small nor too large. The dynamics is
driven by the influx of lymphocytes with randomly generated idiotype from the
bone marrow. Previous work has revealed that the network evolves towards a
highly organized modular architecture, characterized by groups of nodes which
share statistical properties. The structural properties of the architecture can
be described analytically, the statistical properties determined from
simulations are confirmed by a modular mean-field theory. To model the presence
of self we permanently occupy one or several nodes. These nodes influence their
linked neighbors, the autoreactive clones, but are themselves not affected by
idiotypic interactions. The architecture is very similar to the case without
self, but organized such that the neighbors of self are only weakly occupied,
thus providing self-tolerance. This supports the perspective that self-reactive
clones, which regularly occur in healthy organisms, are controlled by
anti-idiotypic clones. We discuss how perturbations, like an infection with
foreign antigen, a change in the influx of new idiotypes, or the random removal
of idiotypes, may lead to autoreactivity and devise protocols which cause a
reconstitution of the self-tolerant state. The results could be helpful to
understand network and probabilistic aspects of autoimmune disorders.
