  In this work we use belief-propagation techniques to study the equilibrium
behaviour of a minimal model for the immune system comprising interacting T and
B clones. We investigate the effect of the so-called idiotypic interactions
among complementary B clones on the system's activation. Our result shows that
B-B interactions increase the system's resilience to noise, making clonal
activation more stable, while increasing the cross-talk between different
clones. We derive analytically the noise level at which a B clone gets
activated, in the absence of cross-talk, and find that this increases with the
strength of idiotypic interactions and with the number of T cells signalling
the B clone. We also derive, analytically and numerically, via population
dynamics, the critical line where clonal cross-talk arises. Our approach allows
us to derive the B clone size distribution, which can be experimentally
measured and gives important information about the adaptive immune system
response to antigens and vaccination.
