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In this work we use belief-propagation techniques to study the equilibriumbehaviour of a minimal model for the immune system comprising interacting T andB clones. We investigate the effect of the so-called idiotypic interactionsamong complementary B clones on the system's activation. Our result shows thatB-B interactions increase the system's resilience to noise, making clonalactivation more stable, while increasing the cross-talk between differentclones. We derive analytically the noise level at which a B clone getsactivated, in the absence of cross-talk, and find that this increases with thestrength of idiotypic interactions and with the number of T cells signallingthe B clone. We also derive, analytically and numerically, via populationdynamics, the critical line where clonal cross-talk arises. Our approach allowsus to derive the B clone size distribution, which can be experimentallymeasured and gives important information about the adaptive immune systemresponse to antigens and vaccination.

The role of idiotypic interactions in the adaptive immune system: a belief-propagation approach