Immune networks: multitasking capabilities near saturation

Pattern-diluted associative networks were recently introduced as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T-and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with N-T T-lymphocytes can manage a number N-B = O(N-T(delta)) of B-lymphocytes simultaneously, with delta < 1. Here we study this model in the extensive load regime N-B = alpha N-T, with a high degree of pattern dilution, in agreement with immunological findings. We use graph theory and statistical mechanical analysis based on replicamethods to show that in the finite-connectivity regime, where each T-lymphocyte interacts with a finite number of B-lymphocytes as N-T -> infinity the T-lymphocytes can coordinate effective immune responses to an extensive number of distinct antigen invasions in parallel. As alpha increases, the system eventually undergoes a second order transition to a phase with clonal cross-talk interference, where the system's performance degrades gracefully. Mathematically, the model is equivalent to a spin system on a finitely connected graph with many short loops, so one would expect the available analytical methods, which all assume locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results are supported by numerical simulations.