Retrieving infinite numbers of patterns in a spin-glass model of immune networks

¹ Dipartimento di Matematica, Sapienza Università di Roma - Rome, Italy
² Gruppo Nazionale per la Fisica Matematica, Sezione di Roma - Rome, Italy
³ Department of Mathematics, King's College University of London - London, UK
⁴ Institute for Mathematical and Molecular Biomedicine, King's College University of London - London, UK
⁵ Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento - Lecce, Italy
⁶ Centro Ennio De Giorgi, Scuola Normale Superiore - Pisa, Italy

Received: 4 December 2016
Accepted: 20 February 2017

Abstract

The similarity between neural and (adaptive) immune networks has been known for decades, but so far we did not understand the mechanism that allows the immune system, unlike associative neural networks, to recall and execute a large number of memorized defense strategies in parallel. The explanation turns out to lie in the network topology. Neurons interact typically with a large number of other neurons, whereas interactions among lymphocytes in immune networks are very specific, and described by graphs with finite connectivity. In this paper we use replica techniques to solve a statistical mechanical immune network model with “coordinator branches” (T-cells) and “effector branches” (B-cells), and show how the finite connectivity enables the coordinators to manage an extensive number of effectors simultaneously, even above the percolation threshold (where clonal cross-talk is not negligible). A consequence of its underlying topological sparsity is that the adaptive immune system exhibits only weak ergodicity breaking, so that also spontaneous switch-like effects as bi-stabilities are present: the latter may play a significant role in the maintenance of immune homeostasis.