Learning cascade in the immune system dynamics : a numerical

The idea of learning cascade, recently introduced to characterize the progressive evolution of the immune system response from low-aanity to high-aanity lymphocites, is explored by means of extensive computer simulations. In a recent work 7] it was proposed that the Immune System (IS) learns how to withstand attacks by foreign invaders, the \Antigens" (Ag for short) by means of a so-called \learning cascade" (LC) in the bit-matching space, namely the abstract space deened by the number of matching bits, m, between the binary strings representing the B-cells and the Ag speciicity respectively. By means of extensive numerical simulations, based on the Celada-Seiden (CS) cellular automaton 2], it was shown that the LC proceeds via the sequential excitation of collective modes in m-space which look like isolated bumps of the B-cell distribution function. This latter is deened as the number of B-cells with m matching bits over the total number of B cells in the system fm = Bm= P l m=0 Bm, where l is the length of the bit-string representing the B-cell speciicity. Upon interpolating these bumps with Gaussian prooles, satisfactory agreement with numerical results was obtained. Besides sheer interpretation of numerical data, the assumption of Gaussian prooles allows to obtain an analytical expression for the entropy gain (or Kullback entropy) 3]. For a transition taking state 1, with gaussian distribution: B (1) m = n (1) 1 p 2 1 exp?(m ? 1) 2 =2 2 1 ] into a gaussian state 2 with parameters n (2) ; 2 ; 2 , the entropy gain, generically deened as G 1;2 = P m f 2 m ln(f 2 m =f 1 m), reads as G 12 = n 2 ln n 2 n 1 + n 2 2 (lnn 12 + ?1 12 ? 1 + 2 12) (1) where n i = P m f i m is the number of individuals belonging to f i , 12 = 2 1 == 2 2 and 12 = (2 ? 1)= p 1. The \mean-eld" quantity 12 is naturally identiied with the degree of learning of the IS, for it is clear that high mean matching numbers correspond to a large population of high aanity (HA) B-cells. The entropy gain G 12 was postulated to represent the information cost required to bring the IS from the low-aanity (LA) state 1 into the high aanity state 2. The analogy …