B Cell–Inspired Algorithms - Immunological Computation: Theory and Applications - page 127

Farmer et al. (1986) reported that the antibodies whose paratopes match
epitopes are amplifi ed at the expense of other antibodies. If the suppression and
stimulation rate are the same (equal to 1 in their work) and k 2

>

0, then every
antibody type will eventually die due to the damping term. However, if k 1

<

1, it
favors the formation of reaction loops; thus, the numbers of loop can gain concen-
tration, fi ghting the damping term. h e number and lengths of the loops increase
as N increases. Antibodies that do not recognize other elements are eventually dis-
carded. Farmer et al. (1987) introduced an equation to describe the change in the
concentration of antigen of type i :

M

dy
dt

∑

i

…

kmc y , for i 1, , n (5.10)

ji j i

4

j 1

h us, Equation 5.10 describes the dynamics of intrinsic antigen elimination.

5.2.1.4 Parisi's Idiotypical Network

To study immunological memory, a simple IN model, which captures most of the
qualitative features, was introduced by Parisi (1990). Parisi's model focuses on the
behavior of the immune system in the absence of external antigens and attempts to
fi nd a global functional description of the IN.
Parisi's model assumes that auto-antibodies of a given antibody are a very large
set of low responder clones and the connectivity of the IN is very high, and this
network cannot be partitioned into subnetworks. h is immune model seems to
have similarity with the Hopfi eld model (Hopfi eld, 1982). Here, a fully connected
network is considered, and a connection weight vector that represents the infl uence
of antibodies on one another is defi ned. h e concentration of any antibody, in the
absence of external antigens, is considered to have only two values, either 0 or 1,
and that the value is greater than 1 in the presence of stimulating antigen. h e
dynamics of the network in discrete time is further analyzed.
A matrix J ik , which codes the eff ect of the k th antibody on the i th antibody
is considered, similar to the synaptic weight matrix in a Hopfi eld network. h e
stimulatory eff ect of the network on the i th antibody is thus given by

N

∑

ht () s Jx t ()

(5.11)

i ij j

j 1

θ [ h i ( t )], where θ ( h ) is a step function defi ned as 0 if h is negative,
otherwise it will be 1. If J ik is positive, then antibody k triggers the production of
antibody i . In contrast, if J ik is negative, then antibody k suppresses the production
of antibody i .

=

with x k ( t )