B Cell–Inspired Algorithms - Immunological Computation: Theory and Applications - page 124

Also, in continuous models, a set of equations are used to predict the concentra-
tion of specifi c antigens and is defi ned as

= ∆ y i = +

dy i / dt antigen elimination natural antigen birth or death (5.2)

where y i is the concentration of antigen of type i at a given time t .
h erefore, one IN model can be distinguished from another by noticing how
each one of these terms represents the dynamic interaction processes. Diff erent
models are also characterized by the way they represent antibodies, including the
number of paratopes, epitopes, and antigens.
A more specifi c form of Equation 5.2 may be given by

∆ x i = − +

internal interactions antibody damping antigen driving (5.3)

where the fi rst term represents the natural dynamics of the idiotypic network as a
result of antibody-antibody interactions; the second term models the reduction of
cells in the absence of stimulation by antigens; whereas the third term represents
the antigenic eff ects.

5.2.1.1 Jerne's Idiotypical Network

Jerne's (1974) model (Weisbuch et al., 1990) introduced the following equation to
describe the change of lymphocytes of a certain type:

N N

dx
dt

∑ ∑

i

x f EKt ( , , ) x g I ( , Kt , ) kkx (5.4)

i j j i j j 1 2 i

j 1 j 1

where the fi rst term represents the total stimulation of lymphocytes of type i by
excitatory signals as a sum of excitatory signals received from stimulating lympho-
cytes. Accordingly, f ( E j , K j , t ) is a measure of excitatory signals from idiotypes in
E j on a type i lymphocyte at time t . K i is a constant associated with the strength of
the a nity between lymphocyte of type i and idiotypes in E j . In a similar fashion,
the second term expresses the total eff ect of inhibitory signals from other lympho-
cytes on a lymphocyte of type I ; thus, I j expresses a lymphocyte whose combining
sites recognize idiotypes on type I cells. In addition, k 1 is the rate at which type i
lymphocytes enter the network and k 2 is a natural death rate of type i lymphocytes
in the absence of antigen.
In this model, a diff erential equation describes the change in the concentration
of lymphocytes of each type. h us, the network presents a dynamic behavior even
in the absence of stimulating antigens. To describe the dynamic behavior of a for-
eign antigen, an additional term needs to be included to represent the interaction
of corresponding type i lymphocyte with external antigens.