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antibodies; T H cells depend on the presence of other helpers (to keep homeostasis) or of
the antigens (implicit presentation is assumed). A network like that in the figure can
schematically represent this simplified model and its evolution in time can be studied by
means of computer simulations. For example, the complete KUT model has five fixed
points in the state space composed by 2 5 =32 points. Fixed points identify the global state
of the immune system: naive, vaccinate, immune, paralyzed, paralyzed and sick.
Many authors followed this simple approach. Weisbuch and Atlan proposed a model
( WA model ) (Weisbuch and Atlan 1988), based on Jerne's theory, to study the special
case of autoimmune diseases, like multiple sclerosis, in which the immune system
attacks the cells of the nervous system. As in the KUT model, this model uses five
binary variables representing killer cells (S 1 ), activated killers (S 2 ), suppressor cells
(S 3 ), helper cells (S 4 ), and activated suppressor cells stimulated by the helpers (S 5 ).
The different types of cells influence each other with a strength that is 1, 0, or -1.
The system evolves according to the following rule: at each time step, the concentra-
tion of one variable is set to unity if the sum of the interactions with the various cell
types is positive, otherwise the concentration is set to zero. This model shows the
existence of only two basins of attraction over 2 5 =32 possible states: the empty state
where all the concentrations are zero and a state where only activated killers disap-
pear whereas the other four concentrations are unity.
These two models have been extensively studied (Stauffer 1989; Pandey and
Stauffer 1990; Atlan and Cohen 1989). Moreover, Pandey and Stauffer further ex-
tended the KUT model by using a probabilistic generalization of the original deter-
ministic cellular automata. Their model tried to provide a possible explanation of the
time delay between HIV infection and the onset of AIDS (Pandey and Stauffer 1990;
Pandey and Stauffer 1989). They represented helper cells (H), cytotoxic cells (S),
virus (V), and interleukin (I). The interleukin molecules produced by helper cells
induce the suppressor cells to kill the virus. The dynamics shows an oscillatory be-
havior followed by a fixed point where the immune system is totally destroyed, simi-
lar to the real onset of the AIDS.
Dayan, Havlin, and Stauffer (1988) studied the WA model on a square lattice in order
to take into account spatial fluctuations of cell concentrations. In their model each lattice
point influences itself and its nearest neighbors with the same rules of the WA model.
Interestingly, this lattice version of the WA model was found to have a different dynam-
ics compared to the original WA model as the number of fixed points is smaller.
Chowdhury and Stauffer (Chowdhury and Stauffer 1992; Chowdhury 1998) pro-
posed a unified model of the immune system that includes, as special cases, the KUT
and WA models. The model describes the immune response to HIV and reproduces
some features of experimental data. Chowdhury and Stauffer also proposed exten-
sions of the original network approach for modeling HIV and cancer (Chowdhury
and Stauffer 1992).
A majority rule cellular automaton was used by Agur (1991) to study the signal
processing in a multilayered network. Chowdhury, Deshpande, and Stauffer (1994)
proposed a model to describe the interaction between various types of immune com-
ponents considering intra- and interclonal interactions.
The interaction among different T H cell subsets (Brass, Bancroft, Clamp, Grencis,
and Else 1994) and the HIV interaction with T cells have been modeled as well
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