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4. If L<S Ab <U , increase the concentration of the antibody by 1, otherwise
decrease it by 1
5. Calculate the simulation S Ag received by each antigen according
6. If L<S Ag , decrease the concentration by 1.
7. Remove any cells whose concentration has reached 0.
3
Experimental Results
We first consider models with complementary anity function. In 2D shape-
space, in order to be consistent with work reported previously in [6,5] and by [2],
experiments are performed on a grid of size 100x100, resulting in 10000 potential
cells. The values of the lower limit L and upper limit U are fixed at L = 100
and U =10 , 000. Previous work shows that interesting network behaviour is
obtained when r = 15, therefore this value is used in these experiments unless
otherwise indicated. (Below this value, percolation does not occur therefore a
network does not emerge; at high r , the high suppressive effect of cells also does
not result in a stable network). Antibody cells are added to the simulation with
concentration 10; antigen cells are added with concentration 1000.
In the simulations with bit-strings, we consider strings of length 13, creating
a space of 8192 possible cells, a potential repertoire size of similar size as the 2D
shape-space. The lower limit L and upper limit U take respectively values 5000
and 10000. T is the anity threshold (similar to r in the 2D shape-space model)
and define the lower limit of complementarity to get stimulation. Regarding the
idiotypic network as just a graph, we may say that a cell A is connected with a
cell B, if the Hamming Distance between A and B is higher than this threshold
T. A high T value imposes a system where an almost perfect complementarity is
needed for stimulation, whereas a low T tolerates very poor complementarity for
the network to pop up. Each combination of parameters gives rise to different
stabilized networks. The size of the stable network will depend primarily on the
Threshold level (T) and the size of the window (U and L). For low specificity
(low T), the network exhibits a high average degree, which may result in a excess
of stimulation depending on the Upper limit of the stimulation window. In this
case of over-stimulation, the network is not able to pop up since the majority of
nodes can hardly remain under the upper limit. The opposite can also happen.
When almost perfect complementarity is needed for stimulation (very high T),
the average degree of the network will be so low that nodes cannot be stimulated
over the lower stimulation limit. So, for an idiotypic network to pop up, an
optimal individual average stimulation value must be found, which depends on
a balance between the cells' initial concentration, upper and lower limit of the
stimulation window, and the anity threshold.
As reported in previously, the 2D shape-space model results in the physical
space being clearly separated by a line of sustained antibody cells into two dis-
tinct regions. In one of these regions, antigens are tolerated; in the other all
antigens are suppressed. The position of these zones, and the resultant ability of
cells to be maintained by the network, emerges only from the network dynamics.
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