A qualitative study of an idiotypic cyclic network.

Abstract Within the network hypothesis proposed by Jerne, the immune response is interpreted as a collective behaviour of different antibody species, interacting through idiotypic recognition. In order to insure the stability of the network, only a few species would be implied in the response to an antigenic challenge. We study a network made up of small cycles of idiotypic units, each element activating the subsequent one and repressing the preceding one. In the recent theoretical models, the kinetics is described by steep sigmoidal functions with a repression threshold lower than the stimulation one. To enable a systematic qualitative analysis of the dynamics, we replace the continuous kinetics by stepfunctions. The antibodies are thus considered as control elements like genes, enzymes or neurones. In order to account for the different thresholds, we use discrete three-level variables. We develop two methods to study the dynamics: the first one, due to Glass, describes the time-evolution of a cycle by a system of piecewise linear (PL) differential equations and the second method is the boolean formalization, applied extensively by Thomas in the field of genetic regulation. These techniques provide complementary informations about the dynamics of the cycle: the PL method establishes a state transition diagram providing all the potential behaviours independently of the parameter values in the model, whereas the purely logical analysis permits a simulation of the trajectories for precise values of the parameters. The state transition diagram presents several steady states. It suggests to interpret the response to an antigenic challenge as a transition from one steady state to another. The multiplicity of the steady states might be associated with the various modes of immune response depending on the doses of antigen injected and on the previous antigenic history of the system.