Analysis of an immune network dynamical system model with a small number of degrees of freedom

We numerically study a dynamical system model of an idiotypic immune network with a small number of degrees of freedom. The model was originally introduced by Varela et al., and it describes antibodies interacting in an organism in order to prepare for the invasion of external antigens. The main purpose of this paper is to investigate the direction of change in the network system when antigens invade it. We investigate three models, the original model, a modified model, and a modified model with a threshold of concentration, above which each antibody can recognize other antibodies. First, we study possible attractors of the networks. In all these models, both chaotic and periodic states exist. In particular, we find peculiar periodic states organized in the network, the differentiating states. In these states, one clone plays the role of switching the clones to be excited. That is, it causes an excited clone to become suppressed and a suppressed clone to become excited. Next, we investigate the response of the system to invasions by antigens. We find that in some cases the system changes in a positive direction when it is invaded by antigens, and the differentiating state can be interpreted as short term memory of such invasion. We also find tolerant behavior. Further, from the investigation of the relaxation times for invasions by antigens, it is found that in a chaotic state, the average response time takes an intermediate value among those in asymmetric periodic states. This suggests a positive aspect of chaos in immune networks.