Talk: Randomly Evolving Idiotypic Networks: Stability Properties
14:00 - 15:00
Johann Müller, M.Sc
University of Leipzig
Max-Planck-Institut für Dynamik und Selbstorganisation (MPIDS)
We consider a minimalistic dynamic model of the idiotypic network of B lymphocytes. A network node represents a population of B lymphocytes of the same specificity (idiotype). The idiotype is encoded by a bit string. Nodes have two possible states: 1 if the idiotype is present in the system, 0 if not. The links of the network connect nodes with complementary and nearly complementary bit strings. Depending on few parameters, the autonomous system evolves under random influx towards patterns of highly organized architecture, where the nodes can be classified into groups according to their statistical properties.
We investigate stability properties of emerging patterns considering their response to two different classes of perturbations. Firstly we study the Lyapunov stability comparing the evolution of closely neighbored initial states. For all static patterns, the Lyapunov exponent is negative. For the so called dynamic pattern we identify a regime where two closely neighbored copies keep a nonzero Hamming distance over time, indicating that the perturabtion is remebered by the system.
Further we investigate perturbation induced transitions between patterns of different architecture and find selection rules which relate the group affiliations of all nodes before and after the transition. These relations determine transition sets that have either strongly different or similar occupation before and after the transition. The occupation of the latter sets as a function of the influx probability hints to an extremal principle. The transition probability is highest, where the change in their occupation is close to zero.