MEMORY IN IDIOTYPIC NETWORKS DUE TO COMPETITION BETWEEN PROLIFERATIONAND DIFFERENTIATION

A model employing separate dose-dependent response functions for proli feration and differentiation of idiotypically interacting B cell clone s is presented. For each clone the population dynamics of proliferatin g B cells, non-proliferating B cells and free antibodies are considere d. An effective response function, which contains the total impact of proliferation and differentiation at the fixed points, is defined in o rder to enable an exact analysis. The analysis of the memory states is restricted in this paper to a two-species system. The conditions for the existence of locally stable steady states with expanded B cell and antibody populations are established for various combinations of diff erent field-response functions (e.g. linear, saturation, log-bell func tions). The stable fixed points are interpreted as memory states in te rms of immunity and tolerance. It is proven that a combination of line ar response functions for both proliferation and differentiation does not give rise to stable fixed points. However, due to competition betw een proliferation and differentiation saturation response functions ar e sufficient to obtain two memory states, provided proliferation prece eds differentiation and also saturates earlier. The use of log-bell-sh aped response functions for both proliferation and differentiation giv es rise to a ''mexican-hat'' effective response function and allows fo r multiple (four to six) memory states. Both a primary response and a much more pronounced secondary response are observed. The stability of the memory states is studied as a function of the parameters of the m odel. The attractors lose their stability when the mean residence time of antibodies in the system is much longer than the B cells' lifetime . Neither the stability results nor the dynamics are qualitatively cha nged by the existence of non-proliferating B cells: memory states can exist and be stable without non-proliferating B cells. Nevertheless, t he activation of non-proliferating B cells and the competition between proliferation and differentiation enlarge the parameter regime for wh ich stable attractors are found. In addition, it is shown that a separ ate activation step from virgin to active B cells renders the virgin s tate stable for any choice of biologically reasonable parameters.