Memory in Idiotypic Networks due to Competition Between Proliferation and Differentiation

A model employing separate dose-dependent response functions for proliferation and differentiation of idiotypically interacting $B$ cell clones is presented. For each clone the population dynamics of proliferating $B$ cells, non-proliferating $B$ cells and free antibodies are considered. An effective response function, which contains the total impact of proliferation and differentiation at the fixed points, is defined in order 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 different field-response functions (e.g., linear, saturation, log-bell functions). The stable fixed points are interpreted as memory states in terms of immunity and tolerance. It is proven that a combination of linear response functions for both proliferation and differentiation does not give rise to stable fixed points. However, due to competition between proliferation and differentiation saturation response functions are sufficient to obtain two memory states, provided proliferation preceeds differentiation and also saturates earlier. The use of log-bell-shaped response functions for both proliferation and differentiation gives rise to a ``mexican-hat'' effective response function and allows for 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 function of the parameters of the model. The attractors loose 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 changed by the existence of non-proliferating $B$ cells: memory states can exist and be stable without non-proliferating $B$ cells. Nevertheless, the activation of non-proliferating $B$ cells and the competition between proliferation and differentiation enlarge the parameter regime for which stable attractors are found. In addition, it is shown that a separate activation step from virign to active $B$ cells renders the virgin state stable for any choice of biologically reasonable parameters.