Interaction of the immune system with a target population of, e.g., bacteria, viruses, antigens, or tumor cells must be considered as a dynamic process. We describe this process by a system of two ordinary differential equations. Although the model is strongly idealized it demonstrates how the combination of a few proposed nonlinear interaction rules between the immune system and its targets are able to generate a considerable variety of different kinds of immune responses, many of which are observed both experimentally and clinically. In particular, solutions of the model equations correspond to states described by immunologists as "virgin state," "immune state" and "state of tolerance." The model successfully replicates the so-called primary and secondary response. Moreover, it predicts the existence of a threshold level for the amount of pathogen germs or of transplanted tumor cells below which the host is able to eliminate the infectious organism or to reject the tumor graft. We also find a long time coexistence of targets and immune competent cells including damped and undamped oscillations of both. Plausibly the model explains that if the number of transformed cells or pathogens exeeds definable values (poor antigenicity, high reproduction rate) the immune system fails to keep the disease under control. On the other hand, the model predicts apparently paradoxical situations including an increased chance of target survival despite enhanced immune activity or therapeutically achieved target reduction. A further obviously paradoxical behavior consists of a positive effect for the patient up to a complete cure by adding an additional target challenge where the benefit of the additional targets depends strongly on the time point and on their amount. Under periodically pulsed stimulation the model may show a chaotic time behavior of both target growth and immune response. (c) 1995 American Institute of Physics.

A heuristic model for the dynamics of recurrent inhibition, emphasizing non-linearities arising from the stoichiometry of transmitter-receptor interactions and time delays due to finite feedback pathway transmission times, is developed and analyzed. It is demonstrated that variation in model parameters may lead to the existence of multiple steady states, and the local stability of these are analyzed as well as the occurrence of switching behaviour between them. As an example of the applicability of this model, parameters are estimated for the hippocampal mossy fibre-CA3 pyramidal cell-basket cell complex. Numerically simulated responses of this system to alterations in presynaptic drive and titration of inhibitory transmitter receptors by penicillin are presented. Numerical simulations indicate the existence of multiple bifurcations between periodic solutions, as well as the existence of bifurcations to chaotic solutions, as presynaptic drive and receptor density are varied. It is hypothesized that the model offers insight into the sequences of events recorded in single CA3 pyramidal cells following the application of penicillin, a specific inhibitory receptor blocking agent.

Dynamical diseases are characterized .by sudden changes in the qualitative dynamics of physiological processes, leading to abnormal dynamics and disease. Thus, there is a natural matching between the mathematical field of nonlinear dynamics and medicine. This paper summarizes advances in the study of dynamical disease with emphasis on a NATO Advanced Research Worshop held in Mont Tremblant, Quebec, Canada in February 1994. We describe the international effort currently underway to identify dynamical diseases and to study these diseases from a perspectiveof nonlinear dynamics. Linearand nonlinear time series analysis combined with analysis of bifurcations in dynamics are being used to help understand mechanisms of pathological rhythms and offer the promise for better diagnostic and therapeutic techniques.

Solution properties of the nonlinear second-order delay-differential equation 2(0 = -ax(t) +f [x(t -3)] are studied where f is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes. KEY WORDS:Nonlinear differential equations of second order with deviating argument; oscillations; periodic solutions.

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