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The ability to form long-term memories is an important function for the nervous system, and the formation process is dynamically regulated through various transcription factors, including CREB proteins. In this paper, we investigate the dynamics of a delay differential equation model for CREB protein activities, which involves two positive and two negative feedbacks in the regulatory network. We discuss the dynamical mechanisms underlying the induction of long-term memory, in which bistability is essential for the formation of long-term memory, while long time delay can destabilize the high level steady state to inhibit the long-term memory formation. The model displays rich dynamical response to stimuli, including monostability, bistability, and oscillations, and can transit between different states by varying the negative feedback strength. Introduction of a time delay to the model can generate various bifurcations such as Hopf bifurcation, fold limit cycle bifurcation, Neimark–Sacker bifurcation of cycles, and period-doubling bifurcation, etc. Increasing the time delay can induce chaos by two routes: quasi-periodic route and period-doubling cascade. [ABSTRACT FROM AUTHOR]

Abstract: Tissue development and homeostasis are thought to be regulated endogenously by control loops that ensure that the numbers of stem cells and daughter cells are maintained at desired levels, and that the cell dynamics are robust to perturbations. In this paper we consider several classes of stochastic models that describe stem/daughter cell dynamics in a population of constant size, which are generalizations of the Moran process that include negative control loops that affect differentiation probabilities for stem cells. We present analytical solutions for the steady-state expectations and variances of the numbers of stem and daughter cells; these results remain valid for non-constant cell populations. We show that in the absence of differentiation/proliferation control, the number of stem cells is subject to extinction or overflow. In the presence of linear control, a steady state may be maintained but no tunable parameters are available to control the mean and the spread of the cell population sizes. Two types of nonlinear control considered here incorporate tunable parameters that allow specification of the expected number of stem cells and also provide control over the size of the standard deviation. We show that under a hyperbolic control law, there is a trade-off between minimizing standard deviations and maintaining the system robustness against external perturbations. For the Hill-type control, the standard deviation is inversely proportional to the Hill coefficient of the control loop. Biologically this means that ultrasensitive response that is observed in a number of regulatory loops may have evolved in order to reduce fluctuations while maintaining the desired population levels. [Copyright &y& Elsevier]

Abstract: Signal transduction is the process by which the cell converts one kind of signal or stimulus into another. This involves a sequence of biochemical reactions, carried out by proteins. The dynamic response of complex cell signalling networks can be modelled and simulated in the framework of chemical kinetics. The mathematical formulation of chemical kinetics results in a system of coupled differential equations. Simplifications can arise through assumptions and approximations. The paper provides a critical discussion of frequently employed approximations in dynamic modelling of signal transduction pathways. We discuss the requirements for conservation laws, steady state approximations, and the neglect of components. We show how these approximations simplify the mathematical treatment of biochemical networks but we also demonstrate differences between the complete system and its approximations with respect to the transient and steady state behavior. [Copyright &y& Elsevier]