On stationary solutions of delay differential equations: a model of local translation in synapses

The results of analytical analysis of stationary solutions of a differential equation with two delayed arguments $\tau_1$ and $\tau_2$ are presented. Such equations are used in modeling of molecular-genetic systems where the delay of arguments appear naturally. Conditions of existence of non-negative solutions are described, and dependence of stability of these solutions on the values of delayed arguments is studied. This stability theory allows to give complete characterization of these solutions for all values of the parameters of the model, and ensures instability of a positive equilibrium point for any values of the delays $\tau_2\ge\tau_1\ge0$ in the case when it is unstable for $\tau_2=\tau_1=0$ (absolute instability). If this positive equilibrium point is stable only for $\tau_2=\tau_1=0$, then this domain $\tau_2\ge\tau_1\ge0$ is the domain of absolute instability as well. For positive equilibrium points which are stable at $\tau_2=\tau_1=0$, we find domains of absolute stability were the equilibrium points remain stable for all values of the parameters $\tau_1$ and $\tau_2$. The domains of relative stability, where these points become unstable for some values of these parameters are also described. We show that when the efficiency of translation, and non-linearity and complexity of its regulation mechanisms grow, the domains of the absolute and relative stability of the positive equilibrium point shrink, while the domains of its instability expand. So, enhanced activity of the local translation system can be a factor of its instability and that of the risk of neuro-psychical diseases related to distortions of plasticity of the synapse and memory, where importance of stability of the proteome in the synapse is postulated.