Dynamics and multi-scale modeling with time delays for three stages of synaptic facilitation

Serotonin-induced facilitation of synaptic transmission at the sensorimotor synapse in Aplysia occurs in three temporal phases: short-term facilitation (STF), intermediate-term facilitation (ITF), and long-term facilitation (LTF). The delayed onset of LTF is considered to be related to the time-consuming processes of transcription and translation in protein synthesis. In this work, we develop a muti-scale mathematical model that involves gene transcription and protein synthesis with time delays, presynaptic membrane potential, and excitatory postsynaptic potential for the three stages of synaptic facilitation. The positive feedback loop that forms the basis of bistability according to our bifurcation analysis may account for the transition between two different states following stimulations. Our model simulates experimental findings on the generation of STF, ITF and LTF under different applications of 5HT stimulation. We further confirm that ITF is dependent on translation, whereas LTF relies on both translation and transcription. In addition, we find that the threshold of the number of repeated pulses of 5HT required for the induction of LTF changes with different ranges of the time delays. Our model could provide a useful framework for exploration of dynamical mechanisms of different temporal domains of synaptic plasticity, as well as guiding investigations into other processes underlying memory formation.