In Diffusive Molecular Communication (DMC), information is transmitted by diffusing molecules. Synaptic signaling is a natural implementation of this paradigm. It is responsible for relaying information from one neuron to another, but also provides support for complex functionalities, such as learning and memory. Many of its features are not yet understood, some are, however, known to be critical for robust, reliable neural communication. In particular, some synapses feature a re-uptake mechanism at the presynaptic neuron, which provides a means for removing neurotransmitters from the synaptic cleft and for recycling them for future reuse. In this paper, we develop a comprehensive channel model for synaptic DMC encompassing a spatial model of the synaptic cleft, molecule re-uptake at the presynaptic neuron, and reversible binding to individual receptors at the postsynaptic neuron. Based on this model, we derive an analytical time domain expression for the channel impulse response (CIR) of the synaptic DMC system. Our model explicitly incorporates macroscopic physical channel parameters and can be used to evaluate the impact of re-uptake, receptor density, and channel width on the CIR of the synaptic DMC system. Furthermore, we provide results from particlebased computer simulation, which validate the analytical model. The proposed comprehensive channel model for synaptic DMC systems can be exploited for the investigation of challenging problems, like the quantification of the inter-symbol interference between successive synaptic signals and the design of synthetic neural communication systems.