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In Diffusive Molecular Communication (DMC), information is transmitted by diffusing molecules. Synaptic signaling, as a natural implementation of this paradigm, encompasses functional components that, once understood, can facilitate the development of synthetic DMC systems. To unleash this potential, however, a thorough understanding of the synaptic communication channel based on biophysical principles is needed. Since synaptic transmission critically depends also on non-neural cells, such understanding requires the consideration of the so-called tripartite synapse. In this paper, we develop a comprehensive channel model of the tripartite synapse encompassing a three-dimensional, finite-size spatial model of the synaptic cleft, molecule uptake at the presynaptic neuron and at glial cells, reversible binding to individual receptors at the postsynaptic neuron, and spillover to the extrasynaptic space. Based on this model, we derive analytical time domain expressions for the channel impulse response (CIR) of the synaptic DMC system and for the number of molecules taken up at the presynaptic neuron and at glial cells, respectively. These expressions provide insight into the impact of macroscopic physical channel parameters on the decay rate of the CIR and the reuptake rate, and reveal fundamental limits for synaptic signal transmission induced by chemical reaction kinetics and the channel geometry. Adapted to realistic parameters, our model produces plausible results when compared to experimental and simulation studies and we provide results from particle-based computer simulations to further validate the analytical model. The proposed comprehensive channel model admits a wide range of synaptic configurations making it suitable for the investigation of many practically relevant questions, such as the impact of glial cell uptake and spillover on signal transmission in the tripartite synapse.