In this chapter, we describe how to create mathematical models of synaptic transmission and integration. We start with a brief synopsis of the experimental evidence underlying our current understanding of synaptic transmission. We then describe synaptic transmission at a particular glutamatergic synapse in the mammalian cerebellum, the mossy fiber to granule cell synapse, since data from this well-characterized synapse can provide a benchmark comparison for how well synaptic properties are captured by different mathematical models. This chapter is structured by first presenting the simplest mathematical description of an average synaptic conductance waveform and then introducing methods for incorporating more complex synaptic properties such as nonlinear voltage dependence of ionotropic receptors, short-term plasticity, and stochastic fluctuations. We restrict our focus to excitatory synaptic transmission, but most of the modeling approaches discussed here can be equally applied to inhibitory synapses. Our data-driven approach will be of interest to those wishing to model synaptic transmission and network behavior in health and disease.
Keywords: AMPA receptor; Chemical synapses; Conductance waveforms; Depletion models; Integrate-and-fire models; Mathematical models; NMDA receptor; Poisson spike trains; Quantal release; Short-term plasticity; Synaptic depression; Synaptic integration; Synaptic transmission; Vesicular release.
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