In neuroscience, it is of paramount interest to understand the principles of information processing in the nervous system, i.e. networks of neurons. Neurons communicate by short and precisely shaped electrical impulses, the so-called action potentials or spikes. It is thus of major interest to understand the principles of the underlying spike generating mechanisms, starting by investigating the dynamics of the membrane potential in a single neuron. After a brief discussion on biophysical neuronal models, we will focus on stochastic Leaky integrate-and-fire (LIF) neuronal models. They are probably some of the most common mathematical representations of single neuron electrical activity. The simplification implies that a spike is represented by a point event modelled by the first passage time to a firing threshold, an upper bound of the membrane voltage. Depending on the underlying assumptions, consecutive interspike intervals may be independent or not, yielding renewal or non-renewal point processes, respectively. After introducing some key LIF models, we will learn how to face the corresponding FPT problem, deriving quantities of interest such as density, mean and variance.
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