"Stochastic modelling and statistics in neuroscience" Mini-course by Massimiliano Tamborrino

Mini-course by Dr. Massimiliano Tamborrino

Institute for Stochastics, Johannes Kepler University Linz

Stochastic Modelling in Neuroscience

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.

Statistics in Neuroscience: inference for stochastic processes and for point processes.

In many signal-processing applications, it is of primary interest to decode/reconstruct the unobserved signal based on some partially observed information. Some examples are all type of recognition (e.g. automatic speech, face, gesture, handwriting), chemistry, genetics and neuroscience (ion channels modelling). From a statistical point of view, this corresponds to perform statistical inference of the underlying model parameters from fully/partially observed stochastic processes (e.g. discrete observations of one or more other coordinates) and (non-renewal) point processes (where each event is the epoch when a coordinate reaches/crosses a certain value, yielding the so-called first-passage-time problem). We will briefly discuss a couple of examples (with application on lung cancer data and visual data) where the underlying likelihood function can be derived, leading to maximum likelihood estimation. Quite often though, the underlying likelihood is unknown or intractable. Among likelihood-free statistical methods, we will focus on Approximate Bayesian Computation. After presenting the method, we will illustrate it on two examples arising from neuroscience, with data corresponding to partially observed stochastic processes.