# Seminar in applied mathematics and statistics

**SPEAKER: **Massimiliano Tamborrino (Institute for Stochastics, Johannes Kepler University Linz, Austria).

**TITLE: **(A bit of modelling and) Statistical inference for non-renewal point processes via Approximate Bayesian Computation. An application to neuroscience.

**ABSTRACT: **In many signal-processing applications, it is of primary interest to decode or reconstruct the unobserved signal based on some partially observed information. Some examples are all type of recognition (e.g. automatic speech, face, gesture, handwriting), genetics, genomics and neuroscience (ion channels modelling). From a statistical point of view, this corresponds to perform statistical inference of the underlying model parameters from partially observed stochastic processes (e.g., discrete observations of one or more other coordinates) and/or (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). Moreover ,due to the increasing complexity of the processes, the underlying likelihoods are often unknown or intractable, so commonly used likelihood-based inference techniques cannot be applied. The problem belongs to the class of intractable-likelihood inference problems, requiring the investigation of new ad hoc mathematical, numerical and statistical techniques to handle it. Here I focus on likelihood-free methods, and in particular on Approximate Bayesian Computation (ABC) method, and I illustrate it in the framework of stochastic modelling of single neuron dynamics.

More specifically, I consider a bivariate stochastic process where available observations are the hitting times of one coordinate to the other, and discuss it in the framework of stochastic modelling of single neuron dynamics. The considered multi-timescale adaptive threshold model is not an ad-hoc model, but can be derived from the detailed Hodgkin-Huxley model, can accurately predict spike times and incorporate the effects of slow K+ currents, usually mediating adaptation. When performing statistical inference of the underlying model parameters, four difficulties arise: none of the two model components is directly observed; the considered process is not of hidden Markov model type; the underlying likelihood is unknown/intractable; consecutive hitting times are neither independent nor identically distributed. I tackle these statistical issues considering a simple acceptance-rejection ABC algorithm. After presenting the ABC method and discussing its criticality and challenges, I illustrate how to use it on the considered model.

References:

[1] M.A. Beaumont, W. Zhang, and D.J. Balding. Approximate Bayesian Computation in Population Genetics. Genetics, 2002, Vol.162(4), pp.2025-2035.

[2] E. Buckwar, M. Tamborrino and I. Tubikanec. Periodogram-Based and Structure-Preserving Approximate Bayesian Computation for Partially Observed Diffusion Processes Admitting an Invariant Measure. A Demonstration on Hamiltonian SDEs, in preparation.

[3] R. Kobayashi, K. Kitano. Impact of slow K+ currents on spike generation can be described by an adaptive threshold model. J. Comput. Neurosci., 2016, 40(3), pp. 347-362.

[4] M. Tamborrino, A. Samson, U. Picchini. Approximate Bayesian Computation for the inference of non-renewal point processes arising from neuroscience, in preparation.

Tea and chocolate will be served in room 04.4.19 after the seminar.

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Upcoming events:

Friday, February 15 at 14.15: Irene Tubikanec

Friday, March 15 at 13.15: Hiroki Masuda

Wednesday, March 27 at 11.15: Kathryn Colborn