# Seminar in applied mathematics and statistics

**SPEAKER: **Irene Tubikanec (Institute for Stochastics, Johannes Kepler University Linz, Austria).

**TITLE: ** Spectral Density-Based and Structure-Preserving Approximate Bayesian Computation for Partially Observed SDEs With an Invariant Measure: *A Demonstration on the Jansen and Rit Neural Mass Model and the FitzHugh-Nagumo Model.*

**ABSTRACT: **Let us consider a n-dimensional stochastic differential equation (SDE) whose solution process is only partially observed through a one-dimensional and parameter-dependent output process admitting an invariant distribution. We aim to infer the parameters from discrete time measurements of the invariant output process. Due to the increased model complexity needed to understand and reproduce the real data, the underlying likelihood is often unknown or intractable. Among several likelihood-free inference methods, we focus on the Approximate Bayesian Computation (ABC) approach.

When applying ABC to stochastic processes, two difficulties arise. First, different realizations from the output process with the same choice of parameters may show a large variability, due to the stochasticity of the model. Second, exact simulation schemes are rarely available for general SDEs and, thus, a numerical integrator for the synthetic data generation within the ABC framework has to be derived. We tackle these issues as follows. To reduce the randomness coming from the underlying model, we propose to use the structural properties of the SDE, namely the existence of the invariant distribution. We map the synthetic data into their estimated invariant density and invariant spectral density, almost eliminating the variability in the data and, thus, making hidden information about the parameters accessible. Since our ABC algorithm is based on the structural property, it can only lead to successful inference when the invariant measure is preserved in the synthetic data simulation. To achieve this, we propose to use a structure-preserving numerical scheme that, differently from the commonly used Euler-Maruyama method, preserves the properties of the underlying SDE.

Here, we illustrate our proposed Spectral Density-Based and Structure-Preserving ABC Algorithm on the stochastic Jansen and Rit Neural Mass Model (JR-NMM) [1] and the stochastic FitzHugh-Nagumo model (FHN) [2]. Both models are ergodic, which results in the output process admitting an invariant measure. The use of numerical splitting schemes guarantees the preservation of the invariant distribution in the synthetic data generation step. With our new approach, we succeed in the simultaneous estimation of three of the most important parameters of the JR-NMM and all four parameters of the FHN model. Finally, we apply our method to fit the JR-NMM to real EEG alpha-rhythmic recordings.

References:

[1] M. Ableidinger, E. Buckwar, H. Hinterleitner, "A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics." In: The Journal of Mathematical Neuroscience 7(8) (2017).

[2] J. R. Leon, A. Samson, "Hypoelliptic stochastic FitzHugh-Nagumo neuronal model: mixing, up-crossing and estimation of the spike rate." In: Annals of Applied Probability 28(4), pp.2243-2274 (2018).

[3] M.A. Beaumont, W. Zhang, D.J. Balding, "Approximate Bayesian computation in population genetics". In: Genetics, 162(4), pp.2025-2035 (2002).

[4] E. Buckwar, M. Tamborrino, 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.

[5] E. Buckwar, A. Samson, M. Tamborrino, I. Tubikanec. "Periodogram-Based and Structure Preserving Approximate Bayesian Computation applied to the hypoelliptic stochastic FitzHugh-Nagumo neuronal model." in preparation.

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

----------

Upcoming events:

Friday, March 15 at 13.15: Hiroki Masuda

Wednesday, March 27 at 11.15: Kathryn Colborn