Optimal control for parameter estimation
in partially observed hypoelliptic stochastic differential equations
used in neuronal modeling
Nightingale Room- Isped
Campus Carreire – Université de Bordeaux
Open to all, in person, in English
University of Bordeaux, Inria Bordeaux Sud-Ouest, Inserm, Bordeaux Population Health Research Center, SISTM Team.
The complex behavior of a neuron activity (brutal membrane potential oscillations, random spiking time) requires stochastic differential equations (SDE)s for proper modeling of its temporal evolution.
Still, in such neuronal models not all variable evolutions are directly affected by stochastic perturbations.
This leads to consider hypoelliptic SDEs characterized by degenerate diffusion coefficients.
Additionally, the statistical problem of parameter estimation often must be treated in a partially observed setting where we only have access to measurements for the variables the less affected by stochasticity. These features often cause the failure of classic estimation procedures based on contrast estimator and the Euler Maruyama discretization scheme as well as dramatically impairs classic stochastic filtering methods used to reconstruct the unobserved states.
All these issues make the estimation problem in hypoelliptic SDEs difficult to solve.
To overcome this, we propose a method based a well-defined cost function no matter the elliptic or hypoelliptic nature of the model. We also bypass the filtering step by considering a control theory perspective.
The unobserved states are estimated by solving deterministic optimal control problems using numerical methods which do not need strong assumptions on the diffusion coefficient conditioning.
Numerical simulations made on different partially observed hypoelliptic SDEs reveal our method produces accurate estimates while dramatically reducing the computational price compared to other methods.