Optimal Design for Estimation in Diffusion Processes from First Hitting Times

We consider the optimal design problem for the Ornstein--Uhlenbeck process with fixed threshold, commonly used to describe a leaky, noisy integrate-and-fire neuron. We present a solution to the problem of devising the best external time-dependent perturbation to the process in order to facilitate the estimation of the characteristic time parameter for this process. The optimal design problem is constrained here by the fact that only the times between threshold crossings from below, known as hitting times, are observable. The optimal control is based on a maximization of the mutual information between the posterior of the unknown parameter given these observations and the distribution of the hitting times. Our approach is based on the adjoint method for computing the gradient of a functional of a solution to a Fokker--Planck partial differential equation with respect to an input function (i.e., to the control). Our method also enables the estimation of other parameters, in the case when more than one param...