Parameter Estimation of the Fitzhugh-Nagumo Model via a Perturbed Accelerated Gradient Descent Algorithm with an n-Dimensional Golden Section Search Method

The Fitzhugh-Nagumo equations is a system of first-order nonlinear ordinary differential equations based on the researches of Fitzhugh and Nagumo et al. This reduced model captures the simplistic essence of neuronal spiking dependent on two variables, the membrane potential and the recovery variable. From a collection of data points, the three parameters can be determined using the process of parameter estimation. This method minimizes the mean-squared difference between the data points and the solutions of the system. Researches by Ramsay et al. have shown the complexity and computational cost of this problem. The minimization may not converge properly using classic algorithms such as gradient descent and conjugate gradient. Hence, we propose a deterministic method that employs an accelerated gradient descent for iterating on the function surface and activates the local search to escape saddle points and local minima. We apply the n-dimensional golden section search as the deterministic local search. It is a novel generalized technique for convex optimization by subsequently enclosing this optimum until convergence. Furthermore, partitioned n-spherical coordinate system is used which creates an adjusted smaller search spaces as an equidistant ball centered on the iterate. For the parameter estimation, the data points used is generated by applying noise to the deterministic solution of the system. The proposed algorithm, in comparison with other gradient-based methods such as the conjugate gradient and the steepest descent, highly performed in terms of its convergence, accuracy, and precision to the true value of the parameters amidst increasing noise level.