Inverse Problem of Class of Delay Differential Equations and Fast Estimation of Parameters

This article introduces an integral representation formula to solve delay nonlinear ordinary differential equations (DDEs) as an improvment to the method proposed by Tyukin I, et al. of solving a class of ordinary differential equations. The integral formula depends on the parameters of the systems explicitly as nonlinear parameterized computable functions. It features both linear and nonlinear equations and show an effective form for estimating unknown parameters. Solutions of DDEs are represented as sums of computable integrals which are implicitly dependent on the initial condition and the unknown parameters. This allows invoking parallel computational streams using Matlab tools of sums to increase the speed of calculations. On the other hand, reducing the dimension of the vector of the unknown parameters proposed by the integral representation method gives faster calculations and high accuracy in advance. Estimating the parameters appears in the model by using the least squares approach. It provides an observation model to determine the most informative data for a specific parameter, and find the best fit model. In the example of Morris-Lecar of neural cells model, the consistency of delay differential equations with the observers of cell’s activation is shown by fitting the observed data to the real data within the high accuracy of estimating the parameters.