# Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh-Rose Model
## 1 Introduction  
This paper proposes a parameter estimation procedure, based on integro-differential (ID) relations in the set-membership framework. These relations are based on differential relations obtained, owing to the Rosenfeld-Groebner algorithm implemented in the package DifferentialAlgebra of Maple (see [1], for more details). The main numerical difficulty in using differential relations provided by elimination algorithms, comes from the presence of derivatives, sometimes of high order, which must be estimated from noisy measurements. Several methods have been used in the literature to obtain new relations less sensitive to the noise (see [2; 3; 4; 5]). In this paper, the main idea is the use of iterative integrals leading to ID relations, sometimes with no derivative. These are used in a bounded-error framework, in order to estimate the model unknown parameters. A bounded-error framework means that all uncertainties (measurement noise, parameters) are considered unknown but belonging to bounded connected sets, and, in the proposed work, intervals. Notice that the least squares method, adapted to the set-membership framework, could be used. However, with intervals, it requires the inversion of interval matrices, and an interval matrix is invertible if all punctual matrices in this interval matrix are invertible, which is a very hard constraint. The dynamic systems considered in this work are given by the following form:  
\[\left\{ \begin{array}{l}\dot{x}(t,\theta)=f(x(t,\theta),\theta)+u(t)g(x(t, \theta),\theta),\\ y(t,\theta)=h(x(t,\theta),\theta).\end{array} \right. \tag{1}\]  
\(x(t,\theta)\in\mathbb{R}^{n}\) represents the state variables, \(y(t,\theta)\in\mathbb{R}^{m}\) represents the model outputs, and \(u(t)\in\mathbb{R}^{l}\) represents the input variables. \(u\) can be considered equal to 0, in the case of uncontrolled systems. The set of model parameters to be estimated are given by \(\theta\in U_{p}\) (\(U_{p}\) is an open subset in \(\mathbb{R}^{n}\)). The functions \(f(.,\theta)\), \(g(.,\theta)\) and \(h(.,\theta)\) are real, rational andanalytic, for every \(\theta\in\mathcal{U}_{p}\) on \(M\) (a connected open subset of \(\mathbb{R}^{n}\), such that \(x(t,\theta)\in M\) for every \(\theta\in\mathcal{U}_{p}\) and every \(t\in[0,T]\)). In the following work, we let \(x_{0}=(x_{0})_{i=1,\ldots,n}\), the vector of initial conditions for \(x(t,\theta)\); note that some components can depend on the parameters to be estimated. Moreover, suppose that \(\theta\in\mathcal{U}_{p}\subseteq[\mathcal{U}_{p}]\), where \([\mathcal{U}_{p}]\) is the smallest connected box belonging to \(\mathcal{U}_{p}\). A box is an interval vector (a vector with intervals components) and may, equivalently, be seen as a Cartesian product of intervals. In this framework, a real interval \([u]=[\underline{u},\overline{u}]\) is a closed and connected subset of \(\mathbb{R}\), where \(\underline{u}\) (respectively, \(\overline{u}\)) represents the lower (respectively the upper) bound of \([u]\). Similarly, an interval matrix \([A]\) is a matrix with interval components (see [6], for more details on interval analysis). Considering the order eliminating, first, the unobserved state variables, then the model outputs and the model parameters, the Rosenfeld-Groebner algorithm applied on system (1) provides input-output polynomials (IO polynomials). Then, using iterative integrals, integro-differential input-output polynomials (ID-IO) can be obtained and used to propose a parameter estimation procedure less sensitive to noise, compared to the initial IO polynomials [7]. Indeed, contrary to the latter, ID-IO polynomials may contain derivatives of lower order. To illustrate the method, the Hindmarsh-Rose (HR) model, resulting from a simplification and a generalization of the Hodgkin-Huxley model, is considered (see [8; 9]). The HR model was developed to better understand neuron activity, from a simpler model that can be studied mathematically (for example [10; 11; 12]). Its particularity is to reproduce different dynamics of neurons.