Explicit Representations Of Periodic Solutions Of Nonlinearly Parameterized Ordinary Differential Equations And Their Applications To Inverse Problems

Developing mathematical models involves joining theory and experimental or observational data. The models often depend on parameters which are not always known or measured. A major task in this process is therefore to determine parameters fitting empirical observations. In this work we consider the fundamental challenge of inferring parameters of systems of ordinary differential equations (ODEs) from the values of their solutions and/or their continuous mappings. To achieve this aim we developed a method for deriving computationally efficient representations of solutions of parametrized systems of ODEs. These representations depend on parameters of the system explicitly, as quadratures of some known parametrized computable functions. The method applies to systems featuring both linear and nonlinear parametrization, and time-varying right-hand side; which opens possibilities to invoke scalable parallel computations for numerical evaluation of solutions for various parameter values. In the core of the methods the idea is to use availability of parallel computational streams offered by modern computational technology and hardware, such as GPUs. This, if used efficiently, drastically reduces the amount of time spent on solving direct problems. This opens up new possibilities for dealing with inverse problems by employing the methods that have not been possible to use to date due to massive computational costs involved. We illustrate our method with parameter estimation problems for classical benchmark models of neuron cells, Hodgkin–Huxley and Morris–Lecar models. These applications enable to assess potential computational advantage, of the method relative to other procedures known in the literature; they also offer new ways to move forward.