Fast Sampling of Evolving Systems with Periodic Trajectories

I.Yu Tyukin,J.M Al-Ameri,T.A Tyukina,Yu.A Korablev,A.N Gorban

doi:10.1051/mmnp/201611406

Abstract

We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are supposed to be of a special class, albeit admitting nonlinear parametrization and nonlinearities. The method enables to represent a given periodic solution as sums of computable integrals and functions that are explicitly dependent on parameters of interest and initial conditions. This allows invoking parallel computational streams in order to increase speed of calculations. Performance and practical implications of the method are illustrated with examples including classical predator-prey system and models of neuronal cells.

Highlights

Evolutionary development and change is a wide-spread phenomenon that is inherent to biological systems
In mathematical biology hereditary characteristics may be imagined, in loose terms, as a model variable determining dynamics of populations that changes over time
The problem is how to find p′ ∈ Rk, x′0 ∈ Rn such that h(t, x(t; t0, x0, p)) = h(t, x(t; t0, x′0, p′)) for all t ∈ [t0, t0 + T ]. This is a standard inverse problem, and many methods for finding solutions to this problem have been developed to date. Despite these methods are based on different mathematical frameworks, they share a common feature: one is generally required to repeatedly find numerical solutions of nonlinear ordinary differential equations (ODEs) over given intervals of time

Summary

Introduction

Evolutionary development and change is a wide-spread phenomenon that is inherent to biological systems. This is a standard inverse problem, and many methods for finding solutions to this problem have been developed to date (sensitivity functions [20], splines [6], interval analysis [15], adaptive observers [19],[5], [9], [12],[24],[25], [8] and particle filters and Bayesian inference methods [1]) Despite these methods are based on different mathematical frameworks, they share a common feature: one is generally required to repeatedly find numerical solutions of nonlinear ordinary differential equations (ODEs) over given intervals of time (solve the direct problem).

Problem Formulation

Main Result

Predator-Prey system

Conclusion