Nonlinear identification of biological clock dynamics

The paper presents an algorithm for identification of nonlinear autonomous systems in sustained periodic oscillations. The algorithm is of least squares type with a very low computational complexity, the latter being a prime target of the paper. The identified model is of second order with both right hand sides modeled by polynomials in the state variables. This choice is motivated by a recent result on identifiability. That result proves that in case the phase portrait of a measured periodic signal does not intersect itself in a 2-dimensional phase plane, then nonlinear autonomous models of a higher order than 2 cannot be uniquely identifiable. In other words, a second order ODE is sufficient for modeling of any periodic signal that does not have a 2-dimensional phase portrait that intersects itself. This is true also in cases where the underlying system that generates the periodic signal has an order that is higher than 2. The new algorithm is applied to identification of the dynamics of a biological clock that keeps track of daily variations known as circadian rhythms. The dynamics is typically created by oscillations in biochemical networks in the cells of many living organisms. The identification problem is shown to be highly non-linear in the paper. Yet good results are obtained in a numerical study that also addresses the effect of noise on the identified model.