Estimating Parameters and Model Transformations by Chaotic Synchronizaton

Henry Abarbanel, Matthew Kennel, Lupcjo Kocarev, Lucas Illing

We attack the problem of finding parameter values in an ordinary differential equation model whose solution
best matches another model output or a given time series. By formulating the problem as a unidirectionally
coupled system of oscillators, we show that generalized synchronization intrinsically regularizes what is
typically be a highly ill-conditioned least-squares optimization problem for chaotic dynamics. This
enables powerful standard optimizing algorithms to search the parameter space efficiently with explicitly
computable gradients. Additionally we allow a general static nonlinear transformation from the space of
the original source to fitted model whose parameters also may be estimated simultaneously with the vector
field. One application is to validate an empirical or semi-empirical small model as a suitable replacement
for a first-principles complex model, for instance a low-degree of freedom model of a neuron which is a
faithful dynamical replica of a very complex first-principles Hodgkin-Huxley neuron.