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.