Correlation does not really make sense in respect to nonlinear variables. Depending on the values of the variables, they will be more or less correlated. So that is meaningless. At best you can talk about a local correlation, where a linear approximation is made, and then correlation could be defined.
So in that context, correlation is simple. You start with the inverse of the local Hessian matrix. Scale that to have unit diagonals, by pre-and post multiplying by the correct diagonal matrix. The off-diagonal terms will be the desired local correlation, as a simple approximation, and valid ONLY locally.