Elimination of the initial value parameters when estimating a system close to Hopf bifurcation
One of the biggest problems when performing system identification of
biological systems is that it is seldom possible to measure more than
a small fraction of the total number of variables. If that is the
case, the initial state, from where the simulation should start, has
to be estimated along with the kinetic parameters appearing in the
rate expressions. This is often done by introducing extra parameters,
describing the initial state, and one way to eliminate them is by
starting in a steady state. We report a generalisation of this
approach to all systems starting on the centre manifold, close to a
Hopf bifurcation. There exist biochemical systems where such data have
already been collected, for example, of glycolysis in yeast. The
initial value parameters are solved for in an optimisation
sub-problem, for each step in the estimation of the other
parameters. For systems starting in stationary oscillations, the
sub-problem is solved in a straight-forward manner, without
integration of the differential equations, and without the problem of
local minima. This is possible because of a combination of a centre
manifold and normal form reduction, which reveals the special
structure of the Hopf bifurcation. The advantage of the method is
demonstrated on the Brusselator.
Gunnar Cedersund
IEE Proceedings - Systems Biology,
2006
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