Milena Anguelova

Brief paper: State elimination and identifiability of the delay parameter for nonlinear time-delay systems

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input-output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to be directly related to the well-characterized identifiability/observability of the other system parameters/states.

An Efficient Method for Structural Identifiability Analysis of Large Dynamic Systems

Ordinary differential equation models often contain a large number of parameters that must be determined from measurements by parameter estimation. For a parameter estimation procedure to be successful, there must be a unique set of parameters that can have produced the measured data. This is not the case if a model is not structurally identifiable with the given set of outputs selected as measurements. We describe the implementation of a recent probabilistic semi-numerical method for testing local structural identifiability based on computing the rank of a numerically instantiated Jacobian matrix (observability/identifiability matrix). To obtain this, matrix parameters and initial conditions are specialized to random integer numbers, inputs are specialized to truncated random integer coefficient power series, and the corresponding output of the state space system is computed in terms of a truncated power series, which then is utilized to calculate the elements of a Jacobian matrix. To reduce the memory requirements and increase the speed of the computations all operations are done modulo a large prime number. The method has been extended to handle parametrized initial conditions and is demonstrated to be capable of handling systems in the order of a hundred state variables and equally many parameters on a standard desktop computer.

Minimal output sets for identifiability

Ordinary differential equation models in biology often contain a large number of parameters that must be determined from measurements by parameter estimation. For a parameter estimation procedure to be successful, there must be a unique set of parameters that can have produced the measured data. This is not the case if a model is not uniquely structurally identifiable with the given set of outputs selected as measurements. In designing an experiment for the purpose of parameter estimation, given a set of feasible but resource-consuming measurements, it is useful to know which ones must be included in order to obtain an identifiable system, or whether the system is unidentifiable from the feasible measurement set. We have developed an algorithm that, from a user-provided set of variables and parameters or functions of them assumed to be measurable or known, determines all subsets that when used as outputs give a locally structurally identifiable system and are such that any output set for which the system is structurally identifiable must contain at least one of the calculated subsets. The algorithm has been implemented in Mathematica and shown to be feasible and efficient. We have successfully applied it in the analysis of large signalling pathway models from the literature.

Brief paper: On analytic and algebraic observability of nonlinear delay systems

The observability of nonlinear delay systems has previously been defined in an algebraic setting by a rank condition on modules over noncommutative rings. We introduce an analytic definition of observability to ensure the local uniqueness of state and initial conditions that correspond to a given input-output behaviour. It is shown that an algebraically observable delay system can be reformulated as a system of ordinary differential equations. Analytic observability is then decided by the local uniqueness of solutions to a boundary value problem for this ODE system.

When retarded nonlinear time-delay systems admit an input-output representation of neutral type

The paper shows that nonlinear retarded time-delay systems can admit an input-output representation of neutral type. This behaviour represents a strictly nonlinear phenomenon, for it cannot happen in the linear time-delay case where retarded systems always admit an input-output representation of retarded type. A necessary and sufficient condition for a nonlinear system to exhibit this behaviour is given, and a strategy for finding an input-output representation of retarded type is outlined. Some open problems that arise consequently are discussed as well. All the systems considered in this work are time-invariant and have commensurable delays.

Input-Output Representation and Identifiability of Delay Parameters for Nonlinear Systems with Multiple Time-Delays

We have analyzed the identifiability of time-lag parameters in nonlinear delay systems using an algebraic framework. The identifiability is determined by the form of the system’s input-output representation. The values of the time lags can be found directly from the input-output equations, if these can be obtained explicitly. Linear-algebraic criteria are formulated to decide the identifiability of the delay parameters when explicit computation of the input-output relations is not possible.

On Retarded Nonlinear Time-Delay Systems That Generate Neutral Input-Output Equations

In this work retarded nonlinear time-delay systems that surprisingly admit an input-output representation of neutral type are discussed. It is shown that such a behaviour is a strictly nonlinear phenomenon, as it cannot happen in the linear time-delay case where the input-output representation of retarded systems is always of retarded type. A necessary and sufficient condition under which nonlinear systems admit a neutral input-output representation is given. Some open problems, like minimality and system transformations, are discussed as well.

When classical nonlinear time-delay state-space systems admit an input-output equation of neutral type

This paper deals with nonlinear retarded time-delay systems that surprisingly admit an input-output representation of neutral type. It is shown that such an unexpected behaviour represents a strictly nonlinear phenomenon, for it cannot happen in the linear time-delay case where retarded systems always admit an input-output representation of retarded type. A necessary and sufficient condition under which the nonlinear systems admit a neutral input-output representation is given and strategies for finding an inputoutput representation of retarded type are briefly outlined. Some open problems that arise, like minimality and system transformations, are discussed as well.