(288e) Sensitivity Analysis for Limit-Cycle Oscillating Hybrid Systems

Sensitivity analysis of dynamic systems provides local information describing the impact of infinitesimal parameter changes on the behavior of the system. The information provided by sensitivity analysis is useful in systems biology analyses such as kinetic parameter estimation, experimental design, analysis of biochemical pathways, and model reduction. Hybrid systems are systems exhibiting both continuous-state and discrete-state dynamics, such as dynamic systems whose state variables exhibit discontinuous jumps, and systems which transition between different physical regimes over time. Thus, limit-cycle oscillating hybrid systems are dynamic systems exhibiting closed and isolated periodic state trajectories, whose state variables evolve under continuous-state dynamics except at discrete events, at which times the state variables and the governing equations may change instantaneously. Examples of such systems are cell cycle models [1], in which a cell's mass instantaneously decreases when casting off a daughter cell via mitosis, but otherwise increases continuously over time, and models of hopping robots [3], in which each period is divided into several separate operating phases.

This work develops a novel theory for local, first-order sensitivity analysis of limit-cycle oscillating hybrid systems, extending both the theory for limit-cycle oscillating continuous systems developed in [4] and [5] and the theory for parametric sensitivity analysis of hybrid systems developed in [2]. The hybrid systems considered in this work are limit-cycle oscillators which may be represented as models in the following form. The dynamics in each particular mode are described by a system of ordinary differential equations involving state variables, time derivatives of at least some of these state variables, and known constant parameters. The system may switch between such modes at discrete events, whose times are described by transition conditions, and whose instantaneous effects on state variables are described by transition functions. The differential equations, transition conditions and transition functions must all satisfy certain regularity conditions.

This work develops methods for the computation of initial condition sensitivities and parametric sensitivities in any such system: accounting exactly for any discontinuities in the sensitivities at events, and exploiting the time-periodicity of the system to yield simplicity in the methods. It is shown that, in general, these initial condition sensitivities and parametric sensitivities are neither periodic nor are they continuous across events, even if the state variables evolve continuously. Analogously to sensitivities of continuous periodic systems [4], it is shown that the initial condition sensitivities of any limit-cycle oscillating hybrid system can be represented as the sum of a time-decaying component and a time-periodic component, so that these initial condition sensitivities become periodic in the long-time limit.

A method is also developed for the decomposition of the parametric sensitivities of state variables into three components, characterizing the influence of parameters on the period of oscillation, state variable amplitudes and relative phases respectively. Evaluation of these sensitivities demonstrates the influence of individual parameters on the overall dynamic behavior of the limit cycle. This result is analogous to one previously developed for continuous dynamic systems [5], but requires a distinct theoretical development.

The methods developed in this work are then applied to several particular models for illustration, including a cell cycle model [1] and a hopping robot model [3].

[1] Chen, K. C., Calzone, L., Csikasz-Nagy, A., Cross, F. R., Novak, B., and Tyson, J. J. Integrative Analysis of Cell Cycle Control in Budding Yeast. Mol. Biol. Cell, 15: 3841-3862, 2004.

[2] Galán, S., Feehrey, W. F., and Barton, P. I. Parametric Sensitivity Functions for Hybrid Discrete/Continuous Systems. Appl. Numer. Math., 31(1): 17-47, 1999.

[3] Koditschek, D. E. , and Bühler, M. Analysis of a Simplified Hopping Robot. Int. J. Rob. Res., 10(6):587?605, 1991.

[4] Rosenwasser, E., and Yusupov, R. Sensitivity of Automatic Control Systems. CRC Press, 2000.

[5] Wilkins, A. K., Tidor, B., White, J., and Barton, P. I. Sensitivity Analysis for Oscillating Dynamical Systems. SIAM J. Sci. Comput., 31(4): 2706-2732, 2009.