Bulletin of Mathematical Biology

, Volume 75, Issue 3, pp 373–392 | Cite as

Dimensionality Reduction of Bistable Biological Systems

  • A. ZakharovaEmail author
  • Z. Nikoloski
  • A. Koseska
Original Article


Time hierarchies, arising as a result of interactions between system’s components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems.


Bistability Time-scales hierarchy Similarity transformation Canonical correlation analysis Dimensionality reduction 



A.Z., Z.N., and A.K. are financially supported by the GoFORSYS Project No. 0313924 funded by the German Federal Ministry of Science and Education.

Supplementary material

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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Center for Dynamics of Complex SystemsUniversity of PotsdamPotsdamGermany
  2. 2.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany
  3. 3.Institute for Biochemistry and BiologyUniversity of PotsdamPotsdamGermany
  4. 4.Max Planck Institute for Molecular Plant PhysiologyPotsdamGermany
  5. 5.Institute of PhysicsHumboldt UniversityBerlinGermany

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