Tropical geometries and dynamics of biochemical networks. Application to hybrid cell cycle models

We use the Litvinov-Maslov correspondence principle to reduce and hybridize networks of biochemical reactions. We apply this method to a cell cycle oscillator model. The reduced and hybridized model can be used as a hybrid model for the cell cycle. We also propose a practical recipe for detecting quasi-equilibrium QE reactions and quasi-steady state QSS species in biochemical models with rational rate functions and use this recipe for model reduction. Interestingly, the QE/QSS invariant manifold of the smooth model and the reduced dynamics along this manifold can be put into correspondence to the tropical variety of the hybridization and to sliding modes along this variety, respectivelyComment: conference SASB 2011, to be published in Electronic Notes in Theoretical Computer Scienc

Zero-Hopf bifurcation in the FitzHugh-Nagumo system

We characterize the values of the parameters for which a zero--Hopf equilibrium point takes place at the singular points, namely, $O$ (the origin), $P_+$ and $P_-$ in the FitzHugh-Nagumo system. Thus we find two $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families we prove the existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point $O$. We prove that exist three $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at $P_+$ and $P_-$ is a zero-Hopf equilibrium point. For one of these families we prove the existence of $1$, or $2$, or $3$ periodic orbits borning at $P_+$ and $P_-$

Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.Comment: to be published in Chao

From First Lyapunov Coefficients to Maximal Canards

Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. A first-order asymptotic expansion of this location was found by Krupa and Szmolyan in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficient in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding use of this normal form. Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur.Comment: preprint version - for final version see journal referenc