Natural extension of fast-slow decomposition for dynamical systems.

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.

Dynamics of plateau bursting depending on the location of its equilibrium.

We present a mathematical analysis of the dynamics that underlies plateau bursting in models of endocrine cells under variation of the location of the (unstable) equilibrium around which these bursting patterns are organised. We focus primarily on the less well-studied case of pseudo-plateau bursting, but also consider the square-wave case. The behaviour of such models is explained using the theory for systems with multiple time scales and it is well known that the underlying so-called fast subsystem organises their dynamics. However, such results are valid only in a sufficiently small neighbourhood of the singular limit that defines the fast subsystem. Hence, the slow variable (intracellular calcium concentration) must be very slow, which is actually not the case for pseudo-plateau bursting. Furthermore, the theoretical predictions are also only valid for parameter values such that the equilibrium is close to a homoclinic bifurcation occuring in the fast subsystem. In the present study, we use numerical explorations to discuss what happens outside this theoretically known neighbourhood of parameter space. In particular, we consider what happens as the equilibrium moves outside a small neighbourhood of the homoclinic bifurcation that occurs in the fast subsystem, and relatively fast speeds are allowed for the slow variable which is controlled by a relatively large value of a parameter ε. The results obtained complement our earlier work [Tsaneva-Atanasova et al. (2010) J Theor Biol264, 1133-1146], which focussed on how the bursting patterns vary with the rate of change ε of the slow variable: we fix ε and move the equilibrium over the full range of the bursting regime. Our findings show that the transitions between different bursting patterns are rather similar for square-wave and pseudo-plateau bursting, provided that the value of ε for the pseudo-plateau-bursting model is chosen so that it is much larger than for the square-wave bursting model. Furthermore, the two families of tonic spiking and plateau bursting, which are generally viewed as two separately generated families, are actually connected into a single family in the two-parameter plane through branches of unstable periodic orbits.

Invariant polygons in systems with grazing-sliding.

The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.