Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. We investigate so-called square-wave and pseudo-plateau bursting patterns found in endocrine cell models that are characterized by a super- or subcritical Hopf bifurcation in the fast subsystem, respectively. By using the polynomial model of Hindmarsh and Rose (Proceedings of the Royal Society of London B 221(1222), 87–102), which preserves the main properties of the biophysical class of models that we consider, we perform a detailed bifurcation analysis of the full fast-slow system for both bursting patterns. We find that both cases lead to the same possibility of two routes to bursting, that is, the criticality of the Hopf bifurcation is not relevant for characterizing the route to bursting. The actual route depends on the relative location of the full-system’s fixed point with respect to a homoclinic bifurcation of the fast subsystem. Our full-system bifurcation analysis reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis.

We present a numerical method for finding and continuing heteroclinic connections of vector fields that involve periodic orbits. Specifically, we concentrate on the case of a codimension-d heteroclinic connection from a saddle equilibrium to a saddle periodic orbit, denoted EtoP connection for short. By employing a Lin's method approach we construct a boundary value problem that has as its solution two orbit segments, one from the equilibrium to a suitable section Σ and the other from Σ to the periodic orbit. The difference between their two end points in Σ can be chosen in a d-dimensional subspace, and this gives rise to d well-defined test functions that are called the Lin gaps. A connecting orbit can be found in a systematic way by closing the Lin gaps one-by-one in d consecutive continuation runs. Indeed, any common zero of the Lin gaps corresponds to an EtoP connection, which can then be continued in system parameters.The performance of our method is demonstrated with a number of examples. First, we continue codimension-one EtoP connections and the associated heteroclinic EtoP cycles in the Lorenz system. We then consider a threedimensional model vector field for the dynamics near a saddle-node Hopf bifurcation with global reinjection to show that our method allows us to complete a complicated bifurcation diagram involving codimension-one EtoP connections. With the example of a four-dimensional Duffing-type system we then demonstrate how a codimension-two EtoP connection can be found by closing two Lin gaps in succession. Finally, we show that our geometric approach can be used to find a codimension-zero heteroclinic connection between two saddle periodic orbits in a four-dimensional vector field.

Electroporation is a physical method of transferring molecules into cells and tissues. It takes advantage of the transient permeabilization of the cell membrane induced by electric field pulses, which gives hydrophilic molecules access to the cytoplasm. This method offers high transfer efficiency for small molecules that freely diffuse through electrically permeabilized membranes. Larger molecules, such as plasmid DNA, face several barriers (plasma membrane, cytoplasmic crowding, and nuclear envelope), which reduce transfection efficiency and engender a complex mechanism of transfer. Our work provides insight into the way electrotransferred DNA crosses the cytoplasm to reach the nucleus. For this purpose, single-particle tracking experiments of fluorescently labeled DNA were performed. Investigations were focused on the involvement of the cytoskeleton using drugs disrupting or stabilizing actin and tubulin filaments as the two relevant cellular networks for particle transport. The analysis of 315 movies (~4,000 trajectories) reveals that DNA is actively transported through the cytoskeleton. The large number of events allows a statistical quantification of the DNA motion kinetics inside the cell. Disruption of both filament types reduces occurrence and velocities of active transport and displacements of DNA particles. Interestingly, stabilization of both networks does not enhance DNA transport.

We present an extension of the theory known as Lin's method to heteroclinic chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based on the construction of a so-called Lin orbit, that is, a sequence of continuous partial orbits that only have jumps in a certain prescribed linear subspace, estimates for these jumps are derived. We use the jump estimates to discuss bifurcation equations for homoclinic orbits near heteroclinic cycles between an equilibrium and a periodic orbit (EtoP cycles).