Global bifurcations involving saddle periodic orbits have recently been recognized as being involved in various new types of organizing centers for complicated dynamics. The main emphasis has been on heteroclinic connections between saddle equilibria and saddle periodic orbits, called EtoP orbits for short, which can be found in vector fields in $\mathbb{R}^3$. Thanks to the development of dedicated numerical techniques, EtoP orbits have been found in a number of three-dimensional model vector fields arising in applications.   We are concerned here with the case of heteroclinic connections between two saddle periodic orbits, called PtoP orbits for short. A homoclinic orbit from a periodic orbit to itself is an example of a PtoP connection, but is generically structurally stable in a phase space of any dimension. The issue that we address here is that, until now, no example of a concrete vector field with a non-structurally stable PtoP connection was known. We present an example of a PtoP heteroclinic cycle of codimension one between two different saddle periodic orbits in a four-dimensional vector field model of intracellular calcium dynamics. We first show that this model is a good candidate system for the existence of such a PtoP cycle and then demonstrate how a PtoP cycle can be detected and continued in system parameters using a numerical setup that is based on Lin's method.

We analyze traveling wave propagation in a simplified model of intracellular calcium dynamics. Despite its simplicity, the model is thought to capture fundamental features of wave propagation in calcium models. We explore aspects of the dynamics of traveling front, pulse, and periodic wave solutions as $J$, a parameter in our model, is varied. We focus on the closed-cell version of the model, which corresponds to a singular limit of the full (open-cell) model. We use our results about the closed-cell model to make conjectures about the nature of wave solutions in the open-cell version of the model. A comparison between the properties of wave solutions of the calcium model and wave solutions of the FitzHugh--Nagumo equations reveals that the calcium model is an excitable system essentially different from the FitzHugh--Nagumo equations. Our analysis suggests that there are two regimes in which the closed-cell model has traveling fronts. In the regime with lower values of $J$ there are two families of traveling fronts, each parametrized by $J$: one with wave speed $s_F(J)$ and one with wave speed $s_B(J)$. For $J$ such that $s_F(J)>s_B(J)$, there is a unique (up to a translation) traveling pulse with wave speed $s_P(J)\in(s_B(J),s_F(J))$, while for $J$ such that $s_F(J)\leq s_B(J)$ there is no traveling pulse. In the regime with higher values of $J$ there are analogous families of traveling fronts and pulses, but in this case the traveling pulses exist when $s_F(J)<s_B(J)$. The stability of the traveling fronts and pulses identified is investigated using the Evans function. The traveling front with wave speed $s_F(J)$ is always stable, while for the traveling front with wave speed $s_B(J)$, we can find numerically a Hopf bifurcation at $s=s_B(J_{\mathrm{HP}})$ dividing the curve $s=s_B(J)$ into stable and unstable sections. The traveling pulse is unstable when it exists. The existence of periodic waves is also investigated.