PDE

1D PDE Barkley Model

Derived from the seminal Hodgkin-Huxley and FitzHugh-Nagumo models, the Barkley model is probably the simplest continuous model for excitable media. It replaces the cubic term in the FitzHugh-Nagumo model with a piece-wise linear term as a simplification that enables fast simulation. The equations for the Barkley model are $$ \begin{aligned} \frac{\partial u}{\partial t}&=\frac{1}{\epsilon}u(1-u) \cdot \left( u - \frac{v+b}{a} \right) + \nabla^2 u\\ \frac{\partial v}{\partial t}&=u-v \\ \end{aligned} $$ where the variable $u$ represents the fast dynamics and controls wave propagation and $v$ represents the slow dynamics controlling refractoriness.

Try the Examples

Examples Menu To explore the potential and modeling features, it is the best to learn by example. Morpheus comes with a range of fully functional example models showcasing a number of model formalism and modeling features. You can open these models from the Examples menu: Use the Examples menu try out some models. There are models showing ordinary differential equations as well as reaction-diffusion systems in 1D, 2D and 3D and cell-based simulations with the cellular Potts model.

Convert to 3D

Now we finally arrive at the 3D model. As an example, we further develop the coupled PDE and CPM model with mechanical interaction. Similar to the 1D to 2D case, it is straighforward to extend the dimensionality to 3D. We define the Lattice Size also in the z direction, i.e. 238,238,238 and set the structure (Lattice class) to cubic. The initial conditions now include the z dimension as well. Three-dimensional initial conditions for u and v.

Conclusion

In this course, we have shown how to convert a 1D PDE model into a 3D multiscale tissue model in Morpheus using 3 steps: Convert a PDE model into a cell-discrete diffusion model, add motility and cell mechanics using cellular Potts sampling, couple intracellular dynamics and tissue mechanics. I hope each step was clear enough to inspire you to experiment around! Our next course is about Drawing Cell Genealogies.

Overview

Convert a 1D PDE model into 2D and 3D multiscale tissue models.

A Multiscale Mini Model

A Word on the Word ‘Multiscale’ First, let’s clarify what we mean by the somewhat hyped term ‘multiscale’. Generally, the term refers to mathematical and computational models that simultaneously describe processes at multiple time and spatial scales. In contrast to the models based on the quasi-steady state assumption that discard interactions between scales, multiscale models describe systems where processes at different scales can influence each other. Therefore, these models should not only describe multiple scales simultaneously, but also allow them to interact.

1D Reaction-Diffusion: Activator-Inhibitor

Morpheus Model ID: M0011 Persistent identifier for this Morpheus model: Copy Introduction The first example models a 1D activator-inhibitor model ( Gierer and Meinhardt, 1972). Space-time plot of 1D reaction diffusion model. Description This 1D PDE model uses a Lattice with linear structure and periodic boundary conditions. The PDE defined two species called Layers: $A$ (activator) and $I$ (inhibitor) with resp.

Example

Let’s go through an example. We’ll construct a model in which an intracellular cell cycle network (ODE) regulates the division of motile cells which (CPM) release a diffusive cytokine (PDE) which, in turn, controls the cell cycle (ODE). Thus, there are 3 sub-models (ODE, CPM and PDE) that interact in a cyclic fashion: Cyclic interaction of ODE, CPM and PDE. Step 1: Intracellular Model (ODE) First, we define a ODE model of a cell cycle that we take directly from Ferrell et al.

2D Reaction-Diffusion: Activator-Inhibitor

Morpheus Model ID: M0012 Persistent identifier for this Morpheus model: Copy Introduction A 2D activator-inhibitor model ( Gierer and Meinhardt, 1972). Stripe pattern generated by 2D Gierer-Meinhardt model. Description This model uses a standard Lattice with square structure and periodic boundary conditions. The PDE defined two species or Layers $A$ (activator) and $I$ (inhibitor) with resp.

Relative Time Scales

At this point, you may be asking yourself: ‘All nice and well, but how can I control the relative time scales between the various models?’ And I’d respond: ‘Great question!’ Morpheus has a number of ways to control the relative time scales of the various sub-models. We can either control the ODE/PDE dynamics in System or control the cellular dynamics in CPM. Controlling ODE Dynamics Within the System element, there is an optional attribute called time-scaling.