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.

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.

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.

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.

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.

1D Reaction-Diffusion: Activator-Inhibitor

Morpheus Model ID: M0011 Open in Morpheus History Persistent Identifier Use this permanent link to cite or share this Morpheus model: Copy Introduction The first example models a 1D activator-inhibitor model (Gierer and Meinhardt, 1972).

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).

2D Reaction-Diffusion: Activator-Inhibitor

Morpheus Model ID: M0012 Open in Morpheus History Persistent Identifier Use this permanent link to cite or share this Morpheus model: Copy Introduction A 2D activator-inhibitor model (Gierer and Meinhardt, 1972).

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: