PDE

‘How can I specify time scales?’

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.

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.

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.

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.

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

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.

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.

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).

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.