1D PDE Barkley Model
Derived from the seminal HodgkinHuxley and FitzHughNagumo models, the Barkley model is probably the simplest continuous model for excitable media. It replaces the cubic term in the FitzHughNagumo model with a piecewise linear term as a simplification that enables fast simulation.
The equations for the Barkley model are
$$
\begin{aligned}
\frac{\partial u}{\partial t}&=\frac{1}{\epsilon}u(1u) \cdot \left( u  \frac{v+b}{a} \right) + \nabla^2 u\\
\frac{\partial v}{\partial t}&=uv \\
\end{aligned}
$$
where the variable $u$ represents the fast dynamics and controls wave propagation and $v$ represents the slow dynamics controlling refractoriness. The parameter $\epsilon$ determines the time scale separation between the two dynamics, larger $a$ gives wider waves, increasing the ratio $\frac{b}{a}$ gives a larger excitability threshold.
Continuous Diffusion
To simulate this model with Morpheus, we first define two Field
s for u
and v
in the Global
section. These define scalar fields, i.e. spatially resolved continuous variables over the whole lattice domain. Second, we define a System
of differential equations (DiffEqn
) for $\frac{\partial u}{\partial t}$ and $\frac{\partial v}{\partial t}$ and enter the reaction term of the equations above. Moreover, we specify the constants e
, a
and b
. What is left is the diffusion rate for u
, which is specified as 0.5
in Field
/Diffusion
.
Check out barkley1d_cont.xml
to try yourself via:
 MorpheusLink or
 Download:
barkley1d_cont.xml
Once the core of the model is set up, we define a linear lattice with periodic
BoundaryConditions
in the Space
section and the simulation duration with StopTime
(here set to 100
) in the Time
section. To visualize the results, you can add a Logger
and a Logger
/Plot
that records and plots the values of u
and v
over space and time.
As initial condition, we let u
be nonzero in a patch (with the location l
) in the middle of the lattice s
:
if( l.x>=s.x/25 and l.x<=s.x/2+5, 1, 0 )
And v
be nonzero in the left part:
if( l.x<=s.x/2 , 1, 0 )
This forces the wave to propagate to the right, as shown below.
Discrete Diffusion
Now, we want to reproduce this continuous spatial model with a discrete spatial model where each lattice site represents a biological cell. Later, we will expand this to 2D and add cellular dynamics using CPM.
In order to emphasize the discrete structure of the system, we first reduce the spatial resolution Space
/Lattice
/Size
from 200
to 50
data points.
Check out barkley1d_discrete.xml
to try yourself via:
 MorpheusLink or
 Download:
barkley1d_discrete.xml
Then, we take our System
as defined in Global
before, but move it to a CellType
in the CellTypes
section. So instead of a scalar field, we now want every cell to compute the reaction part for itself. That is, we convert the PDE to an ODE system.
System
from Global
to CellTypes
.
Furthermore, as you can see from the figure above, we need to adjust three more things in the CellTypes
section:

To spatially couple the ODEs, we use a
NeighborhoodReporter
that looks at the value ofu
in the neighboring cells and returns theaverage
of these values, which is assigned to the newProperty
u_n
. 
In the reaction term of $\frac{\partial u}{\partial t}$, we now add
Du*(0.5*u_n  0.5*u)
that defines the discretized version of diffusion, which can be interpreted as an undirected celltocell transport term. The propertyDu
is then set to the same value (0.5
) as the diffusion rate in the continuous model. 
To specify the initial values of
u
andv
we now put our mathematical expressions, instead of inGlobal
/Field
as described above for the continuous model, in the respectiveProperty
inCellTypes
/CellType
.
To specify these initial values, you can alternatively use the InitProperty
plugin in the CellPopulations
section.
However, since we only need one cell population, we can simply set our initial values directly on the CellType
level.
Besides the initial values, the initial state of the system also includes the spatial distribution of the cells. Therefore, in the section CellPopulations
we finally add the plugin InitCellLattice
which initializes a population of cells in which each lattice site is filled with exactly one cell.
To visualize the behavior of this system, you can use a Gnuplotter
and colorcode the cells using the u
and v
properties and see something like this: