# Simulating neurons or how to solve delay differential equations in R

I discussed earlier how the action potential of a neuron can be modelled via the Hodgkin-Huxely equations. Here I will present a simple model that describes how action potentials can be generated and propagated across neurons. The tricky bit here is that I use delay differential equations (DDE) to take into account the propagation time of the signal across the network.

My model is based on the paper:*Epileptiform activity in a neocortical network: a mathematical model*by F. Giannakopoulos, U. Bihler, C. Hauptmann and H. J. Luhmann. The article presents a flexible and efficient modelling framework for:

- large populations with arbitrary geometry
- different synaptic connections with individual dynamic characteristics
- cell specific axonal dynamics

In its simplest form, for one self-coupled neuron or you could regard this as the synchronised case of several neurons, I have:
\[
\begin{aligned}
\frac{du}{dt}=&\alpha (-u(t) + q g(v(t-T)) + I) \\\\\\
\frac{dv}{dt}=&C (w(t)+v(t)-\frac{1}{3}v(t)^3) + \gamma u(t) \\\\\\
\frac{dw}{dt}=&\frac{1}{C}(a-v(t)-bw(t)) \\\\\\
g(v) = & \frac{1}{1 + \exp(-4v)} \\\\\\
\end{aligned}
\]
The variable *u* describes the total post-synaptic soma potential, which is the sum of all incoming signals *v*. The membrane potential *v* arrives via the dendrites after a time delay of *T* and is either exhibitory (*q > 0*) or inhibitory (*q < 0*). It is transformed from a pre to a post-synaptic signal by the monotonically increasing function *g*.

The membrane potential *v* itself is modelled by FitzHugh-Nagumo differential equations. They can be regarded as a simplified version of the higher dimensional Hodgkin-Huxely model.

The soma potential *u* is for \(0 \lt\alpha\ll 1\) on a much slower time scale than *v*. Thus, with *u* treated as constant we can analyse the FitzHguh-Nagumo (FHN) system. The FHN system has an interesting bifurcation behaviour, using *u* as a parameter. For certain values of *u* the system has a stable equilibrium; increase the value of *u* and the FHN system will go through a Hopf-bifurcation and periodic solutions with a limit cycle attractor appear (the neuron spikes).

Combining the FHN model with the soma potential *u* provides a model which is most fascinating. In my example the resting membrane potential slowly increases the soma potential until the FHN system goes through a Hopf-bifurcation resulting in periodic spiking (bursting), which in in itself pushes the potential *u* down, eventually causing the neuron to rest again.

Introducing the time delay *T* not only makes the model more realistic, it allows me to control the length of the bursting phase as well. A more detailed examination of the system is presented in *Bursting activity in a model of a neuron with recurrent synaptic feedback* by F. Giannakopoulos, C. Hauptmann and A. Zapp.

`dede`

. The function `dede`

is actually easy to use, once you get your head around it. Here is an example of the above model: