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{align} \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{align} $$ 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.

To simulate and analyse the model in R, I use the deSolve package, maintained by Thomas Petzoldt. The package comes with an excellent vignette and the DDE solver

`dede`

. The function `dede`

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