The FitzHugh-Nagumo model, named after Richard FitzHugh and J. Nagumo, who discovered it shortly after another, is a model that describes the reaction of a biological neuron when excited by an external stimulus \(I_{ext}\). The differential equations describing the dynamical system are the following:

$$\begin{pmatrix}\dot{v}\\\dot{w} \end{pmatrix}=\begin{pmatrix}f(v,w)\\g(v,w)\end{pmatrix}= \begin{pmatrix} v-\frac{v^3}{3}-w+I_{e}\\\tau^{-1} (v-a-bw) \end{pmatrix}\tag{FN}$$

Where \(v\) denotes the voltage on the membrane of the neuron and \(w\) is a recovery variable.

It is a special case of the Van-der-Pol Oscillator $$\ddot{x}-\mu (1-x^2)\dot{x}+x = 0$$ which transforms into $$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}\mu (x-\frac{x^3}{3} -y)\\ \frac{x}{\mu}\end{pmatrix}$$ for the Liénard-transformation \(y = x-\frac{x^3}{3}-\frac{\dot{x}}{\mu}\). As one can see, the Van-der-Pol oscillator differs from the FitzHugh-Nagumo Model by the addition of two terms with constants \(a\neq 0,\; b\neq0\) and a rescaling of \(\dot{x}\).

For a system of neurons one can rewrite these equations as a reaction-diffusion system, by treating the discretized system of neurons as a continouus system, where a point in a \(x\times y\) space corresponds to a neuron and holds a value \((v(x,y,t),w(x,y,t))\). The solution to the differential equations (and therefore the reaction to the excitation of a neuron) is a travelling voltage spike on the membrane of the neuron. For a system of neurons that are connected by axons, the spike of the voltage corresponds to a current that flows from neuron to neuron, generating patterns of excited neurons and not excited neurons. In the following simulation the neurons are pixels on a \(300px \times 300px\) canvas who excite nearby neurons.

The key phenomenon that plays a role in pattern formation are Turing Instabilities. When two processes, who stabilize the system themselves are put into a system where they interact, the process of the entire system can turn out to be unstable and exhibit patterns. A Reaction-Diffusion System as (RD) follows a such dynamic. While Reaction and Diffusion lead to a spatially homogenous state, the interaction exhibit Turing patterns for certain parameters.

(RD) can be rewritten into $$\begin{pmatrix} \dot{v}\\ \dot{w}\end{pmatrix} = A_1\begin{pmatrix}v\\w\end{pmatrix} + A_2\begin{pmatrix}v\\w\end{pmatrix} = A\begin{pmatrix}v\\w \end{pmatrix}.$$Where \(A_1 = diag(D_v\Delta,D_w\Delta)\) and \(A_2=\begin{pmatrix}f_v&f_w\\g_v&g_w\end{pmatrix}\). And for a Fourier expansion of \(v\) and \(w\) is \(A_1 = diag(-D_v k^2,-D_w k^2)\). A Turing Instability therefore occurs for $$|A_1|<0, \; |A_2|<0,\; |A|>0.$$ Or explicitly$$|A|=|f_w g_v|-(|f_v|+D_v k^2)(|g_w|-D_w k^2)>0.$$

Click on the canvas to excite neurons around the position of the cursor. Use the sliders to tweak the variables and observe the different patters that the system forms. Vary the parameter \(a\) after a pattern has formed, not before. Pattern presets are starting points, there are many more patterns to be found when varying parameters after setting the starting points and the boundary conditions. Use the RGB sliders to change the colour scheme of the simulation.

These slider will change the parameters of the simulation:

These sliders change the colour scheme of the simulation (RGB):

You can select a preset parameter choices in the drop-down menu. The "wrap"-button activates /deactivates the boundary conditions, the "randomize"-button randomizes the values of each point in the mesh-grid. Different patterns emerge for different boundary conditions/initial conditions.

The algorithm used for solving the problem was Euler Inegration: $$\partial_t f = \frac{f(t+\Delta t)-f(t)}{\Delta t}$$ $$ f_{xy}^{t+1} = f_{xy}^{t}+\Delta t D \cdot \Delta f$$ Where \(\Delta f\) is the laplacian from code above, or explicitly: $$\Delta f = \frac{f_{i+1,j}^t +f_{i-1,j}^t +f_{i,j+1}^t +f_{i,j-1}^t - 4f_{ij}^t}{\Delta x^2} $$ And D is the Diffusion-constant:$$D = \frac{\Delta x^2}{2\Delta t}$$

[2] FitzHugh-Nagumo model, Eugene M. Izhikevich, http://www.scholarpedia.org/article/FitzHugh-Nagumo_model#fig:Nagumo_Circuit.gif

summer term 2020

Vasilios Mitsiioannou, mitsiioannou@itp.uni-frankfurt.de