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Tutorial: your first model

In this tutorial you will build the FitzHugh-Nagumo model of an excitable neuron, simulate it, look at its phase plane and find the current at which it starts to oscillate. It takes about fifteen minutes.

1. Write the equations

Choose File > New model. The editor opens with this model:

v' = v - v^3/3 - w + I
w' = eps*(v + a - b*w)

par I=0.5, a=0.7, b=0.8, eps=0.08
init v=-1, w=1

@ total=200, dt=0.05, xp=v, yp=w
@ xlo=-2.5, xhi=2.5, ylo=-1, yhi=2.5
done

Line by line:

  • v' = ... is the differential equation dv/dt = ... . You can also write dv/dt = ....
  • par declares parameters: numbers you will change while exploring.
  • init gives the initial values of the variables.
  • Lines starting with @ set options: here the integration time (total), the time step (dt) and which variables the phase plane shows.
  • done ends the file.

Press Ctrl+Enter (or Apply in the editor) to load it. If there is a mistake, a red box above the text says what is wrong and on which line.

2. Simulate

The model runs as soon as it loads. The Time Series tab shows v and w over 200 time units. Drag to pan, scroll to zoom, double-click to fit.

To run again after changing something, press Run in the toolbar, or F5. With Live ticked (the default) a change to a parameter reruns the model automatically.

3. The phase plane

Open the Phase Plane tab. The trajectory is drawn in the (v, w) plane.

  • Tick Nullclines. The red curve is where dv/dt = 0, the green one where dw/dt = 0. Where they cross the system has an equilibrium.
  • Tick Flow to see the direction field: which way the state moves from each point.
  • With Click to start on, click anywhere in the plot: a new trajectory starts there. Tick Keep curves in the toolbar to keep the earlier ones.

4. Stability of the equilibrium

Press Equilibrium in the toolbar. Newton's method finds the steady state near the current initial values. The command window prints it with its eigenvalues, and the phase plane marks it: filled means stable, open means unstable.

At I = 0.5 the equilibrium is unstable and the neuron fires repeatedly. Set I to 0 in Workspace > Parameters: the trajectory now settles onto a stable equilibrium. The neuron is at rest.

5. Where does firing start? A bifurcation diagram

Between I = 0 and I = 0.5 the rest state loses stability. AUTO finds exactly where:

  1. Set I = 0 and press Run, so the state sits near the rest state.
  2. Open the Bifurcation window. Choose I as the parameter, set Par max to 2, and press Follow steady states.
  3. The diagram shows the equilibrium as I varies: red is stable, black dashed is unstable. The orange HB labels mark Hopf bifurcations, where oscillations are born.
  4. Click the first HB in the table (or in the plot), then Periodic orbits from this Hopf point. AUTO follows the oscillations. Green is stable, blue dashed unstable, and the curves show each orbit's maximum and minimum.

You have now computed the classic picture of a neuron's onset of firing. The same steps work for any model.

What next

  • Bifurcation analysis with AUTO explains the diagram and its settings in depth.
  • The .ode file reference shows everything a model file can contain: functions, auxiliary outputs, delays, noise, maps and more.
  • The Models window has more than a hundred example models to explore.