FitzHugh-Nagumo model

FitzHugh-Nagumo (FHN) model

For One Cell

This applet shows the voltage produce by a cell using the FHN model as function of time after it is stimulated above its threshold.
The start button initiates the simulation and the reset button brings back the parameters to the initial values.

The FitzHugh-Nagumo model (FitzHugh, 1961) is a generic model for excitable media and can be applied to a variety of systems. FitzHugh called his simplified model the Bon Hoeffer-van der Pol model and derived it in the 1960's as a simplification of the Hodgkin-Huxley equations. The model adiabatically eliminates the h and m gates and retains only a slow variable similar to n, denoted here as v. Because of its simple two-variable form and generality, it has been used widely. The model is able to reproduce many qualitative characteristics of electrical impulses along nerve and cardiac fibers, such as the existence of an excitation threshold, relative and absolute refractory periods, and the generation of pulse trains under the action of external currents. We implement the model as described by the following equations:

In the model, a represents the threshold for excitation, epsilon represents the excitability and beta, gamma and delta are parameters that can change the rest state and dynamics. Since this is a generic model, we keep time in arbitrary units for simplicity.
As in the HH applet, the FHN applet shows the activation produced by two external stimuli. Increasing a from its initial value of 0.1 makes it more difficult for the external stimulus to produce an excitation until at a =0.5, no activation is produced. The reason is that the applet uses 0.5 as the magnitude of the external stimuli. If we instead decrease a to a value such as -0.1, the resting potential becomes unstable. Under these conditions, the system will remain at the resting potential if not perturbed, as can be seen by setting S1 and S2 to times larger than the integration time. However, if an external stimulus is applied by setting S1 to 1, the system will exhibit auto-oscillatory behavior, similar to that seen in the HH model for large values of gNa.
The parameter epsilon is responsible of the different time scale dynamics between the U and v processes and is some times referred as the abruptness of excitation. In the model, the smaller the value of epsilon the faster will be the AP rate of rise and the longer the plateau. Since the rate of rise is directly related to the excitability of the system, in this model decreasing epsilon increases the excitability and the AP duration. These effects can be observed in the applet by setting both S1 and S2 to 1 and by varying epsilon from 0.01 to 0.0001 and increasing the integration time as needed to see the full activation.

The FHN applet also can demonstrate how the interval of time passing between two action potentials can affect the second AP. For instance, set epsilon to 0.0001, S1 to 0 and integration time to 700. Then alter the timing of S2. Notice that a second activation is not produce for S2 below the value 464. When S2 = 464, an activation finally becomes possible, but the duration of the activation is less than half of the previous activation. By increasing time to 2000 and by continuing to increase S2, it can be seen that S2 must be at least 1500 in order for the system to recover its original properties and to produce a second AP as long as the first. The relationship between the duration of an action potential (APD) and the amount of time between the previous activation and the second stimulus (diastolic interval or DI) is known as restitution and is an important characteristic of cardiac tissue. When the heart rate increases, such as during exercise, the lengths of cellular signals that initiate muscular contractions are shortened in a similar way to ensure that filling of the heart chambers and ejection of blood occur efficiently. Without such adaptation, ventricles would not be filled before contracting during faster heart rates. The function that relates APD to DI is known as the restitution curve, and its importance in cardiac dynamics.

For further simulations visualizing the phase space go here.

The FHN model can be found in: Biophys. J. Vol 1 445-466, 1961 by R. FitzHugh see also for example Physica D Vol 41 p173 1990, by M. Courtemanche and W. Skaggs and A.T. Winfree