- Understand the Dynamics of the FitzHugh-Nagumo Model with an App
- The FitzHugh-Nagumo Model
- Fitzhugh-Nagumo model
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- The fitzhugh-nagumo model bifurcation and dynamics pdf995
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Understand the Dynamics of the FitzHugh-Nagumo Model with an App
In , R. Fitzhugh Ref. Nerve cells are separated from the extracellular region by a lipid bilayer membrane. Mineral ions, such as sodium and potassium, and negatively charged protein ions, contained within the cell, maintain the resting potential. When the cell receives an external stimulus, its potential spikes toward a positive value, a process known as depolarization , before falling off again to the resting potential, called repolarization.
In one example, the concentration of the sodium ions at rest is much higher in the extracellular region than it is within the cell. The membrane contains gated channels that selectively allow the passage of ions though them.
The FitzHugh-Nagumo Model
When the cell is stimulated, the sodium channels open up and there is a rush of sodium ions into the cell. However, since the channel gates are voltage driven, the sodium gates close after a while.
The potassium channels then open up and an outbound potassium current flows, leading to the repolarization of the cell. Hodgkin and Huxley explained this mechanism of generating action potential through mathematical equations Ref. While this was a great success in the mathematical modeling of biological phenomena, the full Hodgkin-Huxley model is quite complicated.
On the other hand, the FitzHugh-Nagumo model is relatively simple, consisting of fewer parameters and only two equations. One is for the quantity V , which mimics the action potential, and the other is for the variable W , which modulates V.
The parameter I corresponds to an excitation while a and b are the controlling parameters of the model. The fixed points of the FN model equations are the solutions of the following equation system,.
Note that the V -nullcline is a cubic curve in the VW-plane and the W -nullcline is a straight line. The parameter I simply shifts V -nullcline up or down. The V -nullcline is shown in the figure below in a green color. But since V evolves faster than W , V increases rapidly while W remains virtually unchanged. In the figure, we can see that this results in a near-horizontal part of the V-W curve.
As the curve approaches the V -nullcline, the rate of change of V slows down and W becomes more prominent. The fixed point then attracts this curve and the evolution ends at the fixed point.
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Plot of the VW-plane when the fixed point is on the right side of the V -nullcline. If the fixed point is in the middle, Region 2, then what we have discussed so far holds true. While moving left, the curve crosses the red nullcline from right to left. From this point on, while both V and W diminish, the evolution of V dominates and the curve becomes horizontal once again.
This continues until the curve hits the left part of the V -nullcline. The curve begins to hug the V -nullcline and starts a slow downward journey.
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When it touches the left knee of the V -nullcline, it moves rapidly toward the right part of the V -nullcline. Note that this motion never hits the fixed point and therefore keeps repeating, which we can see in the plot below.
Plot of the VW-plane when the fixed point is in the middle region of the V -nullcline. That leaves us with one last case to discuss — when the fixed point is on the left part, Region 1, of the V -nullcline.
The results should look like the following plot. Note that the analyses we previously performed carry over.
Plot of the VW-plane when the fixed point is on the left side of the V -nullcline. To explore the rich dynamics of the FN model described above, we need to repeatedly change various inputs without changing the underlying model.
As such, a user interface that allows us to easily change the model parameters, perform the simulation, and analyze the new results without having to navigate the Model Builder tree structure to perform these various actions is desirable.
To accomplish this, we can turn to the Application Builder. This platform allows us to create an easy-to-use simulation app that exposes all of the essential aspects of the model, while keeping the rest behind the scenes. With this app, we can rapidly change the parameters via a user-friendly interface and study the results using both static figures and animations. The important parameters of the FN model, i.
The fitzhugh-nagumo model bifurcation and dynamics pdf995
The graphical panels display various quantities of interest such as the waveform for V and W. We display the phase plane diagram in the top-right panel along with the V — and W -nullclines. The position of the fixed point is easily identifiable from that plot.
Once the simulation is complete, we can animate the time trajectories by choosing the animation option from the ribbon toolbar.
To get a summary of the simulation parameters and results, we can select the Simulation Report button. App showing the dynamics of the FitzHugh-Nagumo model when the fixed point is in Region 2.
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We can easily reproduce the cases described in the previous section with our app. The image above, for example, shows what happens when the fixed point is in Region 2. We can easily move the fixed point to either Region 1 or 3 by making the current 0. Note that any other parameters in the app can also be changed to see if other interesting trends emerge.
Bifurcation and Dynamics
The design of your app, from its layout to the parameters that are included, is all up to you. Stay tuned! FitzHugh R. Biophysical Journal. A quantitative description of membrane current and its application to conduction and excitation in nerve.
The Journal of Physiology.
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