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A phase plane provides a visual way of understanding the dynamics of a system of differential equations. In the context of the FitzHugh-Nagumo equations, it allows us to explore how variables "v" and "w" interact over time.

The axes of the phase plane represent the variables: typically, "v" on one axis and "w" on the other. When plotting in this plane, we highlight important features such as nullclines and equilibrium points.

• Nullclines indicate where the change in a variable is zero, helping define areas of stability.
• Equilibrium points, where all changes are zero, can correspond to steady states of the system.

Phase plane analysis helps developers understand how systems evolve over time, determining behavior such as oscillations or steady state convergence. This visual tool is crucial for grasping complex dynamical systems like neural networks described by the FitzHugh-Nagumo model.

Nullclines are the foundation of phase plane analysis in dynamical systems. They are curves on a phase plane where the rate of change of one of the variables is zero.

In the FitzHugh-Nagumo equations:

• The **v-nullcline** is found by solving \((v-a)(1-v)v = w\). This tells us where the derivative of "v" is zero, depending on "w".
• The **w-nullcline** is given by \(w = v\). This is simply a line with slope 1, showing where the derivative of "w" is zero.

Intersections of nullclines in the phase plane identify points where neither variable changes. These overlapping positions are thus crucial, as they hint at potential equilibria.

By understanding these nullclines, you map out the system's potential dynamics and gather insights into stable or unstable areas.

Equilibrium points in a dynamical system are where all changes cease, meaning both \(\frac{dv}{dt} = 0\) and \(\frac{dw}{dt} = 0\). At these points, the system can either remain steady or oscillate around them, depending on stability.

For the FitzHugh-Nagumo model, equilibrium points are the intersections of the nullclines \((v-a)(1-v)v = v\) and \(w = v\). Solving these equations simultaneously provides precise coordinates of equilibria in the phase plane.

• Depending on the chosen parameters \(a\) and \(\varepsilon\), the number and nature of equilibrium points may vary.
• These points are crucial in predicting the behavior of the system over time.

Stability analysis of these points involves examining the surrounding directional field, providing insights into whether the system will return to equilibrium or diverge away.

Directional fields illustrate the trajectory of a system in the phase plane by showing the direction of the vector field, determined by the differential equations.

In practice, you

• Analyze regions around nullclines to determine the sign and behavior of \(\frac{dv}{dt}\) and \(\frac{dw}{dt}\).
• Draw arrows indicating how solutions (trajectories) move in different regions of the phase plane.

For instance, in the FitzHugh-Nagumo equations:

• Above the line \(w = v\), if \(\frac{dw}{dt} < 0\), the trajectories tend downward.
• Below, when \(\frac{dw}{dt} > 0\), they tend upward.

The directional field visually communicates how variables interact and evolve, showing other possible paths the system may take. This empowers students to grasp a complete picture of the system's dynamical behavior.