Math Insight

A simple spiking neuron model, stability of equilibria

Math 2241, Spring 2023
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  1. Consider the Fitzhugh-Nagumo model describing the evolution of the neuron's voltage in response to sodium channels and potassium channels \begin{align*} \diff{v}{t} &= -v(v-a)(v-1) -w\\ \diff{w}{t} &= \varepsilon (v-\gamma w)\\ v(0) &= v_0\\ w(0)&=w_0 \end{align*} where $v_0$ and $w_0$ are the initial conditions, $a$ is a parameter between zero and one-half, $\varepsilon$ is a small positive parameter, and $\gamma$ is a positive parameter. To be concrete, set $a=\frac{1}{4}$, $\varepsilon=0.01$, and $\gamma=8$.
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    1. Calculate the nullclines and plot them on a phase plane. To help you, we show a phase plane plot below. Dotted lines are included as a hint, but they are not intended to be complete. Be sure to label the nullclines.
    2. The equilibria are the points where the two nullclines cross. In this case, there are three equilibria. Calculate the equilibria explicitly by solving the system of equations \begin{align*} -v(v-\tfrac{1}{4})(v-1) -w &= 0\\ \varepsilon (v- 8 w) &= 0\\ \end{align*} and label them on the phase plane. $E_0 = (0,0) \quad$ $E_1 = \qquad \quad$ $E_2 = $
    3. Calculate the derivative of the vector field, i.e. compute the matrix, $J$, of partial derivatives for the system. \[ J(v,w) = \begin{bmatrix} \frac{\partial f}{\partial v} & \frac{\partial f}{\partial w} \\ \frac{\partial g}{\partial v} & \frac{\partial g}{\partial w} \end{bmatrix} \] for the functions $f(v,w) = -v(v-a)(v-1) -w$ and $g(v,w) = \epsilon(v-\gamma w)$.
    4. Evaluate the derivative at each of the equilibrium points. In other words calculate the three matrices $A_0=J(E_0) $, $A_1=J(E_1) $ and $A_2=J(E_2) $
    5. Calculate the determinant and the trace for each matrix above. \[ \det(A_0)= \qquad \qquad \text{tr}(A_0)= \qquad \] \[ \det(A_1)= \qquad \qquad \text{tr}(A_1)= \qquad \] \[ \det(A_2)= \qquad \qquad \text{tr}(A_2)= \qquad \]
    6. Using the values of the determinant and the trace that you calculated, determine the stability of each equilibrium point.
    7. Sketch possible solutions in the phase plane below for initial conditions $(v_0,w_0)$ starting near each of the equilibrium points $E_0 \quad E_1 \quad$ and $E_2$.

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Math 2241, Spring 2023