The firing functions $S_i$ are sigmoidal functions defined by
\begin{align*}
S_i(x) = \frac{M_i}{1+ e^{-(x-\theta_i)/\sigma_i}},
\end{align*}
where $M_i$ is the maximum value, $\theta_i$ is the value of $x$ where $S(x)$ achieves half its maximum, and $\sigma$ determines the steepness of the function.
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What is the equation for the $R_1$-nullcline?
What is the equation for the $R_2$-nullcline?
(Write your answer in terms of the functions $S_1$ and $S_2$, meaning your answer should have an $S_1(\cdot)$ or $S_2(\cdot)$ in it rather than explicitly using the equation for the $S_i$ shown above.)
How do the nullclines depend on the time constants $\tau_1$ and $\tau_2$? (Recall they must be positive.)
Hint
Online, enter $S_1$ as
S_1, $I_1$ as I_1, $W_{12}$ as W_12, etc. If needed, enter $\tau_1$ as tau_1 or τ_1 and $\tau_2$ as tau_2 or τ_2.
You should not be entering parameters like $M_1$, $M_2$, $\theta_1$, etc. of the nonlinear functions $S_1(\cdot)$ and $S_2(\cdot)$. Instead, your answers should just be in terms of the functions $S_1$ and $S_2$. (It will save a lot of typing.)
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Set the nonlinearity parameters to $\theta_1=90$, $M_1=180$, $\sigma_1=30$, $\theta_2=200$, $M_2=220$, and $\sigma_2=30$. Set the inputs to $I_1=-70$ and $I_2=110$ and the coupling strengths to $W_{12}=1.7$ and $W_{21}=0.8$. Set the time constants to $\tau_1=15$ ms and $\tau_2=30$ ms.
Use these parameters for the rest of this problem.
Using these parameters, write the equations for the nullclines. In this case, use the definitions of $S_1(\cdot)$ and $S_2(\cdot)$ in terms of these parameter values.
$R_1$-nullcline:
$R_2$-nullcline:
(Though not required, simplifying the nullclines so that they are as close to the form in the definition of the $S_i(\cdot)$ functions will make graphing them easier. In that form, you can then easily read off the values of $R_2$ or $R_1$ where the nullclines reach their half maximum point.)
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Sketch the nullclines on the phase plane. Show the equilibria on the graph. (At this point, you can't classify them. Leave that part incorrect, as we'll come back to that below.) For each region of the phase plane bounded by the nullclines, plot one direction vector showing the general direction of the solution in that region. For each segment of each nullcline bounded by the equilibria, plot one direction vector showing the direction that solutions must cross that segment of the nullcline. (See Need Help section for instructions on using the applet to sketch the phase plane.)
Equations for the nullclines
`{}`-nullcline:
\[{}\]
`{}`-nullcline:
\[{}\]
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Feedback from applet
step 1, nullcline 1 maximum:
step 1, nullcline 1 orientation:
step 1, nullcline 1 shift:
step 1, nullcline 1 steepness:
step 1, nullcline 2 maximum:
step 1, nullcline 2 orientation:
step 1, nullcline 2 shift:
step 1, nullcline 2 steepness:
step 2, equilibrium locations:
step 2, equilibrium stability:
step 2, equilibrium types:
step 2, number of equilibria:
step 3, number of region vectors:
step 3, one vector per region:
step 3, region vector directions:
step 4, nullcline 1 vector directions:
step 4, nullcline 2 vector directions:
step 4, number of nullcline vectors:
step 4, one vector per nullcline 1 segment:
step 4, one vector per nullcline 2 segment:
Hint
Online, with step=1, use the blue curve with the circle points to represent the $R_1$-nullcline and the green curve with the diamond points to represent the $R_2$-nullcline. Drag the larger of the two points to determine the point at which the nullcline reaches half its maximum (which also then determines the maximum). The smaller point can be moved from side to side to determine the steepness of the curve as well as the direction in which it increases. If the smaller point is placed next to the larger point the nullcline becomes a straight line. You can click the nullcline to flip it.
For this example, the $R_1$-nullcline is simple. To correctly draw the $R_2$-nullcline, you will need to calculate at which value of $R_1$ it reaches half its maximum so that you can correctly place the larger green diamond point. (Some algebra required here.)
To specify the equilibria, change step to 2. Then, drag the point on the $n_e$ slider to reveal the correct number of equilibria. Move the points to the correct locations of the equilibria. Click each equilibrium point so that it cycles through the different equilibrium types. The options for equilibrium types are stable node (SN), stable spiral (SS), center (C), unstable spiral (US), unstable node (UN), are saddle (SD). The equilibrium point also shows the stability graphically, as a filled point represents a stable equilibrium and an unfilled point represents an unstable equilibrium. (Since the stability of a center is not determined by the Jacobian, it shown as an unstable equilibrium.)
To draw the vectors in the regions of the phase plane bounded by the nullclines, change step to 3. Then, use the $n_{rv}$ slider to specify how many of these regional direction vectors you wish to draw. Draw one vector per region. Drag the point at its base to move the vector; drag the point at its head to change its direction.
To draw the vectors in the segments of the nullclines bounded by the equilbria, change step to 4. Then, use the $n_{nv}$ slider to specify how many of these nullcine direction vectors you wish to draw. Draw one vector per segment of the nullcline. Drag the point at its base to move the vector; drag the point at its head to change its direction.
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The next step is the local analysis to classify the equilibria. To do so, we need to calculate the Jacobian matrix of the hand side of the differential equation.
The form of the Jacobian at a generic point could be a bit complicated, depending on how you write it. Here, to show the general form, and to keep the length of the answer manageable, we will insist you write the Jacobian in the following way.
- Don't use parameter values, but instead use the parameters themselves, such as $\tau_1$, $W_{21}$, and $I_2$.
- Don't use the formula for the $S_i()$. Instead, just write you answer in terms of $S_1()$ and $S_2()$, or their derivatives $S_1'()$ and $S_2'()$. This means you answer should not have any $\theta_1$, $M_2$, $\sigma_i$, etc., in it.
Hint
At least one of the entries of the Jacobian will involve the derivative $S_1'(\cdot)$ or $S_2'(\cdot)$. You can just enter it as
S_1'(...) or
S_2'(...), though you'll have to fill in the ... with the argument (in terms of the $I$'s, $R$'s, etc.). Remember the chain rule when calculating the derivative. You are calculating the derivative with respect to $R_1$ or $R_2$, so you have to multiply the derivative of $S$ by the derivative of the argument of $S$ with respect to $R_1$ or $R_2$.
If needed, enter $\tau_1$ as tau_1 or τ_1 and $\tau_2$ as tau_2 or τ_2.
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To evaluate the Jacobian at the equilibria, we need to first calculate the equilibria. Unfortunately, we can't solve for a nice analytic formula for the equilibria. Instead, we'll approximate the equilibria numerically. You can use any computer program to solve for the intersection of the nullclines. The simplest way is to just use the above applet, which shows the values of the equilibria when
step is set to 2.
First of all, how many equilibria are there?
Order the equilibria in terms of their value of the first coordinate. In this order, the equlibria are:
$(R_1^{ss1},R_2^{ss1}) =$
$(R_1^{ss2},R_2^{ss2}) =$
$(R_1^{ss3},R_2^{ss3}) =$
(Include at least 3 significant digits in your response.)
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Classify the first equilibrium, which you above calculated as $(R_1^{ss1},R_2^{ss1}) =＿$.
Evaluate the Jacobian at this equilibrium. Plug in all the values of all the parameters so that each entry of the Jacobian is just a number.
$J(R_1^{ss1},R_2^{ss1})=$
The eigenvalues of the Jacobian evaluated at the first equilibrium are
$\lambda=$
. (Separate eigenvalues by a comma.)
Therefore, the equilibrium is a
.
Be sure to classify the equilibrium in the above phase plane.
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Classify the second equilibrium, which you above calculated as $(R_1^{ss2},R_2^{ss2}) =＿$.
Evaluate the Jacobian at this equilibrium. Plug in all the values of all the parameters so that each entry of the Jacobian is just a number.
The eigenvalues of the Jacobian evaluated at the second equilibrium are
$\lambda=$
. (Separate eigenvalues by a comma.)
Therefore, the equilibrium is a
.
Be sure to classify the equilibrium in the above phase plane.
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Classify the third equilibrium, which you above calculated as $(R_1^{ss3},R_2^{ss3}) =＿$.
Evaluate the Jacobian at this equilibrium. Plug in all the values of all the parameters so that each entry of the Jacobian is just a number.
$J(R_1^{ss1},R_2^{ss1})=$
The eigenvalues of the Jacobian evaluated at the first equilibrium are
$\lambda=$
. (Separate eigenvalues by a comma.)
Therefore, the equilibrium is a
.
Be sure to classify the equilibrium in the above phase plane.
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Sketch a plausible solution $(R_1(t), R_2(t))$ on the phase plane for the initial condition $(R_1(0), R_2(0)) = (106, 90)$. Sketch a plot of the same solution versus time, i.e., $R_1(t)$ and $R_2(t)$ versus time $t$.