# Math Insight

### Two-dimensional neuron problems

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Total points: 10
1. Let $R_1(t)$ be the firing rate of neuron 1 at time $t$ and $R_2(t)$ be the firing rate of neuron 2 at time $t$, where firing rate is measured in spikes/second and time is measured in milliseconds. We let neuron 1 have a connection onto neuron 2, so that the firing rates follow the dynamical system: \begin{align*} \diff{R_1}{t} &= \frac{1}{\tau_1} \left(-R_1 + S_1(I_1)\right)\\ \diff{R_2}{t} &= \frac{1}{\tau_2} \left(-R_2 + S_2(W R_1 + I_2)\right), \end{align*} where $I_1$ and $I_2$ are the input rates to the neurons, $\tau_1$ and $\tau_2$ are positive time constants, and $W$ is the (positive or negative) connection strength from neuron 1 onto neuron 2.

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. A plot of the $S_i(x)$ is shown below, where you can explore the influence of the parameters $M_i$, $\theta_i$, and $\sigma_i$ on the shape of the functions. (Notice how the large point on each function determines $M_i$ and $\theta_i$. You can move the smaller point from side to side to change $\sigma_i$. You can also change the parameters using the control panel.)

Control panel (Show)
1. 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.)

2. Since there is no self-coupling, we can calculate the equilibrium even without giving values for the parameters.

Given the simple form of the $R_1$-nullcline, you already know what the value of $R_1$ must be at the steady state. The steady state firing rate of neuron 1 must be $R_1^{ss} =$

(Just like for the $R_1$-nullcline, specify your answer in terms of $S_1$.)

Now that you know the steady state value of $R_1$, you can plug it into the $R_2$-nullcline to find the steady state firing rate for neuron 2. It is $R_2^{ss} =$

3. As a first step to calculating the stability of the equilibrium, calculate the Jacobian matrix of the right hand side of the differential equation.

$J(R_1,R_2)=$
4. Next, evaluate the Jacobian at the equilibrium. Rather than plug in the above formulas for $R_1^{ss}$ and $R_2^{ss}$, just replace $R_1$ with $R_1^{ss}$ and $R_2$ with $R_2^{ss}$ in the Jacobian.

$J(R_1^{ss},R_2^{ss})=$
5. Rather the calculating the eigenvalues right away, first just calculate the determinant and trace of the Jacobian matrix evaluated at the equilibrium.

$\text{Det } J=$

$\text{Tr }J=$

Given what you know about the signs of $\tau_1$ and $\tau_2$, what is the sign of $\text{Det }J$?

What is the sign of $\text{Tr }J$?

Without knowing anything more, you can already determine if the equilibrium is stable or unstable. The equilibrium is
.

6. To characterize the type of the equilibrium, let's go all the way and calculate the eigenvalues.
$\lambda =$

(Separate the eigenvalues by a comma.)

Therefore, the equilibrium is
.

7. Set the nonlinearity parameters to $\theta_1=80$, $M_1=120$, $\sigma_1=30$, $\theta_2=180$, $M_2=100$, and $\sigma_2=40$. Set the inputs to $I_1=130$ and $I_2=80$ and the coupling strength to $W=0.4$.

For these parameters, what is the steady state?
$R_1^{ss}=$

$R_2^{ss}=$

8. For the above parameters, what what are the nullcline equations?
$R_1$-nullcline:

$R_2$-nullcline:

9. Sketch the nullclines on the phase plane. Mark the equilibrium on the graph and label it by its type. 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.)

Nullcline equations (Show)
Feedback from applet
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:
10. Starting with the initial condition, $R_1(0)=0$ and $R_2(0)=0$, sketch a plausible solution trajectory $(R_1(t), R_2(t))$ on the phase plane. Sketch the corresponding graphs of $R_1(t)$ and $R_2(t)$ versus time.

2. Let $R_1(t)$ be the firing rate of neuron 1 at time $t$ and $R_2(t)$ be the firing rate of neuron 2 at time $t$, where firing rate is measured in spikes/second and time is measured in milliseconds. We let the neurons have a connection onto each other, so that the firing rates follow the dynamical system: \begin{align*} \diff{R_1}{t} &= \frac{1}{\tau_1} \left(-R_1 + S_1(W_{12} R_2 + I_1)\right)\\ \diff{R_2}{t} &= \frac{1}{\tau_2} \left(-R_2 + S_2(W_{21} R_1 + I_2)\right), \end{align*} where $I_1$ and $I_2$ are the external inputs to the neurons, $\tau_1$ and $\tau_2$ are positive time constants, $W_{12}$ is the (positive or negative) connection strength from neuron 2 onto neuron 1, and $W_{21}$ is the (positive or negative) connection strength from neuron 1 onto neuron 2.

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.

1. 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.)

2. 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.)

3. 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.)

Nullcline equations (Show)
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:
4. 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.

$J(R_1,R_2)=$
5. 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.)

6. 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.

7. 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.

$J(R_1^{ss},R_2^{ss})=$

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.

8. 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.

9. 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 $t$.