Note: include at least 3 significant digits if you round your answers.
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Neuron with a single channel. If a neuron had just a single channel, we could model the evolution of its voltage $V$ as
$$C\diff{V}{t} = - g (V - E)$$
where $C$ is the (positive) capacitance of the neuron's membrane, $g$ is the (positive) conductance of the channel (the reciprocal of its resistance) and $E$ is its reversal potential. (We'll measure time $t$ in ms, voltage $V$ in mV, capacitance $C$ in μF, and conductance $g$ in mS.)
For what value of $V$, will the system be at a steady state (or equilibrium), i.e., the change in the voltage, $dV/dt$, will be zero? $V=$
If $V$ is larger than this value, the voltage will be
since the derivative is
. If $V$ is smaller than this value, the voltage will be
since the derivative is
.
Let the channel be a potassium channel with a reversal potential of $E=E_{K} = -95$ mV and a conductance of $g=g_{K} = 1.2$ mS. If we set the capacitance to $C=1$ μF, the equation for $V$ becomes
$$\diff{V}{t} = -1.2 \left(V + 95\right).$$
If when $t=7$ ms, the voltage is $V(7) = -63$ mV, calculate the voltage for all time.
$V(t) = $
Plot $V(t)$ for $7 < t < 20$ on the below graph.
Feedback from applet
Final points of curves:
Initial points of curves:
Number of curves:
Speed profiles of curves:
You should have found that the solution decays exponentially to the reversal potential, with a time constant determined by the conductance.
Hint
If you get confused with all the different parameters, try putting in number for the parameters $C$, $g$ and $E$, for example, substitute $C=2$, $g=4$ and $E=10$. Once you solve it for those numbers, you can just repeat your calculations with keeping the letters $C$, $g$ and $E$.
To use the applet to plot the graph, slide $n_c$ to 1, as you want only one curve. Move the two points to the beginning and end of the curve you wish to draw. Click the curve to change its shape (i.e., its ”speed profile”). The points can only be aligned with the grid, so you will have to round your answers to plot them on the graph.
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Now, we will model both the potassium channel and the sodium channel in the neuron. We assume that the currents from the channels sum linearly so that the combined model for the evolution of the voltage is
$$C\diff{V}{t} = -g_{Na}(V-E_{Na}) - g_{K}(V-E_{K}),$$
where $g_{Na}$ is the sodium channel conductance, $g_K$ is the potassium channel conductance, $E_{Na}$ is the sodium reversal potential, and $E_K$ is the potassium reversal potential. For simplicity, we'll continue to set the capacitance to $C=1$ μF.
For what value of $V$ will the neuron be at a steady state?
$V=$
Hint
Leave $g_{Na}$, $g_K$, $E_{Na}$, and $E_K$ as unspecified parameters, so the answer will be in terms of those four parameters.
Online, enter $g_{Na}$ as
g_Na, enter $g_K$ as
g_K, enter $E_{Na}$ as
E_Na, and enter $E_K$ as
E_K (capitalization matters).
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Imagine that, for this neuron, the sodium reversal potential is $E_{Na} = 60$ mV and the potassium reversal potential is $E_K = -95$ mV. Moreover, when the neuron is at rest, its sodium channel conductance is $g_{Na} = 0.08$ mS and its potassium channel conductance is $g_K = 0.32$ mS.
Calculate the resting potential $E_r$ of this neuron, i.e., the value of the steady state voltage when the neuron is at rest.
$E_r =$
mV
Hint
Since you only are required to keep 3 significant digits, you should be able to round to a simple form for $E_r$.
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An action potential is caused by changes in the sodium and potassium conductances. The conductances changes are triggered by changes in the voltage due to input, but here we'll just manually change the conductances to produce an action potential.
Imagine that the neuron as been at rest for a long time. Therefore, its voltage has reached its steady state, which is $V=$
mV. We'll use that value for the initial condition $V(0)$.
At $t=0$, the action potential is somehow triggered, sending the sodium conductance immediately way up to $g_{Na} = 16.0$ mS, but only for a brief moment, for 1 ms. If $g_K$ remains at $0.32$ mS and $g_{Na} = 16.0$ mS for $0 \le t \le 1$, calculate $V(t)$ for that interval.
$V(t) = $
mV
What is the value of the voltage at $t=1$ ms? $V(1)=$
mV
Hint
You calculated the rest steady state in the previous part. Now, calculate the steady state for the new parameters, then solve the differential equation with those parameters and initial condition being the old rest state.
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At $t=1$, the sodium conductance decreases to zero and the potassium conductance increases to $g_K = 4.0$ mS. Here, let's assume that $V(1) = _$ mV, the answer you entered in the previous part.
Using the value $V(1) = _$ mV for the initial condition at $t=1$, calculate $V(t)$ for $1 \le t \le 4$, assuming that the conductances stayed fixed at $g_{Na}=0$ mS and $g_K=4.0$ mS during this interval.
$V(t) =$
mV
At $t=4$, the voltage is $V(4) = $
mV.
Hint
The tricky part is that you have to use the value of $V(1)$ calculated in part d as your initial condition for this part, which is
$V(1) = _$ mV. If you haven't entered in a value, it will look like $V(1) = _$, and nothing you enter here will be correct. If all went well, $V(1)$ should be a nice looking number when you round to 3 significant digits.
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Finally, for $t \ge 4$, the sodium and potassium conductances return to their resting values $g_{Na}=0.08$ mS and $g_K = 0.32$ mS. Using the value you calculated above, $V(4) = _$ mV, for the initial condition at $t=4$, calculate $V(t)$ for $t \ge 4$.
$V(t)=$
mV.
Hint
Similar to the previous part, you have to use the value of $V(4)$ calculated in part e as your initial condition for this part, which is
$V(4) = _$ mV. If you haven't entered in a value, it will look like $V(4) = _$ mV, and nothing you enter here will be correct. If all went well, $V(4)$ should be a nice looking number when you round to 3 significant digits.
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Sketch the solution for $0 \le t \le 20$.
Feedback from applet
Final points of curves:
Initial points of curves:
Number of curves:
Speed profiles of curves:
Hint
To use the applet to plot the graph, slide $n_c$ to specify the number of pieces you want for your solution. Move the two points on each piece to the beginning and end of the curve you wish to draw. Click the curve to change its shape (i.e., its ”speed profile”). The points can only be aligned with the grid, so you will have to round your answers to plot them on the graph.
This plot requires that you use the correct values for $V(1)$ and $V(4)$; it doesn't depend on the values of $V(1)$ and $V(4)$ that you entered in parts d and e.
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