When we talk about "spikes", we are referring to the change in some property of the neuron over time. When we typically plot a spike, the x-axis represents time. What does the y-axis represent?

In this circuit diagram representing a piece of neuronal cell membrane, the battery, resistor, and capacitor are roughly analogous to the ___, ___, and ___ respectively.

Let's imagine there is another ion that is relevant to determining a neuron's membrane potential in addition to those discussed in the lecture. We'll call the ion $\text{Im}^{+}$ (for Imaginary). The equilibrium potential of $\text{Im}^{+}$ ($E_{Im}$) is -100 mV. Assume the resting potential of the neuron is -65 mV. When specialized $\text{Im}^{+}$ channels open, the $\text{Im}^{+}$ conductance will increase. This will ___ the cell, thus ___ its membrane potential.

Suppose $\text{Im}^{+}$ channels are composed of 5 subunits that open and close independently, as well as an additional "ball-in-socket" gating mechanism. Each of the 5 subunits has a voltage-dependent open probability $u$ and closed probability $(1-u)$, while the ball-in-socket gating mechanism has a voltage-dependent open probability $z$ and closed probability $(1-z)$. Which expression could most likely be used to express the $\text{Im}^{+}$ current across the membrane?

Let's come back to the real-life $\text{Na}^{+}$ (sodium) channel, whose voltage-dependent dynamics are shown above. $m_{\infty}$ and $h_{\infty}$ (the steady states of $m$ and $h$) are shown for different $V$, along with the associated voltage-dependent time constants $\tau_{m}$ and $\tau_{h}$. Remember that $m$ is the probability that any individual channel subunit is open and $h$ is the probability that the additional ball-in-socket gating mechanism is open. If the neuron is at its resting potential (around -65 mV), and we deliver a current injection that decreases the membrane voltage to -75 mV, which of the following will be the first to reach its new steady state?

Refer again to the figure shown above the previous question. Remember that $\text{Na}^{+}$ current depolarizes the cell and is the principal driver for the upward portion of a spike. Both $m$ and $h$ must be high for there to be a lot of $\text{Na}^{+}$ current. $h_{\infty}$ becomes 0 when voltage is close to or greater than -30 mV. How, then, is it possible for the membrane to depolarize beyond $V$ = -30 mV during a spike (spikes peak closer to $V$ = 40 mV)?

True or false: All neural coding can essentially be reduced to variations in firing rate. Thus, precise spike timing is usually unimportant: what matters is a "rate code."

Recall the exponential integrate and fire neuron model, schematized above. How many stable fixed points does the system have?

The FitzHugh-Nagumo model is a 2-dimensional dynamical neuron model. It is defined by the following two differential equations:

$\dfrac{dV}{dt} = V(a - V)(V - 1) - w + I$

$\dfrac{dw}{dt} = bV - cw$

where $V$ is voltage, $w$ represents an outward hyperpolarizing current, $I$ is injected current, and $a$, $b$, and $c$ are constants. Which of the following is an expression for the "$w$-nullcline?"

The above figure is a phase plane based on the FitzHugh-Nagumo model from the previous question. A vector field is shown that gives a sense of the flow of the system. If we observe a spike in this system, the trajectory will travel counterclockwise around the phase plane. The first part, or “upstroke,” of the spike occurs in which of the following regions of the phase plane (regions are labeled in the figure)?

Change the values for the membrane's resistance and capacitance ($R$ and $C$), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after multiplying $R$ by 5?

Does it reach a stable value more quickly or more slowly after dividing $C$ by 10?

Does it reach a stable value more quickly or more slowly after multiplying $R$ by 10 AND dividing $C$ by 10?

An experimental method for calculating a membrane’s time constant $\tau$ (when $R$ and/or $C$ are not known) is to start at zero and record the time at which the membrane potential $V$ reaches a value approximately equal to $0.6321*V_{peak} = 0.6321*IR$, where $I$ is the constant injected current. Check if this method works by injecting different amounts of current $I$ and changing the values for $R$ and $C$. Once you’ve convinced yourself that the experimental $\tau$ appears to be identical to the theoretical $\tau (= RC)$ in all these cases, provide a theoretical justification for why this method works.

To do this, find the solution to the differential equation for the passive membrane:
$\dfrac{\text{d}V}{\text{d}t} = - \dfrac{V}{RC} + \dfrac{I}{C}$

$V(0) = 0$


After solving the differential equation you should be able to use the fact that $V_{peak} = IR$ and $\frac{e - 1}{e} = .6321$ to complete the derivation and show that $V(\tau) = 0.6321*IR$.

Which of these equations is the solution to the given differential equation?

Now let's consider the case that your neuron is not receiving simply a constant input, but a barrage of asynchronous inputs from many other neurons. Model this with the following code by adding a white noise component to the input current (the constant part of the input current is reset to one nA):

MATLAB: intfireNoise.m
Python: intfire_noise.py

Plot the interspike interval distribution, that is, the distribution of the time intervals between consecutive spikes, for a range of different noise amplitudes. Hints: this noise component is already implemented in the code: just switch it on. You can make use of the MATLAB functions diff and hist (in Python np.diff and plt.hist). You will probably want to increase tstop, the length of time you are integrating for, by quite a bit to get a well-sampled histogram. What best describes how this distribution changes as you increase the amplitude of the noise input? (Stay within a range between 0 and 5 nA.)