Current clamp

In the designing current clamp experiments we covered the experimental considerations for running a current clamp experiment. In this chapter we will cover what different measures mean. We will focus on interpretation and how each features relates to the biology of neurons. I tried to write the section with the framing of what is ...?

What is ...?¶

This next section talks about some fundamental features of neurons. These are things you should know even if you want to skip the analysis sections of the book.

...excitability?¶

Excitability could be defined as how likely a neuron would fire when stimulated with some sort of current input. We typically use rheobase and the FI curve as a measure of excitability but, excitability is determined by several factors; membrane resistance, spike threshold, capacitance and the resting membrane potential (RMP) or leak potential (for computational models). Each of these components is explained more in-depth below however, I will go over briefly go over these factors here. Some of these factors can be related using $\Delta V=IR$. We can predict reobase from membrane resistance, spike threshold and the resting membrane potential (RMP) however, it is more complicated than that. If you imagine the cell as a passive resistor then all these factors relate through $\Delta V=IR$, where $\Delta V = V_{threshold}-V_{rest}$, $I$ is the rheobase and $R$ is the membrane resistance. However, when we are looking a cells in-vivo we get synaptic currents that change $\Delta V$ from a previous potential which creates a differential equation. Additionally, excitability in-vivo heavily depends on cell capacitance since the cell membrane acts as a frequency-dependent filter (RC filter) and one second long constant current injections do not occur in-vivo. The cell membrane is also a non-linear resistor, particularly when a cell gets close to the spike threshold which also changes the frequency-dependency of the membrane. It is important to remember, you can get the same excitability from different combinations of spike threshold, resting membrane potential and membrane resistance and that excitability in slice may not translate to excitability in-vivo since we need to account for capacitance and non-linear changes in membrane resistance.

...rheobase?¶

Rheobase is one the key parameters researchers extract from current clamp data. Rheobase is consider the minimal current it takes to get a cell to fire an action potential when using an infinite duration current injection and is related to something called chronaxie Irnich, 2010Irnich, 1980. Rheobase is considered the input gain of a cell; how reactive a cell is to excitatory input. Rheobase as a measure is based off a curve called the Lapicque hyperbolic strength-duration curve where strength is the strength of the current injection and duration is the duration of the current injection. This curve is a decaying exponetial curve and rheobase is essentially the steady state of the duration if the stimulus was infinitely long. In practice we just need a current injection that is long enough that we get to the end of the charging of the membrane. Membrane tau’s are fairly short, ~10-30 ms, so if we have a current injection that reachs the point when the voltage of the amplitude is guaranteed to be 1% of the peak, 30*np.log(1/0.01), we get around ~138 ms for a minumum pulse time. If the current injection is too short, then our measurement will be contaminated by capacitance since the cell membrane acts more as an RC circuit than a pure resistor. Rheobase is the combination of resting membrane potential, spike threshold and membrane resistance. Theoretically rheobase is the related to the voltage-resistance-current relationship $I=V/R$. However, due to nonlinearities in membrane resistance and membrane voltage dynamics as you get close to the spike threshold we cannot assume that equation holds true. Additionally, membrane capacitance can change rheobase due to filtering properties that in turn change the impedance of a neuron. If you go to the chapter Building a model you will see that when we use $I=V/R$ to predict the rheobase, we have to scale our theoretical rheobase ~0.4 which means something else lowers rheobase. When a neuron gets close to the spike threshold voltage-gated sodium channels open lowering the resistance but also depolarizing the cell even more creating a cyclic effect that pushes the neuron towards spiking (bifurcation point in membrane dynamics). If you look at the different neurons you will notice that some neuron’s membrane voltage will continue to climb linearly after the initial charging due to capacitance (MSNs) while other neurons like PV interneurons have a flat steady state. Each of these cells have a different set of ion channels that changes their membrane dynamics. The $I=V/R$ equation assumes that a neuron is a linear integrator when many neurons are not.

Earlier I went over how you could deal with cells that are spontaneously spiking. I have heard some arguments that you cannot even test rheobase in these cells types. If you do not want to inject current to hold the cell at -70 or another membrane voltage then you can specify rheobase as the minimum current needed to suppress spontaneous firing. This might make sense for cells that receive very little excitatory input but a lot of inhibitory input like dopaminergic cells or other spontaneously spiking cells.

...membrane resistance?¶

Membrane resistance is the resistance of the membrane to a current. Resistance is determined by the number of ion channels that are open and the composition of ion channels since the membrane itself is considered impermeable to ions. Membrane resistance is typically measured using $\Delta V$ values around resting membrane potential or the potential you are holding your cell at if you are injecting a steady state current. The change in membrane voltage is the voltage difference between the baseline and the voltage during the current injection. Any resistor will cause a voltage drop that you can measure and calculate resistance using $R = \Delta V/I$ where $\Delta V$ is the difference between baseline and the steady state and $I$ is the current injection amplitude. We always measure the voltage difference across an element such as a resistor. One thing to be aware of is that membrane resistance changes with membrane voltage, that means injecting a slow holding current to hold a cell at a specific $V$ vs letting the cell sit a $V_{rest}$ can change how the membrane resistance is measured since ion channel opening and closing depends on the membrane voltage. Membrane resistance is highly correlated (inversely) with rheobase. Membrane resistance is also important for computing the membrane time constant (see below). There are two ways to conceptualize membrane resistance: chord and slope. This assumes your relationship is linear so you really can only assume this happens close to RMP. The ways you may see membrane resistance measured are:

Slope¶

Slope is using linear regression to measure slope between current and membrane voltage or change in voltage. Slope of the linear regression using only: the negative current injections, the positive and negative current injections, or the rectified current injections (absolute value of current and voltage). Regression of a ramp current injection and the voltage of the cell membrane where the slope is the membrane resistance.

Chord¶

Chord is where we get the change in current and change in voltage at every step. We calculate the instantaneous resistance at each step. Membrane resistance is nonlinear so chord resistance captures that relationship. You can find the nonlinear relationship by using a mixed model or repeated measures ANOVA between delta V (depedent variable) and the current injection (indepedent variable). I have also seen instantaneous membrane resistance vs current injection but that version has depedence between resistance and current so I would not use it. The last thing to note about chord resistance is that is shows you the problem.

One important thing to know about membrane resistance is that it is not constant. Membrane resistance drops as you get close to the spike threshold. Neurons in-vivo (at least pyramidal cells) are typically in a high conductance state, meaning they have lots of ion channels open due to constant synaptic bombardment. Since ion channels control resistance, increased numbers of open channels drops the resistance. This has big implications for the function of a neuron. So then why do we measure membrane resistance assuming it is static? The static membrane resistance is probably a good indicator of overall excitability of a neuron.

...spike threshold?¶

Spike threshold is the voltage threshold at which a spike will occur. This point is called a bifurcation point where there is change in the functional properties of a neuron. This bifurcation point and how it relates to the function of a neuron are studied in the field of dynamical systems. The spike threshold is a not a fixed threshold. It depends on the previous membrane voltage and particularily the change (dV/dt) in membrane voltage Azouz & Gray, 2000. For example, PV interneurons typically often do not spike when using ramp injections due to depolarization block. PV interneurons need a large and sudden (i.e highly synchronous) synpatic/current input. Previous spiking activity also changes the spike threshold of a neuron as we will see in part 2 of the current clamp analysis. The flexibility of the spike threshold is a way for neurons to increase or decrease synchronization, feature selectivity, gain, etc. on the fly.

...the afterhyperpolarization (AHP)?¶

The AHP is the period in which a neuron cannot fire another action potential. The AHP generally starts when the voltage drops below the spike threshold. The occurs due to densisitization of voltage-gated sodium channels, activation of fast and slow K+ channels, and calcium channels. The are several ways you will see the AHP analyzed. One is the maximum (or really the minimum) voltage the AHP hits. Another is the difference the maximum AHP and the spike threshold. This value can be affected by changes in the spike threshold so take care when interpreting it. Lastly, there is the mean/median value or slope of specific segments of the AHP that occur after the AHP has reached is maximum value. The last measure is supposably allows you to measure effects of slow and fast K+ channels on the AHP. The minimum length of the AHP is important because it tells you how long a cell takes to recover before being able to spike again. This “recovery” time is partly measured by the time between spikes, the inter-event interval. The AHP can also be very different between cell types. Some cells like PV interneurons have incredilibly large amplitude but short-time AHPs where as pyramidal cells have smaller but longer AHPs. There are scientists who divide the AHP into short, medium and long portions. There is a after-depolarization that occurs between the short and medium component however, I only occasionally see it and the protocols to get it use a very short (1 ms) current injection. The slow component mostly occurs in trains of multiple spikes Andrade et al., 2012Sah & Louise Faber, 2002. Each component is thought to have a set of ion channels that contributes to the current. The fast component of the AHP occurs right after the peak spike occurs and last for up to 1-5 ms. The fast AHP is thought to be drive by BK potassium channels (large currents) Sah & Louise Faber, 2002. The medium AHP is driven by the SK channels that are voltage-insenstive but open in response to the rise in intracellular calciumSah & Louise Faber, 2002. The medium AHP last for 5-100 ms. The slow AHP occurs 100 ms to several seconds after a neurons fires an action potential. The slow AHP becomes more predominate with repetitive firing. The slow AHP does not dircectly depend on calcium levels even though it tends to increase with calcium levels Andrade et al., 2012. The slow AHP may be why pyramidal neurons show a strong spike adaptation (increase in ISI with each consecutive spike). The problem with measuring the different components is that their timing may differ between neurons and overlap when they start and end. The scientists that discovered they components did so by inhibiting them and studying them under different ionic conditions. I think simple measures using time based cutoffs on the AHP can be a good first pass but you will have to isolate the components to make a strong conclusion if they are altered.

...the FI curve?¶

The FI curve is the input-output curve of a cell and allows you to inspect the different types of gain changes that may occur. You get this by plotting the current injected into the cell on the x-axis and the spike frequency of the cell on the y-axis. The plotting alone does not get you much other than a visualization of how the cell responds to current injections. You can analyze curve by fitting with a sigmoid function to get the additive/subtractive gain (rheobase), input gain (slope) and output gain (maximum firing rate). You can also analyze the FI curve by running a repeated measure ANOVA however, it is harder to interpret the specific features of the curve this way but does not have the issues that curve fitting has.

Understanding gain¶

A core aspect of the FI if there are any gain differences between your conditions. There are three factors we are interested in with the FI curve (slope, max firing rate and current offset) and one nusciance factor (voltage offset).

• Slope: The input gain of a cell, slope of the linear portion of the curve. Sensitivity to input, the more sensitive the neuron is to input the steeper the curve. Neurons with shallow slopes can respond to a wide range of input strength.

• Maximum firing rate: Output gain, the response gain of a cell. The maximum firing rate is a reflection of the biological constraints of the neuron.

• Current offset: Additive/subtractive gain, pretty much just rheobase but is somewhat dependent on the slope since it is measured at the midpoint in linear portion of the slope.

• Voltage offset: This is usually close to zero and considered an unimportant factor for FI curves.

Each of these factors combines to shape a neuron’s output. Neurons with a high maximum firing rate and shallow slope will be able to encode input at a very fine level. Even if the current offset and maximum firing rate are the same, neurons with shallower slope will fire less in response to the same input as the size of the input grows. You can see that excitability is more than just rheobase. If you want to learn more about gain I would recommend looking at Ferguson and Cardin, 2020 Ferguson & Cardin, 2020.

The biological regulation of the FI curve is complex. It depends on the spike threshold, repolarization and AHP. You could try to narrow down what component contributes by looking for differences in each component. Some components may compensate for others such as a change in spike threshold could be offset by a change in on the AHP components (fast, medium and slow) which could result in no difference in the FI curve. It is important to remember the the FI curve shows the computational function of a neuron and interpreting how the biology contributes to changes the FI curve is not always straight forward.

...the firing rate?¶

The firing rate is the number spikes divided by the time of the current injection. If you use a one second current injection this is just the number of spikes. The firing rate on its own is not hugely important but is important for creating the FI curve analysis.

...spike adaptation¶

Spike adaptation is how the interval between spikes changes over a long (1 sec) current injection. Some neurons like layer 5 pyramidal neurons have a very pronounced increased in the inter-spike interval (ISI). Another feature pyramidal neurons have, particularly layer 5 pyramidal neurons, is a scecond spike that occurs very quickly after the first spike. This “doublet” spike is a result of backpropogating dendritic action potentials and can be detected with some spike adaptation measures. Other neurons like PV interneurons have very evenly spaced spikes. Some PV interneurons even stutter spike where they spike in short bursts. Since neurons can have a variety of ways they spike, I think you should choose the spike adaptation measure that is best suited to the neuronal subtype. For pyramidal neurons or other neurons with pronounced increase in ISI with each successive spike we want to use a measure that captures a slowing or speeding up of spiking. For neurons like PV interneurons we are more interested in the spike jitter or variability of the ISI. You may want to find neurons that stutter in which case if you may want to look for high variability in the ISI.

...voltage sag?¶

Voltage sag during a hyperpolarizing current injection are caused by Ih currents which are due to K+ and Na+ ions flowing through hyperpolarization-activated cyclic nucleotide gated (HCN) channels. Voltage sag is most prominent in cells that have HCN channels close to the cell body. Axon and dendritic HCN channels cause less sag. HCN channels help maintain a more depolarized resting membrane potential and can also stabilize the membrane potential so that larger amounts of synaptic input are needed to get the neuron to fire. This function is helpful for pacemaking cells. HCN channels can be expressed on spines, dendrites, cell bodies and axons. HCN channels have a reversal potential around -50 to -20 mV. HCN channels are opened by hyperpolarization below -40 to -60 mV and closed by depolarization. Once the HCN channels are open they will remain open creating an inward current until closed. HCN channels are a tetrameric protein with 4 subunit isoforms, 1-4. These subunits have differences in activation speed where activation speed fast to slowest 1 → 2 → 3 → 4. cAMP can also allow HCN channels to open faster and at more depolarized levels where sensitivity is most to least 4 → 2 → 1 → 3. Lastly HCN channel opening is faster at more negative voltages with voltage sensitivity being different among the channels. One prominent cell type that has very large HCN currents are midbrain dopaminergic neurons. When HCN channels are expressed on the axon, such as in PV interneurons, the HCN channels reduces the effects of hyperpolarizaton thus facilitating fast spiking. When HCN channels are expressed in the spines and dendrites they may help set the level of excitatory input needed to drive an action potential. The sag that we analyze is the HCN channel opening and depolarizing the cell to return the cell to its resting membrane potential. You will see sag calculated a couple different ways.

• Raw sag amplitude which is just the steady state deflection minus the peak deflection

• Sag normalized to peak deflection

• Sag normalized to steady state deflection A couple things to note about the sag amplitude. The sag amplitude will be affected by capacitance, membrane resistance, resting membrane potential and the current you are putting into the cell. The HCN channels can open then start to close technically before the peak voltage deflection is even reached due to the membrane tau (interaction between membrane resistance and capacitance).

...membrane time constant?¶

The membrane time constant relates to the time is takes for the neuron membrane to get a steady state voltage before or after a small current injection and is the product of capacitance and membrane resistance $\tau = C\cdot R$. The time constant can determine the resonant property of a neuron, the preferred frequency at which the neuronal membrane oscillates. Changes in the membrane time constant can indicate a shift in how neurons integrate synaptic input Ratté et al., 2013. Long time constants mean it takes a while for the cell to discharge the current thus it can summate or integrate synaptic potentials over a long period of time but also takes longer to reach the steady state voltage. A large time constant means that a neuron “smooths” input or integrates it which generally means these neurons are more senstive to rate. Small time constants mean a neuron may be more of a coincidence detector were tightly synchronized input is more likey to drive input. If you want to change membrane time constant on the fly the mostly likely way is by changing membrane resistance. This can occur through GPCRs that open or close ion channels through secondary messengers. You can measure the time constant by finding the time at which the voltage gets to ~63% of its steady state voltage or you can curve fit an exponential (or double exponential). Generally you will measure the membrane time constant with small negative or positive current injections. This generally avoids activating HCN channels. The membrane time constant is also used to calculate the capacitance in voltage clamp when running a test pulse.

One thing to note about the membrane time constant is that is typically is much shorted in-vivo than it is in slices. The reason is that membrane resistance changes as the cell gets closer to the spike threshold due to changes in conductance. Neurons in-vivo are typically in high-conductance state, i.e. lots of ion channels (particularly Na+) are open thus reducing membrane resistance and membrane tau. Capacitance is fixed though. So tau and membrane resistance should be correlated.

...resonance?¶

The resonant frequency is the frequency of input which will stimulate the largest voltage change in a neuron or the frequency depedent resistance of a neuron Hutcheon & Yarom, 2000. Resonance is tested by using a small amplitude (50 pA peak to peak) chirp current injection. Not all neurons will have a detectable resonant frequency and some neurons will readily oscillate at a specific frequency without a chirp current injection. The resonant frequency is important for how neurons contribute to the neural computation in a circuit. Lower resonate frequency indicates a neuron will respond more readily to slow fluctuations in input and vice versa for high resonate frequency. Resonance is a way for neurons to gate input. Resonance is formed by a high pass and low pass filter. The low pass filter is generated by the membrane capacitance and is related to the membrane tau. The high pass filter is generated by some voltage-gated currents such as $I_k$ and $I_H$. The time constant of the channels contributing to the high pass filter must have a time constant slower than the membrane Hutcheon & Yarom, 2000. Resonance can be so large that it induces spontaneous spiking. Currents such as the persistant sodium current or the T-type calcium channels are involved in spontaneous spiking. One caveat of resonance is that we measure it with small amplitude current so it does not capture the nonlinear membrane dynamics. To get the impeadance just get the frequency spectra of the membrane voltage and divide it by the frequency spectra of the chirp.

...noise depedent spiking?¶

Noise dependent spiking is tested using white or pink noise current injections. Noise dependent spiking is related to membrane resonance and membrane tau however it accounts for nonlinearities in membrane voltage that occur as a neuron gets close to the spike threshold. The impeadance changes as a neuron nears the spike threshold because the membrane resistance changes. There are a couple ways you can analyze how noise effects spiking. If you are using pink noise you can get the spike triggered noise. This is the noise around or even just before the spike occurs. You can take the FFT of the snippet of noise or you can even use something more exotic like a CWT to get the instaneous frequency distribution around the spike. With the spike triggered average (STA) you can get the frequency spectra and deconvolve out (divide out in the frequency domain) the pink noise frequency spectra to get a neurons nonlinear impeadance. This similar to what you do to test resonance but resonance does not capture the nonlinear dynamics. You can also construct a FI curve. Remember that the FI curve essentially helps us determine the gain of a neuron. How responsive a neuron is to input. You can do the same analysis with pink or white noise. You can also compare your no noise current injections with the noise current injections. Layer 2/3 pyramidal cells have increased output gain where as fast spiking interneurons had increased rheobase Higgs et al., 2006.