1. Review

2. Overview Cable theory Action potential generation
Action potential propagation
Some channel stuff interspersed

3. Cable Theory
Any questions pertaining to cable theory in the past have been conceptual in nature; this stuff gets complicated quickly – being facile with the basic concepts is likely sufficient

4. What does it mean to be non-isopotential?

5. What does it mean to be non-isopotential?
The membrane potential is not the same throughout the cell

6. What does it mean to be non-isopotential?
-This is basically just an application of the cell as an RC circuit stuff Keith talked to you about last week
-Cables/non-isopotential cells function as these RC circuits in series
-Lipid membrane is still responsible for storing charge as the capacitor
-Channels in the membrane are still conduits for current flow/resistance
-What we’re adding is internal/axial resistance – since we have current flowing down a fiber
-These factors essentially sum to determine how voltage travels down the axon…this is cable filtering

7. What are two effects of cable filtering for Vm change in an axon or dendrite?
Thinking back to Steve’s lecture, do you remember what happens to e change in potential as current moves down the axon or dendrite?

8. What are two effects of cable filtering for Vm change in an axon or dendrite?
Vm changes are slowed
Vm changes are reduced
What is meant by “slowed”  time constant increases (takes longer to reach steady state)
What is meant by “reduced”  steady state amplitude is smaller

9. Decay of amplitude with distance along a cable
You can do something like this in large fibers (i.e. squid giant axon, probably calyx of held) – this would be extremely difficult in small fibers
Stimulate the fiber at point A, have recording electrode(s) placed at different distances from point A. Measure the voltage response/membrane potential at each of those points – can compare both the rising phase and steady state amplitude between different time points/distances

10. Dampening and slowing of Vm changes with distance
This graph is normalized to the steady state membrane potential achieved at site of injection
Vx = V0e-x/λ
This essentially just exponential decay – the rate of decay is going to be proportional to your recording position on the fiber relative to how far your electrical charge is able to travel down the fiber
X represents your distance from the site of current injection
Lambda is the length constant – coming up in a slide or two
l = √(rm / ri)

11. Why do these changes occur?

12. Why do these changes occur?
Dampening/reduction occurs because of leak conductance/current escaping across the membrane
Slowing occurs because of membrane capacitance

13. What is the length constant of a cable?

14. What is the length constant of a cable?
Distance an electrical signal will travel along a cable by passive electrical conduction

15. What does a large length constant imply for Vm changes in the cable?

16. What does a large length constant imply for Vm changes in the cable?
Changes in Vm will be better preserved down the length of the axon.

17. What factors dictate the calculation of the length constant?

18. What factors dictate the calculation of the length constant?
The intracellular (axial resistance)
The membrane resistance
Lambda = sqrt(rm/ri)
Does this make sense?

19. Relationship of l to membrane geometrical properties
So, l increases with the square root of fiber radius.

20. How can you increase action potential conduction velocity by manipulating these factors?
Fiber radius
Membrane capacitance
Membrane resistance

21. How can you increase action potential conduction velocity by manipulating these factors?
Fiber radius
Increase
Membrane capacitance
Decrease
Membrane resistance

22. Why might increasing the radius of a cable have diminishing returns?

23. Why might increasing the radius of a cable have diminishing returns?
Increasing the radius decreases the axial resistance, increasing the length constant
Increasing the radius also increases the surface area of the membrane, decreasing membrane resistance and decreasing the length constant
Increasing the radius has a bigger effect on ri than it does on rm (since ri is based on area/volume and therefore has a squared term)
An additional problem is space constraints – you can only increase the radius/size of fibers so much before you run into spacing issues

24. Active membrane

25. Response to current injection
Ena = ~+50mV
AP reaches ~ +40mV
Dominated by Na conductance
When injecting current into the cell you’re just experimentally/artificially changing the membrane potential of the cell
With larger current injections, get nonlinear/nonohmic voltage responses (action potentials)
Note that the peak of the action potential gets up to about mV (far surpasses 0mV).
This is a large deviation from resting potential and highly positive, suggesting that it’s not just the membrane breaking down (initial hypothesis).
+50mV is near the reversal potential of Na (Nernst potential)  suggests that maybe a transient change in the membrane’s selective permeability to Na can be what’s driving the action potential
This is what HHK tested in the giant squid axon preparation by experimentally manipulating the internal/external Na concentrations of their preparation/doing ion substitution with nonpermeable substitutes. In short, their experimental results matched their anticipated results based on simulated Na Nernst potentials based on the concentrations used in their experiments
This provided evidence that Na conductance drives the nonlinear response of the action potential

26. Generating action potentials

27. In a Hodgkin-Huxley model neuron:
Passive membrane/establishing the resting membrane potential depends on LEAK CHANNELS (i.e. channels that have constant conductance for K and Na as a function of voltage) Action potentials (initial depolarization and subsequent repolarization) depend on VOLTAGE-GATED CHANNELS: 1) V-gated Na channel for depolarization 2) V-gated K channel for repolarization

28. Voltage-gated channel contributions to AP generation
1: What conductance(s) contributes to setting Vrest?
1 Vrest

29. Passive current flow at different Vm
𝐼 𝑖𝑜𝑛 = 𝑔 𝑖𝑜𝑛 ( 𝐸 𝑖𝑜𝑛 − 𝑉 𝑚 )
K current
Outward
(hyperpolarizing)
Vrest
ENa EK
Na current
Na/K leak conductance
Membrane is more permeable to K than Na so this pulls the resting membrane potential down towards EK (but not all the way down to -90mV since there is limited permeability to Na)
ECl
ECa
- Vm +
Inward
(depolarizing)

30. Which members of the K channel family are likely participants in background, leak channels that set Vrest?

31. What about leak (background) channels?
Kir1 - 6.x
K2P (TREK, TASK, TWIK)
Likely candidates for the channels that are responsible for the background leak currents (i.e. for the K permeability that largely determines resting membrane potential
IV relationship for inward rectifying K channel (left) is linear at physiological membrane potentials because of a Mg block – the part of the IV relationship that isn’t linear isn’t physiologically relevant
K2P is another family that has a fairly linear IV relationship

32. Channel contributions to AP generation
2: What can potentially account for this initial depolarization?
Theoretical – if you had your pick of any channels that Steve/Yu-Qing talked about, what could cause this?
2 initial depolarization

33. Synaptic input – activation of AMPARs
The point of this slide isn’t to go into glutamate receptors. That will come in another review session – just trying to tie together some concepts.
Synaptic activation of glutamate receptors allows for influx of cations (depending on which receptors are activated and whether they are post-translationally edited)
Around Vrest you’re not going to have much NMDAR-mediated current because of the persistent Mg2+ block – you need to have depolarization in order to remove this block and generate current
AMPAR IV curve is linear – going to have large conductance around Vrest and the channel will be able to open (provided it is gated) to make Na/K conductance available (going to have influx of Na)

34. Functions of different Ca2+ channel sub-classes (LVA)
T-type:
Important at cell body and dendrites of CNS neurons.
Role in boosting synaptic signals from dendrites to cell body.
Rapidly inactivating (though not as fast as Na+ channels).
Activated by small depolarizations.
This is a good slide to memorize – T-type channels tend to be a crowd favorite for questions
LVA refers to low-voltage activated channel – these will open with slight depolarization, allowing inward Ca2+ to further depolarize the cell and open more voltage-gated channels
Gomora et al. (Perez-Reyes) Biophys J 83:229

35. Voltage-gated channel contributions to AP generation
3 rising phase
3: What conductance contributes to the upswing of the action potential? (specifically in neurons)
Why does this ion’s permeability increase?
V-gated Na+ channels

36. Ion substitution isolates the inward and outward components
In neurons V-gated Na channels drive the upswing of the action potential
Early experiments in giant squid axon determined the importance of Na conductance as a function of voltage by manipulating the bath concentration of Na
With depolarization – see an inward current when extracellular concentration is high because Na flows down its concentration gradient. When extracellular concentration is low, the current is outward because Na flows out of the cell down its concentration gradient
You can also just block V-gated Na channels pharmacologically with something like TTX and see that you no longer have an action potential generated

37. As an aside to mention another class of channels: HVA Ca2+ channels drive AP upswing in cardiac pacemaker cells
HVA channels (High-voltage activated)
L-type (Cav )
P/Q-type (Cav 2.1)
N-type (Cav 2.2)
R-type (Cav 2.3)
In cells that don’t have V-gated Na channels, V-gated Ca channels serve essentially the same purpose
Once some threshold membrane potential is reached through the initial depolarization of the membrane, these channels are able to open and generate an inward current
Note: cardiac pacemaker cells do not express V-gated Na channels
Note: this class of Ca2+ channels will come back when we talk about synaptic release of neurotransmitter

38. Voltage-gated channel contributions to AP generation
4 repolarization
4 What two things account for the repolarization of the membrane potential?

39. Voltage-gated channel contributions to AP generation
4 repolarization
4 What two things account for the repolarization of the membrane potential?
Na channels inactivate stopping Na conductance
K channels become activated

40. Channel inactivation Fast inactivation of K+ channels:
N-terminus ball-and-chain model
Fast inactivation of Na+ channels:
III-IV linker hinged lid model
Inactivation of Ca2+ channels:
Voltage-dependent, Ca2+-dependent
S6 and all intracellular segments, auxilliary subunits

41. Calculated conductance for sodium and potassium
Note the relative lag on K channel conductance
Increase in K conductance corresponds temporally with the time at which Na channels are inactivating
Iion = gion (Vm-Eion)

42. Na inactivation and increase in K conductance coincide

43. Voltage-gated channel contributions to AP generation
5 hyperpolarization
5 Why does the membrane potential become hyperpolarized relative to Vrest?

44. Remaining K conductance pulls Vm closer to EK following AP
You still have K conductance while Na channels are inactivated
V-gated K conductance essentially masks leak conductance that’s always at play
You have a concentration gradient for K efflux and channels through which more of it is able to pass
This pulls Vm towards EK. EK is hyperpolarized relative to Vrest

45. If you wanted to generate another action potential from a hyperpolarized membrane potential (and had your choice of channels), what would you use?

46. cAMP modulation of HCN4 channels
HCN channel: hyperpolarization-activated cyclic nucleotide–gated channel
non-selective cation channel, Erev ~ -30 mV
-100 pA
with cAMP
voltage sag
-50 mV
w/o cAMP
-70 mV
membrane potential change
Momin et al, 2008; Emery et al, 2012

47. Do you know what all of these mean?
Activation
Inactivation
Deactivation
Deinactivation

48. Do you know what all of these mean?
Activation – channel opening
Inactivation – channel opens a closed configuration from which it cannot be opened
Deactivation – closing of the channel. Can be activated from this state
Deinactivation – move form inactivated to deactivated

49. Which inactivate and which do not?
Which are open at rest and which are not?

50. Which inactivate and which do not?
Inactivate: A-type (green) & Na (blue)
Which are open at rest and which are not?
Open at rest: A-type
Vertical current here =
instantaneous current; no lag waiting for channel opening

51. At the peak of the action potential, which are true?
GNa = GK
|IK| > |INa|
GK << GNa
ENa = EK
INa = IK

52. At the peak of the action potential
GNa = GK GNa >> GK
|IK| > |INa| INa = -IK or |IK| = |INa|
GK << GNa
ENa = EK ENa & EK do not change during an AP
INa = IK INa = -IK don’t worry if you got this one “wrong.” You had the right idea
At the peak of the action potential, the voltage is no longer changing. Therefore, the sodium and potassium currents must be exactly equal in magnitude at this point and opposite in sign
-IK = INa
The “Physiologist’s Ohm’s law” says that
Iion = Gion(Vm – Eion)
At the peak of the action potential above, Vm = 44mV, which is approximately ENa. This means that the “driving force” term for sodium is extremely small
(Vm – Eion) = (44mV – 50mV) = -6mV
While for potassium
(Vm – Eion) = (44mV – -90mV) = 134mV
Despite having < 1/20th the driving force, INa = -IK. This means that GK must be much less than GNa.

53. What about time? Why the lag?
Na rise time is faster than K rise time
Iion = gion (Vm-Eion)

54. When n is multiplied by itself, then a lag appears
The lag increases with higher powers of n

55. A model for lag closed open
Hodgkin and Huxley proposed that the 4th power relationship would arise if there were 4 particles, each of which must occupy a specific place for the conductance to be active ⇄
closed
open
It is unlikely that you will need to know more than the basics about this – this is more so a further illustration that Na is fast inactivates while K is slow and doesn’t inactivate
Qualitatively/intuitively, you can think about the 4th power relationship in terms of 4 criteria needing to be met for channel opening to occur – channel is open if and only if all 4 criteria are met.
The actual conductance of the channels as a whole depends on how many channels are open.
Mathematically, this can be expressed by:
gK = GK * n4
Where GK is maximal K conductance and n refers to active conductance.
gK grows as more conductance becomes active.
As the membrane depolarizes conductance is slowly activated, making the increase in current flow slow across the membrane
gNa = Gna * m3 * h
h is an inactivation constant – this is binary: either 0 (inactive) or 1 (active)
h essentially overrules the active conductance of the Na channel – all of the conductance of the channel can be active, but as soon as this inactivation term switches to 0, there is no longer any observed conductance
For Na channels, conductance is quickly activated as the membrane depolarizes, so the rise time for Na current is faster. The inactivation phase is slower, stopping Na conductance

56. Gating Model Conductances are “gated”
– Changes in membrane potential physically move charged particles in the membrane and open gates, somehow making the membrane more permeable to specific ions.

57. You don’t actually need to know this
Current due to K conductance
Current due to Na conductance
capacitance
n4 is the gating parameter for K
m3h is the gating parameter for Na
What experimental observations did these exponents account for?

58. You don’t actually need to know this
Current due to K conductance, factoring in lag/time to open channel
Current due to Na conductance, factoring in lag/time to open channel +channel inactivation
capacitance
n4 is the gating parameter for K
m3h is the gating parameter for Na
What experimental observations did these exponents account for?
Recommended for some “light” reading:
This is the original paper where Hodgkin & Huxley figured all this out

59. What experimental observations did the exponents
What experimental observations did the exponents account for in the gating parameter m3 and n4?
The sodium and potassium currents did not change instantaneously with voltage. There was always a small delay.
m and n are between 0 & 1 so raising them to a power makes them smaller

60. The probability of all of the gates doing whatever they do increases with voltage. It is the rate at which they move in response to changes in voltage that is responsible for the shape of the action potential.

61. The sodium conductance has 2 gating components (m,h)
One component for activation that allows current flow (m)
m is cubed. What does this suggest?
A separate one for inactivation that stops the current (h)
The probability of both the activation gate opening and the inactivation gate blocking the current increase with membrane potential
Looking at their equation, what value should m move towards as you increase voltage? What about h?
– If both reach these values, what will the sodium conductance be?

62. The potassium conductance has a single gating component (n)
n is raised to the 4th power. What does this suggest?
Potassium currents do not have an inactivation mechanism
They persist throughout a stimulus

63. Signal propagation

64. What happens to a depolarization/hyperpolarization (from dendrites) as it approaches the spike initiation site? Describe an experimental setup that allows a direct test of this

65. Direct evidence of passive decay
This is hard to do, but if you wanted to see how a signal degrades over time/space, you could stimulate and record at different points on the cell (i.e. out on dendrites, and compare against soma)

66. What is the evidence that action potentials initiate at the axon initial segment?

67. Initiation at distal initial segment
Julian Meeks

68. A high density of sodium channels: substrate for initiation
You’ve learned a bit about cable filtering. You know that membrane potential changes can only travel so far until they are degraded. This is obviously a problem in neurons if depolarizing stimuli/synapses are being received at dendrites and an action potential needs to be generated
The action potential is actually generated at the axon initial segment (some people call it the axon hillock)
This part of the cell has a particularly high densiry of V-gated Na channels – the purpose is SIGNAL AMPLIFICATION. Small depolarizing signals received at the dendrites can be amplified in order to generate an action potential that can be sent down the axon to the axon terminal/for synaptic release

69. What does it mean that action potentials are regenerative?

70. What does it mean that action potentials are regenerative?
Mechanisms (voltage-gated channels) in place along axon to provide active conductance to generate new action potentials along the axon
Action potential propagation does not depend solely on passive current flow
As an aside: distribution of voltage-gated channels along the membrane also prevents back-propagation of the action potential up the axon

71. Propagation by combined passive, active current flow

72. Action potentials propagate by combined passive, active current flow
One way that AP propagation down the axon is supported is by myelination
Myelination helps in a couple of ways:
It increases the size of the fiber – thinking back to the length constant, this supports the signal being able to travel further with less decay
It increases the separation of charge between the axoplasm and external bath. This DECREASES the membrane’s ability to store charge. So, less current is lost to capacitance
Myelination increases the membrane resistance by concentrating the channels at the nodes of ranvier (i.e. breaks in the myelination). So, current is not lost across the membrane either (current is forced down the internal resistance).
Current flow down the myelinated axon is mostly passive (don’t need to deal with the time involved in opening/closing voltage gated channels or expend energy to maintain ion gradients). The exception: Nodes of ranvier serve as modules at which the current can be regenerated. Saltatory conduction (worth memorizing this term)
Na channel inactivation prevents the backpropagation of the action potential up the axon (AP is unidirectional – moves towards the axon terminal)
Myelination is more useful in larger fibers (no real benefit in small fibers).

73. How does myelination improve action potential propagation?

74. How does myelination improve action potential propagation?
Decreases membrane capacitance
Increases membrane resistance
Saltatory conduction

75. What is saltatory conduction?

76. Saltatory conduction Myelination speeds propagation by
Increasing rm, thereby increasing the current through ri
Decreasing cm
Increasing the contribution of the passive component of propagation, thereby consuming less time with active gating of channels (conformational changes).
Also is advantageous metabolically (less ATP used to re- generate ion gradients)