Chapter 12: Nerve Impulses

Biological PhysicsNelson

Updated 1st Edition

Slide 1-1

12 Nerve Impulses

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Biological Question?

• How can a leaky cable carry a sharp signal over

long distances?

• Physical Idea: Nonlinearity in a cell membrane’s

conductance turns the membrane into an

excitable medium, which can transmit waves by

continuously regenerating them ….

• Wow …

12.1 THE PROBLEM OF

NERVE IMPULSES?

12.1.1 Phenomenology of action potential

• This section is historical in nature …

• Three caveats:

1. Stimulation of the cell inputs (typically

dendrites) from cell outputs (typically axon

terminals)

2. Computation of the appropriate output signal

3. Transmission of the output signal (nerve

impulse) along the axon.

• This chapter focuses mainly on the last point ….

12.1.1 Experiment & Electronus

12.1.1 Phenomenology: Action Potential

• Experiments show (see Fig. 12.1) we get

depolarizing and hyperpolarizing propagation

• However, instead of being graded, the action

potential is an all-or-nothing response. That is,

the action potential arises only when the

membrane depolarization crosses a threshold;

subthreshold stimuli give electrotonus, with no

response far from the stimulating point.

• In contrast, above-threshold stimuli create a

traveling wave of excitation, whose peak

potential is independent of the strength of the

the initial stimulus.

Continued

• Action potential moves down axon at a constant

speed (see Figure 12.2b), which can be anywhere

from 0.1 to 120m/s. When progress of action

potential is measured at several distant points, as

in Figure 12.2b, the peak potential is found to be

independent of distance.

• In contrast to the decaying behaviour for

hyperpolarizing or subthreshold stimuli. A single

stimulus suffices to send an action potential all the

way to the end of even the longest axon.

Continued

• Indeed, the entire time course of the action

potential is the same at all distant points (Figure

12.2b). That is, the action potential preserves its

shape as it travels, and that shape is

"stereotyped" (independent of the stimulus)

• After the passage of an action potential, the

membrane potential actually overshoots slightly,

becoming a few millivolts more negative than the

resting potential, and then slowly recovers. This

behaviour is called afterhyperpolarization

• For a certain refractory period after transmitting

an action potential, the neuron is harder to

stimulate than it is at rest

12.1.2 THE CELL MEMBRANE

VIEWED AS AN ELECTRIC

NETWORK

12.1.2 Iconography (see Fig 12.3a)

1. No significant net charge can pile up inside the

individual circuit elements: The charge into one

end of a symbol must always equal the charge

flowing out the other end. Similarly,

2. A junction of three wires implies that the total

current into the junction is zero (Kirchoff’s rules)

3. The electrostatic potential is the same at either

end of a wire and among any set of joined wires

4. The potential changes by a fixed amount across a

battery symbol

5. The potential changes by the variable amount IR

across a resistor symbol

12.1.2 Conductance as Pipes

• There is a physical difference between our

system and ordinary networks, but we can

nevertheless use network ideas because?

• Although we have at least 3 important channels:

Na+, K+ and Cl- are all at the same electrostatic

potential (imagine 3 species in one fluid pipe)

• Instead of Fig. 12.3a we have 12.3b (where we

explain the Capacitance in a few slides time)

• Kirchhoff's loop rule in this diagram implies

current conservation and Δ𝑉 = 𝐼𝑖𝑅𝑖 + 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡,

where ΔV = V2 − V1, 𝐼𝑖 = 𝑗𝑞,𝑖𝐴 & 𝑅𝑖 =1

𝑔𝑖𝐴

Ignore

Vpump

12.1.2 Quasi-steady approximation &

chord conductance

• Can neglect the complication of ion pumping

when studying fast transients like the action

potential: Quasi-steady approximation

• Assume steady state (non-eq) and then

suddenly shut off ion pumps implies

Σ𝑖 𝒱0 − 𝒱𝑖

𝑁𝑒𝑟𝑛𝑠𝑡 𝑔𝑖 = 0

• Then subbing 𝑔𝑡𝑜𝑡 ≡ Σ𝑖𝑔𝑖 leads to chord conductance formula:

𝒱0 =Σ𝑖𝑔𝑖𝑔𝑡𝑜𝑡

𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡

• Evaluating yields 𝒱0=-66 mV only a few millivolts

different from true steady state -72 mV

Capacitance: (C+- C- )e can pile up on each

boundary!

• Membrane capacitance

can happen even with

charge neutrality (see Fig.)

where 𝑞 = 𝐶Δ𝑉

• The capacitive current is

defined as 𝑑 Δ𝑉

𝑑𝑡=

1

𝐶

• This has used Ohmic

hypothesis on a small

patch of membrane

12.1.3 Linear Cable Equation

• Now consider what happens, not radially, but

axially along membrane where uniformity is

gone (see Fig. 12.4)

• Your Turn 12 A (homework) shows we can lump

three pairs of batteries & R’s into one for each

patch of membrane

a. Typical numerical values are on p. 516

b. Equation derived from Fig 12.4:

• Not yet solvable as 3 unknowns?

Figure 12.4 (Schematic; circuit diagram.) Caption: See text.

12.1.3 Linear Cable Equation (LCE)

• Eliminate Ix using

to obtain

• Using Your Turn 12 A: 𝑗𝑞,𝑟 = 𝑉 − 𝒱0 𝑔𝑡𝑜𝑡 ,

defining 𝑣 𝑥, 𝑡 ≡ 𝑉 𝑥, 𝑡 − 𝒱0 , space & time

constants as

we get:

12.1.3 Solution of LCE

• This looks like the diffusion equation: 𝑑𝑐

𝑑𝑡= 𝐷

𝑑2𝑐

𝑑𝑡2

and if we define 𝑤 𝑥, 𝑡 ≡ 𝑒𝑡

𝜏𝑣(𝑥, 𝑡) then the LCE

becomes

𝜆𝑎𝑥𝑜𝑛2

𝜏

𝑑2𝑤

𝑑𝑡2=𝑑𝑤

𝑑𝑡which has solution

• This does not have any travelling wave solutions

so we need something extra?

12.2 SIMPLIFIED

MECHANISM OF ACTION

POTENTIAL

12.2.1 The Puzzle

• How can waves travel without diminution?

• When a system is not in equilibrium, free energy

is not a minimum

• If not in minimum state the system can do work

• The excess free energy is what keeps the signal

from diminishing …

12.2.2 A mechanical analogy

• A nonlinear wave in an excitable medium selects

its waveform and velocity

• Another possible analogy (as opposed to

corrugated roofing) is a bomb fuse …

• The main idea is thus,

– Each segment of axon membrane goes in

succession from resisting change (like chain

segments to the left of the kink in Figure

12.5a) to amplifying it (like segments

immediately to the right of the kink) when

pulled over a threshold by its neighbouring

segment.

A corrugated plate on a slope (a roof?)

12.2.3 A little more history

• Read pages 521-524; however we find:-

– If the membrane could rapidly switch from

being selectively permeable to potassium only

to being permeable mainly to sodium, then

the membrane potential would flip from the

Nernst potential of potassium to that of

sodium, explaining the observed polarization

reversal (see Equation 12.3).

Reprinted by permission from Nature, ©1939, Macmillan Magazines Ltd.

Nobel Prize Winning Work

Effect of sodium content

12.2.4 Time course of action potential

• Given data in Fig. 12.8a) can we find out what

kind of total membrane current 𝑗𝑞,𝑟 is needed?

• Using fact that entire history is known once a

given point is known: V 𝑥, 𝑡 ≡ 𝑉 𝑥, 𝑡 −𝑥

𝜃where

V t ≡ 𝑉 0, 𝑡 is shown in Fig. 12.8a) implies that 𝑑𝑉

𝑑𝑋= −

1

𝜃

𝑑 𝑉

𝑑𝑡′ 𝑡′=𝑡−

𝑥

𝜃

• Rearranging Eq. (12.7) then gives

𝑗𝑞,𝑟 =𝑎𝜅

2𝜃2𝑑2 𝑉

𝑑𝑡2− 𝒞

𝑑 𝑉

𝑑𝑡• Membrane depolarization itself is the trigger that

causes the sodium conductance to increase

Reconstructing the action potential

12.2.5 Voltage gating leads to nonlinear

cable equation with traveling waves

• Once one segment depolarizes, its

depolarization spreads passively to the

neighbouring segment

• Once the neighbouring segment depolarizes by

more than 10m V, the positive feedback

phenomenon described in the previous section

sets in, triggering a massive depolarization

• The process repeats, spreading the depolarized

region

12.2.5 The model

• Assume 𝑔𝑁𝑎+ 𝑣 = 𝑔𝑁𝑎+0 + Bv2 where 𝑔𝑁𝑎+

0 is

the resting conductance per unit area.

• Lumping 𝑔𝑁𝑎+0 into 𝑔𝑡𝑜𝑡

0 we get

𝑗𝑞,𝑟 = Σ𝑖 𝑉 − 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡 𝑔𝑖

0) + (𝑉 − 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡 𝐵𝑣2

= 𝑣𝑔𝑡𝑜𝑡0 + 𝑣 − 𝐻 𝐵𝑣2

where in the 2nd step we used Your Turn 12A

(homework) to group as 𝑣𝑔𝑡𝑜𝑡0

• Solution with 𝑗𝑞,𝑟 = 0 has 𝑣 = 0, 𝑣1, 𝑣2 where

𝑣1/2 =1

2(𝐻 ∓ 𝐻2 −

4𝑔𝑡𝑜𝑡0

𝐵). See Fig. 12.9b)

• v=0 is Ohmic part (region A in Fig. 12.8), above

threshold we get feedback & overshoot

𝑗𝑞,𝑟 = 𝑣𝑔𝑡𝑜𝑡0 + 𝑣 − 𝐻 𝐵𝑣2

Nonlinear Cable Equation (NLCE)

• Subbing in Eq. 12.21 in cable eq. using

𝑣1𝑣2 =𝑔𝑡𝑜𝑡0

𝐵then Eq. 12.17 becomes

• Again shifting variables: 𝑣 𝑥, 𝑡 ≡ 𝑣 𝑥, 𝑡 −𝑥

𝜃and

defining dimensionless quantities: 𝑣 ≡ v/𝑣2, y ≡− 𝜃𝑡/𝜆𝑎𝑥𝑜𝑛, 𝑠 ≡ 𝑣2/𝑣1 & 𝑄 ≡ 𝜏𝜃/𝜆𝑎𝑥𝑜𝑛 we get

which has solution:

Solution of simple nonlinear cable equation

12.3 THE FULL HODGKIN–

HUXLEY MECHANISM AND

ITS MOLECULAR

UNDERPINNINGS

EACH ION CONDUCTANCE

FOLLOWS A CHARACTERISTIC

TIME COURSE WHEN THE

MEMBRANE POTENTIAL CHANGES

12.3.1

Space-clamping

• The conductance determines the current, held at

a fixed, uniform potential drop.

• Highly non-uniform potential, localized pulses,

appear during operation of an axon.

• The metallic wire

was a much better

conductor than the

axoplasm, so its

presence forced

the entire interior to

be at a fixed,

uniform potential.

Voltage-clamping

• So that V is the more natural variable to fix

• In this arrangement the experimenter chooses a “command” value of

the membrane potential; feedback circuitry supplies whatever

current is needed to maintain V at that command value, and reports

the value of that current.

• Forcing a given current

across the membrane,

measuring the resulting

potential drop, and attempting

to recover relation (Section

12.2.5)

• For one membrane current

flux, there are multiple “V”s.

• Our hypothesis: the devices

regulating conductance are

themselves regulated by V ,

not by current flux

Separation of ion currents

• Even with space- and voltage-clamping,

electrical measurements yield only the total

current through a membrane, not the individual

currents of each ion species.

• Ion substitution (Section 12.2.3).

• 𝒱𝑖𝑁𝑒𝑟𝑛𝑠𝑡= the clamped value of V. This ion’s

contribution to the current equals zero,

regardless of the conductance 𝑔𝑖(V )

Results• Immediately after the imposed

depolarization, there is a very short

spike of outward current lasting a few

microseconds. This is the discharge

of the membrane’s

capacitance.(Section 12.1.2)

• An inward sodium current develops

in the first half millisecond. The

sodium conductance peak value can

be calculated and depends on the

selected command potential V.

• After peaking, the sodium

conductance drops to zero, even

though V is held constant.

• The potassium current rises slowly

(in a few milliseconds). Like 𝑔𝑁𝑎+ ,

the potassium conductance rises to a

value that depends on V . Unlike

𝑔𝑁𝑎+, 𝑔𝐾+ holds steady indefinitely at

this value.

• The voltage-gating hypothesis describes the

initial events following membrane depolarization

(points 1–2), which is why it gave a reasonably

adequate description of the leading edge of the

action potential.

• In the later stages, the simple gating hypothesis

breaks down (points 3–4 above), and indeed

here our solution deviated from reality (compare

the mathematical solutions in Figure 12.10 to the

experimental trace in Figure 12.6b).

More realistic gating functions

• After half a millisecond, the spontaneous drop in sodium

conductance begins to drive V back down to its resting

value.

• Indeed, the slow increase in potassium conductance

after the main pulse implies that the membrane potential

will temporarily overshoot its resting value, instead

arriving at a value closer to 𝑉𝑛𝑒𝑟𝑛𝑠𝑡 (see Equation 12.3

and Table 11.1 on page 416). This observation explains

afterhyperpolarization (Section 12.1.1).

• Once the membrane has repolarized, an equally slow

process resets the potassium conductance to its original,

lower value, and the membrane potential returns to its

resting value.

Postscript

• As far as the action potential is concerned, the function

of the cell’s interior machinery is to supply the required

nonequilibrium resting concentration of sodium and

potassium across the membrane.

• P. Baker, Hodgkin, and T. Shaw confirmed this by the

extreme measure of emptying the axon of all its

axoplasm, replacing it by a simple solution with

potassium but no sodium.

THE PATCH-CLAMP TECHNIQUE

ALLOWS THE STUDY OF SINGLE

ION CHANNEL BEHAVIOR

12.3.2

• Hodgkin and Huxley’s measured the behavior of

the membrane conductances under space- and

voltage-clamped conditions

• They suspected the existence of ion channels,

• But they could not see the molecular

mechanisms for ion transport because they were

observing the collective behavior of thousands

ion channels, not the behavior of any individual

channel.

Patch-clamp technique

• Developed by Neher and B. Sakmann in 1975

• One of the first results of patch-clamp recording was an

accurate value for the conductance of individual

channels: A typical value is G ≈ 25 × 10−12Ω−1 for the

open sodium channel.

• Driving potential of V −𝑉𝑁𝑎+𝑛𝑒𝑟𝑛𝑠𝑡≈ 100mV the current

through a single open channel is 2.5 pA.

Results for conductance from patch clamp

a) Mechanism of conduction

• The simplest model for ion channels:

• Each one is a barrel-shaped array of protein

subunits inserted in the axon’s bilayer

membrane (see below), creating a hole through

which ions can pass diffusively.

b) Specificity

• The channel concept suggests that the

independent conductance of the axon

membrane arise through the presence of two

(actually, several) subpopulations of channels,

each carrying only one type of ion and each with

its own voltage-gating behavior

• Sodium Channel

– A channel can accept smaller ions while

rejecting larger ones

– A channel can pass positive ions in

preference to neutral or negative objects

Potassium channel?

• How can a channel pass a large cation, rejecting

smaller ones?

• C. Armstrong and B. Hille in early 1970s. Idea is

that the channel could contain a constriction so

narrow that hydrated ions, have to “undress”

(lose some of their bound water molecules) in

order to pass through

– See next slide

Vestibule & Filtering

Alberts: Molecular Biology Of The Cell 5th Ed.

c) Voltage-gating

• A net positive

charge embedded

in a movable part of

the channel gets

pulled by an

external field. An

allosteric coupling

then converts this

motion into a major

conformational

change, which

opens a gate.

c) Voltage Gating (continued)

• The conformational change is discrete (Chap 9)

• For example, the traces in Figure 12.17b each

show a single channel jumping between a

closed state with zero current and an open state,

which always gives roughly the same current.

c) Voltage gate discreteness?

• Hodgkin & Huxley measured continuous values

in space clamp technique, but on small scales

the switches are discrete (patch clamp)

• Recalling Chap 6 and free energy transitions,

e.g., RNA folding as a two state process we

again find a similar transition closed -> open:

𝑃𝑜𝑝𝑒𝑛 =1

1 + 𝑒Δ𝐹/𝑘𝐵𝑇

• Can’t predict Δ𝐹 without detailed molecular

modelling, but can predict change in Δ𝐹 when

varying V.

• Upon switching, total charge 𝑞 moves distance 𝑙perpendicular to membrane with external field

ℰ ≈ 𝑉/𝑑 then this gives extra contribution to

Δ𝐹 = −𝑞ℰ𝑙 = −𝑞𝑉𝑙/𝑑. This implies Eq. (12.28):

were Δ𝐹0 is unknown internal part: 𝐴 ≡ 𝑒Δ𝐹/𝑘𝐵𝑇

continued

Again theory & experiment agree!!!

d) Kinetics

• Kinetic interpretation of the Boltzmann

distribution (Sec. 6.6.2)

• If the probabilities of occupation are initially not

equal to their equilibrium values, they will

approach those values exponentially (Eq. 6.30)

Ligand-gated ion channels

Ligand-gated ion channels

• The channels studied in Figure 12.19 are

sensitive to the presence of the molecule

acetylcholine, a neurotransmitter

• At the start of each trial, a sudden release of

acetylcholine opens a number of channels

simultaneously.

• Though each channel is either fully open or fully

shut, adding the conductance of many channels

thus gives a total membrane current that roughly

approximates a continuous exponential

relaxation

d) Kinetic continued

• The complex, open-then-shut dynamics of the

sodium channel arises from the all-or-nothing

opening and closing of individual sodium

channels

Observations through ingenious exp

• Observation of two-stage dynamics of the

sodium conductance under sustained

depolarization to two independent, successive

obstructions.

• One of these obstructions opens rapidly upon

depolarization, whereas the other closes slowly.

The second process, called inactivation,

involves a channel-inactivating segment,

loosely attached to the sodium channel by a

flexible tether.

12.4 NERVE, MUSCLE

SYNAPSE: EXTRA READING

Don’t FORGET Your Turn 12A for homework.