Electrical Signals I - 2012

Electrical Signals I Page 1

Electrical Signals I

Wonder of electrical signals Ion channels�magic behind electrical signals

Permeation of ion channels

9/05/2012 & 9/10/2012

Electrical Signals I Page 2

1. What are excitable membranes and why are they exciting?

One of the most ubiquitous forms of communication in biology utilizes the modality of changes in

membrane potential, or electrical signaling. Excitable membranes are those which possess the

intrinsic capability of generating such electrical signals. Their stereotypic electrical response is

called an action potential.

The long sought after and key biological signal turned out to be the entity transmembrane voltage,

defined as Vm = Vi - Vo. Ling electrodes make the measurement possible, as diagrammed below.

The following are three examples illustrating just how important the signal Vm turns out to be.

First, there is the “hot stove example” and the reflex arc comprised of neurons.

Heart muscle

Figure 1. Schematic of classic method for

measuring transmembrane voltage, a lingua

franca of life at the molecular and cellular level.

Figure 2. Flow of information from pain of hot stove to reflexive retraction of limb (left). Heat is converted to

nerve action potentials (right) in afferent neuron. This results in nerve action potentials in interneuron located

in spinal cord. This in turn leads to action potential conducted down motor neuron. This results in activation

of muscle, resulting in retraction of limb out of harm’s way. Absence of this sort of “unconscious reflex

protection” results in limb decimation in diseases like diabetes.

Electrical Signals I Page 3

Second, each beat of the heart is initiated by a much longer, but analogous action potential in heart

cells. Without such action potentials, the heart would be quiescent, and death would ensue.

Third, regulation of a critical biological fuel, glucose in the blood, depends critically upon bursts of

action potentials in beta cells of the pancreas. Remember this as the “late-afternoon chocolate bar

syndrome.”

Despite the fact that action potentials spanning vastly different durations were involved in the three

phenomena described above, the three scenarios all involve the common language of electrical

signals. Moreover, there was a deep-seated hope that the same fundamental mechanisms gave rise

to all of these action potentials. Thus, there might be something akin to a “unified field theory” of

physics, if we could understand the biological basis of electrical signal generation.

Figure 3. Action potential recorded from

beating mammalian heart cell. Note duration

spanning 100s of msec, versus 1-3 msec

duration of action potentials in neurons.

Figure 4. Bursts of action potentials, lasting seconds, mediate negative-feedback control of blood glucose

concentration. Left, transmembrane voltage records of pancreatic beta cell bathed in different concentrations of

blood glucose. Elevated glucose causes depolarization, even leading to bursts of action potentials. These

bursts trigger secretion of a hormone called insulin, which signals other cells to lower blood glucose, a crucial

brain food that must be kept within normal limits. Right, feedback information flow controlling blood glucose

levels. Hypoglycemia (too low blood glucose) causes you to feel drowsy in part because the crucial brain food

of glucose is too low. That accounts for the refreshing rush of a chocolate bar late in the afternoon. The

disease of diabetes fundamentally involves a disturbance of the feedback loop at right.

Electrical Signals I Page 4

2. Ionic channels—the magic behind electrical signals

At the heart of electrical signal generation are transmembrane-spanning proteins called ionic channels. In man-made electrical circuits, transistors and other active components are crucial to

generating interesting electrical tricks; getting a Radio Shack kit with only passive parts like resistors

and capacitors is pretty boring. In regard to biological electrical signals, ionic channels turn out to

be the crucial active, nonlinear components, which undergird the vastly different action potentials in

section 1. Thus, ionic channels are the “magic” underlying electrical signals.

This being the case, we will devote considerable attention to these molecules. Following this, we

will see how they combine in cellular membranes to produce the action potentials used in electrical

signaling.

A. Protein ultrastructure of a prototypical ionic channel

Channels are essentially a very specialized hole in the membrane bilayer. Shown below is a low-

resolution cartoon of the ultrastructure for a nicotinic acetylcholine receptor channel.

Figure 5. Cartoon ultrastructure of a nicotinic acetylcholine receptor channel. When open, these ionic channels

conduct Na+

ions through a central pore. Note the characteristic dimensions: the pore spans the 30-50 angstrom

thickness of the cell membrane. The channel is composed of five subunits, comprised of two alpha subunits,

and single beta, gamma, and delta subunits. The large outer portion of the channel at top (extending into the

extracellular media) is decorated with various sugar molecules (branch-like structures). The smaller

cytoplasmic portions of the channel at bottom are anchored by anchoring proteins (cigar-shaped structures).

Electrical Signals I Page 5

B. Primary sequence and presumed folding pattern of ionic channels

For most ionic channels, however, explicit determination of ultrastructure has not yet been

accomplished. However, it is believed that the overall idea of a fancy “hole in the membrane” will

hold true. In the meantime, we do know the primary sequence of amino acids that comprise just

about all ionic channels. From these we can infer the folding patttern (secondary structure) in the

membrane by hydropathy analysis (“greasy spoon index”), which can be benchmarked against

actual ultrastructure of solved ionic channel structures. The basic primary sequence and presumed

folding pattern of three major classes of voltage-gated channels (Na, K, and Ca channels), the

movers and shakers of action potential generation in the vast majority of cells, are as follows.

i. Na, K, Ca made of four homologous domains

ii. Big: about 2000 DD

iii. Each homologous domain with 6 putative membrane-spanning regions.

Figure 6. Inferred folding pattern (secondary structure) and ultrastructure (right) of voltage-gated ionic channels. Left, the folding pattern

starts with an intracellular amino terminus, and ends with an intracellular carboxy terminus. The folds occur according to four

homologous domains, as numbered by Roman numerals. Each domain seems to have 6 transmembrane spanning regions, as predicted

from hydropathy analysis. Right, inferred way in which homologous domains array themselves at the ultrastructural level to form ionic

channel in 3D. The domains are like the “staves of a barrel,” and the pore would be along the central axis. K channels differ in that each

polypeptide would comprise only one domain. Four domains would come together to comprise a K channel.

I II III IV

I

II

III

IV

Electrical Signals I Page 6

C. Two major performance characteristics of ionic channels

The first characteristic is permeation, defined as how ions get through channels when open. There

are two subcharacteristics:

Selectivity = how picky channels are about which kind of ion they let through. This gives rise to the

"names" of channels, such as Na, K, Ca, ... channels.

Conductivity = property specifying the amplitude of current that will flow through a channel, given a

specified transmembrane voltage gradient, and given a specified [X]i and [X]o, where

X is a permeant ion.

The second major characteristic is gating, defined as how a channel opens and closes. There are

several types of gates, organized according to the signal that controls gate opening or closing:

Voltage-sensitive (Na, K, Ca)

Chemical-sensitive (KATP, nACHR, KCa)

Stretch-sensitive

-sensitive

The selective and exquisite triggering of gates in different channels is crucial to custom action

potential generation.

Figure 7. Cartoon of voltage-gated ionic

channel, illustrating two major performance

characteristics of permeation and gating.

Permeation is a consequence of the narrow

selectivity filter. Gating is a function of the

intracellular gate, which is biased towards

opening or closing by electric fields acting on

the voltage sensor, the latter of which is

somehow coupled to the gate.

Electrical Signals I Page 7

3. Permeation in detail

A. Battery and resistor (BR) model

i. So long as a channel is absolutely selective for one species of ion, the voltage at which zero

current flows is absolutely specified by thermodynamics; it is not model dependent.

Thermodynamics yields the Nernst potential, the voltage difference across the membrane at which

no current will flow through the channel.

From previous lectures (last year) we have that the chemical potential energy is:

ui - uo is the change in energy of the system if one mole of ion X is moved from inside to outside the

membrane.

No current flows (equilibrium) when ui=uo. Hence

It is worth emphasizing how the Nernst potential accords with intuitive deduction of the

transmembrane voltage required to hold net current at zero for a perfectly selective channel, which

in the example below is Na+

.

V F z + P V + ])X([ RT + u = uV F z + P V + ])X([ RT + u = u

oXpooo

iXpioi

ln

ln

X speciesfor potential Nernst = ][X][X

Fz

RT = Vi

o

xx ¸̧

¹

·¨̈©

§¸̧¹

·¨̈©

§ln

Figure 8. Intuitive confirmation of Nernst potential equation, involving balance of diffusive and electrical

forces.

Electrical Signals I Page 8

ii. Away from zero current (away from equilibrium), thermodynamics doesn't tell us what the value

of current should be. It is model dependent. Thermodynamics could tell us the direction of current

flow, and it is not unreasonable for the current-voltage relationship to be linear over a small range.

This could be modeled as a battery (= Nernst potential) in series with a resistor. Selectivity is

reflected in that the reversal potential = Nernst potential; this follows from our assumption that the

channel is perfectly selective for a given ionic species. Conductive properties are embodied in the

value of the resistance.

The equation statement of the two diagrams in Fig. 9 is I = gx (Vm-Vx).

iii. The proof of such a model (away from equilibrium) comes in the form of empirical

verification in many channel types. This simplest of models turns out to be one of the most useful in

constructing models of the cell, and in understanding how action potentials come about.

Figure 9. Battery-resistor model emerges from linear approximation away from Nernst potential.

Electrical Signals I Page 9

iv. The three problems/limitations of the battery-resistor (BR) model.

The first problem is how to account for imperfect selectivity in BR model. The ad hoc modification

is to postulate 2 (or more) independent pathways for ions X and Y, each with its own Nernst

potential battery.

In this case, ichannel = ix + iy = gx (Vm-Vx) + gy (Vm-Vy).

Simplifying this yields, ichannel = (gx + gy) (Vm – [ Vx (gx / (gx + gy)) + Vy (gy / (gx + gy)) ])

This equation has nice intuitive appeal: the reversal potential (Vrev, the entity in the [ ] symbols) is

the weighted sum of the Nernst potentials for ions X and Y, weighted by their relative conductances

gx and gy. This is a useful approximation under one set of ionic conditions.

The second problem comes with the intuitive certainty that conductance is going to vary with

different concentrations of permeant ions. In other words, from intuition based on the bathtub analogy raised in class, it must be that gx = f(Xo, Xi) and gy = f(Yo, Yi). The BR model has no theory

to account for the forms of these functions. As a concrete example of the reality of these functions,

consider the figure below. The bathtub analogy will make it obvious that conductance, in the

presence of only ion X in equal concentrations on the inside and outside, should increase with

concentration of X.

Figure 10. Ad hoc BR model modification to

account for imperfect selectivity of channel to

ions X and Y.

Figure 11.

Consequence

1 of bathtub

analogy

Electrical Signals I Page 10

The third problem is a second related consequence of the bathtub analogy, which arises when the

concentration of a permeant ion X is unequal on the inside and outside. This results in the

phenomenon called rectification. Again, the BR model has no theory or means to account for this

realistic property of channels.

B. Electrodiffusion model (solved with the constant field assumption) = Goldman-Hodgkin-Katz Equation

One of the earliest attempts to address these problems was made by Hodgkin et al using the Nernst-

Planck electrodiffusion equation. We develop this in detail because so much of our "first-order

approximation" logic derives from this model.

In this model, the channel pore is viewed as an isotropic slab of material. The layout is as follows.

Figure 12. Consequence 2 of bathtub analogy logic: prediction of rectification.

Figure 13. Layout and

assumptions of GHK

current equation model.

> >

Electrical Signals I Page 11

This layout can be solved for the GHK current equation as follows. Start with the Nernst-Planck

electrodiffusion equation, which describes current flow through this slab of material (current/area).

ņņņņņņ ņņņņņņņņņņņņņ

n n

a b

Part a in the Nernst-Planck electrodiffusion equation has to do with simple diffusion, tantamount to

Fick's law or random walks. This part is fairly straight forward.

Part b is more unusual. It has to do with the establishment of a certain drift velocity determined by

the balance of friction and electrical force applied to a permeant ion in the channel.

Solving this equation with the boundary conditions and constant-field assumption in Figure 13 (see

Appendix 1), yields the following solution.

Substituting the “permeability coefficient for ion X,” defined as the entity Px = (Dx �E / d), yields the

final equation. This equation is the famous Goldman-Hodgkin-Katz current equation, or GHK

current equation (units of current per cross-sectional area; Dx has units of length2

-sec-1

). Multiplying

by the cross-sectional area of a single channel a gives the single-channel current.

x

x xx,i odX( ) F dZ = + + X( ) J Z FD d RT d

F \FF Fo

§ ·§ ·¨ ¸¨ ¸

© ¹© ¹

exp

1 exp

x mi o2

x xx,i o m

x m

F V-Z - X X( F RT)Z D = J V F RT d V-Z - RT

Eo

§ ·§ ·¨ ¸¨ ¸§ · © ¹¨ ¸¨ ¸ § ·¨ ¸© ¹

¨ ¸¨ ¸© ¹© ¹

exp

1 exp

x mi o2

xxx,i o m

x m

F V-Z - X X( F RT)Z = J VP F RT V-Z - RT

o

§ ·§ ·¨ ¸¨ ¸© ¹¨ ¸

§ ·¨ ¸¨ ¸¨ ¸© ¹© ¹

Electrical Signals I Page 12

It is worth learning how to sketch the predicted behavior of the GHK current equation, as illustrated

in the diagram below. Note the use of extreme-voltage asymptotes to produce the sketch.

With the current equation in hand, we can proceed to engage another famous perspective of this

model, as emodied in the GHK voltage equation. The heart of this latter equation comes from the

realization that we can now predict IT, even when the channel is not perfectly selective. From the

property of independence, we can write

IT = I1 + I2 + ..... + In

where each of the Ii terms on the right are GHK current equations for each of the different types of

permeant ions for an imperfectly selective channel.

To predict the reversal potential (Vrev), set IT = 0, yielding

IT = 0 = INa + IK + ICl

where Na, K, and Cl might be permeant ions for a generic channel in question. Plugging in explicit

GHK current equations for the Na, K, and Cl terms, and solving yields the GHK voltage equation

stated below.

»¼

º«¬

ªCl P + K P + Na PCl P + K P + Na P

FRT- = V

iCloKoNa

oCliKiNarev ln

Figure 14. Asymptote analysis of GHK current equation reveals ability to predict property of rectification.

Electrical Signals I Page 13

C. GHK-like models fit poorly with physical features of ion channel permeation

Problem with GHK model is that it is hard to make the conceptual link between permeabilities and

channel architecture. Moreover, channels are not isotropic slabs, they have discrete features. In

other words, the GHK model is not a very "physical" model. Here we present the Reuter-Stevens

model for two reasons. 1) To emphasize how one might begin to construct a physical model, and 2)

To emphasize that the description of current flow (outside of equilibrium conditions) is inherently

model dependent. There is nothing God-given about the GHK equations.

Some realistic features point to shortcomings in the GHK model. These arise from the first

crystal structure of a “real” ion channel, taken from a watershed paper from Rod

MacKinnon’s laboratory.

From previous mutagenesis studies of K channels, the part of the secondary structure that seemed

important for permeation was the “P-loop” shown as a dark black segment between S5 and S6

transmembrane segments in the figure below.

Figure 15. Permeation

segment of channel

secondary structure, as

suggested by mutagenesis

of K channels.

Electrical Signals I Page 14

In the crystal structure, shown below, we can appreciate the 3D nature of the permeation pathway,

lined in large part by portions of four P loops, each contributed by different subunits.

Three realistic features of ion channel permeation emerge from the crystal structure, and these are all

at odds with the isotropic-slab depiction of the GHK model.

(1) Single-filing of K ions through a narrow pore.

(2) Discrete residence sites (binding pockets) in the channel pore; possibility of saturation, at odds

with independence assumption of GHK model, and at odds with constant diffusion coefficient

through the thickness of the channel incorporated in the GHK model.

(3) Close quarters imply ion-ion electrostatic interaction within the pore; clearly not in agreement

with independence assumption of GHK model.

These features are more clearly depicted by the cartoon abstraction of the crystal structure below.

Figure 16. Crystal structure of Kcsa potassium channel, from bacteria. Upper surface is the “outside” of the

channel, inner surface is the “cytoplasmic” aspect of the channel. Left, ribbon diagram of two opposing

subunits, showing GYG segment of P loops in red. Right, stereo view of cutaway of channel, showing high

density positions of potassium ions (shown in green) in the channel pore.

Figure 17A Cartoon abstraction of permeation

mechanism, inferred from actual crystal

structure of Kcsa channel.

Electrical Signals I Page 15

These physical features (single filing) fit nicely with the functional property of an abrupt size cutoff

of permeant ions in Na channels. This would not be anticipated by the isotropic-slab view of the

GHK mechanism.

Figure 17B Na channel permeability as a

function of various permeant cations with

different diameters. Illustrates abrupt size

cutoff of permeability, consistent with

permeation through a discrete, narrow hole or

pore.

Electrical Signals I Page 16

D. Free-energy barrier (Reuter-Stevens) model, simplest quantitative mechanism with a taste of physical reality

Constructing a fully realistic quantitative model of channel permeation, with complete fidelity to all

known physical features, is an enormous undertaking beyond the capabilities of the fastest

computers (see B Roux (2002) Theoretical and computational models of ion channels. Curr Opin Struct Biol 12:182-189).

However, we can formulate a simple mechanism that gives a “taste” of physical models, called the

barrier, or Reuter-Stevens model. This model serves to introduce the approaches to constructing

more realistic models, and to introduce a very powerful approximation used to describe the discrete

ion or protein movement so prevalent in biology. The latter approximation is called the Eyring rate theory, which we will develop later below.

We begin formulation of the barrier model, by considering what happens when permeant ions (K+

in

the case of K channels) move single-file among discrete positions in the permeation pathway. On

theoretical grounds, we expect there to be some oxygens�from amino acid side chains (glutamate,

aspartate, serine) or even backbone carbonyl groups�to “take the place of” water, the later of which

energetically stabilizes permeant ions in free solution. The free energy analog of this would be

different height free-energy barriers that different ionic species must traverse.

In the barrier model, we quantify the motion of ions across such an energy barrier using an

extremely powerful approximation called Eyring rate theory. According to this theory, the

probability that ion B (in Fig. 18) will jump over the barrier to become ion D, within a small

increment of time 't, is simply given by the expression

k � 't � P(B)

where k is a rate constant pertaining to this transition, and P(B) is the probability that an ion resides

at position B. Not only does the theory posit that a rate constant k exists, the theory also gives the

value of k in terms of the free-energy barrier profile (Fig. 18, right). It turns out that the exact

features of the barrier need not be known, only the activation energy 'Gact, the free energy required

Figure 18 Free-energy

depiction of permeant ion

jumping from one “stable”

position to another.

'Gact

outside inside

Electrical Signals I Page 17

to boost a mole of ions from position B to position C. In terms of this parameter

k = ( kB � T / h ) � exp [-'Gact / (R � T)]

where kB is Boltzmann’s constant, T is the absolute temperature in Kelvin, h is Plank’s constant, R is

the gas constant, and ( kB � T / h ) has units of reciprocal time (like sec-1

).

Hence, and intuitive rationalization of the Eyring functional form is this: First, (kB � T / h) gives the

number of barrier crossing attempts per second that one ion in position B would make to scale

the entry barrier. Second, the fraction of these attempts that result in successful ion leaps over

the barrier peak is given by the exponential factor exp(-'Gact/RT).

For simplicity, the Reuter-Stevens barrier theory assumes that there are no binding sites within the

pore of the channel, only a single barrier that a permeant ion must jump to get clear through the

channel pore. In this sense, the Reuter-Stevens barrier model is unrealistic in that it becomes an

independence model, where doubling the concentration of permeant ion simply doubles the rate of

ion transport through a channel. We don’t have to make this assumption, as Eyring-rate-theory-

based models can easily incorporate binding sites within the permeation pathway, but this is best left

to exercises outside of the lecture.

Even so, the Reuter-Stevens barrier theory is every bit as good as the complicated GHK model in

predicting channel behavior, except that the underlying assumptions are more “physical” in the

barrier theory. Here’s how we get an explicit current equation from the barrier assumption. The

assumption of no binding sites means that the entire model need only consider two rate constants: koi

for B to D transtions, and kio for D to B transitions (see Fig. 18 for orientation). The explicit forms

for these rate constants, including the influence of electrical potential energy on the energy barrier

can be deduced as shown below, taking into account the addition of chemical and electrical potential

energy diagrammed below.

Based on this layout, we have:

This yields the RS current equation, given by [ ( ) ( )]x X io oi= z F k P D - k P BI (current per mole

Figure 19 Summing chemical

and electrical potential free

energy to obtain the free-

energy profile of ion transit in

an ion channel pore. G is a

fractional electrical distance

ranging between 0 and 1.

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

RTV ) - F(1Z +

RTB-

hTk =

RT))-(1V FZ - B(-

hTk = k

RTV FZ-

RTB-

hTk =

RT)V F Z + B(-

hTk = k

mxxBmxxBio

mxxBmxxBoi

GG

GG

expexpexp

expexpexp

Electrical Signals I Page 18

of ion channels), which yields the current flowing through a mole of channels, at steady state.

Channels reach their steady-state throughput in microseconds, so we only consider steady-state

responses for permeation equations.

Substituting the full expressions for the rate constants then yields

The remaining challenge, before this becomes a useful equation, is how to express P(D) and P(B) in

terms of bulk concentrations of ions X (Xi and Xo).

The simplest theory is P(B) = Navogadro � CV � Xo, and P(D) = Navogadro � CV � Xi. Here, CV is the

so-called “capture volume,” a volume (litre) near the mouth of the channel within which

permeant ions are considered to be within a molecular vibration of the entry barrier to the

channel pore. Navogadro � CV is thus the expected number of permeant ions that would be within

the capture volume region, given an bulk permeant ion concentration X of 1 molar (= 1

mole/litre). X � Navogadro � CV thus gives the expected number of ions in the capture volume, and

for smallish X, where the expectation value is close to P(B) or P(D).

Now we can write the RS current equation in its “fully useful form,” with bulk concentrations

for X, as follows Analog of permeability PX

where this current flow through a mole of ion channels. It is worth sketching the predicted

behavior of the RS current equation, as illustrated in the diagram below. Note the use of extreme-

voltage asymptotes to produce the sketch.

¸¹

ᬩ

§¸¹·

¨©§¸

¹

ᬩ

§¸¹·

¨©§¸

¹

ᬩ

§¸¹·

¨©§

)(exp)(expexp BP- RT

V Fz + DP RT

V F z - RTB-

hTk F z = I mXmXxB

xxG

¸¹

ᬩ

§¸¹·

¨©§¸

¹

ᬩ

§¸¹·

¨©§¸

¹

ᬩ

§¸¹·

¨©§��� o

mXi

mXxBVavogadroxx X-

RTV Fz + X

RTV F z -

RTB-

hTkCN F z = I expexpexp

G

Figure 20. Asymptote analysis of RS current equation reveals ability to predict property of rectification.

Electrical Signals I Page 19

Further things to check out with RS model.

1. The Vrev for one ionic species predicted by the Reuter-Stevens current equation is the same as

for the GHK current equation. It has to be since this situation corresponds to thermodynamic

equilibrium. In fact, both equations simplify to the Nernst Potential equation at Vrev. Away

from Vrev, that is away from equilibrium, the Reuter-Stevens and GHK equation differ. This is

as it should be since thermodynamics no longer constrains behavior, and the current flow is then

model dependent.

2. Do both Reuter-Stevens and GHK current equation account for rectification and [Xi] and [Xo]

dependence of limiting conductances?

3. Does the Reuter-Stevens model handle imperfect selectivity? Say cations X and Y are both

permeable. Is there a Reuter-Stevens model prediction for current in this case?

Flaw with all mechanisms thus far in lecture is that they all leave out ion-ion interaction in the

pore, and that there is no ion binding within the pore. Both of these possibilities would lead to

non-independence behavior.

4. Summary

x Electrical signals are a major language of life at the molecular/cellular level.

x Ion channels are the magic behind electrical signaling, knowing their permeation and gating

properties will afford great insight into action potential genesis and signaling.

x Covered overall architecture of ion channels: how K versus Na/Ca channel differ, homologous

domains, transmembrane spanning regions

x Nernst potential: what it means, where it comes from (equilibrium thermodynamics)

x Battery-resistor permeation model: underlying assumptions, how to use it

x The “classic model.” GHK permeation model: underlying assumptions, intuitivebasics of NP

electrodiffusion equation, how to use GHK current equation, how to use GHK voltage

equation. Don’t have know how to derive it. x What is Kcsa?

x The simplest “physical model” is RS barrier model. Know how it comes from Eyring rate

theory, underlying assumptions, how to use RS current equation.

x At the core of the RS barrier model�Eyring rate theory: know this fundamental theory

Next lecture we will put the channels back together and see how they determine membrane

potential. We will also begin to consider gating properties in detail.

Electrical Signals I Page 20

Appendix 1: In-depth derivation of GHK current equation

Part a of the Nernst-Planck electrodiffusion equation comes straight from diffusion theory and will

not need to be derived. Part b comes from the following considerations. Note, in this derivation, l =

d from Figure 13.

J

Electrical Signals I Page 21

J

J

J

J

J

Electrical Signals I Page 22

Appendix 2: Intuitive interrelation between GHK and BR models, classic Eric Young

J

J

J

J