Bioelectric Phenomena

BIOELECTRIC PHENOMENA2014

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3:BIOELECTRIC POTENTIALS

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INTRODUCTION

› In this course we will be examining the behavior of excitable cells, notablynerve and muscle , both descriptively and quantitatively.

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notably nerve and muscle

Potentials and Currents

Where?

1. cells’ interior volume2. across their membranes3. their surrounding conducting volume

• Transmission of information in nerves

• The initiation of contraction in muscles

• Vision

• Hearing

• The heartbeat

• Digestion

• …

INTRODUCTION

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Why?

Their underlying electrical systems share many basic

principles of organization, and how fundamentally

similar they remain in almost all living creatures.

› Ions within the electrolytes

› Both inside and outside of cells

– Sodium

– Potassium

› Charge movement

› Complex

CURRENTS IN SOLUTIONS

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Not a comprehensive study

› The study of bioelectricity ???

› One mole is simply an amount of pure substance in grams that isnumerically equal to its atomic weight.

– A mole includes a fixed number of molecules, namely, Avogadro’s number, whichis 6.0225×1023 molecules.

› Flows: moles per second

› Fluxes: moles per second per unit area (j)

› When ionic movements are described from the perspective of thestudy of electricity, the quantities of ions are expressed in terms ofCoulombs (the forces between charges).

– the magnitude of the charge on one electron: 1.6 × 10−19 Coulombs.

› Flows: Coulombs per second = Amperes.

› Flux (flow per unit area): Amperes per unit area (J)

MOLES AND AMPERES

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Moles per second ↔ Amperes

The conversion factor is called Faraday’s constant, F.

MOLES AND AMPERES

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NERNST–PLANCK EQUATION

The concentrations between the intracellular and extracellularvolumes are especially important to excitable cells.

The Nernst–Planck equation relates the flow of ions to spatialdifferences in concentration or in the electric potential.

It is helpful to think first about these effects separately.

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Concentration The electric potential

NERNST–PLANCK EQUATIONDiffusion and the Diffusion Coefficient

If a drop of blue ink is placed in a beaker of water, then inkmolecules will, on average, move away from the highlyconcentrated region into the surrounding water. The process willcontinue until the ink is uniformly distributed in the water (whichwill become a uniform light blue color).

The process is known as diffusion.

We say that diffusion is in a direction of decreasing concentration.

No electric field is required for diffusion to occur as diffusion isnot an electrical phenomenon.

Rather, diffusion arises as a consequence of the pronouncedrandom motion of molecules that occurs at ordinary temperatures.This random motion scatters the blue ink. At first, more moveaway from the center of the ink drop than toward the center,simply because more ink molecules are near the center.

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NERNST–PLANCK EQUATIONDiffusion and the Diffusion Coefficient

Fick’s law: A quantitative description of diffusion.

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The concentration of some

substance as a function of

position.

Fick’s constant

Diffusion constant

Not quite independent of

concentration but increases

slightly with increases in C.

Fick’s coefficient D is determined from experiment.

????

The number of particles

(ions) moving per unit time

through a cross-section of

unit area.

NERNST–PLANCK EQUATIONElectric Field and Mobility

Because of their charge, ions are subject to electric field forces.

The force exerted by a given field will move ions with a finitevelocity.

This velocity is denoted by up , termed the mobility, which is thevelocity achieved under a unit field for the pth type ion.

If the valence of this ion is Zp , then the ionic flux is given by theproduct of ion concentration and its velocity, namely,

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The electric field

The sign of the force

The mean ion velocity


The ionic mobility (up) depends on the viscosity of the solvent, thesize of the pth ion, and its charge.

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The ion flux per unit area

The units of flux depend on the units of ion concentration. Often flux

is expressed as moles per unit area per second.


The mobility relates the force due to the electric field (−∇Φ) to theionic flux it produces.

With a similar mathematical form, Fick’s constant relates spatialchanges in concentration (∇C), sometimes thought of as the“force” due to diffusion, to the movement of ions down theconcentration gradient.

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Since both flows are impeded by the same molecular processes(collisions with solvent molecules), a physical connection existsbetween parameters up and D. The mathematical description of theconnection was worked out by Einstein, and the resulting equationthus bears his name. Einstein’s equation is

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The absolute temperatureThe gas constant

NERNST–PLANCK EQUATIONFLUX DUE TO DIFFUSION PLUS ELECTRIC FIELD

The total flux when both diffusional and electric field forces arepresent is

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Nernst–Planck Equation

The flux of the pth ion under the influence of diffusion and an electric field.


This flux can be converted into an electric current density whenmultiplied by FZp, the number of charges (Coulombs) carried byeach mole; Jp is the resulting electric current density (in Coulombsper second per cm2 or Amperes per cm2).

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A capital J designates the electric current density while a lowercase jdescribes the flux.

Einstein’s equation


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Transmembrane Potential

Excitable cells are surrounded by a plasma membrane, whosemain function is to control the passage of ions and molecules intoand out of the cell.

If the electrical potential at the inside surface of the membrane ofan excitable cell is compared to the potential at the outsidesurface, then, at rest, a potential difference, called thetransmembrane potential , on the order of 0.1 volt will be found.

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Because the membrane has a resistance (i.e., is not a perfectinsulator), there will be a transmembrane current, Im. Bydefinition, this current is considered to have a positive sign whenit flows across the membrane in the direction from the inside tothe outside.

In this session we will consider explanations for the origin of theresting potential, and explanations of its magnitude and sign. Wewill see that the potential at rest depends on the selectivepermeability of the membrane to the several major ions that arepresent and to the different ionic composition of intracellular andextracellular space.

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One aspect of the membrane that is critical to its electricalfunction is its electrical capacitance.

If we determine the capacitance of such a parallel plane structurewith a high dielectric constant (estimated as k =3, using thevalue for oil), then, using the parallel plane capacitance formula,we get

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High or Low???


The unequal concentration of ions in the intracellular versusextracellular spaces causes diffusion of ions from high to lowconcentrations; the rate of diffusion depends on the difference inconcentration and the membrane permeability.

1. Open channel density

2. Channel resistance

Charged ions accumulate on the membrane because of its capacitance.These charged ions set up an electric field across and internal to themembrane. The electric field will, in turn, exert forces on all chargedparticles lying within the membrane. Consequently any quantitativedescription of ion flow within or across the membrane must take intoaccount both forces of diffusion and electric field.

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At a particular moment, the movement of ions of type p across amembrane will depend on:

1. The relative density of p channels

2. The relative probability of such a channel being open

3. The conductance of the channel

4. The net driving force (diffusional plus electric field) for thision species.

In the following we treat the permeability of the membrane as,in its effect, uniformly distributed, that is, we determine amacroscopic description arising from the averagedmicroscopic channel behavior.

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Concentration Cell.

At equilibrium the electric field force just balances the diffusionforce .

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We assume that quantities vary in the direction perpendicular to the

membrane only.

xxxx


In T = 10ºC

At equilibrium the ion’s rate of movement in one direction, dueto diffusion, is equal to its rate of movement in the oppositedirection, due to the electric field associated with thetransmembrane voltage.

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The Nernst potentials for the major ion species are importantquantities and are usually assigned special symbols, where Ep isdefined to be VP(eq).

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Quiz

Suppose frog muscle had the concentrations of Table 3.1, andsuppose its membrane was permeable to potassium ions butto no others. At what transmembrane potential would there beequilibrium?

If the same membrane were permeable only to sodium ions,at what transmembrane potential would there be equilibrium?

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Answers:

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PARALLEL-CONDUCTANCE MODEL

› The parallel-conductance model embodies the bold mental step of asserting that different ions pass through the membrane independently one of another.

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› Resting Potential: The excitable membrane resting potential is one that brings the membrane into a steady state. It is that value at which the total membrane current is zero (otherwise, we have a changing potential and the membrane is not at rest).

› The model is intended to represent the flow of ions through theirrespective ionic channels in a small area of membrane, oftencalled a membrane element, or a membrane patch.

› Ionic Currents: As shown, the model assumes that thesignificant ions are potassium, sodium, and chloride. Eachbranch determines the contribution to the total transmembranecurrent from a specific ion species, and each branch is thought ofas a macroscopic description of the respective open ionchannels.

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› Assumption: The underlying assumption of the parallel-conductance model is that one can identify segments ofmembrane that are small enough that the transmembrane voltageis the same all across the patch, while at the same time largeenough that the patch encompasses numerous ionic channels, sothat average channel behavior is all that has to be represented.This underlying assumption proves to be true in most situations.

› If the membrane potential is , then the net driving force forpotassium is ( − ),which evaluates the deviation fromequilibrium. The potassium current is, consequently,proportional to − ; the proportionality coefficient hasthe dimensions of a conductance, and is called the potassiumconductance. Conductance is not a constant but variesdepending on how many channels are open at a particularmoment.

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= ( − )

› If > , then the outward diffusional forces are notcompletely equilibrated by the electric field, and a net outwardpotassium flux, hence current, results. Note that isappropriately positive.

› For the sodium ion = ( − )

› Here if > , then because , is positive must bepositive and even larger than . The result is an outwardsodium flux driven by an electric field which exceeds the inwarddiffusional forces.

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› For the chloride ion

= ( − )

› If > , the inward chloride diffusion is not completelyequilibrated and a net influx occurs. Since this influx is of ionswith a negative charge, it constitutes an outward electric current.

› Parallel-conductance models as used in particular contexts arenormally modified in significant ways from the form above. Forexample, Hodgkin and Huxley used a parallel-conductancemodel that had sodium and potassium pathways, and a thirdpathway identified as leakage. They inferred the properties ofthe leakage channel from measurements of the transmembranepotential, rather than by modeling a specific ionic flow.

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› Capacitive Current: To complete the list of contributions to thetransmembrane current we add the capacitive (or displacement)current, which is simply

=

› At rest (that is, at steady state), = 0since

= 0(because

if

was not zero, would be changing and thus not at steady

state). Capacitive current is much more important than onemight at first suspect, because the membrane is very thin andthus highly capacitive.

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› Resting from Steady-State Constraints: As noted, a restingmembrane requires steady-state conditions, namely, zero nettransmembrane current. The membrane current is

= + + +

› At steady state, = 0since

= 0. Thus, at steady state,

= 0 = 0 + + +

› Substituting the expressions for each ionic current gives

( − ) + ( − ) + ( − ) = 0

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› Example for Squid Axon on page 65

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CONTRIBUTIONS FROM CHLORIDE

› Chloride–Potassium Equilibrium

The role of the chloride ion in determining the resting potential appears tobe secondary to that of potassium. This comes about because theintracellular chloride concentration is very small and undergoes a largepercentage change with small amounts of chloride influx or efflux (whichis not true for potassium). Consequently, chloride ion movements can beexpected to occur that accommodate it to the potassium ion ratio to bringboth ion ratios into consonance (i.e., the same Nernst potential), namely,when

! =

" ! "

Since the chloride ion ratio tracks the potassium, one need only follow thelatter to evaluate the resting membrane potential (as a roughapproximation). Changes in resting potential may similarly be thoughtdue solely to the potassium ion ratio, namely,

≈ 25ln(

"

)

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4: CHANNELS

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Channel Structure: Molecular Genetics

› Biological cells are enclosed by a plasma membrane. Thismembrane consists of a lipid bilayer and the hydrophilic polarheads are oriented facing the intracellular and extracellularwater-containing media, while the hydrophobic tails, on theother hand, are internal to the membrane.

› Channel structure by electron microscopy

› Channel structure: molecular genetics

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What is patch-clamp?

Patch-clamping is an electrophysiological technique in which we are able

to CLAMP the VOLTAGE of an isolated piece of cell membrane (or

whole-cell). By clamping the voltage we are able to observe CURRENTS

that flow through ION CHANNELS. It is the current that the patch-clamp

amplifier supplies to hold the voltage steady (clamped) that we measure.

Patch-clamp recording allows us to measure very small currents - in the

pA range (10-12 A).

The patch-clamp circuit

Patch of cell membrane with ion channel

FBR

_

+Amplifier

Technical

The high gain operational amplifier

is connected in the circuit so that the

Current flowing through the ion

channel is measured as a voltage

drop across the feedback resistor

(FBR). The FBR has a resistance of

50 GΩ allowing very small currents

(10-12 A) to be measured.

Patch clampers do it with suction

By increasing the seal resistance we reduce the noise level and increase

temporal resolution.

Patch Clamp

Applications

Cell-attached recording

Cell

Patch-pipette

Cell membrane

External lipid

Internal lipid

Whole-cell recording

Cell

Patch-pipette

Cell membrane

Inside-out recording

Patch-pipette

The internal face of the lipid

bi-layer faces the bath solution

Outside-out recording

Patch-pipette

The external face of the lipid

bi-layer faces the bath solution

Voltage Clamp

Single-Channel Currents

› Examination of patch-clamp current reveals discontinuities thatdirectly reflect the opening and closing of channel gates.

› The current waveform is interpreted as reflecting the openingand closing of a single channel.

› The single-channel record is seen to switch to and from an openor closed state. The time in each state varies randomly, but if theratio of open to closed time is evaluated over a sufficiently longperiod, then this ratio (with some statistical variation) will be thesame over any successive such interval. Such a determinationgives the expected value or probability that the channel will beopen (closed), a value that is independent of time under thesesteady-state conditions.

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› Electrical circuit representation for a single (potassium) channel

The Giant Axon of Squid

Active vs. Passive Membrane

Na K L

In an Active Membrane, some Conductances vary with

respect to time and the membrane potential

Inside

Outside

Currents in an Active Membrane

CLkNaext IIIII +++=

)(

1

V-V :Law sOhm' Na

NaNaNa

Na

Na

NaNa

VVgI

gR

RI

−=

⇒

=

=

)(

)(

dt

dVCI

VVgI

VVgI

C

LLL

kkk

=

−=

−=

Na K L

Inside

OutsideI ext

dt

dVCVVgVVgVVgI LLKKNaNaext +−+−+−= )()()(

extLLKKNaNa IVVgVVgVVgdt

dVC +−+−+−= )()()(

The Potassium Channel (1)

› The potassium has 4 similar sub units

Each subunit can be either “open” or “closed”

(Protein 3D

Configurations)

The channel is open if and only if all 4 subunits are open

The Potassium Channel (2)› The probability of a subunit being open:

› The probability of the channel being open:

› The conductance of a patch of membrane to K+ when all channels are open: (Constant obtained by experiments)

› The conductance of a patch of membrane to K+ when the probability of a subunit being open is n:

n4n

Kg

4 ngg Kk =


dVC +−+−+−= )()()(

extLLKKNaNa IVVgVVngVVgdt

dVC +−+−+−= )()( )( 4

The Sodium Channel (1)

› The potassium has 3 similar fast subunits and a single slowsubunit

Each subunit can be either “open” or “closed”

(Protein 3D

Configurations)

The channel is open if and only if all 4 subunits are open

The Sodium Channel (2)› The probability of a fast subunit being open:

› The probability of a slow subunit being open:

› The probability of the channel being open:

› The conductance of a patch of membrane to Na+ when all channels are open: (Constant obtained by experiments)

› The conductance of a patch of membrane to Na+ :

m

hm3

Nag

hmgg NaNa

3 =


dVC +−+−+−= )()()(

extLLKKNaNa IVVgVVngVVhmgdt

dVC +−+−+−= )()( )( 43

h

The Kinetics of Sodium Channel Subunits

The Kinetics of Sodium Channel Subunits

Comparison of Voltage Dependence of channel kinetics

The mechanism of Action Potential› When membrane Potentialincreases

The fast sodium subunits The fast sodium subunits The fast sodium subunits The fast sodium subunits open rapidlyopen rapidlyopen rapidlyopen rapidly

The slow sodium channel The slow sodium channel The slow sodium channel The slow sodium channel subunit begins to close subunit begins to close subunit begins to close subunit begins to close slowlyslowlyslowlyslowly

The potassium channel The potassium channel The potassium channel The potassium channel

begins to open slowlybegins to open slowlybegins to open slowlybegins to open slowly

The sodium channels The sodium channels The sodium channels The sodium channels openopenopenopen

Influx of sodium IonsInflux of sodium IonsInflux of sodium IonsInflux of sodium Ions

Further increase of the Further increase of the Further increase of the Further increase of the membrane potentialmembrane potentialmembrane potentialmembrane potential

The sodium channels The sodium channels The sodium channels The sodium channels closeclosecloseclose

Influx of sodium Ions Influx of sodium Ions Influx of sodium Ions Influx of sodium Ions TerminateTerminateTerminateTerminate

Efflux of Potassium ionsEfflux of Potassium ionsEfflux of Potassium ionsEfflux of Potassium ions

membrane membrane membrane membrane Potential Potential Potential Potential Decreases Decreases Decreases Decreases

The mechanism of Action Potential

V

t

Action Potential

(spike)

After-Hyperpolarization

The role of each subunit

› Opening of the fast sodium subunits (m) cause the initial increase of the

voltage

V

t

Closure of the slow sodium subunit (h) causes the decline of the voltage to the Closure of the slow sodium subunit (h) causes the decline of the voltage to the Closure of the slow sodium subunit (h) causes the decline of the voltage to the Closure of the slow sodium subunit (h) causes the decline of the voltage to the resting pointresting pointresting pointresting pointOpening of the potassium subunits (n) cause the hyperpolarizationOpening of the potassium subunits (n) cause the hyperpolarizationOpening of the potassium subunits (n) cause the hyperpolarizationOpening of the potassium subunits (n) cause the hyperpolarization

ClosureClosureClosureClosure of the potassium subunits (n) cause the set back to restof the potassium subunits (n) cause the set back to restof the potassium subunits (n) cause the set back to restof the potassium subunits (n) cause the set back to rest

Question

• Action Potential Generation has a voltage threshold: If themembrane voltage increases more than the threshold, an actionpotential will be initiated; otherwise the membrane voltage setsback quickly to the resting potential.

• Now a question should be ask:

Is it always true?

Anode Break

Figure describes what happens after the terminationof a 2-msec hyperpolarization and the suddenrestoration of normal transmembrane potential. Justprior to release of the hyperpolarization, the valueof h is elevated while m and n are reduced.However, m rapidly regains its normal valuefollowing restoration of normal Vm, since τm isrelatively short. The result, based on τm << τn, τh,is that there is a depressed n, normal m, andelevated h. All three combine to promote INa > IK.The consequence can be the initiation of excitation.It is here that excitation can be initiated by theprocess just described [after a prolongedstimulation (hyperpolarization)]; this accounts forthe name given it: anode break excitation.

MACROSCOPIC CHANNEL KINETICS

If the rate constant for switching from a closed to an open state is α while

that for switching from an open to a closed state is β, then the average

behavior is described by

we expect α and β to depend on the transmembrane potential (only)

and therefore to be constant when the potential is fixed

MACROSCOPIC CHANNEL KINETICS

CHANNEL STATISTICS

Suppose the total number of channels is N and under steady-state

conditions an average number No are open and an average numberNc are closed. Under these circumstances the probability, p, of a

single channel being open is

The probability, q, of a channel being closed is given by the ratio

CHANNEL STATISTICS

As an illustration, if the aforementioned channels were all

potassium channels then the individual open-channel current is

For N channels with p being the probability of a channel being

open, the macroscopic current IK is given by

The macroscopic membrane conductance GK is given by

THE HODGKIN–HUXLEY MEMBRANE MODEL

For the potassium channel, Hodgkin and Huxley assumed that it

would be open only if four independent subunits of the channels

(which they called “particles”) had moved from a closed to an open

position. Letting n be the probability that such a particle is in the

“open” position, then

is the probability pK of the potassium channel being open.

The movement of the particle from closed to open was assumed to

be described by a first order process with rate constant αn, while the

rate constant for going from open to closed is βn. Consequently,


The maximum conductance of N potassium channels occurs when

they are all open and is


The maximum conductance of N potassium channels occurs when

they are all open and is

5ACTION POTENTIALS

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EXPERIMENTAL ACTION POTENTIALS

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EXPERIMENTAL ACTION POTENTIALS

If the amplitude of the stimulus pulse is relatively small, then noresponse is detected in the recording circuit.

As the stimulus strength is increased, a point is reached at which aresponse (a nerve action potential) is suddenly seen.

As the stimulus is further increased in strength, the response remainsunchanged. For this reason the action potential is described as all ornone, i.e., the action potential waveform remains unchanged at thehigher level of transthreshold stimulus amplitude.

Action potentials are nonlinear. If the stimulus current is reduced byhalf, then no action potential occurs. Conversely, if the stimuluscurrent is doubled, the action potential deflection remains largelyunchanged, but the latency is markedly reduced.

Noteworthy AP Attributes

Action potentials have a number of noteworthy characteristics. These unusual characteristics include thresholding, differential response by diameter, and latency, as described in the following.

Threshold responseThreshold responseThreshold responseThreshold response

Fiber diameterFiber diameterFiber diameterFiber diameter

Threshold due to an external stimulating source is lower in a large fiber, roughly inversely proportional to the square root of the fiber diameter.

LatencyLatencyLatencyLatency

The latency until the appearance of an action potential for either fiber as a function of separation between stimulating and recording electrodes, if examined, would show a linear dependency, confirming the assertion of uniformity of propagation. The velocity of propagation within each fiber is proportional to the square root of its diameter.

Subthreshold Responses

If one examines the responsesto the subthreshold pulses, it isseen that they are essentiallythose expected from a (passive)RC network. In fact, theresponses in figure can besimulated from fixed, lumped,RC elements. For thehyperpolarizing condition thiscorrespondence is exactlycorrect, but for depolarizingstimulation a deviation fromstrictly passive behavior beginsto arise beyond 50–80% ofthreshold.

Subthreshold Responses

Note the presence (or absence)of a mirror image in figure (as inparts a and b). Fordepolarization to potentialslying between 50 and 100% ofthreshold, the response is not amirror image of thehyperpolarizing response forstimuli of the same magnitudeand duration. Under theseconditions the depolarizationresponse is not fully passive.

Linear and nonlinear responses to stimulusThe voltage measured at 0.29msec following the stimulus isplotted. The voltage isexpressed as a fraction of thepeak action potentialamplitude and is shown as afunction of stimulusamplitude.

One notes that the relationshipbetween the stimulus and theresulting transmembranevoltage is linear for allhyperpolarizing stimuli. Thisobservation supports a linearpassive model.

For depolarizing stimuli of greater magnitude the behavior becomesnonlinear, and an active system description is required.

Voltage Clamp Currents

The initial flowof ionic current arising from a transthreshold voltage step is due to the sodium ion influx.

As we have seen, the activation process is characterized by a rapid increase in sodium permeability.

????

Illustrative example

Voltage-clamp measurements

Strategies for Na/K Ion Separation

› A separation of the sodium and potassium currents is necessaryin order to model the behavior of each ionic component alone.

› Hodgkin and Huxley performed a series of voltage-clampexperiments for increasingly depolarizing values. For each valueof clamp voltage two experiments were performed.

› The first was for normal composition seawater and the secondwith a low-sodium seawater (replace 90% sodium chloride bycholine chloride while potassium and remaining chloride ionsare unchanged).

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Key assumptions

› The Hodgkin and Huxley ion separation procedure is based onthree assumptions:

1. Early current is sodium current alone.

2. Outside Na affects .

3. Ionic Independence.

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Strategies for Na/K Ion Separation

6

IMPULSE PROPAGATION

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7ELECTRICAL STIMULATION OF EXCITABLE TISSUE

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INTRODUCTION

In designing systems for stimulation, a qualitative understandingtogether with mathematical descriptions of responses tostimulation are essential. The response of excitable cells tonaturally occurring or artificial stimuli is a subject of greatimportance in understanding natural function of nerve and muscle,because most stimuli are produced by the natural system itself.Both electric and magnetic field stimulation are used in researchinvestigations and in clinical diagnosis, therapy, and rehabilitation.This section focuses primarily on responses to electrical stimuli,which are more frequent, and examines several biologicalpreparations.

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INTRODUCTION

Analysis of stimuli focuses especially on mathematicalrelationships between the current applied as the stimulus and theresulting transmembrane potentials. Such knowledge, whenquantitative, then allows one to draw quantitative conclusionsabout the strength and duration of stimuli that will result intransmembrane potentials above the threshold level in new orfuture situations, as well as those already explored experimentally.

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INTRODUCTION

› The initial part of this section provide the simple mathematicsgiving the transmembrane voltages produced by a constantcurrent stimulus, in a spherical cell. These current–voltageequations then are manipulated to produce strength–durationcurves. A few real cells can be idealized as spherical, and theidealization is useful and interesting because, in a spherical cell,the response to stimulus depends only on time.

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SPHERICAL CELL STIMULATION

› An analysis of the response of a spherical cell to an intracellularsubthreshold stimulating current shows that the intracellularregion is isopotential, to a good approximation. If one picturesthe cell placed within an extensive extracellular region, then theextracellular volume also will be virtually isopotential.Consequently, all points on the cell membrane elements willhave very nearly the same transmembrane potential. (Thetransmembrane potential has to be uniform because allintracellular potentials are nearly the same, and all extracellularpotentials are nearly the same).

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Spherical Cell’s Response to a Current Step

› Because the intracellular and extracellular regions are essentiallyisopotential, all membrane elements are electrically in parallel.Thus the entire cell can be represented by a single lumped-RCcircuit, and both R and C will be constant under subthresholdconditions.

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› For a membrane surface area A, we have from

( =(

)

= )

› Here, (is the specific leakage resistance (Ω+,-), Cm is the

specific membrane capacitance (./

01), while R and C are the

total membrane resistance (Ω) and capacitance (23).

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› The transmembrane potential developed in the cell is readily foundfrom the equivalent circuit and is

4 = 5((1 − 7!

8)

› Rewriting for a stimulus just strong enough and long enough to reach athreshold voltage level 9with stimulus duration T, we have

9 = :(1 − 7!

98 )

› In this Eq. time constant ; = ( = (and stimulus strength : = 5(.

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› Note that parameter S is the steady-state voltage approached by4as t → ∞. The quantity S can be thought of as a measure ofthe depolarizing strength of the applied stimulus current 5; infact, it is the maximum depolarization that can be producedpassively by 5.

› τ & A???

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Strength–Duration

› It is well known experimentally and theoretically that asstimulus strength S is increased, a shorter stimulus duration T isneeded to reach a particular transmembrane voltage. To examinethe correspondence mathematically, suppose that thetransmembrane voltage threshold needed for initiate activation isfixed at 4= 9, and a stimulus strength S greater than 9isused. The consequence will be that membrane voltage 9will bereached with a shorter stimulus duration, T, than T →∞.

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Strength–Duration

› What stimulus duration T is necessary?

798 =

1

1 −9

<

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Weiss–Lapicque equation

› Rearranging previous Eq. in a different way, one sees that therelationship between stimulus strength S and threshold voltage9can be written as

: = 9

1

1 −7!

98

› Division on both sides by the membrane resistance R leads to

= =>?

@!"A

BC

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Rheobase

› D is named the rheobase, while =is the minimum currentrequired to reach threshold with stimulus duration T.

› The rheobase, D, is the minimum stimulus intensity that stillproduces a threshold value of transmembrane voltage, as thestimulation duration grows long (conceptually, as T → ∞). 9isthe strength at rheobase, or rheobase voltage.

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Rheobase

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Chronaxie

› The pulse duration when the stimulus strength S is twicerheobase is called chronaxie. Chronaxie, E0 , can be foundanalytically, since at chronaxie : = 29. Multiplying throughby the term in parentheses, we have

E = 2E(1 − 7!

9F8 )

7!

908 =

1

2

E0 = ;ln2 = 0.693;

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Chronaxie

› Chronaxie is significant as a practical time period required toreach the threshold voltage when using a practical stimulusstrength. In a comparison of different membranes or the samemembrane under different conditions, chronaxie provides anominal measure of excitability.

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Electric stimulus

› Electric stimulus properties:

1. Pulse duration D in milliseconds (ms)

2. Current I in milliamps (mA)

3. Charge Q= ID in microcoulombs (µC)

4. Energy K = LMNOin millijoules (mJ)

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Electric stimulus

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Electric stimulus

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Comparison of chronaxies of different excitable tissues

› Myelinated nerve

› Unmyelinated nerve

› Skeletal muscle

› Smooth muscle

› Cardiac muscle

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What is the significance of the Chronaxie?

Given that two nerves have the same

Rheobase, the Chronaxie (the stimulus

duration corresponding to twice the

rheobase) can give an indication of

their relative excitabilities. In the

strength-duration curve to the right,

nerve B is the more excitable.

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How would the strength-duration curve for a set of slow fibres (not very

excitable) compare to the strength-duration curve for a set of quick

fibres (very excitable)?

The curve for the slower fibres would be

shifted to the right, indicating that for a

given stimulus strength, a longer stimulus

duration would be needed to bring the

slower fibres to threshold.

Documents

Bioelectric Phenomena