lecture slides week 4 slides new

7/27/2019 Lecture Slides Week 4 Slides NEW

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Bioelectricity week 4


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Bioelectricity -- Week 4

Bioelectricity week 4:

A) Hodgkin-Huxley Membrane ModelB) HH Numerical calculations

Train system:Integrate the cars.


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Bioelectricity Week4_2

Hodgkin-Huxley Membrane Model

HH Numerical Calculations.Train system:Build the cars. Put train together.

1. Introduction to week 4

2. What is the problem?

3. HH replacement for Rm4. The equation for each pathway

5. Changes in n, m, h6. Equations for alphas and betas7. Problem session , gNa

8.Putting it all together.

9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, a Vm step

12. Week 4 conclusions


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Active tissue the problemwhere does the active response come from?

PassiveActive


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Where does the AP come from? algorithm?

How canVm be

getting

bigger?

Im = Ic + Ir

Ic = Im Ir

Cm dVm / dt = Im IrIm is Istim and then zero

Ir is Vm / Rm

So,

dVm = ( Istim Vm/Rm) *dt / Cm


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Hodgkin and Huxley Giant Squid

Recorded data from thenerve axon of the giant

squid

Drawing of giant squid on a basketball court


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Hodgkin and Huxley Squid Axon, Electrodes

Inserted electrodes insideand outside, along the axisto measure voltages and

currents across the

membrane

Sketch of axon with electrodes


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Hodgkin and Huxley Squid Axon, Electrodes

Changed the composition ofextracellular solution, so as

to isolate the action of

individual ions

Sketch of axon with solution


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Hodgkin and Huxley Squid Axon, Model

Created a mathematicaltheory to unify the

experimental findings andprovide a mechanism for the

creation of action potentials.

This mathematical theory,somewhat updated, is

presented here.

Received Nobel PrizeSingle current through resistor replaced by 3individual ionic currents

Each ionic current describedmathematically, including changes with Vm,

and, for Na+, changes with time.


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Path to Duke University Law School


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9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, a Vm step



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Replacing Rm with Ionic Currents

Hodgkin-Huxleymodel: Replace Rm

with individual ionic

pathways.

Ir is replaced by Iion Iion has a path for each

of 3 ions Each ionic pathway has

a battery and a variableresistance


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Stored energy now comes into the model

Each ionic pathwayhas a battery

The batteries havea voltage equal tothe Nernst potential

for that ion.

Each ionic pathway also has a

variable resistance.


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Stored energy now comes into the model

Each ionic pathway

also has a variable

resistance.

Each variable resistance is

written as its reciprocal, a

conductance, hence, for

example, gK.

So what is gK, gNa, or gL?


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Duke University, Science Drive


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9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, Vm steps



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The equations for the K pathway.

n is the probability of an nparticle being open


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K channel sketch, with n


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The equations for the Na pathway.

m is the probability of an mparticle being open, and

similarly forh


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Sketch of an Na channel, with m and h


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The equations for the L pathway.

There are no probabilitieshere J


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Duke University, Science Drive


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5. Changes in n, m, h6. Equations for alphas and betas7. Problem session , I_Na





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Changes in n, m, h

The rates of change ofn, m, hare given by these differential

equations:

Keep in mind that all the alphas and betas change when Vm changes.


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Changes in n, m, h: estimates.

The rates of change ofn, m, h are givenby these differential equations:

Often changes in n,m,h are

computed by turning theseequations around into

approximations, where dt is now

a finite interval and dn, dm, dh

are estimates:

Keep in mind that all the alphas and betas change when Vm changes.


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Dukes Sanford Institute of Public Policy


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First a matter of terminology:

What is the difference between Vm and vm?sketch

vm with a lower-case vis thetrans-membrane voltage

relative to the trans-

membrane baseline voltage.

Vm with an upper-case Visthe trans-membrane voltage,

absolute.

Note that vm=Vm-Vr, whereVris the resting trans-

membrane voltage.


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First a matter of terminology:

What is the difference between Vm and vm?comments

vm with a lower-case visthe trans-membranevoltage relative to the

baseline voltage. Vm with an upper-case V

is the trans-membranevoltage, absolute.

Note that vm=Vm-Vr,where Vris the restingVoltage.

vm is easier to measure, as it isnot affected by DC drift, an issue

with instrumentation. Baseline vmis by definition equal to zero.

Vm is the right Vm. The baselinevalue of Vm will be -70mV, more or

less, rather than 0 mV.

Everyone recognizes the difference

between Vm and vm, but the specificnotation is not always that used here.


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Alphas and Betas - 0

The alpha functions aremeasurements of how fast the n,m, h particles open.

The beta functions aremeasurements of how fast then,m,h particles close.

The HH expressions are in permsec units.

On the train on hears,Doors are opening

Doors are closing


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The alpha and beta functions given by HH aremathematical encapsulations of experimental

measurements.

In the HH model, they are specific to squid axon. Even though they are presented as mathematical

functions, they are experimental results, given in

a mathematical expression.


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There is an alpha (turn on) function and a beta(turn off) function.

There are alphas and betas for each of the 3probabilities n, m, h. Each function is notably

different from the others.

So there are 6 functions in all


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In coding the functions for computer evaluation,attention has to be given to assigning the correct

function value for values of the argument vm whenthe functions denominator goes to zero.

HH did their numerical calculations manually. J


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Alphas and betas use vm

The alpha and betafunctions are written

in terms of vm, thevoltage as measured

against the baseline.


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Equation for alpha_n, beta_n

expressed here in C language code

//nu = (10-vm)/10;

if (u > 0){au = u;} else{au = -u;}if (au > 1.E-7){

an=.01*(10.-vm)/(exp((10-vm)/10)-1);}

else{ an = 0.1; }

bn=0.125*exp(-vm/80);


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Equation for alpha_m, beta_m

//mu = (25-vm)/10;

if (u > 0){au = u;} else{au = -u;}if (au > 1.E-7){

am=0.1*(25-vm)/(exp((25-vm)/10)-1);}

else {am = 1.;}

bm=4*exp(-vm/18);


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Equation for alpha_h, beta_h

//hah=0.07*exp(-vm/20);

bh=1/(exp((30-vm)/10)+1);return 0;

Hallelujah hallelujah no probabilities here J


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HH Parameters and State Variables

Istim = 0.Im = 0.

Istim=0Im = 0.


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Flowers gone to seed


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Problem Session, I_ion, QThe tissue is active. Using the HH standardparameter values, using the state variables

set 1, and with no stimulus current at thistime, what is:

a) IKb) Inac) ILd) If these currents were maintained without

change for 50 usec, what would be the

change in Vm?


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Problem Session, I_ion, a

Using the HH standardparameter values, and

using the state variablesset 1, what is:

a) IK


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Problem Session, dVm eq

Using the HH standard parametervalues, and using the state

variables set 1, what is:

d) If these currents weremaintained without change for50 usec, what would be thechange in Vm?

Notes: (1) for this question compute the

change in one big step. Normally manysmaller steps would be better.

(2) usec is an abbreviation for microseconds.

dVm = ( Istim Vm/Rm) *dt / Cm


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Problem Session, dVm numerical

Using the HH standardparameter values, and

using the state variablesset 1, what is:

the change in Vm?


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Science Drive Woods, Duke University


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9. Changes in n, m, h and Vm

10. Numerical calculations, time and space11. Problem session, Vm steps



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Putting it all together - 1

Hodgkin andHuxley were

trying to

explain why

the big

response

occurs, in

active tissue,

after a small

stimulus.


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The provided amechanism by

replacing the

resistancecurrent, in the

passive model,

with 3

components of

ionic current, in

an active model.


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. The

totalcurrentof these3components iscalledtheioniccurrent.


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The ioniccurrentsdepend on

probabilitiesof particleopenings,and theseprobabilitiesaredesignatedn, m, h


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If one knows Vm, n, m, h

at one time, and

if one knows Im andIstim at this time,

then one can projectnew values for Vm, n, m,h at a subsequent time,

To a goodapproximation, if thetime interval is short.

And


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Finding individual ionic currents is nice, from aphysiological perspective. It gives understanding of themechanisms at work.

Being able to project forward, from one time to the next,for all the state variables Vm, n, m, h, is absolutelyessential.

That time shift, repeated over and over, is what allowsone to move forward, step by step, through a timeinterval. Doing that allows one to figure out what thesequence of events is, in an action potential.


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Duke University, grass


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Changes in n,m,h and Vm

Istim, Im plus Vm and n,m,h

Vm[1] = Vm + dVm n[1], m[1], h[1]

Get alphas and betas

dn / dt =dm / dt =

dh /dt =

Im = ic + iion

Iion = Ina + IK + IL

dVm = dt*( Im - Iion )/Cm


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Duke University sidewalk with cracks


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Numerical

calculations

Vm

gNa

gK


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Sitting there


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Na+ comes in


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K+ goes out


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Duke University Science Drive Tree


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Problem session: Vm steps, Q

Begin with the standard conditions. The tissue is active. At thismoment the state variables are those of set 1.

A) What is Vm after one time step, if each time step is 50 usec? B) What are n,m, h after one time step? C) What is Vm after 2 time steps?


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Problem session: Vm steps, A

Begin with the standard conditions and state variables set 1. A) What is Vm after one time step (Vm1), if each time step is 50

usec?


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Problem session: Vm steps, B

Begin with the standard conditions and state variables set 1. B) What are n,m, h after one time step (n1, m1, h1) of 50 usec?


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Problem session: Vm steps, C

Begin with the standard conditions and state variables set 1. C) What is Vm after 2 time steps (Vm2)?


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Duke University Science Drive Bench


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12. Week 4 in review


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Week 4 in Review -1

We started the week with a mystery. Where did the action potentialcome from? What was the mechanism?


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Week 4 in Review -2


The HH model provided the answer. The resistive current in thepassive model was replaced by 3 ionic currents. Each had aprobabilistic basis.

.


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Week 4 in Review -3


The HH model provided the answer. The resistive current in thepassive model was replaced by 3 ionic currents. Each had aprobabilistic basis.

With the HH model, it was possible to move from one time to thenext, and thereby to trace out the time history of the trans-

membrane potential, and each of its component currents.


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Week 4 in Review -4

From the perspective of energy, we began theweek with no apparent source of energyfor the big voltages changes seen in the active

tissues response.


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Week 4 in Review -5

From the perspective ofenergy, we began the

week with no

source ofenergy for the bigvoltages changes seen

in the active tissues

response.

Once again:

Out of gas

L


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Week 4 in Review -6

The Hodgkin-Huxleymodel showed how

the membrane uses

the energy

stored in themembranethat comes from the

ionic concentrationdifferences of K, Na,

and L ions.


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Week 4 in Review -7

The Hodgkin-Huxley modellinked the membrane model

to the energy stored in the

membrane that came from

the ionic concentration

differences of K, Na, and L.

Vm changes no longer haveto be energized by Istim.

Bye-bye

Istim,

Idont

needyou

any

moreNot like pulse transmission in coaxial cable


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Duke University, Camel with Knut Schmidt-Nielsen

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