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7/27/2019 Lecture Slides Week 4 Slides NEW
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Bioelectricity week 4
7/27/2019 Lecture Slides Week 4 Slides NEW
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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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Bioelectricity Week4_3
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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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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Bioelectricity Week4_4
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, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_5
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_6
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Alphas and Betas - 1
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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Alphas and Betas - 2
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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Alphas and Betas - 3
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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Bioelectricity Week4_7
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_8
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm
10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Putting it all together - 2
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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Putting it all together - 3
. The
totalcurrentof these3components iscalledtheioniccurrent.
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Putting it all together - 4
The ioniccurrentsdepend on
probabilitiesof particleopenings,and theseprobabilitiesaredesignatedn, m, h
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Putting it all together - 5
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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Putting it all together - 6
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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Bioelectricity Week4_9
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm
10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_10
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm
10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_11
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm
10. Numerical calculations, time and space11. Problem session, Vm steps
12. Week 4 conclusions
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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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Bioelectricity Week4_12
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 , I_Na
8.Putting it all together.
9. Changes in n, m, h and Vm
10. Numerical calculations, time and space11. Problem session, Vm steps
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
We started the week with a mystery. Where did the action potentialcome from? What was the mechanism?
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
We started the week with a mystery. Where did the action potentialcome from? What was the mechanism?
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