11
Neuron Module
Quantitative Physiology IIOrgan SystemsBMEN E4002
Professor Morrison
2
Todays Overview
General purpose of the nervous system General structures (at neuron level) Introduction to myelin Difference between PNS and CNS Structure of a peripheral nerve Simple electrical model of a cell
What is the purpose of the nervous system?
3
Why a Neuron Module?
What are the applications? Neural engineering
q Restoration of lost function Cochlear implants (~100,000 in use) Brain computer interface Replace cognitive / higher order processing
q Treat diseases Deep brain stimulators
Parkinsons, epilepsy, even severe depression http://www.neuro.jhmi.edu/DBS/cases.htm
2
5
Dendrite Soma Hillock Axon Presynaptic Terminal
q Bouton
Copyright 2002 Elsevier Science (USA)All rights reserved
10.1
7
Information Flow
Copyright 2002 Elsevier Science (USA)All rights reserved
10.9
Excitatory Post Synaptic Potential: EPSPAction Potential: AP
8
Myelin
Copyright 2002 Elsevier Science (USA)All rights reserved
10.12
CNS: OligodendrocytesPNS: Schwann cells
3 Central nervous system (CNS)q Dense and complex connectionsq Designed for computation
Peripheral nervous system (PNS)q Transport of information to/from peripheryq Mechanically active environments
Specialized structures Mechanical protection
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10
Peripheral Nerve
Copyright 2002 Elsevier Science (USA)All rights reserved 10.12
11
MotivationStimulus current
I
Vm
Hyperpolarizingstimulus
I
Depolarizingstimulus
Response
Vm
Copyright 2002 Elsevier Science (USA)All rights reserved7.2
4
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Motivation
Why do we want to model neuron behavior?q To gain a deeper understand of dataq To better understand normal functionq To better understand disease statesq To identify underlying causes of pathologyq To develop treatments / cures
Epilepsy
14
Epilepsy
Coordinated and repetitive dischargesq Of large populations of cells
Bursting
q Capture normal behavior with a modelq Alter the model to produce pathology
Models help explain experimental data Models can guide new experiments
15
Phospholipids
3-5nm Thick
Copyright 2002 Elsevier Science (USA)All rights reserved2.1
516
Capacitor Model
Copyright 2002 Elsevier Science (USA)All rights reserved 6.9
17
Parallel Plate Capacitor
Passive Element 1q Membrane capacitance: Cm
Q = charge (Coulombs) Vm is membrane voltage C: Farads m
m VQC
mNC
2
18
Membrane Capacitance
Rearrange
Differentiate wrt time (assuming Cm const.)
Implies Cm dictates speed
mm V
QC mmVCQ
dtdVCI
IdtdQ
dtdVC
dtdQ
mm
mm
6
19
Membrane Resistance
Passive element 2: Membrane resistance: Rm : Ohms Leak current is Ohmic
q Obeys V=IRq Alternatively I= g V
g conductance = 1/R (Siemens)
Rm
20
Passive Membrane Model
Vm is measured: Inside - Outside
RmCmIin
Vm
Extracellular
Intracellular
21
Circuit Equations
Kirchoffs Current Law (KCL) Kirchoffs Voltage Law (KVL)
KCL: Sum of current into a node = 0
0 ji
722
Kirchoffs Current Law
Requires a sign conventionq Current into a node is positive
i1
i2
i3
i4
04321 iiii
23
Kirchoffs Voltage Law
Sum of the voltages in a loop = 0
Voltage dropacross nodes
Algebraic sumq Assign +/- to each end of voltage drop
0 jv v2
v3v4
v1
24
Kirchoffs Voltage Law
+
-
v2
v3v4
v1
-
+
-
+-
+
04321 vvvv
8
25
Passive Model
Use KCL to sum I in top node
Define:units?
First order ODE
m
mmmin R
Vdt
dVCI 0
mmCR W
minmm RIV
dtdV W
mNCoulombC
CoulombsmNR
2
2
RmCm
Vm
Extracellular
Intracellular
26
Separate variables and integrate
W
W
W
t
minm
minm
mminm
DeRIV
DtRIV
dtdVRIV
c
x ln
11
27
Integration constant from ICq At t=0, Vm(0) = 0
W
W
t
minm
t
minm
eRItV
DeRIV
1)(
928
I = 0; t < 0I = Iin; t t 0
At t = 0; Vm(0) = 0
W
t
minm eRItV 1)(
Iin
I(t)
Vm(t)IinRm
Speed DW
Vm(0)
63%
t=W
29
At t = 0; Vm(0) = 0
Iin
I(t)
Vm(t)
I = 0; t < 0I = Iin; 0 d t < toI = 0; to d t
Vm(to)
W
ot
minom eRItV 1)(
to
0 d t < to
30
Discharge
Reexamine the system nowq No current source: Iin = 0
RmCm
Vm
Extracellular
Intracellular
W
W
W
W
W
t
m
m
mm
m
m
mm
DeV
DtV
dtdVV
dtVdV
Vdt
dV
c
x
ln
11
10
31
Initial Condition
Find D from ICq This model is valid from to
What is Vm(to)?
W
ot
minom eRItV 1)(
to
Vm(to)
32
IC for Discharge
WW
W
W
tt
omom
t
om
t
om
eetVttV
etVD
DetV
o
o
o
x t
)()(
)(
)(
33
Substitute for Vm(to)
Collect terms
Wot
minom eRItV 1)(
WW)(
1)(oo ttt
minom eeRIttV
x
t
11
34
Iin
I(t)
Vm(t)
I = 0; t < 0I = Iin; 0 d t < toI = 0; to d t
Vm(to)
to
W
t
minom eRIttV 1)0(
WW)(
1)(oo ttt
minom eeRIttV x
t
35
Typical values
Rm and Cm related to membrane areaq Unit capacitance
q Unit resistance
2
1cm
Fc P
22000 cmr x:
Rm RmRmRm
37
NaK PUMP
Extracellularspace
ATPase
Cytosol
ADP+
E subunitD subunit
Pi
Na+
K+
ATP
Copyright 2002 Elsevier Science (USA)All rights reserved3.8; 5-8
12
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Potential Energy Storage
Ion gradients store energyq Concentration potentialq Electrical potential
> @> @o
imx X
XzFRTVU ln
40
Nernst Potential Potential of the concentration gradient
q Nernst potential
q E is measured Ein Eout same as Vmq [ ] = concentration
R = ideal gas constant = 8.314 J/K/Mole T = absolute temperature in K (C=273) z = algebraic charge of the ion F = Faradays constant 96,500 Coulomb/mole
> @> @i
ox X
XzFRTE ln > @> @o
ix X
XzFRTE ln
41
Nernst Potential CalculationIon [Intracellular] [Extracellular]Na+ 50mM 400mMK+ 400mM 10mMCl- 40mM 540mM
mV F
RT E: ENa outin 5450400ln
1
mV F
RT E: EK outin 9640010ln
1
mV F
RT E: ECl outin 6840540ln
1
13
44
Driving Potential
Vm Exq Vm ~ -70mVq Na+: Vm ENa = -124mVq K+: Vm EK = 26mVq Cl-: Vm ECl = -2mV
Sign determines directionq With respect to the voltage drop (Vm ENa)
I =V/R
45
Q: How can ions cross the membrane?q Membrane core - hydrophobicq Ions polar
A: Embedded channel proteinsq Form hydrophilic pores
Extracellular space
Cytosol
K+ channel Na+ channel Ca2+ channel Cl- channel Copyright 2002 Elsevier Science (USA)All rights reserved6.9
46
Ohmic Current
Ion movement generates a currentq Assume it is an Ohmic current
I = V * g INa: (Vm ENa ) * gNa IK: (Vm EK ) * gK ICl: (Vm ECl ) * gCl
Add these to a new cell model
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47
Cm
Vm
Extracellular
Intracellular
ENa
gNa
EK
gK
ECl
gCl
48
Resting Membrane Potential
At Vm, I = 0 Apply KCL
q Current out of node is +
q V is constant:
q And ICl ~ Ileak (IL)
dtdVCIII mmClKNa 0
0 dt
dVm
Cm
Vm
Extracellular
Intracellular
ENa
gNa
EK
gK
ECl
gCl
49
Substitute definitions for each channel
Rearrange
)()()(0 LmLKmKNamNa EVgEVgEVg
LKNa
LLKKNaNam ggg
EgEgEgV
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50
Ion [In] [Out] Conductance
Na+ 12mM 120mM
K+ 120mM 4mM
Cl- 4mM 120mM
05.0 gg Na
5.0 gg K
45.0 ggCl
LKNa
LLKKNaNam ggg
EgEgEgV
> @> @i
ox X
XzFRTE ln
> @> @
> @> @
> @> @4
120ln45.0120
4ln5.012
120ln05.0F
RTF
RTF
RTVm
mVVm 81
52
Action Potential
Copyright 2002 Elsevier Science (USA)All rights reserved7.1&2
53
AP Regenerative Conduction
Copyright 2002 Elsevier Science (USA)All rights reserved7.2
16
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AP Threshold
Copyright 2002 Elsevier Science (USA)All rights reserved7.1&2
55
Stimulus Intensity & Duration
Combine RiRaCm
Iin
Vm
Extracellular
Intracellular
Em
56
Apply KCL to the top node
Define
a
mmmmin R
EVdt
dVCI
0
maCR W
mmmain EVdtdVRI W0
RaCmIin
Vm
Extracellular
Intracellular
Em
17
57
Rearrange and separate
Integrate
dtRIEV
dV
RIEVdt
dV
ainmm
m
ainmmm
W
W
1
DtRIEV ainmm c Wln
Wt
ainmm eDRIEV
RaCmIin
Vm
Extracellular
Intracellular
Em
58
Apply initial conditions Vm(0) = Em Find D
DRIeDRIEV
EVt
ain
ainmm
mm
0
0
Wt
ainmm eDRIEV
RaCmIin
Vm
Extracellular
Intracellular
Em
59
Substitute
Collect terms
Wt
ainainmm eRIRIEV
m
t
ainm EeRItV
W1)(
18
60
For a given Vthq How are stimulus intensity and time related?q Rearrange
W
t
a
mthin
eR
EVI1
m
t
ainm EeRItV
W1)(
61
Rheobase
I
t
Ij
tj
W
t
a
mthin
eR
EVI1
a
mth
REV
f
The minimum level of current capable of generating an AP if applied for an infinite time
m
t
ainm EeRItV
W1)(
62
Adding to the model Cannot reproduce the Action Potential
q Hodgkin Huxley neuron model Invented the Voltage Clamp
Holds the cell voltage constant Measures the necessary current
1963 Nobel Prize in Physiology / Medicine
0
1
1Im(mA/cm2)
Outwardcurrent
TOTAL IONIC CURRENT
Inwardcurrent
20
80Vm(mV)
Copyright 2002 Elsevier Science (USA)All rights reserved
7.5
19
63
Hodgkin Huxley Assumptions
Ions are separated by the membrane Current flow is Ohmic Ions flow through channels
q Their conductances are variableq Function of Vm and time
Add these to the model
64
Cm
Vm
Extracellular
Intracellular
ENa
gNa
EK
gK
EL
gL
Hodgkin Huxley Model
INa IK ILIT
IC
65
Apply KCL
Current out of bottom node as positive
LKNam
mT IIIdtdV
CI 0
LKNa
LLKKNaNam ggg
EgEgEgE
)( jmjj EVgI
> @> @i
oj X
XzFRTE ln
Cm
Vm
Extracellular
Intracellular
ENa
gNa
EK
gK
EL
gL
INa IK IL IT
IC
20
66
H&H assumedq Changes in Vm were due to
Time dependent changes in conductances
Increasing gi will drive Vm to Ei
LKNa
LLKKNaNam ggg
EgEgEgE
67
Calculated Nernst potentials earlier
Leak current is mainly Cl current
mVEmVEmVE
L
K
Na
689654
68
Conductance Changes in AP
70
0
Vm
t
1) Increase gNa to drive Vm to ENa (+54mV)2) Increase gK to drive Vm to EK (-96mV)3) Reset gK and gNa to drive voltage back to Vm
gNa gKReset
Copyright 2002 Elsevier Science (USA)All rights reserved7.2
LKNa
LLKKNaNam ggg
EgEgEgE
21
69
Experimental Requirements
Needed to measure gNa and gKq Functions of both time and Vmq Measure INa and IKq Calculate g from
Km
KK
Nam
NaNa
jmjj
EVI
g
EVI
g
EVgI
)(
71
Experimental Methods
Experiments to understand physiologyq Drive innovation and technology development
Devised a method to hold Vm constantq Constant in timeq Constant in space along the axon
Space ClampVoltage
72
Experimental Methods
Needed to measure the current necessary to hold Vm at a desired levelq Feedback Amplifier
Second wire to apply a current Feedback circuitry to measure current
for a desired VmVoltage Current
22
73
Experimental Methods
With this set-upq Measured the membrane current to different
voltage steps
t (msec)
I
74
Separate Current
Needed to separate IK from INaq Today wed use pharmacology
Tetrodotoxin TTX to block Na channels Tetraethyl ammonium TEA to block K channels
None available - ? Separate mathematically
q Experiments in normal sea waterq Experiments in sea water with reduced [Na]
76
Experimental Paradigm
Classic experimentsq Nobel prize winning work!
Step and hold voltage at V Measure current through the membrane Calculate gNa and gK
23
77
gK
t (msec)
1/:
510254055
90'V (mV
70
gNa
t (msec)
1/:
78
Model Equations
Hodgkin and Huxley devised a model Fit the model to the data Examine K channel first
q Formulated the concept of a gate To explain the data That is fit the experiment to a model
q Probability it is open = nq Assumed 4 gates in the K channel
79
K Channel Conduction Gates
Probability of one gate being opened = n
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
24
82
K Channel
Examine K channel firstq Concept of a gateq Probability it is open = nq Assumed 4 gates in the K channelq All must be open for conduction
Reproduces the sigmoidal shape of the curve
83
K Channel
Assume each gate operates independently Probability that a channel is open
q n*n*n*n = n4 The proportion of open channels in a
populationq n4
4ngg KK
84
Time Dependence
Gates can transition from open to closedq Rate constant associated with each transition
q Write a differential equation for nCOCO
n
n
moD
E
)1()( nndtdn
nn DE
25
85
ndtdn
nnn EDD
ndtdn
nn ED
nnn EDW
1
21)( cectn nt
W
Homogeneous solution
Define
86
Apply an initial conditionq n(0) = 0 q c1 = -c2
Solve for a particular solution
q i.e. the system comes to a steady state
0o
fo
dtdnt
87
ndtdn
nnn EDD
nnnn EDD0
nn
nn EDD f
26
88
nn
nn EDD f
11)( cectn nt
W
f nc1
)1()( nt
entn W
f
89
The model was then fit to the dataq Normalized gK and took the 4th rootq Determined nf from the dataq Fit a first order exponential to find Wn
q Calculated Dn and En from their definitions
)1()( nt
entn W
f
nn
nn EDD f nnn ED
W 1
90
Captures time dependency of the channel Estimates an Dn and En valid for one Vm
q Need to find Dn and En for other voltages
gK
t (msec)
1/:
510254055
90'V (mV
70
27
91
Determined Dn and En for many 'Vm Fit a smooth function through the points
q Empirical functions onlyq Capture voltage dependence
80
1010
125.0
1
1001.0
Q
Q
E
QD
e
e
n
n
mV
Rat
e C
onst
ant (
1/m
sec) D gate
E gate
Q = V-Vrest
92
Na Channel
Kinetics are more complexq Activation followed by inactivation
q Proposed two kinds of gates Activation: m Inactivation: h
gNa
t (msec)
1/:
hmgg NaNa3
93
Na Channel Conduction Gates
Probability of an open channel = m3h
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Closed Open ClosedClosed
28
94
Na Channel Kinetics
COCO
COCO
h
h
m
m
mo
mo
D
E
D
E
)()1(
)()1(
hhdtdh
mmdtdm
hh
mm
ED
ED
95
Similar procedure as for the K channel Fit function to the data
q Dm, Em, Dh, Eh are all functions of Vm
1
107.0
4
1
251.0
1030
20
18
1025
Q
Q
Q
Q
E
D
E
QD
e
e
e
e
h
h
m
mQ = V-Vrest
96
Complete H&H Model
Passive properties of the membrane Nernst potentials of each ion Time / voltage dependent conductances
> @> @i
oj X
XzFRTE ln
KmKNamNaLmLmm EVgEVgEVgdtdVCI LmLmm EVgdtdVCI
29
97
Complete H&H Model
Time / voltage dependent conductances KmKNamNaLmLmm EVngEVhmgEVgdt
dVCI 43
)1()( nndtdn
nn DE
)()1(
)()1(
hhdtdh
mmdtdm
hh
mm
ED
ED
1
107.0
4
1
251.0
1030
20
18
1025
Q
Q
Q
Q
E
D
E
QD
e
e
e
e
h
h
m
m
80
1010
125.0
1
1001.0
Q
Q
E
QD
e
e
n
n
Q = V-Vrest
98
Example
Discover a new organism extremophileq 90C
q What are the Nernst potentials for Gd3+ & P2-?q Communicates by a P channel
What is the theoretical maximum potential when this channel opens?
q What is the resting membrane potential?
Ion [In] (mM) [Out] (mM) g (Siemens)Gd+++ 30 520 12P-- 450 12 2
99
Example
Discovered a cuboid cell: 100Pm per edgeq Em = -40mVq rm = 2500 :xcm2q cm = 5PF/cm2q Vth = -5mv
What is the minimum current injection to get the cell to fire in 5ms?
30
103
Numerical Simulation Results
DepolarizingStimulus
HHSimTutorial
membrane voltage (mV)
-80
-60
-40
-20
0
20
40
60
0 5 10 15 20 25
01020
0 5 10 15 20 25
104
AP Characteristics
AP is an all or nothing phenomenonq No half amplitude APq Magnitude and duration are fixed
Absolute Refractory Periodq A second stimulus cannot elicit an AP
If close in time to the first stimulus
Both can be explained by the model
105
AP Initiation
AP begins when Vm > Vth Two opposing currents
q INa depolarizingq IK hyperpolarizing
Vth corresponds to INa > IKq Initiates a positive feedbackq Stopped when Na channels shut down (h)
31
106
GateActivation
0
0.25
0.5
0.75
1
0 5 10 15 20 25
m h n
-100
-50
0
50
0 5 10 15 20 25
)1()( nt
entn W
f
ENa
EK
107
Absolute Refractory Period
Two stimuliq Two APs
Two stimuliq Only one AP?
108
Absolute Refractory Period
h must resetq Wh
0
0.25
0.5
0.75
1
0 5 10 15 20 25
m h n
32
109
HW 2q Will explore the H&H model in more detailq Using a numerical simulation
Links are in the homework Detailed instructions in the homework
115
H&H model reproduces the APq Captures experimental dataq Leads to testable predictions
Limitations of what weve modeled so far?q Treats the neuron as a single compartment
No spatial information Cant represent a realistic neuron
0 ww
xVm
116
Actual Neuron Morphology
Duke-Southampton archive of neuronal morphology
33
117
Passive Conduction Model the passive spread of signal down a
neuronal process Processes are not perfectly insulated
conductorsq Signal gets attenuated as it travelsq There is a finite resistance between the
intracellular and extracellular spaces
118
Why model neuron structure? Important for modeling networks
q Networks are the basis for computationq Capture network behavior
Understand higher order functions Ocular dominance columns (p. 370)
q If we understand network behavior We can design a replacement circuit
Replace defective neuron networks Repair damaged neuron networks
Berger et al., Restoring lost cognitive function, IEEE Eng Med Biol Mag. 24: 30-44, 2005
Cohen and Nicolelis, Reduction of single-neuron firing uncertainty by cortical ensembles during motor skill learning, J Neurosci 24: 3574-3582, 2004
119
Passive Conduction
Signal is attenuated Signal becomes spread out
70
80
1 2 3 4
2 34
Stimulus current
I 1I
V
V
7.2Copyright 2002 Elsevier Science (USA)All rights reserved
34
120
Signal Attenuation
Insulation is not perfect Current loss through the membrane
Injection
7.22Copyright 2002 Elsevier Science (USA)All rights reserved
121
Passive Conduction
Constant velocity of propagation
70
80
1 2 3 4
2 34
Stimulus current
I 1I
V
V
t
xx1
t3t2t1
x3x2
Slope = conduction velocity
122
Cable Theory
First developed by William Thomson,1855q University of Glasgowq Later Lord Kelvin of absolute 0qK fameq Describe conduction in the
Trans-Atlantic Telegraph Cable, was knighted for itq Applied to neurons by
Hodgkin and Rushton (1946) Rall (1957-1969)
q B&B pp. 207-211
35
123
Cable Theory Assumptions
Need a new model for the neuron
For the neuronal structure (process)q Uniform cylindrical coreq Length >> diameterq Uniform membrane propertiesq Uniform core properties
124
Definitions
ro: external resistance per axial length
ri: internal resistance per axial length
cm: capacitance of membrane per unit length
rm: resistance across the membrane times unit length
cmFP
cmx:
cm:
cm:
125
Extracellular fluid
ro ro ro ro ro ro
Cytoplasm
Membrane
ri ri ri ri ri ri
rm cm
Cable Model
Simple model of the plasma membraneq Linked in series with two resistances
Internal and external resistance
Copyright 2002 Elsevier Science (USA)All rights reserved
V(t,x)
7.22
36
126
Components
ro
ri
rm cm
Intracellular
Extracellular
127
Currents
ro
ri
rm
im
cm
io+dioio
ii+diiii
128
Voltages
ro
ri
rm cm
Vo+dVoVo
Vi+dViVi
37
129
Membrane Current
rm
imcm
dt
VVdcr
VVi oimm
oim
Vo+dVo
Vi+dVi
dt
dVVdVVdcr
dVVdVVi ooiimm
ooiim
0limodx
130
ro
ri
rm
im
cm
io+dioio
ii+diiii
External Current
Use KCL on top and bottom nodesq Top node
dxdii
dxidiidxidii
om
mo
omoo
00
131
ro
ri
rm
im
cm
io+dioio
ii+diiii
Internal Current
Use KCL on top and bottom nodesq Bottom node
dxdii
didxidiidxii
im
im
iimi
0)(0
38
132
Axial Currents
V=IR Extracellular current
ro
ri
ioVo+dVoVo
ooo
ooooo
ridx
dVdxriVdVV
133
Axial Currents
V=IR Intracellular current ro
riii Vi+dViVi
iii
iiiii
ridxdV
dxridVVV
)(
134
Subtract the currents
oi ii
ooiioi riri
dxdV
dxdV
ooii
oi riridx
VVd
39
135
Differentiate with respect to x
ooii
oi riridx
VVd
dx
riddx
riddx
VVddxd ooiioi )()(
dxdir
dxdir
dxVVd
dxd o
oi
ioi
136
dxdii om dx
dii im dxdi
dxdi io
dxdir
dxdir
dxVVd
dxd o
oi
ioi
oiioi rrdxdi
dxVVd
dxd
137
dxdi
dxVVd
dxd
rrioi
oi
1
dxdii im
m
oi
oi
idx
VVddxd
rr
1
40
138
Membrane Current
rm
imcm
Vo
Vi
dt
VVdcr
VVi oimm
oim
m
oi
oi
idx
VVddxd
rr
1
139
Rearrange:
Define:
tVVc
rVV
xVV
xrroi
mm
oioi
oi ww
ww
ww
1
tVVrc
xVV
rrrVV oimmoi
oi
moi w
www
2
2
oi VV Vthq How does it get propagated?
By depolarizing the next piece of membrane How does the next piece of membrane get to Vth?
Active InactiveInactive
48
177
-+
----------- --- -++++ + ++++++++++
- -
+ +
Passive propagation of the depolarizationq Cable equation
q Channel opening generates local depolarizationq Depolarizes the next segment
By passive propagationq Until Vth is reached
Then action potential is regeneratedCopyright 2002 Elsevier Science (USA) 7.21
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Propagation of action potential in spaceq Dependent on passive axon properties
Regeneration of action potentialq Requires voltage activated channels
179
Myelination
Myelinated or unmyelinated axons
Copyright 2002 Elsevier Science (USA)All rights reserved 10.12
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Myelin Diseases
Myelin plays a critical physiological role in neuronal activity
Myelin damage causes severe disruption of nerve functionq Multiple sclerosis, amyotrophic lateral
sclerosis (ALS, Lou Gehrigs disease), progressive multifocal leukoencephalopathy
We can explain the pathology through our models and make predictions of function
181Extracellular fluid
ro ro ro ro ro ro
Cytoplasm
Membrane
ri ri ri ri ri ri
rm cm
Myelin Sheath
Composed of many layersq Plasma membrane (phospholipids)q Wrapped tightly around the axon
Produced by Schwann Cells in the PNS Produced by Oligodendrocytes in the CNS
q Hundreds of layers Acts as an insulator
q Decreases leak currents to the outside
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Myelin
Use our passive model againq One layer of myelin
How are the layers arranged?q In series or parallel?
RC
RC
RC
RC
RC
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183
Arranged in Series
What are the implications for the model? Examine the capacitance first
q Assume each layer has capacitance Cq Capacitors in series:
q For n identical C
...1111
321 CCCCT
nCCT
184
Each myelin layer adds to the axon sizeq Radius a = n * layer thickness
J
Jan
na
aCCT
J
185
Implications for the resistanceq Arranged in series
q How does the myelin affect the cable equation
RaR
RnRRRRR
T
T
T
J
...321
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186
Space constant: O
Ri does not change due to myelin sheath Rm is now different
i
m
RRa
2~O
mT
mT
RaR
RnR
J
187
aRRaa
RRa
i
m
i
m
v
OJO
O
2~
2~
O
a
Myelinated
Un-myelinated
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Time constant: W mm CR W
mT RaR J a
CCTJ
mm CaRa JJW x
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189
Effects of Myelin
Improves conduction byq Increasing the space constantq No effect on time constant
q Assuming material properties of myelin Equal those of regular plasma membranes
Not entirely true Myelin resistance is greater Myelin capacitance is lower
194
Myelinated Nerve Structure
Myelin is not continuousq Nodes of Ranvier
Separate myelin sheaths
q No ion channels beneath myelin sheath Concentrated at the Nodes of Ranvier
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AP jumps from node to nodeq Saltatory conductionq Possible because of the increased insulation
Myelin Decreased loss of signal
Copyright 2002 Elsevier Science (USA)All rights reserved 7.21
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Implications for diseases and injuriesq Myelinated nerveq Channels concentrated at nodesq What happens if it becomes unmyelinated?
Injury Multiple Sclerosis Amyotrophic Lateral Sclerosis
q What happens to information transfer?
197
Normal condition
Injured or diseased condition
q Will the nerve continue to conduct?Copyright 2002 Elsevier Science (USA)All rights reserved 7.21
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Re-state the problemq Will the next Node of Ranvier
Reach Vth to re-initiate the action potential?
q How can we calculate the voltage As a function of distance?
Assume steady state applies
q Cable equation
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200
x
Vth?
201
Assume that
How long can the gap beq For conduction to be maintained?
oth eV < 1
othV < %37
202
Use our steady state solution:Ox
o e< Vth
242
The timing and magnitude of the APq Determined by the gate kineticsq Well described by the Hodgkin Huxley model
AP propagates down the axon Releases neurotransmitter at the synapse
q Vesicle fusion is Ca++ dependent Neurotransmitter activates receptors
q On the post-synaptic side