Neuron Basics

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


10.9

Excitatory Post Synaptic Potential: EPSPAction Potential: AP

8

Myelin


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

9

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

13

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


516

Capacitor Model


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

39

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

14

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

15

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


16

54

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)


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


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


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


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


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

178

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


49

180

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

182

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

50

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

51

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

188

Time constant: W mm CR W

mT RaR J a

CCTJ

mm CaRa JJW x

52

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

195

AP jumps from node to nodeq Saltatory conductionq Possible because of the increased insulation

Myelin Decreased loss of signal


53

196

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

199

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

54

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

Documents

Neuron Basics