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NBE-E4120 Cellular Electrophysiology Lecture 4
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Small perturbations of Vm: Linearity of Vm vs. current
• Crab axon immersed in oil
• Delay in voltage compared to current
• Charging of the membrane
• At small currents behaviour linear
• Current injection does not affect
membrane impedance
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Electrical Properties of Excitable Membranes
Equivalent circuit representation
 Biological membranes  electrical components and circuits
 Membrane specific capacitance Cm  1 F/cm2
 Membrane specific resistance Rm, [Rm] = ·m2
 Rm (Vm, t, [neurotransmitters] etc.)
 Membrane specific conductance Gm = 1/Rm, [Gm] = S·m-2
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 Membrane current density (total) J (A/cm2):
J  J i  JC 
Vm  Er
dV
dV
 C m  Gm Vm  E r   C m m
Rm
dt
dt
ionic capacitive
resting potential of the cell
 For each ion species:
J i  gi (Vm  Ei )
specific membrane conductance
refers to ion species i
specific conductance for ion species i
 E.g. for Na+:
J Na  g Na (Vm  E Na )
 For the whole membrane:
Vrev  Er
(Er = resting potential)
 But for currents carried by single ions:
Vrev  Ei
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Ionic conductances
 Conductance gi (for each ion type)
 Driving force (Vm – Ei )
J i  gi (V  Ei )
J i  gi (Vm  E i )
No concentration dependence?
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Ionic conductances
 Conductance gi (for each ion type)
 Driving force (Vm – Ei )
J i  gi (Vm  E i ),
RT c o
Ei 
ln i
zi F c
Concentration dependence of ionic currents
comes from Nernst potential
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Ionic conductances
 Conductance gi (for each ion type)
 Driving force (Vm – Ei )
J i  gi (Vm  Ei ),
RT c o
Ei 
ln i
zi F c
 For passive individual ionic currents:
1. When Vm  Ei , J i  f (Vm , E i , t )  0


2. When Vm E i , J i  f (Vm , E i , t ) 0


 Only one equilibrium potential for each ion

holds also when several types of passive conductances present
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The parallel conductance model
 For K+, Na+ and Cl-:
J  Cm
dVm
 g K (VM  E K )  g Na (Vm  E Na )  gCl (Vm  ECl )
dt
 Steady-state:
V
g K E K  g Na E Na  gCl ECl
g K  g Na  gCl
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Current-voltage relations
 Most common way to analyze membrane conductances
 Usually non-linear
 Current proportional to the number of ion channels open
linear range
non-linear range
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Isopotential cell (spherical)
Assumptions:
 The cell is isopotential and ohmic
(Rm constant, Rm  Rm(V))
 Resting membrane potential Vrest = 0
 Current injection into a cell:
 Current:
I m  Cm
dVm Vm

dt
Rm
 Step current response:

Vm  I m Rm 1  e  t / m

 m  Rm C m
 Input resistance:

Rinput 
Vm ()
I0
For a spherical cell (radius a):
Injected current
Rinput 
I m Rm
I m  4 a 2

Rm
4 a 2
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Non-isopotential cell (cylindrical)
Axons, sections of dendrites
 Propagating current has to flow in the resistive cytoplasm and all along load
the cell membrane to a new membrane potential
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CABLE THEORY:
Cells are not generally isopotential: current flow in cylindrical
geometries (cells) can be described by cable theory
Isopotentiality can be achieved by recording geometry, in which
case cable theory is not needed
Examples of isopotential recording geometries:
 Voltage-clamp of the squid giant axon:
 Sucrose gap -method:
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Complex geometries (e.g. dendritic trees) can be approached
mathematically
Role of cable theory in electrophysiology:
 stimulation of cells
 quantitative determination of electrical properties of long cells
 extracellular recordings

origins of extracellular signals (EEG, EKG, ERG, EMG)
 clinical recordings
The cable theory is a general theory for a ”core conductor” with certain
properties.
No neurophysiological or -anatomical knowledge is necessary for the
derivation of the theory.
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Key assumptions of 1-dimensional cable theory:
1. The cable (axon) is a linear, infinitely long tube.
2. The membrane parameters (ri , ro ja cm) are linear and do not depend on membrane
potential.
3. The current flow is along the distance of the cable. Then V = V(x,t).
4. The resistance of the extracellular conductor ro is zero (in the textbook the effect of
ro is included).
For unit
length of a
cylinder:
ri,o axial resistances
[ri,o] = /m
rm membrane resistance
[rm] = ·m
cm membrane capasitance
[cm] = F/m
ii axial current
[ii] = A
im membrane current
[im] = A/m
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 Axial voltage drop generates an axial current (Ohm’s law):
Vm   ii ri x
When
x  0 
Vm ( x , t )
  ii ri
x
 The value of the axial current can change only if the membrane
current is im  0:
ii   im x
When x  0 
ii
  im
x
 2Vm 
ii
 Vm
 By differentiation of Vm/x:
(
)

(

i
r
)


r
(
)  ri im
i i
i
2
x x
x
x
x
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 Membrane current: im  iionic  ic 
Vm
V
 cm m
rm
t
 Combining with the differential equation:

Vm Vm
1  2Vm

c

m
2
ri x
t
rm
Cable equation (linear)
 With parameters that are not geometry specific (cyl. radius a):
 Ri   a 2ri

Vm Vm
a  2Vm
R

2

ar


C

 m
m
m
2
2 Ri x
t
Rm
C  c / 2 a
m
 m
 For a length x, axial resistance = ri x and membrane resistance = rm /x
 These are equal when ri x  rm / x  x  rm / ri  
space constant,
length constant
 
2
2
Vm
x
2
m
Vm
 Vm
t
 m  rm cm
time constant
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L-transformation of the cable equation:
 Normalization of variables
X  x/
T  t /m
 Cable equation:
 Vm
L-transformation:

L( f (t ))  F ( s )   e  st f (t )dt
0
Vm
 Vm  0
2
T
X
  2Vm   2Vm
L  2  
2
 X  X
L  Vm   sVm  V (0, x )
 T 
2
L1 ( F ( s ))  f (t )


L( f '(t ))   e  st
0


 e  st f (t ) 0  s  e  st f (t )dt
0,
0
L Vm   Vm
 sL( f (t ))  f (0)
 2Vm

 ( s  1)Vm  0
2
X
 General solution:
d f (t )
dt
dt
Note: No more partial
differential equation
Vm  Ae  X
s 1
 Be X
s 1
,
A and B from boundary conditions
 Now Vm  Vm ( X , s), by inverse transformation Vm  Vm ( X ,T ) 
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Solutions of cable equations
Infinite cable, response to constant internal current:

Easy experimentally
 Solution:
Vm  Ae  X

s 1
 Be X
s 1
Conditions:
1. Vm  0, when X  
2. Injection of constant current I0 at X = 0, when T  0
1.)  B  0
 Vm  Ae  X
1 Vm
1 Vm

2.) ii  
ri x
ri  X
by derivation
 ii 
s 1
Ltransformation
A
s  1e X
ri 

s 1
ii  
1 Vm
ri  X
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A from the boundary condition ii  I 0 / 2 at X  0
 ii ( X  0) 
I0
2s
( L(1)  1 / s )

A
i

s  1e X
 i
ri 

At X  0 :
A s  1 0
e
ri 
s 1

I0
2s
ri  I 0
2s s  1
r I
 V m  i 0 e  X s 1
2s s  1
r I 
 X

 X

 Vm  i 0 e  X erfc 
 T   e X erfc 
 T 
4 
2 T

2 T


A
 Electrotonic distance in the cable : X = x/
 Electrotonic length of the cable : L = l/
 A cable is long, when its electrotonic length  2
s 1 


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Error-function and complementary error-function 2

 e x
Properties:
erf ( x ) 
2

x
e
 y2
dy
0
d
2  x2
erf ( x ) 
e
dx

erf (0)  0,
erf ()  1
erf symmetrinen:
erf   x   erf  x 
erfc ( x )  1  erf ( x )
1

2

2


e
x
2
y
 e dy
0
 y2
dy
x
erfc ()  0, erfc (0)  1,
erfc ()  2
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General solution of a biological ”cable problem”
Typical number of
pages in a textbook
2
 partial differential equations describing the system
1
 transformation into dimensioless form
1
Internal regularities of the system  equations
1
 L-transformation and general solution
200
 special solutions with different initial and boundary conditions
model
1.
2.
3.
4.
5.
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Infinite cable, current injection, steady-state solution:
Vm 
ri  I 0   X
 X
 X
 X

e
erfc

T

e
erfc

T




4 
2 T

2 T

Kun T  , erfc
 T 0


ja erfc  T  2
In cells with large
diameter  ~ mm;
usually in m range
rI  
 vm ( x , )  i 0 e 
2
x
Input resistance:
V ( x  0, t  )
RN  m
I0
ri 
(infinite cable)
2
r r
 mi
Semi-infinite cable: RN  2 RN
2
Rm Ri / 2
Rm Ri / 2
1 Rm Ri
RN 

 2 
3
3
2 2 a  a
a2
2 a 2
Input resistance measured at the point of current injection

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Infinite cable, internal current step, time behaviour:
ri  I 0   X
 X
 X
 X

e
erfc

T

e
erfc

T




4 
2 T

2 T

At X  0 :
Vm 
ri  I 0 
erfc  T  erfc T 


4
r I
 i 0 1  erf  T  1  erf T 


4
r I
 i 0 erf T
2

Vm 



 
Charging of membrane
capacitance slower at
longer distances
 

 
erf symmetrinen:
erf   x   erf  x 
Spherical cell
Sphrical cell or cable with
an axial electrode
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 Elsewhere: