Dr. Aslı AYKAÇ

a=∆V/∆t 0       t dt dV t V Lim

ma

F



the equation could be arranged as first order differential equation

dt dv m

F  or

ma

F



the equation could be arranged as first order differential equation

dt dv m F  or dt dx v  _since

(

)

dt

dx

dt

d

m

dt

x

d

m

F





as a second order differential equation important 2 2 2

)

(

dt

dx

dt

x

d

_

A

x

y

dx

dy









1

A

x

y

dx

dy









1

is the general solution İf additional conditions are given

0

0 

 için y x

the condition is satisfied only

0



A

x

y



R

V

I



CV

Q



R

V

I



CV

Q



as a definition rate of change in the amount of charged particle per unit time is the current

dt

dv

C

dt

dQ

I

_C





R

V

I



r c s

I

I

I





E

E

E





R

I

I

R

I

E





(



)

C

E

I

R

R

dt

dE

_

_

)

1

(

t

/ RmCm

m

s

R

e

I

E







IsRm

E

t







IsRm

E

E

E













1

a

R



)

1

(

R

e

I

E





N

R

E





)

1

(

R

e

I

E





-the maximal amplitude of the passive membrane potential is defined by the input resistance of the cell.

IsRm

E

t









-Membrane capacitance Cm prolongs the time course of the electrical signals (τ_m = R_mC_m).

)

1

(

R

e

I

E





-Membrane capacitance is proportional to the surface area while the input -resistance is inversely proportional.

2 4 a Rm I E R Rinput S N     



Axon is tubular in shape



The whole membrane is homegenous.



Physical properties are constant and are not

dependent on voltage



Axonal currents are unidirectional (radial

currents are ignored)



Extracellular solution is very conductive, its

 Vm (x ve t) changes as a function of time and

distance

 Voltage change is in the form of a reduction  Rate of change is related to r_ii_i

 Axoplasmic current i_i will get smaller by distance

i ii r dx t x dVm( , ) _{ } m i dx di   i m i i r dx di r dx Vm d    2 2 rm Vm dt dVm Cm i i i_m  _c  _r  

rm

Vm

dt

dVm

Cm

dx

Vm

d

r





1

ri

a

Ri





Rm



2



a

rm

Cm



cm

/

2



a

with refer to the resistivity of a 1 cm2 _{membrane, and}

1cm3 _axoplasm

Ri specific intracellular resistivity (Ω-cm) Rm specific membrane resistance (Ω-cm2₎ Cm specific membrane capacitance (F/cm2₎

for an axon in any shape

ri intracellular resistivity (Ω/cm) rm membrane resistance (Ω-cm) cm membrane capacitance (F/cm) Considering the tubular shape of the axon

Ri aRm r r i m 2  



rm

Vm

dt

dVm

Cm

dx

Vm

d

r





1



/

x

X



t

t

T



/

0 2 2    Vm dT dVm dX Vm d





T

Vm



X



riIo

e

2

)

,

(





2

)

,

(

x

riIo

e

Vm





X

X _e

Vo

Ri aRm r r i m 2   





a



)

( T

erf

2 1 ) , ( ) , ( _  X Vm X T Vm   5 . 0 2 5 . 0 2     t t x veya T X m dt dx   2 / 1 2 ) 2 ( 2 RmRiCm a t dt dx m     

a





 As in the spherical cell input resistance defines the

amplitude of the pasive potential response

 Length constant λ defines the amplitude of the

propagated electrical signal.

 Length constant becomes longer as the diameter of

the axon increases

 As in the spherical cell membrane capacitance

prolongs the time course of the passive signals (τm = RmCm).

 Rate of passive spread is faster in axons with large

λgreen =2,24 mm λred =1,58 mm lgreen=100 μm L=l/λ=0,1/2.24=0,045 lred=500 μm L=0,5/1.58=0,32

İf both dendrites are depolirized to the potential level of Vo, as can be

calculated by the the equation

96 % of Vo will propagate to the

green soma. However only 72% of Vo will arrive to the red soma

 / x X _e Vo V _{ }

d s d s

I

I

G

G

_



