Nens220, Lecture 3 Cables and Propagation

Cable theory

• Developed by Kelvin to describe properties of current flow in transatlantic telegraph cables.

• The capacitance of the “membrane” leads to temporal and spatial differences in transmembrane voltage.

From Johnston & Wu, 1995

Current flow in membrane patch RC circuit

m=Cm*Rm

And now in a system of membrane patches

Components of current flow in a neurite

xdt

dVcI mc

Lcm III

xVVgI LmL )(

mg normalized leak conductance per unit length of neurite

mc normalized membrane capacitance per unit length of neurite

xr

xVxxVxI

i

)()(

)(

xr

xxVxVxxI

i

)()(

)(

0)()( xxIxIII LCir normalized internal resistance per unit length of neurite

Solving Kirchov’s law in a neurite

0)()( xxIxIII LC

0)()()()(

)(

xr

xxVxV

xr

xVxxVxVVgx

dt

dVc

iiLmm

^

Lg xdt

dVcI mc

Lcm III

xVVgI LmL )(

xr

xVxxVxI

i

)()(

)(

xr

xxVxVxxI

i

)()(

)(

Final derivation of cable equation

divide by x and approach limit x -> 0

0)(1

)(2

2

x

xV

rVVg

dt

dVc

iLmm

^

Lg

0)(1

)(2

2

x

xV

grVV

dt

dV

miL

divide by gm

21 migr

membrane space constant, is membrane time constant

0)(

)(2

22

x

xVVV

dt

dVL

0)()()()(

)(

xr

xxVxV

xr

xVxxVxVVgx

dt

dVc

iiLmm

Cable properties, unit properties• For membrane, per unit area

– Ri = specific intracellular resistivity (~100 -cm) – Rm = specific membrane resistivity (~20000 -cm2)

• Gm =specific membrane conductivity (~0.05 S/cm2)– Cm = specific membrane capacitance (~ 1 F/cm2)

• For cylinder, per unit length:– ri = axial resistance (units = /cm)

• Intracellular resistance ()= resistivity (Ricm) * length (l, cm)/ cross sectional area (πr2, cm2)

• Resistance per length (ri) = resistivity / cross sectional area = Ri/πr2 (/cm) – For 1 m neurite (axon) = 100 -cm/(π*.00005 cm2) = ~13G/cm = 1.3 G/mm = 1.3M/m– For 5 m neurite (dendrite) = 100 -cm/(π*.00025cm2) = ~ 500 M/cm = 50 M/mm = 50k/m

– rm = membrane resistance (units: cm, divide by length to obtain total resistance) • Rm2πr. Probably more intuitive to consider reciprocal resistance, or conductance:

– In a neurite total conductance is Gm2πrl, i.e. proportional to membrane area (circumference * length) • Normalized conductance per unit length (gm) = Gm2πr (S/cm)

– For 1 m neurite (axon) = 0.05 S/cm2(2π*.00005cm) = ~16 nS/cm ~ 1.6pS/m» (equivalent normalized membrane resistance, rm obtained via reciprocation is ~60 Mohm-cm)

– For 5 m neurite (dendrite) = 0.05 S/cm2(2π*.00025cm) = ~80 nS/cm ~ 8pS/m» (rm ~ 13 Mohm-cm)

– cm = membrane capacitance (units: F/cm)• Derived as for gm, normalized capacitance per unit length = Cm2πr (F/cm)

– For 1 m neurite (axon) = 1 F/cm2(2π*.00005cm) = ~300 pF/cm ~ 30 fF/m– For 5 m neurite (dendrite) = 1 F/cm2(2π*.00025cm) = ~1.6 nF/cm ~160 fF/m

Cable equation• Solved for different boundary conditions

– Infinite cylinder– Semi infinite cylinder (one end)– Finite cylinder

i

m

i

m

m

R

rR

r

r

VmT

V

X

Vm

2

t/ T and , x/ X where

0

m

2

2

For 1 m neurite (axon) =sqrt(64e6/13e9) = 0.07 cm, 700 m

scales with square root of radius

For 5 m neurite (dendrite) =sqrt(13e6/79e9) = 0.16 cm, 1600 m

Electrotonic decay

Electrotonic decay in a neuron

Electrotonic decay in a neuron with alpha synapse

Compartmental models

• Can be developed by combining individual cylindrical components

• Each will have its own source of current and EL via the parallel conductance model

• Current will flow between compartments (on both ends) based on V and Ri

Reduced models of cells with complex morphologies

• Rall analysis

• Bush and Sejnowski

Collapsing branch structures

• From cable theory– conductance of a cable =

• (/2) (RmRi)-1/2(d)3/2

• When a branch is reached the conductances of the two daughter branches should be matched to that of the parent branch for optimal signal propagation

• This occurs when the sum of the two daughter g’s are equal to the parent g, which occurs when

• d03/2 = d13/2 + d23/2

• This turns out to be true for many neuronal structures

Bush and Sejnowski

Using Neuron

• Go to neuron.duke.edu and download a copy

• Work through some of the tutorials

Preview: dendritic spike generation

Stuart and Sakmann, 1994, Nature 367:69