Cable Properties Properties of the nerve, axon, cell body and dendrite

affect the distance and speed of membrane potential Passive conduction properties = cable properties Signal becomes reduced over distance depending

on the cable properties Current (I) – amount of charge moving past a point at

a given time A function of the drop in voltage (V) across the

circuit and the resistance (R) of the circuit Voltage – energy carried by a unit charge Resistance – force opposing the flow of electrical

current Ohm’s law: V = IR

Passive flow of currentA current traveling down a copper wire

Voltage Decreases With Distance

Conduction with decrement Due to resistance Intracellular fluid: high resistance decrement Extracellular fluid: high resistance

decrement Membrane: high resistance decrement

K+ leak channels (always open): some + charge leaks out current

Few K+ leak channels + charge leak out high membrane resistance

Cable Properties

Each area of axon consists of an electrical circuit • Three resisters: extracellular fluid (Re), the membrane

(Rm), and the cytoplasm (Rc)

• A capacitor (Cm) – stores electrical charge; two conducting materials (ICF and ECF) and an insulating layer (phospholipids)

Cable Properties

1. Loss of current across membrane (through rest channels)

loss of current across membrane results in membrane potential dropping with distance

dependent on the internal resistance (ri) and the membrane resistance (rm)

the length or space constant (λ) describes this property λ = distance (mm) at which V = 1/e V0 or the distance at which V

has decreased to 37% the relationship between the voltage at any distance (x) from the

applied (or original) voltage is :

 

Vx = Vo e-x/λ

Cable Properties2. Loss of current (charge) due to capacitance properties

of the membrane

cell membrane acts as a capacitor 2 conducting sheets separated by an insulating material

- the closer the sheets the better the capacitor lipid bilayer is 7 nm thick therefore = excellent capacitor it takes time and current (charge) to charge the membrane capacitor as current drops over the length of the nerve takes longer and longer to

charge the capacitor the time constant describes this effect τ is the time it takes to reach 63% of the final voltage (msec) τ = Rm x Cm the smaller the capacitance properties the less the current

loss and the faster the nerve impulse travels the larger the capacitance properties the more current loss and the

slower the nerve impulse time constants range from 1 to 20 msec.

 

Length Constant () Distance over which change in membrane potential will decrease

by 37% (1/e) where e = 2.718 dependent on the internal (ri) and membrane resistance (rm)  is largest when rm is high and ri is low

ro is usually low and constant λ = square root of (rm/ri) if the membrane resistance is large then the longer the impulse

will travel along the nerve before reaching 37% of original if the internal resistance is large then the shorter the impulse

will travel along the nerve before reaching 37% of original giant axon of squid (1mm diameter) λ = 13 mm mammalian nerve fiber (1 micron diameter)

λ = 0.2 mm

)/( orrr im

im rr /

Conduction Speed

• rm is inversely proportional to surface area: diameter surface area leak channels resistance

• ri is inversely proportional to volume: diameter volume resistance

• Effect of resistance rm conduction speed

ri conduction speed

• Do not cancel each other out: rm is proportional to radius, ri is proportional to radius2

• Therefore, net effect of increasing radius of the axon is to increase the speed of conduction

im rr /

Conduction Speed

Figure 5.25

Speed of Conduction and Capacitance

Capacitance – quantity of charge needed to create a potential difference between two surfaces of a capacitor

Depends on three features of the capacitor• Material properties: generally the same in cells• Area of the two conducting surfaces: area

capacitance• Thickness of the insulating layer: thickness

capacitance

Speed of Conduction and Capacitance

Time constant () - time needed to charge the capacitor; = rmcm

Low rm or cm low capacitor becomes full faster faster depolarization faster conduction

Conduction Speed

Two ways to increase speed: myelin and increasing the diameter of the axon

Table 5.3

Axon diameter

• increased axon diameter in axons increases action potential velocity - i.e. giant axon of squid = 1 mm diameter = huge!

• why does increasing the diameter of an axon increase the speed of an action potential?  

• rm, ri and cm are all related to the radius of a fiber • rm ~ ½ π radius • ri ~ 1/π radius2

• cm ~ radius - increase diameter of a fiber rm and ri decrease, but ri decreases faster, therefore benefit as the internal resistance decreases faster relative to the membrane resistance- therefore the distance the membrane potential can travel is increased by an increased diameter  

Axon diameter, cont….

• the length constant is increased - giant axon of squid (1 mm dia.) λ = 13mm- mammalian nerve fiber (1 micron dia.) λ  = 0.2mm

• - increase in fiber diameter also increases cm, but this increase is proportional to the increase in the radius while the decrease in

ri is proportional to the radius2

- therefore internal resistance decreases faster than the capacitance of the membrane- the decrease in ri speeds up the current transfer to the next region of the nerve and threshold is reached sooner

Giant Axons

• Easily visible to the naked eye• Not present in mammals

Figure 5.24

Myelin Increases Conduction Speed

membrane resistance: act as insulators current loss through leak channels membrane resistance

capacitance: thickness of insulating layer capacitance time to constant of membrane conduction speed

Nodes of Ranvier are needed to boost depolarization

Myelin Increases Conduction Speed

passive spread of the depolarizing current between the nodes is the rate limiting step on an action potential

depends on how much current is lost due the three cable properties

1. if the internal membrane resistance (ri) is high - current spread is not as far, speed of the action potential is slower

2. if the membrane resistance (rm) is low- current is lost and so current spread is slower and the action potential slows down

myelin increases rm so that little current is lost, passive spread of the current is further

3. if the membrane capacitance (cm) is high - the longer and more charge it takes to charge the capacitor and the slower the action potential

myelin decreases cm so that less current is lost in charging the capacitor and more is available to spread down the axon