1. Aim of this class
2. A first order approximation of neuronal biophysics
1. Introduction
2. Electro-chemical properties of neurons
3. Ion channels and the Action Potential
4. The Hodgkin-Huxley model
5. The Cable equation
6. Multi-compartmental models
Contents
The Cable equation
• Describes the propagation of signals in electrical cables, and in this case it will be applied to dendrites and axons
Case study: Simultaneous intracellular recordings from soma and dendrite
A) An action potential is produced in the soma
B) A set of axon fibers is stimulated to produce a compound excitatory post-synaptic potential
What are the differences and how do you explain them ?
• The longitudinal resistance of an axon or dendrite is:
)/( 2axrR LL
with rL - intracellular resistivity (m)
Δx - segment length
a - segment radius
• The intracellular resistivity depends on the ionic composition of the intracellular milieu (and on the distribution of organelles)
xr
txVaI
LL
),(2
• The longitudinal current through such a segment is:
where ΔV(x,t) is the voltage gradient across the segment
),(),(),( txVtxxVtxV
• Currents flowing in the increasing direction of x are defined to be positive
x
txV
r
aI
LL
),(2
• In the limit 0x :
• Besides the longitudinal currents, there are several membrane currents flowing in/out of the segment:
do you understand the formula ?
)(2222
em
rightLleftLm iixa
x
V
r
a
x
V
r
a
t
Vxca
• Applying the principle of charge conservation for the previous cable segment we get:
• Divide the above by xa2 such that the r.h.s. is in the limit 0x
x
V
r
a
xx
V
r
a
x
V
r
a
x LleftLrightL
2221
emL
m iix
Va
xart
Vc
2
2
1
• Under the assumption that rL does not vary with position the cable
equation is obtained:
• The radius of the cable is allowed to vary to simulate the tapering
of dendrites
• Boundary conditions required for V(x,t) and xtxV /),(
• Linear cable approximation: Ohmic membrane current im
mrestm rVVi /)(
emL
m irxr
a
tc
2
2
2
• Use change of variables restVV
• And multiply by rm
emm irxt
2
22
mmm rc
with membrane time constantto get:
and electrotonic length
L
m
r
ar
2(in the linear cable approximation)
• Steady state (A) and transient (B) solutions to the linear cable equation:
Multi-compartmental models
• To calculate the membrane potential dynamics of a neuron, the cable equation has to be discretized and solved numerically
• The membrane potential dynamics of a single isolated compartment
is described by:
A
Ii
dt
dVc e
mm
injected current through electrode
surface area of compartment
membrane currents due to ion-channels / membrane area
specific membrane capacitance (Fm2)
)()( 11,11,
VVgVVg
A
Ii
dt
dVc e
mm
• Several compartments coupled in a non-branching manner:
• The Ohmic coupling constants between two compartments with same length and radii:
2', 2 Lr
ag
L
• Next time you see a neuron, you should see this: