Neurophysics

Adrian Negrean

- part 2 -

adrian.negrean@cncr.vu.nl

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: