Modeling the Axon

Noah Weiss & Susan Koons

Neuron Anatomy

Ion Movement

Neuroscience: 3ed

Biological Significance of Myelination

Neuroscience: 3ed

Biological Significance of Myelination

Neuroscience: 3ed

Biological Significance of Myelination

Neuroscience: 3ed

Circuit Notation

• Resistors: Linear or non-linear

F(V,I)=0 V=IR

I=f(V) V = h(I)

• Capacitors:

• Pumps:

CC

dVC I

dt

dA VAdt

Circuit Laws

• Kirchhoff’s Current Law:The principle of conservation of electric charge implies that:

The sum of currents flowing towards a point is equal to the sum of currents flowing away from that point.

i1

i2 i3

i1 = i2 + i3

Circuit Laws

• Kirchhoff’s Voltage Law The directed sum of the electrical potential differences around any closed circuit must be zero. (Conservation of Energy)

VR1 + VR2 + VR3 + VC =0R1

R2

R3

Circuit Model

Circuit Model

• Neurons can be modeled with a circuit model– Each circuit element has an IV characteristic– The IV characteristics lead to differential

equation(s)• Use Kirchhoff’s laws and IV characteristics to get the

differential equations

Equations- Circuit Model

• Solve for and use

• To find use the current law:– Additionally, define the absolute current– Assume a linear resistor with (small) resistance γ in series

with the pumps

• Use Kirchhoff’s laws to get:

AI

CI

1 2 AA A I

1 2SI A A

Cext Na K KC A

A S C A

S A C A

Na Na Na NaC

dVC I I f V E I

dtI I V I

I I V I

I V E h I

CC

dVC I

dt

Reducing Dimensions

• Assume the “N” curve doesn’t interact with the “S” curve– All three parts of “N” are

within primary branch of “S”

– Also, let ε = 0:

Cext Na K KC A

A S C A

S A C A

Na Na NaC

dVC I I f V E I

dtI I V I

I I V I

I g V E

I

V

KNa

Reducing Dimensions

• Substitute the 4th equation into the 1st

• Nullclines: Set the derivatives equal to zero– Nontrivial nullcline in the 2nd and 3rd equations are

same– Re-arrange and obtain the following:

Nullcline equations

ext Na Na K KA C C

C A

I I g V E f V E

V I

Cext Na Na K KC C A

A S C A

S A C A

dVC I g V E f V E I

dtI I V I

I I V I

Resting Potential

• Let – Analyze the nullclines: vector field directions– Assume C<<1: singular perturbation– nullcline intersects nullcline in primary

branch

0extI

IA

Vc

AI

CV

IA nullclineVC nullcline

Action Potential Conditions

• Increase to shift the nullcline upward• To get an action potential:

extIC

V

Action Potential Conditions

• The “N” curve has 2 “knee” points at

• The “S” curve is merely linear by assumption (i.e. is constant)

• Some algebra shows that must satisfy:

1 2 and V V

1

K C 1 2 1 C 1 1 C 2

2 1 2 1 2 C

0, if

if

if

C

C

V V

f V GV G G V V V V V

G G V V V V

Nullcline equations

ext Na Na K KA C C

C A

I I g V E f V E

V I

Nag

extI

Na Na NaC 2 1 2 1 C 1 21 with ,V g G g E V G G V V V

extI >=

Circuit Equations of a Node

Cext Na K KC A

A S C A

S A C A

Na Na Na NaC

dVC I I f V E I

dtI I V I

I I V I

I V E h I

Multiple Nodes

Inside the cell

Outside the cell

Multiple Nodes

• Recall the equations for one node:– There is no outgoing

current

• Consider a second node that is not coupled to the first node– It should have the same

equation (but with different currents)

Cext Na K KC A

AS C A

SA C A

NaNa Na NaC

dVC I I f V E I

dtdI

I V Idt

dII V I

dtdI

V E h Idt

22 2 2 2 2 2

22 2 2

22 2 2

22 2 2 2

Cext Na K KC A

AS C A

SA C A

NaNa Na NaC

dIdt

dIdtdI

dt

dVC I I f V E I

dt

I V I

I V I

V E h I

Multiple Nodes

• Couple the nodes by adding a linear resistor between them

1 2 11 1 1 1

11

1 1 1

11 1 1

11 1 1

2 2 12 2 2 2

12

2 2 2

2

C C Cext Na K KC A

AS C A

SA C A

NaNa Na NaC

C C CNa K KC A

AS C A

S

dV V VC I I f V E I

Rdt

dII V I

dtdI

I V IdtdI

V E h Idt

dV V VC I f V E I

Rdt

dII V I

dtdI

2 2 2

22 2 2

A C A

NaNa Na NaC

I V IdtdI

V E h Idt

Current between the nodes

The General Case (N nodes)

• This is the general equation for the nth node

• In and out currents are derived in a similar manner:

1n

n n n n n nCout inNa K KC A

nn n nAS C A

nn n nSA C A

nn n nNa

Na Na NaC

dVC I I f V E I I

dtdI

I V Idt

dII V I

dtdI

V E h Idt

1 1

1

1

if 1

if 1

if 1

0 if

extn n nout C C

n

n nC Cn nin

I nI V V

nR

V Vn NI R

n N

Results

Forcing current

C=.1 pF

Results

C=.1 pF

Results

C=.1 pF

Results

C=.1 pF

Results

C=.01 pF

Results

C=.01 pF

Results

C=.7 pF

Results

C=.7 pF

Transmission Failure

(x10 pF)

Transmission Failure

(x10 pF)

(ms)

The Importance of Myelination

The Importance of Myelination

The Importance of Myelination

The Importance of Myelination

(ms)

(x1

00

mV

)

The Importance of Myelination(x

10

0 m

V)

(ms)

The Importance of Myelination- Myelinated Axon

Conclusions

• Myelination matters! Myelination decreases capacitance and increases conductance velocity

• If capacitance is too high, the pulse will not transmit

• First model that shows a pulse that travels down the entire axon without dying out