noah weiss & susan koons. neuroscience: 3ed
Post on 19-Dec-2015
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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