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Computational Biology, Part H
Neuronal Modeling
Robert F. Murphy
Copyright 1996, 1999-2004.All rights reserved.
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Basic Neurophysiology
An imbalance of charge across a membrane
is called a membrane potential
The major contribution to membrane
potential in animal cells comes from
imbalances in small ions (e.g., Na, K)
The maintenance of this imbalance is anactive process carried out by ion pumps
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Basic Neurophysiology
The cytoplasm of most cells (including
neurons) has an excess of negative ions over
positive ions (due to active pumping ofsodium ions out of the cell)
By convention this is referred to as a
negative membrane potential (insideminus outside)
Typical resting potential is -50 mV
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Basic Neurophysiology
Ion pumps require energy (ATP) to carry
ions across a membrane up a concentration
gradient (theygenerate a potential)
Ion channels allow ions to flow across a
membrane down a concentration gradient
(they dissipate a potential)
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Basic Neurophysiology
A cell is said to be electrically polarized
when it has a non-zero membrane potential
A dissipation (partial or total) of the
membrane potential is referred to as a
depolarization, while restoration of the
resting potential is termed repolarization
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Basic Neurophysiology
Ion channels can switch between open and
closed states
If an ion channel can switch its state due to
changes in membrane potential, it is said to
be voltage-sensitive
A membrane containing voltage-sensitiveion channels and/or ion pumps is said to be
an excitable membrane
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Basic Neurophysiology
An idealized neuron consists of
soma orcell body
contains nucleus and performs metabolic functions
dendrites
receive signals from other neurons through synapses
axonpropagates signal away from soma
terminal branches
form synapses with other neurons
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Basic Neurophysiology
The junction between the soma and the axon
is called the axonhillock
The soma sums (integrates) currents
(inputs) from the dendrites
When the received currents result in a
sufficient change in the membrane potential,a rapid depolarization is initiated in the axon
hillock
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Basic Neurophysiology
The depolarization is caused by opening of
voltage-sensitive sodium channels that
allow sodium ions to flow into the cell
The sodium channels only open in response
to a partial depolarization, such that a
threshold voltage is exceeded
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Basic Neurophysiology
As sodium floods in, the membrane
potential reverses, such that the interior is
now positive relative to the outside
This positive potential causes voltage-
sensitive potassium channels to open,
allowing K+ ions to flow out The potential overshoots (becomes more
negative than) the resting potential
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Basic Neurophysiology
The fall in potential triggers the sodium
channels to close, setting the stage for
restoration of the resting potential bysodium pumps
This sequential depolarization, polarity
reversal, potential overshoot andrepolarization is called an action potential
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Action Potential
-80
-60
-40
-20
0
20
40
60
Voltage(mV)
0
10
20
30
40
0 2 4 6 8 10Time (ms)
G(Na)
G(K)
0
50
100
150
Stimulus
(uA)
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Basic Neurophysiology
The depolarization in the axon hillock
causes a depolarization in the region of the
axon immediately adjacent to the hillock
Depolarization (and repolarization)
proceeds down the axon until it reaches the
terminal branches, which releaseneurotransmitters to stimulate neurons
with which they form synapses
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Basic Neurophysiology
These sequential depolarizations form a
traveling wave passing down the axon
Note that while a signal is passed down theaxon, it is not comparable to an electrical
signal traveling down a cable
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Basic Neurophysiology
Current flows in an electrical cable
are in the direction that the signal is propagating
consist of electrons
Current flows in a neuron
are transverse to the signal propagation
consist of positively-charged ions
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The Hodgkin-Huxley Model
Based on electrophysiological
measurements of giant squid axon
Empirical model that predicts experimentaldata with very high degree of accuracy
Provides insight into mechanism of action
potential
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The Hodgkin-Huxley Model
Define
v(t) voltage across the membrane at time t
q(t) net charge inside the neuron at t
I(t) current of positive ions into neuron at t
g(v) conductance of membrane at voltage v
Ccapacitance of the membrane
Subscripts Na, K and L used to denote specific
currents or conductances (L=other)
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The Hodgkin-Huxley Model
Start with equation for capacitor
v(t) q( t)
C
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The Hodgkin-Huxley Model
Consider each ion separately and sum
currents to get rate of change in charge and
hence voltagedq
dt INa IK IL
INa
gNa
(v vNa
)
IK gK(v vK )
IL gL (v vL )
dv
dt
1
CgNa
(v)(v vNa
) gK
(v)(v vK
) gL
(v vL
)
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The Hodgkin-Huxley Model
Central concept of model: Define three state
variables that represent (or control) the
opening and closing of ion channelsm controls Na channel opening
h controls Na channel closing
n controls K channel opening
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The Hodgkin-Huxley Model
Define relationship of state variables to
conductances of Na and K
gNa gNam3h
gK gKn4
0 m, n, h 1
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The Hodgkin-Huxley Model
Can write differentials form,n,h with
respect to t
Gives set of four coupled, non-linear,ordinary differential equations
Must be integrated numerically
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Hodgkin-Huxley Gates
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-60
-40
-20
0
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40
60
Voltage(mV)
0
50
100
150
Stimulus
(uA)
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10Time (ms)
m gate (Na)
h ga te (Na)
n gate (K)
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Interactive demonstration
(Integration of Hodgkin-Huxley equations
using Maple)
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Interactive demonstration
(Fitzhugh-Nagamo simplification)