Computational Biology, Part 20Neuronal Modeling
Computational Biology, Part 20Neuronal Modeling
Robert F. MurphyRobert F. Murphy
Copyright Copyright 1996, 1999, 2001. 1996, 1999, 2001.
All rights reserved.All rights reserved.
Basic NeurophysiologyBasic Neurophysiology
An imbalance of charge across a membrane An imbalance of charge across a membrane is called a is called a membrane potentialmembrane potential
The major contribution to membrane The major contribution to membrane potential in animal cells comes from potential in animal cells comes from imbalances in small ions (e.g., Na, K)imbalances in small ions (e.g., Na, K)
The maintainance of this imbalance is an The maintainance of this imbalance is an activeactive process carried out by ion pumps process carried out by ion pumps
Basic NeurophysiologyBasic Neurophysiology
The cytoplasm of most cells (including The cytoplasm of most cells (including neurons) has an excess of negative ions over neurons) has an excess of negative ions over positive ions (due to active pumping of positive ions (due to active pumping of sodium ions out of the cell)sodium ions out of the cell)
By convention this is referred to as a By convention this is referred to as a negative membrane potential negative membrane potential (inside (inside minus outside)minus outside)
Typical Typical resting potential resting potential is -50 mVis -50 mV
Basic NeurophysiologyBasic Neurophysiology
Ion pumps Ion pumps require energy (ATP) to carry require energy (ATP) to carry ions across a membrane ions across a membrane upup a concentration a concentration gradient (they gradient (they generate generate a potential)a potential)
Ion channels Ion channels allow ions to flow across a allow ions to flow across a membrane membrane downdown a concentration gradient a concentration gradient (they (they dissipatedissipate a potential) a potential)
Basic NeurophysiologyBasic Neurophysiology
A cell is said to be electrically A cell is said to be electrically polarizedpolarized when it has a non-zero membrane potentialwhen it has a non-zero membrane potential
A dissipation (partial or total) of the A dissipation (partial or total) of the membrane potential is referred to as a membrane potential is referred to as a depolarizationdepolarization, while restoration of the , while restoration of the resting potential is termed resting potential is termed repolarizationrepolarization
Basic NeurophysiologyBasic Neurophysiology
Ion channels can switch between Ion channels can switch between openopen and and closedclosed states states
If an ion channel can switch its state due to If an ion channel can switch its state due to changes in membrane potential, it is said to changes in membrane potential, it is said to be be voltage-sensitivevoltage-sensitive
A membrane containing voltage-sensitive A membrane containing voltage-sensitive ion channels and/or ion pumps is said to be ion channels and/or ion pumps is said to be an an excitable membraneexcitable membrane
Basic NeurophysiologyBasic Neurophysiology
An idealized An idealized neuronneuron consists of consists of somasoma or or cell bodycell body
contains nucleus and performs metabolic functionscontains nucleus and performs metabolic functions
dendritesdendrites receive signals from other neurons through receive signals from other neurons through synapsessynapses
axonaxon propagates signal away from somapropagates signal away from soma
terminal branchesterminal branches form form synapsessynapses with other neurons with other neurons
Basic NeurophysiologyBasic Neurophysiology
The junction between the soma and the The junction between the soma and the axon is called the axon is called the axonaxon hillockhillock
The soma sums (“integrates”) currents The soma sums (“integrates”) currents (“inputs”) from the dendrites(“inputs”) from the dendrites
When the received currents result in a When the received currents result in a sufficient change in the membrane sufficient change in the membrane potential, a rapid depolarization is initiated potential, a rapid depolarization is initiated in the axon hillockin the axon hillock
Basic NeurophysiologyBasic Neurophysiology
The depolarization is caused by opening of The depolarization is caused by opening of voltage-sensitive sodium channels that voltage-sensitive sodium channels that allow sodium ions to flow into the cellallow sodium ions to flow into the cell
The sodium channels only open in response The sodium channels only open in response to a partial depolarization, such that a to a partial depolarization, such that a threshold voltage threshold voltage is exceededis exceeded
Basic NeurophysiologyBasic Neurophysiology
As sodium floods in, the membrane As sodium floods in, the membrane potential reverses, such that the interior is potential reverses, such that the interior is now positive relative to the outsidenow positive relative to the outside
This positive potential causes voltage-This positive potential causes voltage-sensitive potassium channels to open, sensitive potassium channels to open, allowing Kallowing K++ ions to flow out ions to flow out
The potential overshoots (becomes more The potential overshoots (becomes more negative than) the resting potentialnegative than) the resting potential
Basic NeurophysiologyBasic Neurophysiology
The fall in potential triggers the sodium The fall in potential triggers the sodium channels to close, setting the stage for channels to close, setting the stage for restoration of the resting potential by restoration of the resting potential by sodium pumpssodium pumps
This sequential depolarization, polarity This sequential depolarization, polarity reversal, potential overshoot and reversal, potential overshoot and repolarization is called an repolarization is called an action potentialaction potential
Action PotentialAction Potential
-80
-60
-40
-20
0
20
40
60
Voltage (mV)
0
10
20
30
40
0 2 4 6 8 10Time (ms)
Conductance (mS/cm2)
G(Na)
G(K)
0
50
100
150
Stimulus (uA)
Basic NeurophysiologyBasic Neurophysiology
The depolarization in the axon hillock The depolarization in the axon hillock causes a depolarization in the region of the causes a depolarization in the region of the axon immediately adjacent to the hillockaxon immediately adjacent to the hillock
Depolarization (and repolarization) Depolarization (and repolarization) proceeds down the axon until it reaches the proceeds down the axon until it reaches the terminal branches, which release terminal branches, which release neurotransmittersneurotransmitters to stimulate neurons to stimulate neurons with which they form synapseswith which they form synapses
Basic NeurophysiologyBasic Neurophysiology
These sequential depolarizations form a These sequential depolarizations form a traveling wavetraveling wave passing down the axon passing down the axon
Note that while a signal is passed down the Note that while a signal is passed down the axon, it is axon, it is notnot comparable to an electrical comparable to an electrical signal traveling down a cablesignal traveling down a cable
Basic NeurophysiologyBasic Neurophysiology
Current flows in an electrical cableCurrent flows in an electrical cable are in the direction that the signal is are in the direction that the signal is
propagatingpropagating consist of electronsconsist of electrons
Current flows in a neuronCurrent flows in a neuron are transverse to the signal propagationare transverse to the signal propagation consist of positively-charged ionsconsist of positively-charged ions
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Based on electrophysiological Based on electrophysiological measurements of giant squid axonmeasurements of giant squid axon
Empirical model that predicts experimental Empirical model that predicts experimental data with very high degree of accuracydata with very high degree of accuracy
Provides insight into mechanism of action Provides insight into mechanism of action potentialpotential
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
DefineDefine v(t) v(t) voltage across the membrane at time voltage across the membrane at time tt q(t) q(t) net charge inside the neuron at net charge inside the neuron at tt I(t) I(t) current of positive ions into neuron at current of positive ions into neuron at tt g(v) g(v) conductance of membrane at voltage conductance of membrane at voltage vv CC capacitance of the membranecapacitance of the membrane Subscripts Na, K and L used to denote specific Subscripts Na, K and L used to denote specific
currents or conductances (L=“other”)currents or conductances (L=“other”)
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Start with equation for capacitorStart with equation for capacitor
v(t ) =q(t)C
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Consider each ion separately and sum Consider each ion separately and sum currents to get rate of change in charge and currents to get rate of change in charge and hence voltagehence voltage
dq
dt= I Na + I K + I L
I Na =gNa (v−vNa )I K =gK (v−vK )I L =gL (v−vL )
dv
dt=
−1
CgNa (v)(v−vNa ) + gK (v)(v−vK ) + gL (v−vL )[ ]
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Central concept of model: Define three state Central concept of model: Define three state variables that represent (or “control”) the variables that represent (or “control”) the opening and closing of ion channelsopening and closing of ion channels mm controls Na channel opening controls Na channel opening hh controls Na channel closing controls Na channel closing nn controls K channel opening controls K channel opening
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Define relationship of state variables to Define relationship of state variables to conductances of Na and Kconductances of Na and K
g Na = gNam3h
gK = gK n4
0 ≤m,n,h≤1
The Hodgkin-Huxley ModelThe Hodgkin-Huxley Model
Can write differentials for Can write differentials for m,n,h m,n,h with with respect to respect to tt
Gives set of four coupled, non-linear, Gives set of four coupled, non-linear, ordinary differential equationsordinary differential equations
Must be integrated numericallyMust be integrated numerically
Hodgkin-Huxley GatesHodgkin-Huxley Gates
-80
-60
-40
-20
0
20
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)
Gate param
value
m gate (Na)
h gate (Na)
n gate (K)
Interactive demonstrationInteractive demonstration
(Integration of Hodgin-Huxley equations (Integration of Hodgin-Huxley equations using Maple)using Maple)
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