Neuronal Modeling

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


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


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)


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


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


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


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


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


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


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)


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


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


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


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”)


Start with equation for capacitorStart with equation for capacitor

v(t ) =q(t)C


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 )[ ]


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


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


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

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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)

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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Neuronal Modeling