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Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
Neuronal Modeling
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Neuronal Modeling

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  • 1. Computational Biology, Part 20 Neuronal Modeling Robert F. MurphyRobert F. Murphy CopyrightCopyright ©© 1996, 1999, 2001.1996, 1999, 2001. All rights reserved.All rights reserved.
  • 2. Basic Neurophysiology An imbalance of charge across a membraneAn imbalance of charge across a membrane is called ais called a membrane potentialmembrane potential The major contribution to membraneThe major contribution to membrane potential in animal cells comes frompotential 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 anThe maintainance of this imbalance is an activeactive process carried out by ion pumpsprocess carried out by ion pumps
  • 3. Basic Neurophysiology The cytoplasm of most cells (includingThe cytoplasm of most cells (including neurons) has an excess of negative ions overneurons) has an excess of negative ions over positive ions (due to active pumping ofpositive 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 aBy convention this is referred to as a negative membrane potentialnegative membrane potential (inside(inside minus outside)minus outside) TypicalTypical resting potentialresting potential is -50 mVis -50 mV
  • 4. Basic Neurophysiology Ion pumpsIon pumps require energy (ATP) to carryrequire energy (ATP) to carry ions across a membraneions across a membrane upup a concentrationa concentration gradient (theygradient (they generategenerate a potential)a potential) Ion channelsIon channels allow ions to flow across aallow ions to flow across a membranemembrane downdown a concentration gradienta concentration gradient (they(they dissipatedissipate a potential)a potential)
  • 5. Basic Neurophysiology A cell is said to be electricallyA 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 theA dissipation (partial or total) of the membrane potential is referred to as amembrane potential is referred to as a depolarizationdepolarization, while restoration of the, while restoration of the resting potential is termedresting potential is termed repolarizationrepolarization
  • 6. Basic Neurophysiology Ion channels can switch betweenIon channels can switch between openopen andand closedclosed statesstates If an ion channel can switch its state due toIf an ion channel can switch its state due to changes in membrane potential, it is said tochanges in membrane potential, it is said to bebe voltage-sensitivevoltage-sensitive A membrane containing voltage-sensitiveA membrane containing voltage-sensitive ion channels and/or ion pumps is said to beion channels and/or ion pumps is said to be anan excitable membraneexcitable membrane
  • 7. Basic Neurophysiology An idealizedAn idealized neuronneuron consists ofconsists of somasoma oror cell bodycell body contains nucleus and performs metabolic functionscontains nucleus and performs metabolic functions dendritesdendrites receive signals from other neurons throughreceive signals from other neurons through synapsessynapses axonaxon propagates signal away from somapropagates signal away from soma terminal branchesterminal branches formform synapsessynapses with other neuronswith other neurons
  • 8. Basic Neurophysiology The junction between the soma and theThe junction between the soma and the axon is called theaxon is called the axonaxon hillockhillock The soma sums (“integrates”) currentsThe soma sums (“integrates”) currents (“inputs”) from the dendrites(“inputs”) from the dendrites When the received currents result in aWhen the received currents result in a sufficient change in the membrane potential,sufficient change in the membrane potential, a rapid depolarization is initiated in thea rapid depolarization is initiated in the axon hillockaxon hillock
  • 9. Basic Neurophysiology The depolarization is caused by opening ofThe depolarization is caused by opening of voltage-sensitive sodium channels thatvoltage-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 responseThe sodium channels only open in response to a partial depolarization, such that ato a partial depolarization, such that a threshold voltagethreshold voltage is exceededis exceeded
  • 10. Basic Neurophysiology As sodium floods in, the membraneAs sodium floods in, the membrane potential reverses, such that the interior ispotential 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 outions to flow out The potential overshoots (becomes moreThe potential overshoots (becomes more negative than) the resting potentialnegative than) the resting potential
  • 11. Basic Neurophysiology The fall in potential triggers the sodiumThe fall in potential triggers the sodium channels to close, setting the stage forchannels to close, setting the stage for restoration of the resting potential byrestoration of the resting potential by sodium pumpssodium pumps This sequential depolarization, polarityThis sequential depolarization, polarity reversal, potential overshoot andreversal, potential overshoot and repolarization is called anrepolarization is called an action potentialaction potential
  • 12. Action 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)
  • 13. Basic Neurophysiology The depolarization in the axon hillockThe depolarization in the axon hillock causes a depolarization in the region of thecauses 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 theproceeds down the axon until it reaches the terminal branches, which releaseterminal branches, which release neurotransmittersneurotransmitters to stimulate neuronsto stimulate neurons with which they form synapseswith which they form synapses
  • 14. Basic Neurophysiology These sequential depolarizations form aThese sequential depolarizations form a traveling wavetraveling wave passing down the axonpassing down the axon Note that while a signal is passed down theNote that while a signal is passed down the axon, it isaxon, it is notnot comparable to an electricalcomparable to an electrical signal traveling down a cablesignal traveling down a cable
  • 15. Basic Neurophysiology Current flows in an electrical cableCurrent flows in an electrical cable are in the direction that the signal isare 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
  • 16. The Hodgkin-Huxley Model Based on electrophysiologicalBased on electrophysiological measurements of giant squid axonmeasurements of giant squid axon Empirical model that predicts experimentalEmpirical model that predicts experimental data with very high degree of accuracydata with very high degree of accuracy Provides insight into mechanism of actionProvides insight into mechanism of action potentialpotential
  • 17. The Hodgkin-Huxley Model DefineDefine v(t)v(t) ≡≡ voltage across the membrane at timevoltage across the membrane at time tt q(t)q(t) ≡≡ net charge inside the neuron atnet charge inside the neuron at tt I(t)I(t) ≡≡ current of positive ions into neuron atcurrent of positive ions into neuron at tt g(v)g(v) ≡≡ conductance of membrane at voltageconductance of membrane at voltage vv CC ≡≡ capacitance of the membranecapacitance of the membrane Subscripts Na, K and L used to denote specificSubscripts Na, K and L used to denote specific currents or conductances (L=“other”)currents or conductances (L=“other”)
  • 18. The Hodgkin-Huxley Model Start with equation for capacitorStart with equation for capacitor v(t ) = θ(τ) Χ
  • 19. The Hodgkin-Huxley Model Consider each ion separately and sumConsider each ion separately and sum currents to get rate of change in charge andcurrents to get rate of change in charge and hence voltagehence voltage dq dt = ΙΝα + ΙΚ + ΙΛ ΙΝα = γ Να (ϖ− ϖΝα ) ΙΚ = γ Κ (ϖ− ϖΚ ) ΙΛ = γ Λ (ϖ− ϖΛ ) δϖ δτ = −1 Χ γ Να (ϖ)(ϖ− ϖΝα ) + γ Κ (ϖ)(ϖ− ϖΚ ) + γ Λ (ϖ− ϖΛ )[ ]
  • 20. The Hodgkin-Huxley Model Central concept of model: Define three stateCentral concept of model: Define three state variables that represent (or “control”) thevariables that represent (or “control”) the opening and closing of ion channelsopening and closing of ion channels mm controls Na channel openingcontrols Na channel opening hh controls Na channel closingcontrols Na channel closing nn controls K channel openingcontrols K channel opening
  • 21. The Hodgkin-Huxley Model Define relationship of state variables toDefine relationship of state variables to conductances of Na and Kconductances of Na and K gNa = γ Ναµ 3 η γ Κ = γ Κ ν4 0 ≤ µ , ν, η ≤ 1
  • 22. The Hodgkin-Huxley Model Can write differentials forCan write differentials for m,n,hm,n,h withwith respect torespect 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
  • 23. Hodgkin-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)
  • 24. Interactive demonstration (Integration of Hodgin-Huxley equations(Integration of Hodgin-Huxley equations using Maple)using Maple)

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