General

442 views
398 views

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
442
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
10
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • Check direction for figures
  • http://www.computationalcellbiology.net/
    http://en.wikipedia.org/wiki/Hodgkin-Huxley_model
  • General

    1. 1. Computational Biology, Part 20 Stochastic Modeling / Neuronal Modeling Arvind Rao, Robert F. Murphy, Shann-Arvind Rao, Robert F. Murphy, Shann- Ching Chen, Justin NewbergChing Chen, Justin Newberg CopyrightCopyright ©© 2004-2009.2004-2009. All rights reserved.All rights reserved.
    2. 2. Stochastic Modeling in Biology
    3. 3. Stochastic Modeling in Biology Why? A: BetterWhy? A: Better resolutionresolution in speciesin species amountsamounts When? A: BiochemicalWhen? A: Biochemical kineticskinetics, gene, gene expression stochasticity in cellsexpression stochasticity in cells How? A:How? A: SSASSA Case studies:Case studies: Chemical master equationChemical master equation Biochemical kineticsBiochemical kinetics Gene networksGene networks
    4. 4. Why?Why? Recall Chemical kinetics examplesRecall Chemical kinetics examples 1.1. In differential/difference equations, weIn differential/difference equations, we examined regimes where number ofexamined regimes where number of molecules of reactants were always largemolecules of reactants were always large enough to have followed mass actionenough to have followed mass action kinetics.kinetics. 2.2. Think “Law of Large Numbers”Think “Law of Large Numbers”
    5. 5. Why Are Stochastic Models Needed? •• Much of the mathematical modeling ofMuch of the mathematical modeling of biochemical/gene networks represents gene expressionbiochemical/gene networks represents gene expression deterministicallydeterministically •• Deterministic models describe macroscopic behavior;Deterministic models describe macroscopic behavior; but many cellular constituents are present in smallbut many cellular constituents are present in small numbersnumbers •• Considerable experimental evidence indicates thatConsiderable experimental evidence indicates that significant stochastic fluctuations are presentsignificant stochastic fluctuations are present •• There are many examples when deterministic modelsThere are many examples when deterministic models areare not adequatenot adequate
    6. 6. Stochastic Chemical Kinetics
    7. 7. The Chemical Master Equation
    8. 8. Challenges in the solution of CME
    9. 9. Exploiting Underlying Biology
    10. 10. Monte Carlo Simulations: Stochastic Simulation Algorithm
    11. 11. Gillespie Algorithm Gillespie algorithm allows a discrete and stochasticGillespie algorithm allows a discrete and stochastic simulation of a system with few reactants because everysimulation of a system with few reactants because every reaction is explicitly simulated.reaction is explicitly simulated. a Gillespie realization represents a random walk that exactlya Gillespie realization represents a random walk that exactly represents the distribution of the Master equation.represents the distribution of the Master equation. The physical basis of the algorithm is the collision ofThe physical basis of the algorithm is the collision of molecules within a reaction vessel (well mixed).molecules within a reaction vessel (well mixed). all reactions within the Gillespie framework must involve atall reactions within the Gillespie framework must involve at most two molecules. Reactions involving three molecules aremost two molecules. Reactions involving three molecules are assumed to be extremely rare and are modeled as a sequenceassumed to be extremely rare and are modeled as a sequence of binary reactions.of binary reactions.
    12. 12. 1.1. InitializationInitialization: Initialize the number of molecules in the: Initialize the number of molecules in the system, reactions constants, and random number generators.system, reactions constants, and random number generators. 2.2. Monte Carlo StepMonte Carlo Step: Generate random numbers to determine: Generate random numbers to determine the next reaction to occur as well as the time interval. Thethe next reaction to occur as well as the time interval. The probability of a given reaction to be chosen is proportionalprobability of a given reaction to be chosen is proportional to the number of substrate molecules.to the number of substrate molecules. 3.3. UpdateUpdate: Increase the time step by the randomly generated: Increase the time step by the randomly generated time in Step 1. Update the molecule count based on thetime in Step 1. Update the molecule count based on the reaction that occurred.reaction that occurred. 4.4. IterateIterate: Go back to Step 1 unless the number of reactants is: Go back to Step 1 unless the number of reactants is zero or the simulation time has been exceeded.zero or the simulation time has been exceeded. Algorithm Summary
    13. 13. SSA AdvantagesAdvantages Low memory requirementLow memory requirement Computation is not O(exp(N))Computation is not O(exp(N)) DisadvantagesDisadvantages Convergence is slowConvergence is slow Little insightLittle insight
    14. 14. References Daniel T. Gillespie (1977). "Exact Stochastic Simulation of CoupledDaniel T. Gillespie (1977). "Exact Stochastic Simulation of Coupled Chemical Reactions".Chemical Reactions". The Journal of Physical ChemistryThe Journal of Physical Chemistry 8181 (25):(25): 2340-2361.2340-2361. doidoi::10.1021/j100540a00810.1021/j100540a008.. Daniel T. Gillespie (2007). “Stochastic Simulation of ChemicalDaniel T. Gillespie (2007). “Stochastic Simulation of Chemical Kinetics".Kinetics". Annu. Rev. Phys. Chem. 2007.58:35-55.Annu. Rev. Phys. Chem. 2007.58:35-55. 10.1146/annurev.physchem.58.032806.10463710.1146/annurev.physchem.58.032806.104637 D.Wilkinson (2009), “Stochastic modelling for quantitative descriptionD.Wilkinson (2009), “Stochastic modelling for quantitative description of heterogeneous biological systems.” Nature Reviews Genet. Febof heterogeneous biological systems.” Nature Reviews Genet. Feb 2009; 10(2):122-33.2009; 10(2):122-33. Slides adapted from:Slides adapted from: http://www.cds.caltech.edu/~murray/wiki/images/d/d9/Khammash_master-http://www.cds.caltech.edu/~murray/wiki/images/d/d9/Khammash_master- Gillespie:Gillespie: http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Gillespie_algorithmGillespie_algorithm
    15. 15. Neuronal Modeling: The Hodgkin Huxley Equations
    16. 16. 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 maintenance of this imbalance is anThe maintenance of this imbalance is an activeactive process carried out by ion pumpsprocess carried out by ion pumps
    17. 17. 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)
    18. 18. Basic neurophysiology Example electrochemical gradients (left)Example electrochemical gradients (left) Example ion channel (right)Example ion channel (right) Johnston & Wu,Johnston & Wu, Foundations of CellularFoundations of Cellular NeurophysiologyNeurophysiology, 5, 5thth ed.ed.
    19. 19. 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
    20. 20. Basic neuro- physiology 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
    21. 21. 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
    22. 22. Basic neuro- physiology Electrical signals regulate localElectrical signals regulate local calciumcalcium concentrationsconcentrations Synaptic vesiclesSynaptic vesicles fuse with the axon membrane,fuse with the axon membrane, andand neurotransmittersneurotransmitters are released into the spaceare released into the space between axon and dendrite in a process mediatedbetween axon and dendrite in a process mediated by calcium ionsby calcium ions Binding of neurotransmitters to dendrite causesBinding of neurotransmitters to dendrite causes influx of sodium ions that diffuse into somainflux of sodium ions that diffuse into soma
    23. 23. Basic electrophysiology 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
    24. 24. Action potential in neurons http://highered.mcgraw-hill.com/olc/dl/120107/bio_d.swfhttp://highered.mcgraw-hill.com/olc/dl/120107/bio_d.swf http://bcs.whfreeman.com/thelifewire/content/chp44/4402002.htmlhttp://bcs.whfreeman.com/thelifewire/content/chp44/4402002.html Sodium (Na+)(Na+) Potassium (K+)K+) insideinside outsideoutside Cell MembraneCell Membrane timetime Voltagedifference(inside–outside)Voltagedifference(inside–outside)
    25. 25. Action potential is linked to ion channel conductances -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) G is channelG is channel conductance.conductance. High conductanceHigh conductance allows for ions toallows for ions to pass throughpass through channel easier.channel easier.
    26. 26. The Hodgkin-Huxley model Based onBased on electrophysiologicalelectrophysiological measurements of giantmeasurements of giant squid axonsquid axon Empirical model thatEmpirical model that predicts experimental datapredicts experimental data with very high degree ofwith very high degree of accuracyaccuracy Provides insight intoProvides insight into mechanism of actionmechanism of action potentialpotential http://www.mun.ca/biology/desmid/brian/BIOL2060/BIOLhttp://www.mun.ca/biology/desmid/brian/BIOL2060/BIOL 2060-13/1310.jpg2060-13/1310.jpg
    27. 27. 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”) (I(INaNa , I, IKK , I, ILL )) (g(gNaNa , g, gKK , g, gLL ))
    28. 28. The Hodgkin-Huxley model Note:Note: Conductance isConductance is 1/R1/R, where, where RR isis resistanceresistance EE indicatesindicates membranemembrane potential,potential, EExx areare equilibriumequilibrium potentialspotentials ExperimentsExperiments show onlyshow only ggNaNa andand ggKK vary withvary with time whentime when stimulus isstimulus is appliedapplied
    29. 29. The Hodgkin-Huxley model Start with equation for capacitorStart with equation for capacitor v(t ) = q(t) C
    30. 30. 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 [ ])())(())(( 1 )( )( )( )( LLKKNaNa LLL KKK NaNaNa LKNa vvgvvvgvvvg Cdt dv vvgI vvgI vvgI III dt dq −+−+− − = −= −= −= ++−=
    31. 31. 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
    32. 32. 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 = gNa m3 h gK = gK n4 0 ≤ m, n, h ≤ 1 mm nnhh Q: How wereQ: How were the powersthe powers determined?determined? A: SmartA: Smart guessingguessing
    33. 33. The Hodgkin-Huxley model Define empirical differential equations toDefine empirical differential equations to model behavior of each gatemodel behavior of each gate dn dt =αn (v)(1− n)−βn (v)n αn(v) =0.01(v +10) (e(v+10)/10 −1) βn(v) = 0.125ev /80
    34. 34. The Hodgkin-Huxley model Define empirical differential equations toDefine empirical differential equations to model behavior of each gatemodel behavior of each gate dm dt =αm (v)(1− m)−βm(v)m αm(v) = 0.1(v + 25) (e(v+25)/10 −1) βm(v) = 4ev /18
    35. 35. The Hodgkin-Huxley model Define empirical differential equations toDefine empirical differential equations to model behavior of each gatemodel behavior of each gate dh dt =αh (v)(1− h)−βh (v)h αh (v) = 0.07ev /20 βh (v) = 1 (e(v+30)/10 +1)
    36. 36. The Hodgkin-Huxley model 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 Constants (Constants (gg in mmho/cmin mmho/cm22 andand vv in mV)in mV) gNa =120 gK = 36 gL = 0.3 vNa =−115 vK =12 vL =−10.5989
    37. 37. 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) Gateparam value m gate (Na) h gate (Na) n gate (K) -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)
    38. 38. Interactive demonstration (Integration of Hodgkin-Huxley equations(Integration of Hodgkin-Huxley equations using Maple)using Maple)
    39. 39. Interactive demonstration > Ena:=55: Ek:=-82: El:= -59: gkbar:=24.34: gnabar:=70.7:> Ena:=55: Ek:=-82: El:= -59: gkbar:=24.34: gnabar:=70.7: > gl:=0.3: vrest:=-69: cm:=0.001:> gl:=0.3: vrest:=-69: cm:=0.001: > alphan:=v-> 0.01*(10-(v-vrest))/(exp(0.1*(10-(v-vrest)))-> alphan:=v-> 0.01*(10-(v-vrest))/(exp(0.1*(10-(v-vrest)))- 1):1): > betan:=v-> 0.125*exp(-(v-vrest)/80):> betan:=v-> 0.125*exp(-(v-vrest)/80): > alpham:=v-> 0.1*(25-(v-vrest))/(exp(0.1*(25-(v-vrest)))-1):> alpham:=v-> 0.1*(25-(v-vrest))/(exp(0.1*(25-(v-vrest)))-1): > betam:=v-> 4*exp(-(v-vrest)/18):> betam:=v-> 4*exp(-(v-vrest)/18): > alphah:=v->0.07*exp(-0.05*(v-vrest)):> alphah:=v->0.07*exp(-0.05*(v-vrest)): > betah:=v->1/(exp(0.1*(30-(v-vrest)))+1):> betah:=v->1/(exp(0.1*(30-(v-vrest)))+1): > pulse:=t->-20*(Heaviside(t-.001)-Heaviside(t-.002)):> pulse:=t->-20*(Heaviside(t-.001)-Heaviside(t-.002)): > rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) + >> gkbar*n^4*(V-Ek) + gl*(V-El)gkbar*n^4*(V-Ek) + gl*(V-El) + pulse(t))/cm:+ pulse(t))/cm: > rhsn:=(t,V,n,m,h)-> 1000*(alphan(V)*(1-n) - betan(V)*n):> rhsn:=(t,V,n,m,h)-> 1000*(alphan(V)*(1-n) - betan(V)*n): > rhsm:=(t,V,n,m,h)-> 1000*(alpham(V)*(1-m) - betam(V)*m):> rhsm:=(t,V,n,m,h)-> 1000*(alpham(V)*(1-m) - betam(V)*m): > rhsh:=(t,V,n,m,h)-> 1000*(alphah(V)*(1-h) - betah(V)*h):> rhsh:=(t,V,n,m,h)-> 1000*(alphah(V)*(1-h) - betah(V)*h):
    40. 40. Interactive demonstration > inits:=V(0)=vrest,n(0)=0.315,m(0)=0.042, h(0)=0.608;> inits:=V(0)=vrest,n(0)=0.315,m(0)=0.042, h(0)=0.608; > sol:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric,{V(t),n(t),m(t),h(t)},type=numeric, output=listprocedure);output=listprocedure); > Vs:=subs(sol,V(t));> Vs:=subs(sol,V(t)); > plot(Vs,0..0.02);> plot(Vs,0..0.02); > sol20:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol20:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric);{V(t),n(t),m(t),h(t)},type=numeric); > with(plots):> with(plots):
    41. 41. Interactive demonstration >> J:=odeplot(sol20,[V(t),n(t)],0..0.02):J:=odeplot(sol20,[V(t),n(t)],0..0.02): > display({J});> display({J}); > pulse:=t->-2*(Heaviside(t-.001)-Heaviside(t-.002)):> pulse:=t->-2*(Heaviside(t-.001)-Heaviside(t-.002)): > rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) + gkbar*n^4*(V-Ek) + gl*(V-El)+ pulse(t))/cm:gkbar*n^4*(V-Ek) + gl*(V-El)+ pulse(t))/cm: > sol2:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol2:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric);{V(t),n(t),m(t),h(t)},type=numeric); > K:=odeplot(sol2,[V(t),n(t)],0..0.02,color=green):> K:=odeplot(sol2,[V(t),n(t)],0..0.02,color=green): > display({J,K});> display({J,K});
    42. 42. Interactive demonstration > L:=odeplot(sol20,[V(t),n(t)],0..0.02,numpoints=400,> L:=odeplot(sol20,[V(t),n(t)],0..0.02,numpoints=400, color=blue):color=blue): > display({J,L});> display({J,L}); > odeplot(sol20,[V(t),m(t)],0..0.02,numpoints=400);> odeplot(sol20,[V(t),m(t)],0..0.02,numpoints=400); > odeplot(sol20,[V(t),h(t)],0..0.02,numpoints=400);> odeplot(sol20,[V(t),h(t)],0..0.02,numpoints=400); > odeplot(sol20,[m(t),h(t)],0..0.02,numpoints=400);> odeplot(sol20,[m(t),h(t)],0..0.02,numpoints=400); > a:=0.7; b:=0.8; c:=0.08;> a:=0.7; b:=0.8; c:=0.08; > rhsx:=(t,x,y)->x-x^3/3-y;> rhsx:=(t,x,y)->x-x^3/3-y; > rhsy:=(t,x,y)->c*(x+a-b*y);> rhsy:=(t,x,y)->c*(x+a-b*y); > sol2:=dsolve({diff(x(t),t)=rhsx(t,x(t),y(t)),> sol2:=dsolve({diff(x(t),t)=rhsx(t,x(t),y(t)), diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1},diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1}, {x(t),y(t)},type=numeric, output=listprocedure);{x(t),y(t)},type=numeric, output=listprocedure); > xs:=subs(sol2,x(t)); ys:=subs(sol2,y(t));> xs:=subs(sol2,x(t)); ys:=subs(sol2,y(t)); > K:=plot([xs,ys,0..200],x=-3..3,y=-2..2,color=blue):> K:=plot([xs,ys,0..200],x=-3..3,y=-2..2,color=blue): > J:=plot({[V,(V+a)/b,V=-2.5..1.5],[V,V-V^3/3,V=-2.5..2.2]}):> J:=plot({[V,(V+a)/b,V=-2.5..1.5],[V,V-V^3/3,V=-2.5..2.2]}): > plots[display]({J,K});> plots[display]({J,K});
    43. 43. Virtual Cell - Hodgkin-Huxley Versions of the models in “ComputationalVersions of the models in “Computational cell biology” by Fall et al have beencell biology” by Fall et al have been implemented in Virtual Cellimplemented in Virtual Cell These are available as Public modelsThese are available as Public models TheThe Hodgkin-Huxley ModelHodgkin-Huxley Model is a scientificis a scientific model that describes how action potentials inmodel that describes how action potentials in neurons are initiated and propagatedneurons are initiated and propagated Within Virtual Cell, use File/Open/BiomodelWithin Virtual Cell, use File/Open/Biomodel Then open SharedThen open Shared Models/CompCell/Hodgkin-HuxleyModels/CompCell/Hodgkin-Huxley
    44. 44. Biochemical Species • Potassium • Sodium • Potassium Channel Inactivation Gate-closed "n_c" • Potassium Channel Inactivation Gate-open "n_o" • Sodium Channel Inactivation Gate-closed "h_c" • Sodium Channel Inactivation Gate-open "h_o" • Sodium Channel Activation Gate-open "m_o" • Sodium Channel Activation Gate-closed "m_c" Compartments • Extracellular • Plasma Membrane • Cytosol
    45. 45. m_o:: Sodium Channel Activation Gate-open

    ×