List of Equations

APPENDIX S.A

List of Equations

S.A.1. Neuron

1.01 Reversal Potential of K+ channel (Hodgkin & Huxley, 1952)
vk = 26.70*log[Ko][KiN]
1.02 Reversal Potential of Na+ channel (Hodgkin & Huxley, 1952)
vNa= 26.70*log[Nao][Nai]
1.03 Conductance of K+ channels (Hodgkin & Huxley, 1952)
gk = gKMax*nN4
1.04 Conductance of Na+ channels (Hodgkin & Huxley, 1952)
gNa= gNaMax*m3*hN
1.05 Activity of Na+/K+ ATPase pump (Modified, Kager et al., 2000 )
Apump= 1+KmK[Ko]-2*1+KmNa[Nai]-3*1+KATP[ATP]-1
1.06 K+ current of Na+/K+ ATPase pump (Kager et al., 2000)
Ikp=-2*Imax*Apump
1.07 K+ current of Na+/K+ ATPase pump (Kager et al., 2000)
INap=3*Imax*Apump
1.08 K+ channel current (Hodgkin & Huxley, 1952)
IK=gK*v-vK
1.09 Na+ channel current (Hodgkin & Huxley, 1952)
INa=gNa*v-vNa
1.10 Total K+ current (Kager et al., 2000)
IKTot=IK+IKp
1.11 Total Na+ current (Kager et al., 2000)
INaTot=INa+INap
1.12 Change in [K+] in neuron (Kager et al., 2000)
d[KiN]dt=-IKTot*sF*Vi*1000
1.13 Change in [Na+] in neuron (Kager et al., 2000)
d[Nai]dt=-INaTot*sF*Vi*1000
1.14 Total ionic conductance of neuron (Hodgkin & Huxley, 1952)
gTot=gNa+gK+gl
1.15 Steady state neuronal membrane potential
vinf=gNa*vNa+gK*vK+gl*vl+ iappgTot
1.16 Time constant for neuronal membrane potential (Hodgkin & Huxley 1952)
τv=cmgTot
1.17 Update rule for membrane potential (Hodgkin & Huxley 1952)
v(t+Δt)=vinf+(v(t)-vinf)*exp-Δtτv
1.18 Activation rate constant for gating variable m at membrane potential v (Hodgkin & Huxley 1952)
αm= 0.1*v+351-exp-v+3510
1.19 Inactivation rate constant for gating variable m at membrane potential v (Hodgkin & Huxley 1952)
βm= 4*exp-0.0556*v+65
1.20 Activation rate constant for gating variable n at membrane potential v (Hodgkin & Huxley 1952)
αnN= 0.01*v+501-exp-v+5010
1.21 Inactivation rate constant for gating variable n at membrane potential v (Hodgkin & Huxley 1952)
βnN= 0.125*exp-v+6080
1.22 Activation rate constant for gating variable h at membrane potential v (Hodgkin & Huxley 1952)
αhN= 0.07*exp-0.05*v+60
1.23 Activation rate constant for gating variable h at membrane potential v (Hodgkin & Huxley 1952)
βhN=11+exp-0.1*v+30
1.24 Time constant for gating variable m (Hodgkin & Huxley 1952)
τm=1αm+βm
1.25 Time constant for gating variable h (Hodgkin & Huxley 1952)
τhN=1αh+βh
1.26 Time constant for gating variable n (Hodgkin & Huxley 1952)
τnN=1αn+βn
1.27 Steady state value of gating variable m (Hodgkin & Huxley 1952)
minf =αmαm+βm
1.28 Steady state value of gating variable h (Hodgkin & Huxley 1952)
hNinf=αhαm+βm
1.29 Steady state value of gating variable n (Hodgkin & Huxley 1952)
nNinf=αnαn+βn
1.30 Update rule for gating variable m (Hodgkin & Huxley 1952)
m(t+Δt)=minf+m(t)-minf*exp-Δtτm
1.31 Update rule for gating variable h (Hodgkin & Huxley 1952)
hN(t+Δt)=hinf+hN(t)-hinf*exp-Δtτh
1.32 Update rule for gating variable n (Hodgkin & Huxley 1952)
nN(t+Δt)=ninf+nN(t)-ninf*exp-Δtτn
1.33 Change of [Ca2+] in neuron (Lee et al., 2009)
d[CaN]dt=-[CaN]+[CaNo]+KCaN*ICaτCa
1.34 Release probability of vesicles (Lee et al., 2009)
Prel=Prel,max*[CaN]4[CaN]4+ Krel,1/24
1.35 Recovery rate from empty to releasable state (Lee et al., 2009)
Krec=Krec0+Krec0,max-Krec0*[CaN][CaN]+Krec,1/2
1.36 Ratio of releasable vesicles (Lee et al., 2009)
dRreldt=Krec*1-Rrel-Prel*ICa*Rrel
1.37 Synaptic glutamate concentration (Modified fron Lee et al., 2009)
dGlutdt=n*Ntot*Rrel*Prel*ICaNA –GluτG
1.38 Neuron Vmax hexokinase (Mangia et al.,2009)
nVmh = GlycolyticRatio*nn2.95nRoi
1.39 Neuron Vmax lactate consumption (Mangia et al.,2009)
nVml = 1.22*OxidativeRationRLoi
1.40 Neuronal lactate oxidation (Mangia et al.,2009)
NLOxi=1*nVml*NLnVKml+NLnV
1.41 Neuronal glucose oxidation (Mangia et al.,2009)
NGO=NGnV*nVmh*112Kmh+NGnV
1.42 Neuronal glucose utilization (Mangia et al.,2009)
NGUt=NGnV*nVmhKmh+NGnV
1.43 Conversion of neuronal glucose to lactate (Mangia et al.,2009)
NG2L= 2*NGUt
1.44 Change in neuronal glucose concentration (Mangia et al.,2009)
dNGdt= +GI2N–NO–NGUt
1.45 Change in neuronal lactate concentration (Mangia et al.,2009)
dNLdt= +LI2N–NLOxi+NG2L
1.46 Rate of ATP production
VATP=NLOxi*1015*.53*17*180*10-960
1.47 Change in neuronal ATP concentration
d[ATP]dt=1*VATPVi*10-15-INap+IKp*sF*Vi*109*10-6

S.A.2. Astrocyte

1.48 Ratio of bound to total glutamate receptors in synapse (Bennett et al., 2008)
ρ =[Glu]Kglut+[Glu]
1.49 Ratio of activated to total G-protein receptors (Bennett et al., 2008)
G =ρ+δKg+ρ+δ
1.50 Change in IP3 concentration in astrocyte (Modified from Bennett et al., 2008)
d[IP3]dt=rh*G-Kdeg*[IP3]
1.51 Rate of Ca2+ concentration change due to pump uptake into ER (Bennett et al., 2008)
Jpump=Vmax*CaA2CaA2+Kp2
1.52 Gating variable of Ca2+ activated IP3 receptors (Bennett et al., 2008)
dhAdt=kon*Kinh-[CaA]+Kinh*hA
1.53 Rate of Ca2+ concentration change due to release through IP3 channels (Bennett et al., 2008)
JIP3=Jmax*IP3IP3+KiA*[CaA][CaA]+Kact*hA3*1-[CaA][CaA]+CaERA
1.54 Rate of Ca2+ concentration change due to leakage from ER (Bennett et al., 2008)
Jleak=Pl*1-[CaA]CaERA
1.55 Cytosolic Ca2+ concentration in astrocyte (Bennett et al., 2008)
dCaAdt=βcyt*JIP3-Jpump+Jleak
1.56 Rate of production of EET (Modified from Bennett et al., 2008)
d[EET]dt= VEET*ramp[CaA]-CaMinA- .2*[EET][EET]+18
1.57 Astrocytic Vmax lactate consumption (Mangia et al.,2009)
aVml = 0.24*OxidativeRatioaRLoi
1.58 Astrocytic Vmax hexokinase (Mangia et al.,2009)
aVmh = GlycolyticRatio*naRoi
1.59 Astrocytic lactate oxidation (Mangia et al., 2009)
ALOxi= 1*aVml*ALaVKml+ALaV
1.60 Astrocytic glucose oxidation (Mangia et al., 2009)
AGO=aVmh*AGaV*112Kmh+AGaV
1.60 Astrocytic glucose oxidation (Mangia et al., 2009)
1.61 Astrocytic glucose utilization (Mangia et al., 2009)
AGUt=aVmh*AGaVKmh+AGaV
1.62 Conversion of astrocytic glucose to lactate (Mangia et al.,2009)
AG2L= 2*AGUt
1.63 Change in astrocytic glucose concentration (Mangia et al.,2009)
dAGdt=+GB2A-GA2I-AO-AGUt
1.64 Change in astrocytic lactate concentration (Mangia et al.,2009)
dALdt=+LB2A-LA2I-ALOxi+AG2L

S.A.3. Vessel

1.65 Smooth muscle membrane potential
Vm=5-8011+e-2EET
1.66 Change in vessel radius
r=rmin+rmax-rminVmax-VmVmax-Vmin
1.67 Glucose flux from blood
BGF=r-rminrmax-rmin[Glc]B
1.68 Lactate flux from blood
BLF=r-rminrmax-rmin[Lac]B
1.69 Glucose flux from blood to endothelium (Mangia et al., 2009)
GlcB=SG*K+EGeV-EGeV*K+SGK2*eRoo+eRoi*K*SG+eRio*K*EGeV+eRee*SG*EGeV
1.70 Glucose flux from endothelium to basal lamina (Mangia et al., 2009)
GE2B=EGeV*K+BGbV-BGbV*K+EGeVK2*eRoofe+eRoife*K*BGbV+eRiofe*K*EGeV+eReefe*BGbV*EGeV
1.71 Lactate flux from blood to endothelium (Mangia et al., 2009)
LacB=SL*eKL+ELeV-ELeV*eKL+SLeKL2*eRLoo+eRLoi*eKL*SL+eRLio*eKL*ELeV+eRLee*SL*ELeV
1.72 Lactate flux from endothelium to basal lamina (Mangia et al., 2009)
LE2B=ELeV*eKL+BLbV-BLbV*eKL+ELeVeKL2*eRLoofe+eRLoife*eKL*BLbV+eRLiofe*eKL*ELeV+eRLeefe*BLbV*ELeV
1.73 Glucose flux through endothelium (Mangia et al., 2009)
dEGdt=+BGF-GE2B
1.74 Glucose flux through basal lamina (Mangia et al., 2009)
dBGdt=+GE2B-GB2A-GB2I
1.75 Lactate flux through endothelium (Mangia et al., 2009)
dELdt=-LE2B+BLF
1.76 Lactate flux through basal lamina (Mangia et al., 2009)
dBLdt=+LE2B-LB2A-LB2I

S.A.4. Interstitium

1.77 Buffering of interstitial K+
dK+updt=k1Kbuffer-k2BufferKo
1.78 Change in interstitial K+ concentration (Kager et al., 2000)
dKodt=IKTot*sF*Ve*1000+dK+updt
1.79 Change in interstitial Na+ concentration (Kager et al., 2000)
dNaodt=INaTot*sF*Ve*1000
1.80 Glucose flux through interstitial space (Mangia et al., 2009)
dIGdt=+GA2I-GI2N+GB2I
1.81 Lactate flux through interstitial space (Mangia et al., 2009)
dILdt=-LI2N+LA2I+LB2I
1.82 Glucose flux from basal lamina to astrocyte (Mangia et al., 2009)
GB2A=BGbV*K+AGaV-AGaV*K+BGbV12.5*K2*aRoo+aRoi*K*BGbV+aRio*K*AGaV+aRee*BGbV*AGaV
1.83 Glucose flux from astrocyte to interstitial space (Mangia et al., 2009)
GA2I=AGaV*K+IGiV-IGiV*K+AGaV1*(K2*aRoo+aRoi*K*IGiV+aRio*K*AGaV+aRee*IGiV*AGaV)
1.84 Glucose flux from interstitial space to neuron (Mangia et al., 2009)
GI2N=IGiV*nK+NGnV-NGnV*nK+IGiVnK*nK*nRoo+nRoi*nK*IGiV+nRio*nK*NGnV+nRee*IGiV*NGnV
1.85 Glucose flux from basal lamina to interstitial space (Mangia et al., 2009)
GB2I=BGbV*1e15*kdiff-IGiV*1e15*kdiff
1.86 Lactate flux from basal lamina to astrocyte (Mangia et al., 2009)
LB2A=BLbV*aKL+ALaV-ALaV*aKL+BLbV12.5*aKL2*aRLoo+aRLoi*aKL*BLbV+aRLio*aKL*ALaV+aRLee*BLbV*ALaV
1.87 Lactate flux from astrocyte to interstitial space (Mangia et al., 2009)
LA2I=1*ALaV*aKL+ILiV-ILiV*aKL+ALaVaKL2*aRLoo+aRLoi*aKL*ILiV+aRLio*aKL*ALaV+aRLee*ILiV*ALaV
1.88 Lactate flux from interstitial space to neuron (Mangia et al., 2009)
LI2N=ILiV*nKL+NLnV-NLnV*nKL+ILiVnKL*nKL*nRLoo+nRLoi*nKL*ILiV+nRLio*nKL*NLnV+nRLee*ILiV*NLnV
1.89 Lactate flux from basal lamina to interstitial space (Mangia et al., 2009)
LB2I=BLbV*1e15*kdiff-ILiV*1e15*kdiff

APPENDIX S.B

List of Constants

gKMax=0.36mmhomm2 / Peak Potassium Conductance / Hodgkin & Huxley, 1952
gNaMax=1.20mmhomm2 / Peak Sodium Conductance / Hodgkin & Huxley, 1952
gl=0.003mmhomm2 / Peak Leakage Conductance / Hodgkin & Huxley, 1952
vl=-54.387 mV / Leakage Membrane Potential / Hodgkin & Huxley, 1952
cm=0.01 µFmm2 / Membrane Capacitance / Hodgkin & Huxley, 1952
KmK=3.5 mM / Kager et al. 2000
KmNa=10 mM / Kager et al. 2000
Imax=0.013*100mAcm2 / Maximum Na/K ATPase Pump Current / Kager et al. 2000
KCaN=0.120mMms / Intracellular Calcium Concentration Gain per Action Potential / Lee et al. 2009
KATP=170 mM / Fitted to Kager et al. 2000
τCa=250 ms / Fitted to Lee et al. 2009
F=96500Cmol / Faraday’s Constant
Krel,12=0.009 mM / Calcium Sensitivity Regarding Transmitter Release / Lee et al. 2009
Prel,Max=0.9 / Maximum Probability of Release / Lee et al. 2009
Krec0=2.2*10-2 ms-1 / Initial Recovery Rate of Synaptic Vesicle from Empty to Releasable State / Lee et al. 2009
Krec,Max=2.2*10-2 ms-1 / Maximum Recovery Rate of Synaptic Vesicle from Empty to Releasable State / Lee et al. 2009
Krec,12=20*10-3 mM / Calcium Sensitivity Regarding Transmitter Release / Lee et al. 2009
s=1586 µm2 / Total Surface Area of Neuron / Kager et al. 2000
Vi=2160 µm3 / Total Internal Volume of Neuron / Kager et al. 2000
OxidativeRatio =.184 / Mangia et al. 2009
GlycolyticRatio =.0845 / Mangia et al. 2009
KGlu=1 mM / Dissociation Constant / Bennett et al. 2008
Kg=8.82 / G-Protein Dissociation Constant / Bennett et al. 2008
Kdeg=1.251000 s-1 / IP3 Degradation Rate / Bennett et al. 2008
CaERA=400µM / Ca2+ concentration in ER / Bennett et al. 2008
Vmax=201000µMms / Maximum Pumping Rate of Ca2+ into ER / Bennett et al. 2008
Kp=0.24µM / Pump Dissociation Constant / Bennett et al. 2008
Jmax=28801000µMms / Maximum Ca2+ Channel Current / Bennett et al. 2008
KiA = 0.03µM / IP3 Channel Kinetic Parameter / Bennett et al. 2008
Kact = 0.17µM / IP3 Channel Kinetic Parameter / Bennett et al. 2008
kon=21000μMms / IP3 Channel Kinetic Parameter / Bennett et al. 2008
CaMinA=0.1µM / Minimum Ca2+ Concentration for EET Production / Bennett et al. 2008
βcyt=0.0244 / Endogenous Buffer Parameter / Bennett et al. 2008
Kinh=0.1µM / IP3 Channel Kinetic Parameter / Bennett et al. 2008
Pl=0.08041000µMms / Determined by Steady State Balance Condition / Bennett et al. 2008
Kml = 2 / Mangia et al. 2009
Kmh = 0.045 / Mangia et al. 2009
Veet=3*10-13*6.022*1023*10-6*6*25*25*10-12*100 µMolms / EET Production Rate / Bennett et al. 2008
aRoi=5e14.4sm mol / Mangia et al. 2009
n=1 (#) / Mangia et al. 2009
aRLoi=6.6e13sm mol / Mangia et al. 2009
aVml=0.24*OxidativeRatioaRLoi / Mangia et al. 2009
aVmh=GlycolyticRatio*naRoi / Mangia et al. 2009
SG=5.5mM / Serum Glucose Concentration / Mangia et al. 2009
rmin=8 µm / Minimum Vessel Radius
rmax=30µm / Maximum Vessel Radius
eKL=8 / Mangia et al. 2009
SL=1mM / Serum Lactate Concentration / Mangia et al. 2009
fe=1 (#) / Mangia et al. 2009
eRoi=13e13.4sm mol / Mangia et al. 2009
eRio=13e13.4sm mol / Mangia et al. 2009
eRee =13e13.4sm mol / Mangia et al. 2009
eRoo=eRoi+eRio-eRee / Mangia et al. 2009
eRLoi=2.8e15sm mol / Mangia et al. 2009
eRLio = 2e15sm mol / Mangia et al. 2009
eRLee=2e15sm mol / Mangia et al. 2009
eRLoo=2.8e15sm mol / Mangia et al. 2009
Ve=0.15*2160 µL / Extracellular Volume / Kager et al. 2000
k1=.0008 / Kager et al. 2000
k2=.00081+expKo-15-1.09 / Kager et al. 2000
Nao=140 mM / Extracellular Na+ Concentration
Ko=3 mM / Extracellular K+ Concentration
Nai=5 mM / Neuronal Cytosolic Na+ Concentration
KiN=200 mM / Neuronal Cytosolic K+ Concentration
m=0.0530 / Initial Probability of Activation of Na+ Ion Channel / Hodgkin & Huxley, 1952
hN=0.5960 / Initial Probability of Inactivation of Na+ Ion Channel / Hodgkin & Huxley, 1952
nN=0.3177 / Initial Probability of Activation of K+ Ion Channel / Hodgkin & Huxley, 1952
v=-65 mV / Resting Membrane Potential of Neuron / Hodgkin & Huxley, 1952
CaNo=0.0047 mM / Initial Neuronal Ca2+ Concentration / Lee et al. 2009
ATP=20 mM / Initial Neuronal ATP Concentration unless mentioned otherwise
buffer=500 mM / Initial K+ Buffer Capacity of Astrocyte / Kager et al. 2000
kbuffer=0 mM / Initial K+ Buffered / Kager et al. 2000
Rrel=1 (#) / Remaining Ratio of Vesicles Releasable / Lee et al. 2009
τG=1.2 ms / Time constant for clearance of synaptic glutamate / Clements et al. 1992
glu=0 mM / Initial Synaptic Glutamate Concentration / Bennett et al. 2008
Ica=0 ms-1 / Inflow Calcium Current as Dirac Delta Function / Kager et al. 2000
rh=2*625*1000*10 µMolms / IP3 Production Rate / Bennett et al. 2008
IP3=0.01µM / Initial Astrocytic IP3 Concentration / Bennett et al. 2008
CaA=0.05µM / Initial Astrocytic Ca2+ Concentration / Bennett et al. 2008
hA=KinhCaA+Kinh / Initial Probability of Ca2+ Occupying its Inhibitory Binding Site / Bennett et al. 2008
EET=0 µM / Initial Extracellular EET Concentration / Bennett et al. 2008
δ=Kg*Kdeg*IP3rh-Kdeg*IP3 / Bennett et al. 2005
iV=68e-15*.18 L / Interstitial Volume / Mangia et al. 2009
nV=68e-15*.495 L / Neuronal Volume / Mangia et al. 2009
bV=68e-15*.015 L / Basal Lamina Volume / Mangia et al. 2009
aV=68e-15*.29 L / Astrocyte Volume / Mangia et al. 2009
eV=68e-15*.0189 L / Endothelium Volume / Mangia et al. 2009
nKL=0.7 / Mangia et al. 2009
kdiff=1 / Mangia et al. 2009
nK=4 / Mangia et al. 2009
aKL=5 / Mangia et al. 2009
K=8 / Mangia et al. 2009
nRoi=4.4e13.4sm mol / Mangia et al. 2009
nRio=3.2e13.4sm mol / Mangia et al. 2009
nRee=3.2e13.4sm mol / Mangia et al. 2009
nRoo=4.4e13.4sm mol / Mangia et al. 2009
aRoi=5e14.4sm mol / Mangia et al. 2009
aRio=3.8e14.4sm mol / Mangia et al. 2009
aRee=3.8e14.4sm mol / Mangia et al. 2009
aRoo=aRio+aRoi-aRee / Mangia et al. 2009
nRLoi=2e14sm mol / Mangia et al. 2009
nRLio=1e13sm mol / Mangia et al. 2009
nRLee=0.1e14sm mol / Mangia et al. 2009
nRLoo=nRLio+nRLoi-nRLee / Mangia et al. 2009
aRLoi=6.6e13sm mol / Mangia et al. 2009
aRLio=3.3e13sm mol / Mangia et al. 2009
aRLee=1e13sm mol / Mangia et al. 2009
aRLoo=aRLoi+aRLio-aRLee / Mangia et al. 2009
IG/iV=1.45 mM / Initial Concentration of Glucose in Interstitium / Mangia et al. 2009
IL/iV=1 mM / Initial Concentration of Lactate in Interstitium / Mangia et al. 2009
NG/nV=1.26 mM / Initial Concentration of Glucose in Neuron / Mangia et al. 2009
NL/nV=1 mM / Initial Concentration of Lactate in Neuron / Mangia et al. 2009
AG/aV=1.42 mM / Initial Concentration of Glucose in Astrocyte / Mangia et al. 2009
AL/aV=0.96 mM / Initial Concentration of Lactate in Astrocyte / Mangia et al. 2009
EG/eV=3.44 mM / Initial Concentration of Glucose in Endothelium / Mangia et al. 2009
BG/bV=1.87 mM / Initial Concentration of Glucose in Basal Membrane / Mangia et al. 2009
EL/eV=1 mM / Initial Concentration of Lactate in Endothelium / Mangia et al. 2009

1