#### A mathematical model for recurrent spreading depolarizations #### C. Conte, R. Lee, M. Sartar, D. Terman #### J. Computational Neuroscience # Gas constant, temperature, Faraday's constant p R=8310 p Temp=310.0 p F=96485 frt=R*Temp/F phi(v)=v/frt ################ #### Neuron ################ # Fast sodium p pna=1e-05 p pnal=2e-09 p thm=-34.,sigm=5. minf(v)=1./(1.+exp(-(v-thm)/sigm)) ina(v,n,nae,nai)=PNa*(minf(v)^3*(1-n)+pnal)*F*phi(v)*(Nae*exp(-phi(v))-Nai)/(exp(-phi(v))-1) # NaP p pnap=3e-08 p taubar=10000 p thmp=-40,sigmp=6 p thhp=-48,sighp=-6 p vt=-49,sig=6 p phihp=.05 minfp(v)=1./(1.+exp(-(v-thmp)/sigmp)) hinfp(v)=1./(1.+exp(-(v-thhp)/sighp)) tauhp(v)=taubar/cosh((v-vt)/(2*sig)) inap(v,hp,nae,nai)=PNap*minfp(v)*hp*F*phi(v)*(Nae*exp(-phi(v))-Nai)/(exp(-phi(v))-1) # IK p pk=7e-05 p pkl=4e-07 p thn=-55.,sgn=14. p taun0=.05,taun1=.27,thnt=-40,sn=-12 p phin=.8 ninf(v)=1./(1.+exp(-(v-thn)/sgn)) taun(v)=taun0+taun1/(1+exp(-(v-thnt)/sn)) ik(v,n,ke,ki)=PK*(n^4+pkl)*F*phi(v)*(ke*exp(-phi(v))-ki)/(exp(-phi(v))-1) # NMDA p pnmda=3e-06 p thetat=-10,trise=2,tdecay=1,alphan=.5 binf(v)=1/(1+exp(-(v-thetat)/16.13)) p pca=3 sinfg(x)=1./(1.+exp(-(x-thg)/sigmag)) p thg=.01 p sigmag=.001 inanmda[1..25]=Pnmda*sg[j]*binf(v[j])*F*phi(v[j])*(Nae[j]*exp(-phi(v[j]))-Nai[j])/(exp(-phi(v[j]))-1) iknmda[1..25]=Pnmda*sg[j]*binf(v[j])*F*phi(v[j])*(ke[j]*exp(-phi(v[j]))-ki[j])/(exp(-phi(v[j]))-1) icanmda[1..25]=pca*2*Pnmda*sg[j]*binf(v[j])*F*phi(2*v[j])*(cae[j]*exp(-phi(2*v[j]))-cai[j])/(exp(-phi(2*v[j]))-1) inmda[1..25]=inanmda[j]+iknmda[j]+icanmda[j] # Cl leak p pcl=2e-07 icl[1..25]=-Pcl*F*phi(-v[j])*(cle[j]*exp(-phi(-v[j]))-cli[j])/(exp(-phi(-v[j]))-1) # Neuron Na-K ATPase p imax=5 ipump[1..25]=imax/(((1+2/ke[j])^2)*(1+7.7/nai[j])^3) inapump[1..25]=3*ipump[j] ikpump[1..25]=-2*ipump[j] # Diffusion kdiff[1...1]=dk*(ke[j+1]-ke[j]) kdiff[2..24]=dk*(ke[j+1]+ke[j-1]-2*ke[j]) kdiff[25..25]=dk*(ke[j-1]-ke[j]) nadiff[1...1]=dna*(nae[j+1]-nae[j]) nadiff[2..24]=dna*(nae[j+1]+nae[j-1]-2*nae[j]) nadiff[25..25]=dna*(nae[j-1]-nae[j]) gdiff[1...1]=dg*(glut[j+1]-glut[j]) gdiff[2..24]=dg*(glut[j+1]+glut[j-1]-2*glut[j]) gdiff[25..25]=dg*(glut[j-1]-glut[j]) p dk=.002 p dna=.001333 p dg=.0021 # Differential equations v[1..25]'= -(ina(v[j],n[j],nae[j],nai[j])+inap(v[j],hp[j],nae[j],nai[j])+ik(v[j],n[j],ke[j],ki[j])+icl[j]+ipump[j]+inmda[j]) n[1..25]'= phin*(ninf(v[j])-n[j])/taun(v[j]) hp[1..25]'=phihp*(hinfp(v[j])-hp[j])/tauhp(v[j]) sg[1..25]'=-sg[j]/tdecay+alphan*sinfg(glut[j])*(1-sg[j]) ##################### #### Astrocytes #################### p pnaa=.015e-06 inaa(v,nae,nai)=PNaa*F*phi(v)*(Nae*exp(-phi(v))-Nai)/(exp(-phi(v))-1) p pka=6e-06 ika(v,ke,ki)=PKa*F*phi(v)*(ke*exp(-phi(v))-ki)/(exp(-phi(v))-1) p imaxa[1..4]=.08 p imaxa[5..9]=.12 p imaxa[10..14]=.16 p imaxa[15..19]=.2 p imaxa[20..25]=.24 ipumpa(ke,nai,imaxa)=imaxa/(((1+2/ke)^2)*(1+10/nai)^3) # Na-glut co-transporter p hr=.5 p gnagl=3e-05 enagl[1..25]=(frt/2)*ln(((nae[j]/naia[j])^3)*(kia[j]/ke[j])*(glut[j]/glui[j])*hr) inagl[1..25]=gnagl*(va[j]-enagl[j]) aux inagl[1..25]=inagl[j] # Differential Equation va[1..25]'=-(ika(va[j],ke[j],kia[j])+inaa(va[j],nae[j],naia[j])+ipumpa(ke[j],naia[j],imaxa[j])+inagl[j]) ######################### #### Ion Concentrations ######################## # Volumes(um^3) and Areas (um^2) p S=922 p voln=2160 p delta=.05 vole=delta*voln p Sa=1600 p vola=2000 # Conversion factors from uA/cm^2 to mM/ms c1=10.0*s/(F*voln) c2=10.0*s/(F*vole) c3a=10*sa/(F*vola) c3e=10*sa/(F*vole) # Na+, K+ & Cl- ke[1..25]'=c2*(ik(v[j],n[j],ke[j],ki[j])+ikpump[j]+iknmda[j])+kdiff[j]+c3e*(ika(va[j],ke[j],kia[j])-2*ipumpa(ke[j],naia[j],imaxa[j])-inagl[j])+gkb*(k0-ke[j]) nai[1..25]'=-c1*(ina(v[j],n[j],nae[j],nai[j])+inap(v[j],hp[j],nae[j],nai[j])+inapump[j]+inanmda[j]) nae[1..25]'=c2*(ina(v[j],n[j],nae[j],nai[j])+inap(v[j],hp[j],nae[j],nai[j])+inapump[j]+inanmda[j])+nadiff[j]+c3e*(inaa(va[j],nae[j],naia[j])+3*ipumpa(ke[j],naia[j],imaxa[j])+(3)*inagl[j])+gnab*(na0-nae[j]) ki[1..25]'=-c1*(ik(v[j],n[j],ke[j],ki[j])+ikpump[j]+iknmda[j]) kia[1..25]'=-c3a*(ika(va[j],ke[j],kia[j])-2*ipumpa(ke[j],naia[j],imaxa[j])-inagl[j]) naia[1..25]'=-c3a*(inaa(va[j],nae[j],naia[j])+3*ipumpa(ke[j],naia[j],imaxa[j])+(3)*inagl[j]) cle[1..25]'=-c2*icl[j]+gclb*(cl0-cle[j]) cli[1..25]'=c1*icl[j] p gnab=1e-05 p na0=140 p gkb=2e-05 p k0=4 p cl0=110 p gclb=6e-06 # Glutamate glut[1..25]'=c3e*inagl[j]+gdiff[j] glui[1..25]=(1/c3e)*(gltot-c3a*glut[j]) gltot=c3e*glui0+c3a*glut0 p glui0=1 p glut0=.001 # Ca2+ cae[1..25]'=c2*(icanmda[j])+gcab*(cae0-cae[j]) cai[1..25]'=-fi*c1*(icanmda[j])-fi*(e*(jserca[j]-jerout[j]))/(vc*minute*second) her[1..25]'=(dinh-(cai[j]+dinh)*her[j])/tau caer[1..25]'=fi*(e*(jserca[j]-jerout[j]))/(ve*minute*second) jerout[1..25]=(vip3*((ip3/(ip3+dip3))^3)*((cai[j]/(cai[j]+dact))^3)*(her[j]^3)+vleak)*(caer[j]-cai[j]) jserca[1..25]=vserca*cai[j]^2/(kserca^2+cai[j]^2) p gcab=3e-05 p cae0=3 p minute=60 p second=1000 p vg=1.0 p pcytosol=0.5 p dcytosol=75 p per=0.10 p der=1000 p fi=.01 e=vg*per*der vc=vg*pcytosol ve=vg*per p vip3=3000 p vleak=0.01 p dip3=0.25 p dinh=.5 p dact=1 p tau=4 p vserca=110 p kserca=0.4 p ip3=.3 i v[1..25]=-80 i n[1..25]=0.15 i hp[1..25]=0.9 i sg[1..25]=.00 i va[1..25]=-80 i kia[1..25]=130 i naia[1..25]=2 i ke[1..25]=4 i nai[1..25]=2 i nae[1..25]=135 i ki[1..25]=140 i cli[1..25]=6 i cle[1..25]=110 i glut[1..25]=.0001 i cae[1..25]=3 i cai[1..25]=.0126 i her[1..25]=0.9754 i caer[1..25]=10 ### XPP settings @ dt=.5,total=50000,meth=qualrk,tolerance=.0000001 @ xp=t,yp=v1,xlo=0,xhi=50000,ylo=-80,yhi=0.,bound=500000,maxstor=1500000 done