TITLE Calcium ion accumulation and diffusion : The internal coordinate system is set up in PROCEDURE coord_cadifus() : and must be executed before computing the concentrations. : The scale factors set up in this procedure do not have to be recomputed : when diam or DFree are changed. : The amount of calcium in an annulus is ca[i]*diam^2*vol[i] with : ca[0] being the second order correct concentration at the exact edge : and ca[NANN-1] being the concentration at the exact center NEURON { SUFFIX cadifus2 USEION ca READ cao, cai, ica WRITE cai, ica GLOBAL vol, Buffer0 RANGE ipump } DEFINE NANN 4 UNITS { (mol) = (1) (molar) = (1/liter) (mM) = (millimolar) (um) = (micron) (mA) = (milliamp) FARADAY = (faraday) (10000 coulomb) PI = (pi) (1) } PARAMETER { DFree = .6 (um2/ms) diam (um) cao (mM) ica (mA/cm2) k1buf = 50 (/mM-ms) k2buf = 5 (/ms) k1=1.e10 (um3/s) k2=50.e7 (/s) : k1*50.e-3 k3=1.e10 (/s) : k1 k4=5.e6 (um3/s) : k1*5.e-4 area (um2) } CONSTANT { volo=1 (liter)} ASSIGNED { cai (mM) vol[NANN] (1) : gets extra cm2 when multiplied by diam^2 ipump (mA/cm2) last_ipump (mA/cm2) } STATE { ca[NANN] (mM) <1.e-5> : ca[0] is equivalent to cai CaBuffer[NANN] (mM) Buffer[NANN] (mM) pump (mol/cm2) <1.e-15> pumpca (mol/cm2) <1.e-15> } LOCAL totpump, kd,totbuf INITIAL { totpump=0.2 pump=totpump/(1+1.e-18*k4*cao/k3) pumpca=2.e-22 ipump=0 totbuf=1.2 kd=k2buf/k1buf FROM i=0 TO NANN-1 { ca[i] = cai CaBuffer[i] =(totbuf*ca[i])/(kd+ca[i]) Buffer[i] = totbuf - CaBuffer[i] } } BREAKPOINT { SOLVE state METHOD sparse last_ipump=ipump ica = ipump } LOCAL coord_done INITIAL { if (coord_done == 0) { coord_done = 1 coord() } : note Buffer gets set to Buffer0 automatically : and CaBuffer gets set to 0 (Default value of CaBuffer0) as well FROM i=0 TO NANN-1 { ca[i] = cai } } LOCAL frat[NANN] : gets extra cm when multiplied by diam PROCEDURE coord() { LOCAL r, dr2 : cylindrical coordinate system with constant annuli thickness to : center of cell. Note however that the first annulus is half thickness : so that the concentration is second order correct spatially at : the membrane or exact edge of the cell. : note ca[0] is at edge of cell : ca[NANN-1] is at center of cell r = 1/2 :starts at edge (half diam) dr2 = r/(NANN-1)/2 :half thickness of annulus vol[0] = 0 frat[0] = 2*r FROM i=0 TO NANN-2 { vol[i] = vol[i] + PI*(r-dr2/2)*2*dr2 :interior half r = r - dr2 frat[i+1] = 2*PI*r/(2*dr2) :exterior edge of annulus : divided by distance between centers r = r - dr2 vol[i+1] = PI*(r+dr2/2)*2*dr2 :outer half of annulus } } LOCAL dsq, dsqvol : can't define local variable in KINETIC block or use : in COMPARTMENT KINETIC state { COMPARTMENT i, diam*diam*vol[i]*1(um) {ca CaBuffer Buffer} COMPARTMENT (1.e10)*area {pump pumpca} COMPARTMENT (1.e15)*volo {cao} ~ ca[0] << (-(ica-last_ipump)*PI*diam*frat[0]*1(um)/(2*FARADAY)) FROM i=0 TO NANN-2 { ~ ca[i] <-> ca[i+1] (DFree*frat[i+1]*1(um), DFree*frat[i+1]*1(um)) } dsq = diam*diam*1(um) FROM i=0 TO NANN-1 { dsqvol = dsq*vol[i] ~ ca[i] + Buffer[i] <-> CaBuffer[i] (k1buf*dsqvol,k2buf*dsqvol) } ~ca[0] + pump <-> pumpca ((1.e-11)*k1*area, (1.e7)*k2*area) ~pumpca <-> pump + cao ((1.e7)*k3*area, (1.e-11)*k4*area) ipump = 2*FARADAY*(f_flux-b_flux)/area cai = ca[0] } COMMENT At this time, conductances (and channel states and currents are calculated at the midpoint of a dt interval. Membrane potential and concentrations are calculated at the edges of a dt interval. With secondorder=2 everything turns out to be second order correct. ENDCOMMENT