: \$Id: bg.inc,v 1.3 1996/04/05 23:20:18 billl Exp \$ TITLE Borg-Graham Channel Model COMMENT Modeling the somatic electrical response of hippocampal pyramidal neurons, MS thesis, MIT, May 1987. Each channel has activation and inactivation particles as in the original Hodgkin Huxley formulation. The activation particle mm and inactivation particle hh go from on to off states according to kinetic variables alpha and beta which are voltage dependent. The form of the alpha and beta functions were dissimilar in the HH study. The BG formulae are: alpha = base_rate * Exp[(v - v_half)*valence*gamma*F/RT] beta = base_rate * Exp[(-v + v_half)*valence*(1-gamma)*F/RT] where, baserate : no affect on Inf. Lowering this increases the maximum value of Tau basetau : (in msec) minimum Tau value. chanexp : number for exponentiating the state variable; e.g. original HH Na channel use m^3, note that chanexp = 0 will turn off this state variable erev : reversal potential for the channel gamma : (between 0 and 1) does not affect the Inf but makes the Tau more asymetric with increasing deviation from 0.5 celsius : temperature at which experiment was done (Tau will will be adjusted using a q10 of 3.0) valence (z) : determines the steepness of the Inf sigmoid. Higher valence gives steeper sigmoid. vhalf : (a voltage) determines the voltage at which the value of the sigmoid function for Inf is 1/2 vmin, vmax : limits for construction of the table. Generally, these should be set to the limits over which either of the 2 state variables are varying. vrest : (a voltage) voltage shift for vhalf ENDCOMMENT INDEPENDENT {t FROM 0 TO 1 WITH 1 (ms)} NEURON { RANGE gmax, g, i GLOBAL erev, Inf, Tau, Mult, Add, vmin, vmax, vrest } : end NEURON CONSTANT { FARADAY = 96489.0 : Faraday's constant R= 8.31441 : Gas constant } : end CONSTANT UNITS { (mA) = (milliamp) (mV) = (millivolt) (umho) = (micromho) } : end UNITS COMMENT ** Parameter values should come from files specific to particular channels PARAMETER { erev = 0 (mV) gmax = 0 (mho/cm^2) vrest = 0 (mV) mvalence = 0 mgamma = 0 mbaserate = 0 mvhalf = 0 mbasetau = 0 mtemp = 0 mq10 = 3 mexp = 0 hvalence = 0 hgamma = 0 hbaserate = 0 hvhalf = 0 hbasetau = 0 htemp = 0 hq10 = 3 hexp = 0 cao (mM) cai (mM) celsius (degC) dt (ms) v (mV) vmax = 100 (mV) vmin = -100 (mV) } : end PARAMETER ENDCOMMENT ASSIGNED { i (mA/cm^2) g (mho/cm^2) Inf[2] : 0 = m and 1 = h Tau[2] : 0 = m and 1 = h Mult[2] : 0 = m and 1 = h Add[2] : 0 = m and 1 = h } : end ASSIGNED STATE { m h } INITIAL { mh(v) if (usetable==0) { m = Inf[0] h = Inf[1] } else { m = Add[0]/(1-Mult[0]) h = Add[1]/(1-Mult[1]) } } BREAKPOINT { LOCAL hexp_val, index, mexp_val SOLVE states hexp_val = 1 mexp_val = 1 : Determining h's exponent value if (hexp > 0) { FROM index=1 TO hexp { hexp_val = h * hexp_val } } : Determining m's exponent value if (mexp > 0) { FROM index = 1 TO mexp { mexp_val = m * mexp_val } } : mexp hexp : Note that mexp_val is now = m and hexp_val is now = h g = gmax * mexp_val * hexp_val iassign() } : end BREAKPOINT : ASSIGNMENT PROCEDURES : Must be overwritten by user routines in parameters.multi : PROCEDURE iassign () { i = g*(v-erev) ina=i } : PROCEDURE iassign () { i = g*ghkca(v) ica=i } :------------------------------------------------------------------- : I suppose we have 2 choices, to use the DERIVATIVE function or : to explicitly state m+ and h+. If you were to use the DERIVATIVE : function, then you will do as follows: : DERIVATIVE deriv { : m' = (-m + minf) / mtau : h' = (-h + hinf) / htau : } : Else, since m' = (m+ - m) / dt, setting the 2 equations together, : we can solve for m+ and eventually get : : m+ = (m * mtau + dt * minf) / (mtau + dt) : and same for h+: : h+ = (h * htau + dt * hinf) / (htau + dt) : This is the one we will use, so ... PROCEDURE states() { : Setup the mh table values mh (v*1(/mV)) m = m * Mult[0] + Add[0] h = h * Mult[1] + Add[1] VERBATIM return 0; ENDVERBATIM } :------------------------------------------------------------------- : NOTE : 0 = m and 1 = h PROCEDURE mh (v) { LOCAL a, b, j, mqq10, hqq10 TABLE Add, Mult DEPEND dt, hbaserate, hbasetau, hexp, hgamma, htemp, hvalence, hvhalf, mbaserate, mbasetau, mexp, mgamma, mtemp, mvalence, mvhalf, celsius, mq10, hq10, vrest, vmin, vmax FROM vmin TO vmax WITH 200 mqq10 = mq10^((celsius-mtemp)/10.) hqq10 = hq10^((celsius-htemp)/10.) : Calculater Inf and Tau values for h and m FROM j = 0 TO 1 { a = alpha (v, j) b = beta (v, j) Inf[j] = a / (a + b) VERBATIM switch (_lj) { case 0: /* Make sure Tau is not less than the base Tau */ if ((Tau[_lj] = 1 / (_la + _lb)) < mbasetau) { Tau[_lj] = mbasetau; } Tau[_lj] = Tau[_lj] / _lmqq10; break; case 1: if ((Tau[_lj] = 1 / (_la + _lb)) < hbasetau) { Tau[_lj] = hbasetau; } Tau[_lj] = Tau[_lj] / _lhqq10; if (hexp==0) { Tau[_lj] = 1.; } break; } ENDVERBATIM Mult[j] = exp(-dt/Tau[j]) Add[j] = Inf[j]*(1. - exp(-dt/Tau[j])) } } : end PROCEDURE mh (v) :------------------------------------------------------------------- FUNCTION alpha(v,j) { if (j == 1) { if (hexp==0) { alpha = 1 } else { alpha = hbaserate * exp((v - (hvhalf+vrest)) * hvalence * hgamma * FRT(htemp)) } } else { alpha = mbaserate * exp((v - (mvhalf+vrest)) * mvalence * mgamma * FRT(mtemp)) } } : end FUNCTION alpha (v,j) :------------------------------------------------------------------- FUNCTION beta (v,j) { if (j == 1) { if (hexp==0) { beta = 1 } else { beta = hbaserate * exp((-v + (hvhalf+vrest)) * hvalence * (1 - hgamma) * FRT(htemp)) } } else { beta = mbaserate * exp((-v + (mvhalf+vrest)) * mvalence * (1 - mgamma) * FRT(mtemp)) } } : end FUNCTION beta (v,j) :------------------------------------------------------------------- FUNCTION FRT(temperature) { FRT = FARADAY * 0.001 / R / (temperature + 273.15) } : end FUNCTION FRT (temperature) :------------------------------------------------------------------- FUNCTION ghkca (v) { : Goldman-Hodgkin-Katz eqn LOCAL nu, efun nu = v*2*FRT(celsius) if(fabs(nu) < 1.e-6) { efun = 1.- nu/2. } else { efun = nu/(exp(nu)-1.) } ghkca = -FARADAY*2.e-3*efun*(cao - cai*exp(nu)) } : end FUNCTION ghkca()