COMMENT ----------------------------------------------------------------------------- Simple synaptic mechanism derived for first order kinetics of binding of transmitter to postsynaptic receptors. A. Destexhe & Z. Mainen, The Salk Institute, March 12, 1993. ----------------------------------------------------------------------------- During the arrival of the presynaptic spike (detected by threshold crossing), it is assumed that there is a brief pulse (duration=Cdur) of neurotransmitter C in the synaptic cleft (the maximal concentration of C is Cmax). Then, C is assumed to bind to a receptor Rc according to the following first-order kinetic scheme: Rc + C ---(Alpha)--> Ro (1) <--(Beta)--- where Rc and Ro are respectively the closed and open form of the postsynaptic receptor, Alpha and Beta are the forward and backward rate constants. If R represents the fraction of open gates Ro, then one can write the following kinetic equation: dR/dt = Alpha * C * (1-R) - Beta * R (2) and the postsynaptic current is given by: Isyn = gmax * R * (V-Erev) (3) where V is the postsynaptic potential, gmax is the maximal conductance of the synapse and Erev is the reversal potential. If C is assumed to occur as a pulse in the synaptic cleft, such as C _____ . . . . . . Cmax | | _____| |______ . . . 0 t0 t1 then one can solve the kinetic equation exactly, instead of solving one differential equation for the state variable and for each synapse, which would be greatly time consuming... Equation (2) can be solved as follows: 1. during the pulse (from t=t0 to t=t1), C = Cmax, which gives: R(t-t0) = Rinf + [ R(t0) - Rinf ] * exp (- (t-t0) / Rtau ) (4) where Rinf = Alpha * Cmax / (Alpha * Cmax + Beta) and Rtau = 1 / (Alpha * Cmax + Beta) 2. after the pulse (t>t1), C = 0, and one can write: R(t-t1) = R(t1) * exp (- Beta * (t-t1) ) (5) There is a pointer called "pre" which must be set to the variable which is supposed to trigger synaptic release. This variable is usually the presynaptic voltage but it can be the presynaptic calcium concentration, or other. Prethresh is the value of the threshold at which the release is initiated. Once pre has crossed the threshold value given by Prethresh, a pulse of C is generated for a duration of Cdur, and the synaptic conductances are calculated accordingly to eqs (4-5). Another event is not allowed to occur for Deadtime milliseconds following after pre rises above threshold. The user specifies the presynaptic location in hoc via the statement connect pre_GLU[i] , v.section(x) where x is the arc length (0 - 1) along the presynaptic section (the currently specified section), and i is the synapse number (Which is located at the postsynaptic location in the usual way via postsynaptic_section {loc_GLU(i, x)} Notice that loc_GLU() must be executed first since that function also allocates space for the synapse. ----------------------------------------------------------------------------- GLUTAMATE SYNAPSE (AMPA-Kainate receptors) Parameters estimated from whole cell recordings of synaptic currents on Cochlear neurons (Raman & Trussel, Neuron 9: 173-186, 1992) as well as from sharp electrode EPSP's recordings in thalamocortical neurons (LGN) (Crunelli et al. J. Physiol. 384: 603, 1987). ----------------------------------------------------------------------------- ENDCOMMENT INDEPENDENT {t FROM 0 TO 1 WITH 1 (ms)} NEURON { POINT_PROCESS NMDA POINTER pre RANGE C, R, R0, R1, g, gmax, lastrelease NONSPECIFIC_CURRENT i GLOBAL Cmax, Cdur, Alpha, Beta, Erev, Prethresh, Deadtime, Rinf, Rtau } UNITS { (nA) = (nanoamp) (mV) = (millivolt) (umho) = (micromho) (mM) = (milli/liter) } PARAMETER { Cmax = 1 (mM) : max transmitter concentration Cdur = 1.1 (ms) : transmitter duration (rising phase) Alpha = 10 (/ms mM) : forward (binding) rate Beta = 0.0125 (/ms) : backward (unbinding) rate Erev = 0 (mV) : reversal potential Prethresh = 0 : voltage level nec for release Deadtime = 0 (ms) : mimimum time between release events gmax (umho) : maximum conductance eta = 0.33 (/mM) mag = 1 (mM) gamma = 0.06 (/mV) } ASSIGNED { v (mV) : postsynaptic voltage i (nA) : current = g*(v - Erev) g (umho) : conductance C (mM) : transmitter concentration R : fraction of open channels R0 : open channels at start of release R1 : open channels at end of release Rinf : steady state channels open Rtau (ms) : time constant of channel binding pre : pointer to presynaptic variable lastrelease (ms) : time of last spike } INITIAL { R = 0 C = 0 Rinf = Cmax*Alpha / (Cmax*Alpha + Beta) Rtau = 1 / ((Alpha * Cmax) + Beta) lastrelease = -9e9 } BREAKPOINT { SOLVE release g = (gmax * R)/(1 + eta * mag * exp( - (gamma * v))) i = g*(v - Erev) } PROCEDURE release() { LOCAL q :will crash if user hasn't set pre with the connect statement q = ((t - lastrelease) - Cdur) : time since last release ended : ready for another release? if (q > Deadtime) { if (pre > Prethresh) { : spike occured? C = Cmax : start new release R0 = R lastrelease = t } } else if (q < 0) { : still releasing? : do nothing } else if (C == Cmax) { : in dead time after release R1 = R C = 0. } if (C > 0) { : transmitter being released? R = Rinf + (R0 - Rinf) * exptable (- (t - lastrelease) / Rtau) } else { : no release occuring R = R1 * exptable (- Beta * (t - (lastrelease + Cdur))) } VERBATIM return 0; ENDVERBATIM } FUNCTION exptable(x) { TABLE FROM -10 TO 10 WITH 2000 if ((x > -10) && (x < 10)) { exptable = exp(x) } else { exptable = 0. } }