TITLE Fluctuating conductances COMMENT ----------------------------------------------------------------------------- Fluctuating conductance model for synaptic bombardment ====================================================== THEORY Synaptic bombardment is represented by a stochastic model containing two fluctuating conductances g_e(t) and g_i(t) descibed by: Isyn = g_e(t) * [V - E_e] + g_i(t) * [V - E_i] d g_e / dt = -(g_e - g_e0) / tau_e + sqrt(D_e) * Ft d g_i / dt = -(g_i - g_i0) / tau_i + sqrt(D_i) * Ft where E_e, E_i are the reversal potentials, g_e0, g_i0 are the average conductances, tau_e, tau_i are time constants, D_e, D_i are noise diffusion coefficients and Ft is a gaussian white noise of unit standard deviation. g_e and g_i are described by an Ornstein-Uhlenbeck (OU) stochastic process where tau_e and tau_i represent the "correlation" (if tau_e and tau_i are zero, g_e and g_i are white noise). The estimation of OU parameters can be made from the power spectrum: S(w) = 2 * D * tau^2 / (1 + w^2 * tau^2) and the diffusion coeffient D is estimated from the variance: D = 2 * sigma^2 / tau NUMERICAL RESOLUTION The numerical scheme for integration of OU processes takes advantage of the fact that these processes are gaussian, which led to an exact update rule independent of the time step dt (see Gillespie DT, Am J Phys 64: 225, 1996): x(t+dt) = x(t) * exp(-dt/tau) + A * N(0,1) where A = sqrt( D*tau/2 * (1-exp(-2*dt/tau)) ) and N(0,1) is a normal random number (avg=0, sigma=1) IMPLEMENTATION This mechanism is implemented as a nonspecific current defined as a point process. PARAMETERS The mechanism takes the following parameters: E_e = 0 (mV) : reversal potential of excitatory conductance E_i = -75 (mV) : reversal potential of inhibitory conductance g_e0 = 0.0121 (umho) : average excitatory conductance g_i0 = 0.0573 (umho) : average inhibitory conductance std_e = 0.0030 (umho) : standard dev of excitatory conductance std_i = 0.0066 (umho) : standard dev of inhibitory conductance tau_e = 2.728 (ms) : time constant of excitatory conductance tau_i = 10.49 (ms) : time constant of inhibitory conductance Gfluct2: conductance cannot be negative REFERENCE Destexhe, A., Rudolph, M., Fellous, J-M. and Sejnowski, T.J. Fluctuating synaptic conductances recreate in-vivo--like activity in neocortical neurons. Neuroscience 107: 13-24 (2001). (electronic copy available at http://cns.iaf.cnrs-gif.fr) A. Destexhe, 1999 ----------------------------------------------------------------------------- ENDCOMMENT INDEPENDENT {t FROM 0 TO 1 WITH 1 (ms)} NEURON { POINT_PROCESS Gfluct2 RANGE g_e, g_i, E_e, E_i, g_e0, g_i0, g_e1, g_i1 RANGE std_e, std_i, tau_e, tau_i, D_e, D_i NONSPECIFIC_CURRENT i } UNITS { (nA) = (nanoamp) (mV) = (millivolt) (umho) = (micromho) } PARAMETER { dt (ms) E_e = 0 (mV) : reversal potential of excitatory conductance E_i = -75 (mV) : reversal potential of inhibitory conductance g_e0 = 0.0121 (umho) : average excitatory conductance g_i0 = 0.0573 (umho) : average inhibitory conductance std_e = 0.0030 (umho) : standard dev of excitatory conductance std_i = 0.0066 (umho) : standard dev of inhibitory conductance tau_e = 2.728 (ms) : time constant of excitatory conductance tau_i = 10.49 (ms) : time constant of inhibitory conductance } ASSIGNED { v (mV) : membrane voltage i (nA) : fluctuating current g_e (umho) : total excitatory conductance g_i (umho) : total inhibitory conductance g_e1 (umho) : fluctuating excitatory conductance g_i1 (umho) : fluctuating inhibitory conductance D_e (umho umho /ms) : excitatory diffusion coefficient D_i (umho umho /ms) : inhibitory diffusion coefficient exp_e exp_i amp_e (umho) amp_i (umho) } INITIAL { g_e1 = 0 g_i1 = 0 if(tau_e != 0) { D_e = 2 * std_e * std_e / tau_e exp_e = exp(-dt/tau_e) amp_e = std_e * sqrt( (1-exp(-2*dt/tau_e)) ) } if(tau_i != 0) { D_i = 2 * std_i * std_i / tau_i exp_i = exp(-dt/tau_i) amp_i = std_i * sqrt( (1-exp(-2*dt/tau_i)) ) } } BREAKPOINT { SOLVE oup if(tau_e==0) { g_e = std_e * normrand(0,1) } if(tau_i==0) { g_i = std_i * normrand(0,1) } g_e = g_e0 + g_e1 if(g_e < 0) { g_e = 0 } g_i = g_i0 + g_i1 if(g_i < 0) { g_i = 0 } i = g_e * (v - E_e) + g_i * (v - E_i) } PROCEDURE oup() { : use Scop function normrand(mean, std_dev) if(tau_e!=0) { g_e1 = exp_e * g_e1 + amp_e * normrand(0,1) } if(tau_i!=0) { g_i1 = exp_i * g_i1 + amp_i * normrand(0,1) } } PROCEDURE new_seed(seed) { : procedure to set the seed set_seed(seed) VERBATIM printf("Setting random generator with seed = %g\n", _lseed); ENDVERBATIM }