FIGURE 1.4-13 Systems block diagram of the voltage-clamp system of Figure 1.4-12. Gm represents the parallel behavior of gK, gNa, and gL.
Referring to the systems block diagram in Figure 1.4-13 based on the circuit of Figure 1.4-12, it is easy to write the overall transfer function for Vm:
The (1/2) gain factor comes from letting R1 = R2. If (1/2)Kv Gc > Gm(max), then the dc gain of the closed-loop system ^ -1, and its time constant tc ^ Cm/[(1/2)KvGc] sec. Gm is the sum of [gL + gNa(Vm) + gK(Vm)] and varies with Vm and time. The closed-loop system xcl is much smaller than [Cm/Gm]min. The voltage clamp controller computes:
It is easy to include the voltage clamp in the HH patch model of Section 1.4.1. Note that Vm = -(v + 70) mV. Assume that the transconductance amplifier (VCVS) is ideal. That is, it has a zero Norton output conductance, and infinite bandwith (at least compared with the neuron membrane patch). The op amp is ideal except for finite gain, Kv, and the microelectrode is perfectly neutralized so Vm is measured exactly. The short Simnon program HHVCLAMP.t is listed below.
continuous system HHVCLAMP " 6/04/99 " Run w/ Euler integration w/ dt = 0.0001.
STATE v n m h " v in mV. v is depolarization if < 0.
DER dv dn dm dh " v = Vmo - Vm. Vm is actual transmembrane V.
" HH membrane patch ODE.
" Ionic current densities. Jk = gko* (n^4) * (v - Vk) Jna = gnao*(m^3)*h*(v - Vna) Jl = glo* (v - Vl) Jc = Cm*dv
dn = - n* (an + bn) + an dm = - m* (am + bm) + am dh = - h* (ah + bh) + ah
K+ activation parameter ODE. Na+ activation parameter ODE. Na+ inactivation para-meter ODE.
" VOLTAGE-DEPENDENT PARAMETER FUNCTIONS:
Vm = -(v + 70) Jksc = Jk/10 Jnasc = Jna/10 Jcsc = 10*Jc Vmsc = Vm/70
" Scaled current densities
" Scaled capacitor current density. " Scaled membrane voltage.
" VOLTAGE CLAMP CONTROL ALGORITHM: Jin = Kc* (Vs - v - 70)
" CONSTANTS: zero:0 Kc:1.E4 Vk: 12 Vna:-115 VL:-10.613 glo:0.3
Controller gain, Kc mV
" INPUTS: Jin < 0 depolarizes membrane. Jin inward is + charges inward "Vs1 = IF t > to THEN Vso ELSE 0 " Delayed voltage pulse input.
delt:14 " Duration of voltage pulse, ms.
Vs = 70 + ramp ramp = IF t > to THEN Kr*(t - to) ELSE 0 Kr:-3
" INITIAL CONDS. (Also final values for v = 0.)
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