A Dynamic Clamp on Every Rig

Breadboard electronics. The five parts of the system illustrated in Fig. 1B are shown schematically. A, Rail splitter power supply. An 18-V power supply (wall adaptor) is split by a TLE 2426 rail splitter IC into 9 V, –9 V, and ground. The capactors are C₁ = 200 μF and C₂ = 1 μF. The dark numbers refer to the pins of the TLE 2426 IC. B, Follower, voltage divider, and differential amplifier circuits to map the range −9 to 9 V onto the range 0 to 3.3 V. The resistor values are R₁ = 2200 Ω, R₂ = 470 Ω, R₃ = 4700 Ω, R₄ = 22,000 Ω, R₅ = 10,000 Ω, and R₆ = 100 Ω. C, Connections to the Teensy 3.6 microcontroller. The output of the previous circuit is fed to pin A0, and the output of pin DAC0 is fed to the next circuit. D, Differential amplifier circuit to transform the output of the microcontroller (0–3.3 V, representing the dynamic clamp current) into a range (±9 V) expected by the intracellular amplifier. The resistor values are R₇ = 4700 Ω, R₈ = 22,000 Ω, R₉ = 10,000 Ω, and R₁₀ = 10,000 Ω. E, Summing amplifier. The voltage command from the DAQ board (representing the current clamp’s specified current) is added to the voltage command from the Teensy microcontroller. The sum is sent to the command input of the intracellular amplifier. Resistors R₁₁, R₁₂, R₁₃, R₁₅, and R₁₆ are 10,000 Ω; resistor R₁₄ is 3300 Ω; and R₁₇ is 4700 Ω.

Calibrating and testing the system. A, Although the electronic components were not ideal, the input-output characteristics of the system were highly linear. Left, the voltages measured by the microcontroller’s analog input in response to a range of membrane potentials. Right, the currents injected into a model cell in response to a range of voltages sent out by the microcontroller’s analog output. B, To test the system, a model cell was attached to the intracellular amplifier’s headstage. Shown is the model’s equivalent circuit. C, The system’s speed approaches 100 kHz and depends only weakly on the number of conductances being simulated. We recorded the durations of 51,200 time steps (at microsecond resolution) for each of three dynamic clamp configurations: shunt conductance only; shunt, HCN, and sodium conductances together; and shunt, HCN, sodium, Ornstein–Uhlenbeck, and EPSC conductances all together. Shown in the three figures at left are the resulting histograms; the number at the top of each is mean ± SD. The temporal jitter in all cases was 1–2 μs. To check the temporal latency, sinusoidal voltages (5 kHz) were fed to the system’s input (replacing V_IN of Fig. 2B) and the resulting outgoing current commands (V_AMP of Fig. 2E) were measured for a shunt conductance (2 nS). Both the input signal and the output signal were sampled at 100 kHz. The latency between input and output was roughly 10 μs.

Shunt conductance. Adding 5 nS of shunt conductance to the model cell reduced and quickened the voltage deflections to a family of current steps (upper left). The shunt currents added by the dynamic clamp system (measured directly by the DAQ board) closely matched what they should have been given the recorded membrane potentials (numerical; upper right). The inset shows a probability histogram of the difference (error) between the measured and numerical currents. Varying the amplitude of the shunt conductance affected the input resistance and time constant (measured) as expected given the numerical values of the model cell resistance and the shunt conductance amplitudes (numerical; lower panels).

HCN conductance. A, The conductance was modeled by a single-activation gate that had a steady-state value s_inf(V) and a time constant τ_s(V). B, Current steps (–100 to 40 pA) were injected into a model cell without (CTL) and with (+HCN) the addition of 2 nS HCN conductance. Note that adding the simulated conductance resulted in the appearance of a sag potential (left). At right, the currents injected by the dynamic clamp system (measured, directly by the DAQ board) for the eight steps are plotted together with the result of numerically integrating the Hodgkin–Huxley equation using the fourth-order Runge–Kutta method (numerical). The top histogram (“steps error”) of D shows how good the agreement between the measured and predicted (numerical) currents was. C, The model cell (Fig. 3B) is essentially an “RC circuit.” In response to a time-varying input, it acts like a low-pass filter. This can be seen (left) by its response to a chirp stimulus (black); the voltage deflection steadily decreases as the frequency increases. Addition of 4 nS HCN conductance transforms the system into a bandpass filter, with a resonant frequency. Again, the agreement between the current injected by the dynamic clamp system (measured) and the expected current given by numerical integration of the Hodgkin–Huxley equations (numerical) was excellent. D, Histograms of the error between the measured and expected currents for step currents (top) and the chirp current (bottom).

Sodium conductance. A, The conductance was modeled using both an activation gate m(V_m,t) and an inactivation gate h(V_m,t). The steady-state and kinetic values of the two gates are plotted. The total sodium current was given by g_Nam ³h(V_m – E_Na), where g_Na is the maximal sodium conductance and E_Na is the sodium reversal potential (50 mV). B, Comparison of different numerical integration methods. In a simulation, the voltage was stepped instantaneously from –70 mV to 0 mV at a time t = 20 ms. The resulting (inward) sodium current was calculated using the forward Euler method (time step 200, 100, or 12 μs), the fourth-order Runge–Kutta method (time step 10 μs), or an exact analytical calculation (exact). For the Euler simulation of 12 μs, the time step was also jittered by 2 μs (standard deviation). C, At left, the response of the model cell to slow current ramps (5, 10, and 20 pA/s) is plotted in the absence (CTL) and presence (+ Na) of an added sodium conductance (g_Na = 20 nS). In the middle, the close agreement between the sodium currents produced by the dynamic clamp system (measured) and those expected from precise numerical integration of the Hodgkin–Huxley equations (numerical) is demonstrated. At right is a histogram of the error (difference) between the measured and expected currents. D, The model cell was subjected to brief current steps. Without a sodium conductance, the responses showed pure exponential growth, as expected of an RC circuit. With a sodium conductance (g_Na = 80 nS), the responses showed nonlinear behavior above a threshold (>25 pA). The sodium currents (measured directly and expected from numerical integration) are plotted in the middle. Not only do the currents agree with each other, but they show a striking threshold behavior. At right (top), the error (difference) between the measured current and the expected current is plotted for the largest current step (100 pA). The shape of the error is consistent with what would be expected from a small offset in the baseline membrane potential. At right (bottom) is the difference between the fourth-order Runge–Kutta (time step 10 μs) estimate of the current for a baseline of –70 mV and the estimate for a baseline of –69.5 mV.

Excitatory postsynaptic currents. A, EPSCs were triggered at fixed times with stimulation frequencies of 10, 20, and 50 Hz. Note how the potentials summate for the higher frequencies. B, The agreement between the currents injected by the dynamic clamp system (measured) and the expected currents given by precise Runge–Kutta numerical integration of the two-stage kinetic scheme we used for EPSCs (Compte et al., 2000) was excellent at all frequencies. C, Histogram of the difference (error) between the measured and expected currents. Data from all three frequencies were combined in this histogram.

Synaptic background activity. A, In the point conductance model of Destexhe et al., (2001), synaptic background activity is modeled by two noisy conductance trains. One represents excitatory input, and the other represents inhibitory input; both are generated by Ornstein–Uhlenbeck processes. Each train is normally distributed (middle) and is correlated at short times (right; the power spectrum goes like 1/f ² for higher frequencies). B, Without background activity, the model cell has a flat membrane potential (left) that is almost constant (middle); its input resistance is large (right). Adding in vivo–like background activity depolarizes the membrane, introduces membrane potential fluctuations, and reduces the input resistance. The vertical scale of the middle histogram is truncated so that the membrane potential distribution in the active state is easier to see. C, Varying the mean (g, in nanoSiemens) and standard deviation (s, in nanoSiemens) of the excitatory and inhibitory conductances produced the predicted changes in total conductance (top) and membrane potential fluctuations (bottom). The added conductance is expected to equal the sum of gE and gI. The numerical estimates of standard deviation were calculated by simulating the model cell. When sE and sI were varied, gE and gI were held fixed at 4 nS.