Frequency-dependent responses of neuronal models to oscillatory inputs in current versus voltage clamp

Action potential generation in neurons depends on a membrane potential threshold and therefore on how subthreshold inputs influence this voltage. In oscillatory networks, for example, many neuron types have been shown to produce membrane potential (V_m) resonance: a maximum subthreshold response to oscillatory inputs at a nonzero frequency. Resonance is usually measured by recording V_m in response to a sinusoidal current (I_app), applied at different frequencies (f), an experimental setting known as current clamp (I-clamp). Several recent studies, however, use the voltage clamp (V-clamp) method to control V_m with a sinusoidal input at different frequencies [V_app(f) ] and measure the total membrane current (I_m). The two methods obey systems of differential equations of different dimensionality, and while I-clamp provides a measure of electrical impedance [Z(f) = V_m(f) / I_app(f) ], V-clamp measures admittance [Y(f) = I_m(f) / V_app(f) ]. We analyze the relationship between these two measurement techniques. We show that, despite different dimensionality, in linear systems the two measures are equivalent: Z= Y^{- 1}. However, nonlinear model neurons produce different values for Z and Y^{- 1}. In particular, nonlinearities in the voltage equation produce a much larger difference between these two quantities than those in equations of recovery variables that describe activation and inactivation kinetics. Neurons are inherently nonlinear, and notably, with ionic currents that amplify resonance, the voltage clamp technique severely underestimates the current clamp response. We demonstrate this difference experimentally using the PD neurons in the crab stomatogastric ganglion. These findings are instructive for researchers who explore cellular mechanisms of neuronal oscillations.