A modified cable formalism for modeling neuronal membranes at high frequencies

Intracellular recordings of cortical neurons in vivo display intense subthreshold membrane potential (Vm) activity. The power spectral density (PSD) of the Vm displays a power-law structure at high frequencies (>50 Hz) with a slope of about -2.5. This type of frequency scaling cannot be accounted for by traditional models, as either single-compartment models or models based on reconstructed cell morphologies display a frequency scaling with a slope close to -4. This slope is due to the fact that the membrane resistance is “short-circuited” by the capacitance for high frequencies, a situation which may not be realistic. Here, we integrate non-ideal capacitors in cable equations to reflect the fact that the capacitance cannot be charged instantaneously. We show that the resulting “non-ideal” cable model can be solved analytically using Fourier transforms. Numerical simulations using a ball-and-stick model yield membrane potential activity with similar frequency scaling as in the experiments. We also discuss the consequences of using non-ideal capacitors on other cellular properties such as the transmission of high frequencies, which is boosted in non-ideal cables, or voltage attenuation in dendrites. These results suggest that cable equations based on non-ideal capacitors should be used to capture the behavior of neuronal membranes at high frequencies.

💡 Research Summary

The paper addresses a long‑standing discrepancy between intracellular recordings of cortical neurons and the predictions of classic cable theory. In vivo, the subthreshold membrane potential (Vm) exhibits a power‑law power spectral density (PSD) at frequencies above roughly 50 Hz, with an exponent close to –2.5. Traditional single‑compartment or morphologically realistic cable models, however, predict a much steeper decay (≈ –4) because at high frequencies the membrane resistance is effectively short‑circuited by the capacitance, leading to an impedance that scales as ω⁻¹ and a voltage that scales as ω⁻².

To reconcile theory with experiment, the authors introduce a “non‑ideal” capacitor into the cable equations. Real biological membranes do not charge instantaneously; they possess dielectric losses that can be modeled as a series resistance (the Stroud resistance) in parallel with the ideal capacitor. This adds a finite time constant τₘ = Rₛ·Cₘ, so the current‑voltage relationship becomes I = Cₘ·dV/dt + V/Rₛ rather than the pure capacitive term alone. Consequently, the complex admittance of the membrane is
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