Power Laws from Linear Neuronal Cable Theory: Power Spectral Densities of the Soma Potential, Soma Membrane Current and Single-Neuron Contribution to the EEG

Power laws, that is, power spectral densities (PSDs) exhibiting behavior for large frequencies f, have been observed both in microscopic (neural membrane potentials and currents) and macroscopic (electroencephalography; EEG) recordings. While complex network behavior has been suggested to be at the root of this phenomenon, we here demonstrate a possible origin of such power laws in the biophysical properties of single neurons described by the standard cable equation. Taking advantage of the analytical tractability of the so called ball and stick neuron model, we derive general expressions for the PSD transfer functions for a set of measures of neuronal activity: the soma membrane current, the current-dipole moment (corresponding to the single-neuron EEG contribution), and the soma membrane potential. These PSD transfer functions relate the PSDs of the respective measurements to the PSDs of the noisy input currents. With homogeneously distributed input currents across the neuronal membrane we find that all PSD transfer functions express asymptotic high-frequency power laws with power-law exponents analytically identified as for the soma membrane current, for the current-dipole moment, and for the soma membrane potential. Comparison with available data suggests that the apparent power laws observed in the high-frequency end of the PSD spectra may stem from uncorrelated current sources which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. While the PSD noise spectra at low frequencies may be dominated by synaptic noise, our findings suggest that the high-frequency power laws may originate in noise from intrinsic ion channels. The significance of this finding goes beyond neuroscience as it demonstrates how power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation.Author Summary: The common observation of power laws in nature and society, that is, quantities or probabilities that follow distributions, has for long intrigued scientists. In the brain, power laws in the power spectral density (PSD) have been reported in electrophysiological recordings, both at the microscopic (single-neuron recordings) and macroscopic (EEG) levels. We here demonstrate a possible origin of such power laws in the basic biophysical properties of neurons, that is, in the standard cable-equation description of neuronal membranes. Taking advantage of the mathematical tractability of the so called ball and stick neuron model, we demonstrate analytically that high-frequency power laws in key experimental neural measures will arise naturally when the noise sources are evenly distributed across the neuronal membrane. Comparison with available data further suggests that the apparent high-frequency power laws observed in experiments may stem from uncorrelated current sources, presumably intrinsic ion channels, which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. The significance of this finding goes beyond neuroscience as it demonstrates how power laws power-law exponents α may arise from a simple, linear physics equation.