On 1/f^alpha power laws originating from linear neuronal cable theory: power spectral densities of the soma potential, transmembrane current and single-neuron contribution to the EEG

Power laws, that is, power spectral densities (PSDs) exhibiting 1/f^alpha behavior for large frequencies f, have commonly been observed in neural recordings. Power laws in noise spectra have not only been observed in microscopic recordings of neural membrane potentials and membrane currents, but also in macroscopic EEG (electroencephalographic) 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 1/f^alpha power laws. The corresponding power-law exponents are analytically identified as alpha_inf^I = 1/2 for the soma membrane current, alpha_inf^p = 3/2 for the current-dipole moment, and alpha_inf^V = 2 for the soma membrane potential. These power-law exponents are found for arbitrary combinations of uncorrelated and correlated noisy input current (as long as both the dendrites and the soma receive some uncorrelated input currents). Comparison with available data suggests that the apparent 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 ...