Power laws from linear neuronal cable theory: power spectral densities of the soma potential, soma membrane current and single-neuron contribution to the EEG.
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With homogeneously distributed input currents across the neuronal membrane we find that all PSD transfer functions express asymptotic high-frequency 1/f(α) power laws with power-law exponents analytically identified as α∞(I) = 1/2 for the soma membrane current, α∞(p) = 3/2 for the current-dipole moment, and α∞(V) = 2 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 (1/f) 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.
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Affiliation: Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway; Letten Centre and GliaLab, Institute of Basic Medical Sciences, University of Oslo, Oslo, Norway; Centre for Molecular Medicine Norway, University of Oslo, Oslo, Norway.
ABSTRACT
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Power laws, that is, power spectral densities (PSDs) exhibiting 1/f(α) 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 1/f(α) power laws with power-law exponents analytically identified as α∞(I) = 1/2 for the soma membrane current, α∞(p) = 3/2 for the current-dipole moment, and α∞(V) = 2 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 (1/f) 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 1/f(α) power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation. Related in: MedlinePlus |
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Mentions: In Figs. 4, 5, and 6 we show color plots of for the soma current , current-dipole moment , and soma potential , respectively, both for cases with uncorrelated and correlated inputs. The depicted results are found by numerically evaluating Eq. 116 based on the expressions for listed in Eqs. 81–89. Note that since our model is linear, the log-log derivative is independent of the amplitude . Thus, with either completely correlated or completely uncorrelated input, the dimensionless parameters , , and span the whole parameter space of the model. The 2D color plots in Figs. 4–6 depict as function of and for three different values of the electronic length ( = 0.25, 1, and 4), i.e., spanning the situations from a very short dendritic stick () to a very long stick (). Electrotonic lengths greater than produced plots that were indistinguishable by eye from the plots for . The thin black contour line denotes the transition between the low- and intermediate-frequency regimes (), whereas the thick black contour line denotes the transition between the intermediate- and high-frequency regimes (). |
View Article: PubMed Central - PubMed
Affiliation: Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway; Letten Centre and GliaLab, Institute of Basic Medical Sciences, University of Oslo, Oslo, Norway; Centre for Molecular Medicine Norway, University of Oslo, Oslo, Norway.