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Power laws from linear neuronal cable theory: power spectral densities of the soma potential, soma membrane current and single-neuron contribution to the EEG.

Pettersen KH, Lindén H, Tetzlaff T, Einevoll GT - PLoS Comput. Biol. (2014)

Bottom Line: 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.

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.

ABSTRACT
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.

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Dependence of PSDs on biophysical parameters for uncorrelated input.PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (row 3) for the ball and stick model with uncorrelated white-noise input currents homogeneously distributed throughout the membrane. The input density is two inputs per square micrometer, and the input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and a specific membrane capacitance . The values of  printed in the legends describe the powers of the slopes at 1000 Hz. The upper  corresponds to the low value of the parameter varied (green), the middle  corresponds to the default parameter (red), while the lower  corresponds to the high value of the parameter varied (blue).
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pcbi-1003928-g007: Dependence of PSDs on biophysical parameters for uncorrelated input.PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (row 3) for the ball and stick model with uncorrelated white-noise input currents homogeneously distributed throughout the membrane. The input density is two inputs per square micrometer, and the input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and a specific membrane capacitance . The values of printed in the legends describe the powers of the slopes at 1000 Hz. The upper corresponds to the low value of the parameter varied (green), the middle corresponds to the default parameter (red), while the lower corresponds to the high value of the parameter varied (blue).

Mentions: The 2D color plots in Figs. 4–6 depicting the slopes of the PSDs of the transfer functions , give a comprehensive overview of the power-law properties of the ball and stick model as they are given in terms of the three key dimensionless parameters , , and . To get an additional view of how the model predictions depend on biophysical model parameters, we plot in Figs. 7 and 8 PSDs, denoted , for a range of model parameters for the soma current, current-dipole moment and soma potential when the neuron receives homogeneous white-noise current input across the dendrite and/or the soma. We focus on biophysical parameters that may vary significantly from neuron to neuron: the dendritic stick length , the specific membrane resistance , the dendritic stick diameter , and the soma diameter . The specific membrane resistance may not only vary between neurons, but also between different network states for the same neuron [40], [41].


Power laws from linear neuronal cable theory: power spectral densities of the soma potential, soma membrane current and single-neuron contribution to the EEG.

Pettersen KH, Lindén H, Tetzlaff T, Einevoll GT - PLoS Comput. Biol. (2014)

Dependence of PSDs on biophysical parameters for uncorrelated input.PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (row 3) for the ball and stick model with uncorrelated white-noise input currents homogeneously distributed throughout the membrane. The input density is two inputs per square micrometer, and the input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and a specific membrane capacitance . The values of  printed in the legends describe the powers of the slopes at 1000 Hz. The upper  corresponds to the low value of the parameter varied (green), the middle  corresponds to the default parameter (red), while the lower  corresponds to the high value of the parameter varied (blue).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4230751&req=5

pcbi-1003928-g007: Dependence of PSDs on biophysical parameters for uncorrelated input.PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (row 3) for the ball and stick model with uncorrelated white-noise input currents homogeneously distributed throughout the membrane. The input density is two inputs per square micrometer, and the input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and a specific membrane capacitance . The values of printed in the legends describe the powers of the slopes at 1000 Hz. The upper corresponds to the low value of the parameter varied (green), the middle corresponds to the default parameter (red), while the lower corresponds to the high value of the parameter varied (blue).
Mentions: The 2D color plots in Figs. 4–6 depicting the slopes of the PSDs of the transfer functions , give a comprehensive overview of the power-law properties of the ball and stick model as they are given in terms of the three key dimensionless parameters , , and . To get an additional view of how the model predictions depend on biophysical model parameters, we plot in Figs. 7 and 8 PSDs, denoted , for a range of model parameters for the soma current, current-dipole moment and soma potential when the neuron receives homogeneous white-noise current input across the dendrite and/or the soma. We focus on biophysical parameters that may vary significantly from neuron to neuron: the dendritic stick length , the specific membrane resistance , the dendritic stick diameter , and the soma diameter . The specific membrane resistance may not only vary between neurons, but also between different network states for the same neuron [40], [41].

Bottom Line: 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.

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.

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus