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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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Schematic illustration of the ball and stick neuron model and its filtering properties.(A) Schematic illustration of the ball and stick neuron model with a single input at a given position . The lumped soma is assumed iso-potential and located at . (B) Frequency-dependent current-density envelopes of return currents for a ball and stick neuron with input at . The somatic return currents are illustrated as current densities from a soma section with length  placed below the stick. For 1 Hz, 10 Hz, 100 Hz and 1000 Hz the amplitudes of the somatic return currents are about 1/7.3, 1/7.5, 1/22 and 1/3100 of the input current, respectively. Parameters used for the ball and stick neuron model: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and specific membrane capacitance of . This parameter set is the default parameter set used in the present study, see Table 1. (C) Representative log-log plot for a PSD when input is homogeneously distributed across the entire neuron model. The low frequency (lf), intermediate frequency (if) and high frequency (hf) regimes are stipulated. The regimes are defined relatively to  describing the asymptotic value of the exponent of the respective power-law transfer functions (,  or ), with both uncorrelated and correlated input (‘all’ types of input) onto both the soma and the stick.
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pcbi-1003928-g003: Schematic illustration of the ball and stick neuron model and its filtering properties.(A) Schematic illustration of the ball and stick neuron model with a single input at a given position . The lumped soma is assumed iso-potential and located at . (B) Frequency-dependent current-density envelopes of return currents for a ball and stick neuron with input at . The somatic return currents are illustrated as current densities from a soma section with length placed below the stick. For 1 Hz, 10 Hz, 100 Hz and 1000 Hz the amplitudes of the somatic return currents are about 1/7.3, 1/7.5, 1/22 and 1/3100 of the input current, respectively. Parameters used for the ball and stick neuron model: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and specific membrane capacitance of . This parameter set is the default parameter set used in the present study, see Table 1. (C) Representative log-log plot for a PSD when input is homogeneously distributed across the entire neuron model. The low frequency (lf), intermediate frequency (if) and high frequency (hf) regimes are stipulated. The regimes are defined relatively to describing the asymptotic value of the exponent of the respective power-law transfer functions (, or ), with both uncorrelated and correlated input (‘all’ types of input) onto both the soma and the stick.

Mentions: The ball and stick neuron [27] consists of a dendritic stick attached to a single-compartment soma, see Fig. 3A. Here we envision the stick to be a long and thin cylinder with diameter d and length l. The membrane area of the soma is set to be , corresponding to the surface area of a sphere with diameter , or equivalently, the side area of a cylindrical box with diameter and height .


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)

Schematic illustration of the ball and stick neuron model and its filtering properties.(A) Schematic illustration of the ball and stick neuron model with a single input at a given position . The lumped soma is assumed iso-potential and located at . (B) Frequency-dependent current-density envelopes of return currents for a ball and stick neuron with input at . The somatic return currents are illustrated as current densities from a soma section with length  placed below the stick. For 1 Hz, 10 Hz, 100 Hz and 1000 Hz the amplitudes of the somatic return currents are about 1/7.3, 1/7.5, 1/22 and 1/3100 of the input current, respectively. Parameters used for the ball and stick neuron model: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and specific membrane capacitance of . This parameter set is the default parameter set used in the present study, see Table 1. (C) Representative log-log plot for a PSD when input is homogeneously distributed across the entire neuron model. The low frequency (lf), intermediate frequency (if) and high frequency (hf) regimes are stipulated. The regimes are defined relatively to  describing the asymptotic value of the exponent of the respective power-law transfer functions (,  or ), with both uncorrelated and correlated input (‘all’ types of input) onto both the soma and the stick.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003928-g003: Schematic illustration of the ball and stick neuron model and its filtering properties.(A) Schematic illustration of the ball and stick neuron model with a single input at a given position . The lumped soma is assumed iso-potential and located at . (B) Frequency-dependent current-density envelopes of return currents for a ball and stick neuron with input at . The somatic return currents are illustrated as current densities from a soma section with length placed below the stick. For 1 Hz, 10 Hz, 100 Hz and 1000 Hz the amplitudes of the somatic return currents are about 1/7.3, 1/7.5, 1/22 and 1/3100 of the input current, respectively. Parameters used for the ball and stick neuron model: stick diameter , somatic diameter , stick length mm, specific membrane resistance , inner resistivity m and specific membrane capacitance of . This parameter set is the default parameter set used in the present study, see Table 1. (C) Representative log-log plot for a PSD when input is homogeneously distributed across the entire neuron model. The low frequency (lf), intermediate frequency (if) and high frequency (hf) regimes are stipulated. The regimes are defined relatively to describing the asymptotic value of the exponent of the respective power-law transfer functions (, or ), with both uncorrelated and correlated input (‘all’ types of input) onto both the soma and the stick.
Mentions: The ball and stick neuron [27] consists of a dendritic stick attached to a single-compartment soma, see Fig. 3A. Here we envision the stick to be a long and thin cylinder with diameter d and length l. The membrane area of the soma is set to be , corresponding to the surface area of a sphere with diameter , or equivalently, the side area of a cylindrical box with diameter and height .

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