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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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Normalized power spectral densities (PSDs) for the soma current, the current-dipole moment (i.e., EEG contribution) and the soma potential for a ball and stick neuron and a pyramidal neuron.A homogeneous density of noisy input currents is applied throughout the neural membrane. Columns 1 (ball and stick neuron) and 2 (pyramidal neuron) show PSDs for white-noise input, the blue and green lines correspond to uncorrelated and correlated input currents, respectively. Note that there is no green line in the two upper rows, since a homogeneous density of correlated inputs throughout the neuron gives no net soma current or dipole moment. An ensemble of PSDs from 20 single input currents for the ball and stick neuron and 107 single input currents for the pyramidal neuron is shown in grey. The results for the most distal synapses are shown in dark grey and the results for the proximal synapses in light grey, corresponding to the color shown in the filled circles at the respective neuron morphology (between columns 1 and 2). Column 3 illustrates how different power-law spectra of the input currents change the output PSDs: the blue, pink and brown lines express the PSD for uncorrelated white (constant), pink () and Brownian noise input (), respectively. The values of  in legends denote estimated power-law exponents at 1000 Hz, i.e., the negative discrete log-log derivative, . In the rightmost column the values of  correspond to pink noise input, for Brownian noise input and white-noise input the values are ‘+1’ and ‘−1’ with respect to the pink input, respectively, as indicated by the brown ‘+’ and the blue ‘−’. The ball and stick neuron was simulated with 200 dendritic segments (corresponding to the default parameters listed in Table 1), while the pyramidal neuron was simulated with 3214 dendritic segments. Broken lines correspond to the ball and stick neuron, whole lines to the pyramidal neuron.
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pcbi-1003928-g002: Normalized power spectral densities (PSDs) for the soma current, the current-dipole moment (i.e., EEG contribution) and the soma potential for a ball and stick neuron and a pyramidal neuron.A homogeneous density of noisy input currents is applied throughout the neural membrane. Columns 1 (ball and stick neuron) and 2 (pyramidal neuron) show PSDs for white-noise input, the blue and green lines correspond to uncorrelated and correlated input currents, respectively. Note that there is no green line in the two upper rows, since a homogeneous density of correlated inputs throughout the neuron gives no net soma current or dipole moment. An ensemble of PSDs from 20 single input currents for the ball and stick neuron and 107 single input currents for the pyramidal neuron is shown in grey. The results for the most distal synapses are shown in dark grey and the results for the proximal synapses in light grey, corresponding to the color shown in the filled circles at the respective neuron morphology (between columns 1 and 2). Column 3 illustrates how different power-law spectra of the input currents change the output PSDs: the blue, pink and brown lines express the PSD for uncorrelated white (constant), pink () and Brownian noise input (), respectively. The values of in legends denote estimated power-law exponents at 1000 Hz, i.e., the negative discrete log-log derivative, . In the rightmost column the values of correspond to pink noise input, for Brownian noise input and white-noise input the values are ‘+1’ and ‘−1’ with respect to the pink input, respectively, as indicated by the brown ‘+’ and the blue ‘−’. The ball and stick neuron was simulated with 200 dendritic segments (corresponding to the default parameters listed in Table 1), while the pyramidal neuron was simulated with 3214 dendritic segments. Broken lines correspond to the ball and stick neuron, whole lines to the pyramidal neuron.

Mentions: In the present study the idealized ball and stick neuron model will be treated analytically, while simulation results will be presented for a reconstructed layer V pyramidal neuron from cat visual cortex [34] (Fig. 2). Both the ball and stick model and the reconstructed layer V neuron model are purely passive, ensuring that linear theory can be used. The input currents are distributed throughout the neuron models with area density in the dendrite and in the soma. The input currents share statistics, i.e., they all have the same PSD, denoted , and a pairwise coherence . The coherence is zero for uncorrelated input and unity for perfectly correlated input.


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)

Normalized power spectral densities (PSDs) for the soma current, the current-dipole moment (i.e., EEG contribution) and the soma potential for a ball and stick neuron and a pyramidal neuron.A homogeneous density of noisy input currents is applied throughout the neural membrane. Columns 1 (ball and stick neuron) and 2 (pyramidal neuron) show PSDs for white-noise input, the blue and green lines correspond to uncorrelated and correlated input currents, respectively. Note that there is no green line in the two upper rows, since a homogeneous density of correlated inputs throughout the neuron gives no net soma current or dipole moment. An ensemble of PSDs from 20 single input currents for the ball and stick neuron and 107 single input currents for the pyramidal neuron is shown in grey. The results for the most distal synapses are shown in dark grey and the results for the proximal synapses in light grey, corresponding to the color shown in the filled circles at the respective neuron morphology (between columns 1 and 2). Column 3 illustrates how different power-law spectra of the input currents change the output PSDs: the blue, pink and brown lines express the PSD for uncorrelated white (constant), pink () and Brownian noise input (), respectively. The values of  in legends denote estimated power-law exponents at 1000 Hz, i.e., the negative discrete log-log derivative, . In the rightmost column the values of  correspond to pink noise input, for Brownian noise input and white-noise input the values are ‘+1’ and ‘−1’ with respect to the pink input, respectively, as indicated by the brown ‘+’ and the blue ‘−’. The ball and stick neuron was simulated with 200 dendritic segments (corresponding to the default parameters listed in Table 1), while the pyramidal neuron was simulated with 3214 dendritic segments. Broken lines correspond to the ball and stick neuron, whole lines to the pyramidal neuron.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003928-g002: Normalized power spectral densities (PSDs) for the soma current, the current-dipole moment (i.e., EEG contribution) and the soma potential for a ball and stick neuron and a pyramidal neuron.A homogeneous density of noisy input currents is applied throughout the neural membrane. Columns 1 (ball and stick neuron) and 2 (pyramidal neuron) show PSDs for white-noise input, the blue and green lines correspond to uncorrelated and correlated input currents, respectively. Note that there is no green line in the two upper rows, since a homogeneous density of correlated inputs throughout the neuron gives no net soma current or dipole moment. An ensemble of PSDs from 20 single input currents for the ball and stick neuron and 107 single input currents for the pyramidal neuron is shown in grey. The results for the most distal synapses are shown in dark grey and the results for the proximal synapses in light grey, corresponding to the color shown in the filled circles at the respective neuron morphology (between columns 1 and 2). Column 3 illustrates how different power-law spectra of the input currents change the output PSDs: the blue, pink and brown lines express the PSD for uncorrelated white (constant), pink () and Brownian noise input (), respectively. The values of in legends denote estimated power-law exponents at 1000 Hz, i.e., the negative discrete log-log derivative, . In the rightmost column the values of correspond to pink noise input, for Brownian noise input and white-noise input the values are ‘+1’ and ‘−1’ with respect to the pink input, respectively, as indicated by the brown ‘+’ and the blue ‘−’. The ball and stick neuron was simulated with 200 dendritic segments (corresponding to the default parameters listed in Table 1), while the pyramidal neuron was simulated with 3214 dendritic segments. Broken lines correspond to the ball and stick neuron, whole lines to the pyramidal neuron.
Mentions: In the present study the idealized ball and stick neuron model will be treated analytically, while simulation results will be presented for a reconstructed layer V pyramidal neuron from cat visual cortex [34] (Fig. 2). Both the ball and stick model and the reconstructed layer V neuron model are purely passive, ensuring that linear theory can be used. The input currents are distributed throughout the neuron models with area density in the dendrite and in the soma. The input currents share statistics, i.e., they all have the same PSD, denoted , and a pairwise coherence . The coherence is zero for uncorrelated input and unity for perfectly correlated input.

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