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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 input-output relationship between transmembrane currents (input) and the different measurement modalities (output).The transmembrane currents are illustrated by synaptic currents and channel currents. A synaptic current is commonly modeled by means of exponentially decaying functions (synaptic kernel) triggered by incoming spike trains, whereas a channel current typically is modeled by a channel switching between an open state (o), letting a current with constant amplitude through the channel, or a closed state (c). The input currents are filtered by the neuronal cable, resulting in a low-pass filtered output current in the soma with a power spectral density (PSD) designated . The PSDs of the other measurement modalities studied here, i.e., the soma potential () and the current-dipole moment giving the single-neuron contribution to the EEG (), are typically even more low-pass filtered, as illustrated by the PSDs plotted in the lower right panel.
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pcbi-1003928-g001: Schematic illustration of the input-output relationship between transmembrane currents (input) and the different measurement modalities (output).The transmembrane currents are illustrated by synaptic currents and channel currents. A synaptic current is commonly modeled by means of exponentially decaying functions (synaptic kernel) triggered by incoming spike trains, whereas a channel current typically is modeled by a channel switching between an open state (o), letting a current with constant amplitude through the channel, or a closed state (c). The input currents are filtered by the neuronal cable, resulting in a low-pass filtered output current in the soma with a power spectral density (PSD) designated . The PSDs of the other measurement modalities studied here, i.e., the soma potential () and the current-dipole moment giving the single-neuron contribution to the EEG (), are typically even more low-pass filtered, as illustrated by the PSDs plotted in the lower right panel.

Mentions: Lack of sufficient statistical support have questioned the validity of identified power-law behaviors, and as a rule of thumb, it has been suggested that a candidate power law should exhibit an approximately linear relationship in a log-log plot over at least two orders of magnitude [1]. Further, a mechanistic explanation of how the power laws arise from the underlying dynamics should ideally be provided [1]. In the present paper we show through a combination of analytical and numerical investigations how power laws in the high-frequency tail of PSDs naturally can arise in neural systems from noise sources homogeneously distributed throughout neuronal membranes. We further show that the mechanism behind microscopic (soma potential, soma current) power laws will also lead to power laws in the single-neuron contribution (current-dipole moment) to the EEG. Moreover, we demonstrate that if all single-neuron contributions to the recorded EEG signal exhibit the same power law, the EEG signal will also exhibit this power law. We find that for different measurement modalities different power-law exponents naturally follow from the well-established, biophysical cable properties of the neuronal membranes: the soma potential will be more low-pass filtered than the corresponding current-dipole moment determining the single-neuron contribution to the EEG [23], [24], and as a consequence, the power-law exponent α will be larger for the soma potential than for the single-neuron contribution to the EEG [25] (see illustration in Fig. 1).


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 input-output relationship between transmembrane currents (input) and the different measurement modalities (output).The transmembrane currents are illustrated by synaptic currents and channel currents. A synaptic current is commonly modeled by means of exponentially decaying functions (synaptic kernel) triggered by incoming spike trains, whereas a channel current typically is modeled by a channel switching between an open state (o), letting a current with constant amplitude through the channel, or a closed state (c). The input currents are filtered by the neuronal cable, resulting in a low-pass filtered output current in the soma with a power spectral density (PSD) designated . The PSDs of the other measurement modalities studied here, i.e., the soma potential () and the current-dipole moment giving the single-neuron contribution to the EEG (), are typically even more low-pass filtered, as illustrated by the PSDs plotted in the lower right panel.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003928-g001: Schematic illustration of the input-output relationship between transmembrane currents (input) and the different measurement modalities (output).The transmembrane currents are illustrated by synaptic currents and channel currents. A synaptic current is commonly modeled by means of exponentially decaying functions (synaptic kernel) triggered by incoming spike trains, whereas a channel current typically is modeled by a channel switching between an open state (o), letting a current with constant amplitude through the channel, or a closed state (c). The input currents are filtered by the neuronal cable, resulting in a low-pass filtered output current in the soma with a power spectral density (PSD) designated . The PSDs of the other measurement modalities studied here, i.e., the soma potential () and the current-dipole moment giving the single-neuron contribution to the EEG (), are typically even more low-pass filtered, as illustrated by the PSDs plotted in the lower right panel.
Mentions: Lack of sufficient statistical support have questioned the validity of identified power-law behaviors, and as a rule of thumb, it has been suggested that a candidate power law should exhibit an approximately linear relationship in a log-log plot over at least two orders of magnitude [1]. Further, a mechanistic explanation of how the power laws arise from the underlying dynamics should ideally be provided [1]. In the present paper we show through a combination of analytical and numerical investigations how power laws in the high-frequency tail of PSDs naturally can arise in neural systems from noise sources homogeneously distributed throughout neuronal membranes. We further show that the mechanism behind microscopic (soma potential, soma current) power laws will also lead to power laws in the single-neuron contribution (current-dipole moment) to the EEG. Moreover, we demonstrate that if all single-neuron contributions to the recorded EEG signal exhibit the same power law, the EEG signal will also exhibit this power law. We find that for different measurement modalities different power-law exponents naturally follow from the well-established, biophysical cable properties of the neuronal membranes: the soma potential will be more low-pass filtered than the corresponding current-dipole moment determining the single-neuron contribution to the EEG [23], [24], and as a consequence, the power-law exponent α will be larger for the soma potential than for the single-neuron contribution to the EEG [25] (see illustration in Fig. 1).

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