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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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Slopes  for the PSD transfer function for the soma potential for a ball and stick neuron in terms of its dimensionless parameters.Row 1 corresponds to correlated input currents solely onto the dendrite. Row 2 corresponds to input currents solely onto soma, either correlated () or uncorrelated (). In row 3 uncorrelated input currents are applied homogeneously across the dendrite. Row 5 corresponds to uncorrelated input currents with equal density, , throughout the neuron. Row 6 shows results for correlated input currents with equal density, , throughout the neuron. The dimensionless parameter  is plotted along the vertical axes, while the dimensionless frequency  is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is , in the middle column  and the right column . The horizontal white line express the default value of the parameter ,  (soma diameter , stick diameter , length constant  mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant  ms. The thin black line denotes  and the thicker black line denotes , with  denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for , given by the color bar to the right.
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pcbi-1003928-g006: Slopes for the PSD transfer function for the soma potential for a ball and stick neuron in terms of its dimensionless parameters.Row 1 corresponds to correlated input currents solely onto the dendrite. Row 2 corresponds to input currents solely onto soma, either correlated () or uncorrelated (). In row 3 uncorrelated input currents are applied homogeneously across the dendrite. Row 5 corresponds to uncorrelated input currents with equal density, , throughout the neuron. Row 6 shows results for correlated input currents with equal density, , throughout the neuron. The dimensionless parameter is plotted along the vertical axes, while the dimensionless frequency is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is , in the middle column and the right column . The horizontal white line express the default value of the parameter , (soma diameter , stick diameter , length constant mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant ms. The thin black line denotes and the thicker black line denotes , with denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for , given by the color bar to the right.

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 ().


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)

Slopes  for the PSD transfer function for the soma potential for a ball and stick neuron in terms of its dimensionless parameters.Row 1 corresponds to correlated input currents solely onto the dendrite. Row 2 corresponds to input currents solely onto soma, either correlated () or uncorrelated (). In row 3 uncorrelated input currents are applied homogeneously across the dendrite. Row 5 corresponds to uncorrelated input currents with equal density, , throughout the neuron. Row 6 shows results for correlated input currents with equal density, , throughout the neuron. The dimensionless parameter  is plotted along the vertical axes, while the dimensionless frequency  is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is , in the middle column  and the right column . The horizontal white line express the default value of the parameter ,  (soma diameter , stick diameter , length constant  mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant  ms. The thin black line denotes  and the thicker black line denotes , with  denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for , given by the color bar to the right.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003928-g006: Slopes for the PSD transfer function for the soma potential for a ball and stick neuron in terms of its dimensionless parameters.Row 1 corresponds to correlated input currents solely onto the dendrite. Row 2 corresponds to input currents solely onto soma, either correlated () or uncorrelated (). In row 3 uncorrelated input currents are applied homogeneously across the dendrite. Row 5 corresponds to uncorrelated input currents with equal density, , throughout the neuron. Row 6 shows results for correlated input currents with equal density, , throughout the neuron. The dimensionless parameter is plotted along the vertical axes, while the dimensionless frequency is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is , in the middle column and the right column . The horizontal white line express the default value of the parameter , (soma diameter , stick diameter , length constant mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant ms. The thin black line denotes and the thicker black line denotes , with denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for , given by the color bar to the right.
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 ().

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