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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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Suggested scenario for generation of soma-potential noise in the in vivo situation with a combination of  membrane current sources, presumably due to intrinsic ion channels, and synaptic current sources.Both sources are assumed uncorrelated and homogeneously spread out across a ball and stick neuron. (A) Excerpt of real-time soma potential following injection of synaptic noise through an exponential synapse (white noise filtered through Eq. (117), blue line),  noise, putatively from intrinsic ion channel (white noise filtered through a  filter, red line), and sum of both (black line). (B) Histogram over soma potential for the three situations in A (50 s period with a sampling rate of 10 kHz). (C) Soma-potential PSDs for five cases: the three cases in A (; exponential synapse, Eq. (117); sum of  and exponential synapse) as well as alpha-function synapse (Eq. 118, green line) and sum of alpha-function synapse and  (magenta line). All traces are normalized to the value of the summed PSDs for  noise and exponential synapse for the lowest depicted frequency (0.1 Hz). (D) Locally (in frequency) estimated power-law coefficient , i.e., Eq. (116). The noise amplitudes are set so that soma-potential noise from (i) the  current noise input has a standard deviation of  = 0.6 mV (as seen in in vitro experiments [19]; frequencies between 0.2 and 100 Hz included in the noise variance sum) and (ii) total noise (synaptic+) a standard deviation of  = 2.5 mV (similar to in vivo experiments reported in Fig. 11 in [18]). Parameters used for the ball and stick neuron model is the default values (cf. caption of Fig. 3 and Table 1) except for the membrane resistance which has been reduced to  to mimic an expected high conductance in an in vivo state [21]. The synaptic time constant is set to  ms for the exponential synapse (Eq. 117) and  ms for the alpha-function synapse (Eq. 118).
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pcbi-1003928-g009: Suggested scenario for generation of soma-potential noise in the in vivo situation with a combination of membrane current sources, presumably due to intrinsic ion channels, and synaptic current sources.Both sources are assumed uncorrelated and homogeneously spread out across a ball and stick neuron. (A) Excerpt of real-time soma potential following injection of synaptic noise through an exponential synapse (white noise filtered through Eq. (117), blue line), noise, putatively from intrinsic ion channel (white noise filtered through a filter, red line), and sum of both (black line). (B) Histogram over soma potential for the three situations in A (50 s period with a sampling rate of 10 kHz). (C) Soma-potential PSDs for five cases: the three cases in A (; exponential synapse, Eq. (117); sum of and exponential synapse) as well as alpha-function synapse (Eq. 118, green line) and sum of alpha-function synapse and (magenta line). All traces are normalized to the value of the summed PSDs for noise and exponential synapse for the lowest depicted frequency (0.1 Hz). (D) Locally (in frequency) estimated power-law coefficient , i.e., Eq. (116). The noise amplitudes are set so that soma-potential noise from (i) the current noise input has a standard deviation of  = 0.6 mV (as seen in in vitro experiments [19]; frequencies between 0.2 and 100 Hz included in the noise variance sum) and (ii) total noise (synaptic+) a standard deviation of  = 2.5 mV (similar to in vivo experiments reported in Fig. 11 in [18]). Parameters used for the ball and stick neuron model is the default values (cf. caption of Fig. 3 and Table 1) except for the membrane resistance which has been reduced to to mimic an expected high conductance in an in vivo state [21]. The synaptic time constant is set to ms for the exponential synapse (Eq. 117) and ms for the alpha-function synapse (Eq. 118).

Mentions: From recordings of the PSD of the soma potential in hippocampal cell culture for frequencies up to 500 Hz, a value of of about 2.4 was estimated at the high-frequency end [17]. Here synaptic blockers were applied, and the resulting noise level was small. Similar power-law exponents, i.e., 2.4 and 2.5, were estimated in slice experiments from rat somatosensory cortex for frequencies up to (only) 100 Hz [19], [20]. In these experiments synaptic blockers were generally not used, and the noise level was found to have a standard deviation about a factor two larger than in the cell culture study of [17]. In [19] it was shown that with synapses blocked, the noise in the frequency interval between 15 Hz to 35 Hz was reduced with about a factor two. For the soma current, the experiments are fewer, but a power law with was observed in experiments on hippocampal cell culture for frequencies up to 500 Hz [17]. For cultures and slices we expect synaptic noise to play a minor role for frequencies above a few hundred hertz, where intrinsic ion-channel noise, presumably largely uncorrelated, is expected to dominate (see also Fig. 9 showing how this might be the case even in an in-vivo like situation where the synaptic noise has much larger overall power than the ion-channel noise).


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)

Suggested scenario for generation of soma-potential noise in the in vivo situation with a combination of  membrane current sources, presumably due to intrinsic ion channels, and synaptic current sources.Both sources are assumed uncorrelated and homogeneously spread out across a ball and stick neuron. (A) Excerpt of real-time soma potential following injection of synaptic noise through an exponential synapse (white noise filtered through Eq. (117), blue line),  noise, putatively from intrinsic ion channel (white noise filtered through a  filter, red line), and sum of both (black line). (B) Histogram over soma potential for the three situations in A (50 s period with a sampling rate of 10 kHz). (C) Soma-potential PSDs for five cases: the three cases in A (; exponential synapse, Eq. (117); sum of  and exponential synapse) as well as alpha-function synapse (Eq. 118, green line) and sum of alpha-function synapse and  (magenta line). All traces are normalized to the value of the summed PSDs for  noise and exponential synapse for the lowest depicted frequency (0.1 Hz). (D) Locally (in frequency) estimated power-law coefficient , i.e., Eq. (116). The noise amplitudes are set so that soma-potential noise from (i) the  current noise input has a standard deviation of  = 0.6 mV (as seen in in vitro experiments [19]; frequencies between 0.2 and 100 Hz included in the noise variance sum) and (ii) total noise (synaptic+) a standard deviation of  = 2.5 mV (similar to in vivo experiments reported in Fig. 11 in [18]). Parameters used for the ball and stick neuron model is the default values (cf. caption of Fig. 3 and Table 1) except for the membrane resistance which has been reduced to  to mimic an expected high conductance in an in vivo state [21]. The synaptic time constant is set to  ms for the exponential synapse (Eq. 117) and  ms for the alpha-function synapse (Eq. 118).
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Related In: Results  -  Collection

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pcbi-1003928-g009: Suggested scenario for generation of soma-potential noise in the in vivo situation with a combination of membrane current sources, presumably due to intrinsic ion channels, and synaptic current sources.Both sources are assumed uncorrelated and homogeneously spread out across a ball and stick neuron. (A) Excerpt of real-time soma potential following injection of synaptic noise through an exponential synapse (white noise filtered through Eq. (117), blue line), noise, putatively from intrinsic ion channel (white noise filtered through a filter, red line), and sum of both (black line). (B) Histogram over soma potential for the three situations in A (50 s period with a sampling rate of 10 kHz). (C) Soma-potential PSDs for five cases: the three cases in A (; exponential synapse, Eq. (117); sum of and exponential synapse) as well as alpha-function synapse (Eq. 118, green line) and sum of alpha-function synapse and (magenta line). All traces are normalized to the value of the summed PSDs for noise and exponential synapse for the lowest depicted frequency (0.1 Hz). (D) Locally (in frequency) estimated power-law coefficient , i.e., Eq. (116). The noise amplitudes are set so that soma-potential noise from (i) the current noise input has a standard deviation of  = 0.6 mV (as seen in in vitro experiments [19]; frequencies between 0.2 and 100 Hz included in the noise variance sum) and (ii) total noise (synaptic+) a standard deviation of  = 2.5 mV (similar to in vivo experiments reported in Fig. 11 in [18]). Parameters used for the ball and stick neuron model is the default values (cf. caption of Fig. 3 and Table 1) except for the membrane resistance which has been reduced to to mimic an expected high conductance in an in vivo state [21]. The synaptic time constant is set to ms for the exponential synapse (Eq. 117) and ms for the alpha-function synapse (Eq. 118).
Mentions: From recordings of the PSD of the soma potential in hippocampal cell culture for frequencies up to 500 Hz, a value of of about 2.4 was estimated at the high-frequency end [17]. Here synaptic blockers were applied, and the resulting noise level was small. Similar power-law exponents, i.e., 2.4 and 2.5, were estimated in slice experiments from rat somatosensory cortex for frequencies up to (only) 100 Hz [19], [20]. In these experiments synaptic blockers were generally not used, and the noise level was found to have a standard deviation about a factor two larger than in the cell culture study of [17]. In [19] it was shown that with synapses blocked, the noise in the frequency interval between 15 Hz to 35 Hz was reduced with about a factor two. For the soma current, the experiments are fewer, but a power law with was observed in experiments on hippocampal cell culture for frequencies up to 500 Hz [17]. For cultures and slices we expect synaptic noise to play a minor role for frequencies above a few hundred hertz, where intrinsic ion-channel noise, presumably largely uncorrelated, is expected to dominate (see also Fig. 9 showing how this might be the case even in an in-vivo like situation where the synaptic noise has much larger overall power than the ion-channel noise).

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