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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 
PLoS Computational Biology  2014;10(11):e1003928.
Power laws, that is, power spectral densities (PSDs) exhibiting 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 power laws with power-law exponents analytically identified as for the soma membrane current, for the current-dipole moment, and 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 () 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 power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation.
Author Summary
The common observation of power laws in nature and society, that is, quantities or probabilities that follow distributions, has for long intrigued scientists. In the brain, power laws in the power spectral density (PSD) have been reported in electrophysiological recordings, both at the microscopic (single-neuron recordings) and macroscopic (EEG) levels. We here demonstrate a possible origin of such power laws in the basic biophysical properties of neurons, that is, in the standard cable-equation description of neuronal membranes. Taking advantage of the mathematical tractability of the so called ball and stick neuron model, we demonstrate analytically that high-frequency power laws in key experimental neural measures will arise naturally when the noise sources are evenly distributed across the neuronal membrane. Comparison with available data further suggests that the apparent high-frequency power laws observed in experiments may stem from uncorrelated current sources, presumably intrinsic ion channels, which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. The significance of this finding goes beyond neuroscience as it demonstrates how power laws power-law exponents α may arise from a simple, linear physics equation.
doi:10.1371/journal.pcbi.1003928
PMCID: PMC4230751  PMID: 25393030
2.  Frequency Dependence of Signal Power and Spatial Reach of the Local Field Potential 
PLoS Computational Biology  2013;9(7):e1003137.
Despite its century-old use, the interpretation of local field potentials (LFPs), the low-frequency part of electrical signals recorded in the brain, is still debated. In cortex the LFP appears to mainly stem from transmembrane neuronal currents following synaptic input, and obvious questions regarding the ‘locality’ of the LFP are: What is the size of the signal-generating region, i.e., the spatial reach, around a recording contact? How far does the LFP signal extend outside a synaptically activated neuronal population? And how do the answers depend on the temporal frequency of the LFP signal? Experimental inquiries have given conflicting results, and we here pursue a modeling approach based on a well-established biophysical forward-modeling scheme incorporating detailed reconstructed neuronal morphologies in precise calculations of population LFPs including thousands of neurons. The two key factors determining the frequency dependence of LFP are the spatial decay of the single-neuron LFP contribution and the conversion of synaptic input correlations into correlations between single-neuron LFP contributions. Both factors are seen to give low-pass filtering of the LFP signal power. For uncorrelated input only the first factor is relevant, and here a modest reduction (<50%) in the spatial reach is observed for higher frequencies (>100 Hz) compared to the near-DC () value of about . Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs: the low-frequency () LFP power can here be an order of magnitude or more larger than at 60 Hz. Moreover, the low-frequency LFP components have larger spatial reach and extend further outside the active population than high-frequency components. Further, the spatial LFP profiles for such populations typically span the full vertical extent of the dendrites of neurons in the population. Our numerical findings are backed up by an intuitive simplified model for the generation of population LFP.
Author Summary
The first recording of electrical potential from brain activity was reported already in 1875, but still the interpretation of the signal is debated. To take full advantage of the new generation of microelectrodes with hundreds or even thousands of electrode contacts, an accurate quantitative link between what is measured and the underlying neural circuit activity is needed. Here we address the question of how the observed frequency dependence of recorded local field potentials (LFPs) should be interpreted. By use of a well-established biophysical modeling scheme, combined with detailed reconstructed neuronal morphologies, we find that correlations in the synaptic inputs onto a population of pyramidal cells may significantly boost the low-frequency components and affect the spatial profile of the generated LFP. We further find that these low-frequency components may be less ‘local’ than the high-frequency LFP components in the sense that (1) the size of signal-generation region of the LFP recorded at an electrode is larger and (2) the LFP generated by a synaptically activated population spreads further outside the population edge due to volume conduction.
doi:10.1371/journal.pcbi.1003137
PMCID: PMC3715549  PMID: 23874180
4.  LFPy: a tool for biophysical simulation of extracellular potentials generated by detailed model neurons 
Electrical extracellular recordings, i.e., recordings of the electrical potentials in the extracellular medium between cells, have been a main work-horse in electrophysiology for almost a century. The high-frequency part of the signal (≳500 Hz), i.e., the multi-unit activity (MUA), contains information about the firing of action potentials in surrounding neurons, while the low-frequency part, the local field potential (LFP), contains information about how these neurons integrate synaptic inputs. As the recorded extracellular signals arise from multiple neural processes, their interpretation is typically ambiguous and difficult. Fortunately, a precise biophysical modeling scheme linking activity at the cellular level and the recorded signal has been established: the extracellular potential can be calculated as a weighted sum of all transmembrane currents in all cells located in the vicinity of the electrode. This computational scheme can considerably aid the modeling and analysis of MUA and LFP signals. Here, we describe LFPy, an open source Python package for numerical simulations of extracellular potentials. LFPy consists of a set of easy-to-use classes for defining cells, synapses and recording electrodes as Python objects, implementing this biophysical modeling scheme. It runs on top of the widely used NEURON simulation environment, which allows for flexible usage of both new and existing cell models. Further, calculation of extracellular potentials using the line-source-method is efficiently implemented. We describe the theoretical framework underlying the extracellular potential calculations and illustrate by examples how LFPy can be used both for simulating LFPs, i.e., synaptic contributions from single cells as well a populations of cells, and MUAs, i.e., extracellular signatures of action potentials.
doi:10.3389/fninf.2013.00041
PMCID: PMC3893572  PMID: 24474916
local field potential; extracellular potential; biophysics; forward modeling; compartmental modeling; detailed morphology; spike waveform; Python
9.  The Local Field Potential Reflects Surplus Spike Synchrony 
Cerebral Cortex (New York, NY)  2011;21(12):2681-2695.
While oscillations of the local field potential (LFP) are commonly attributed to the synchronization of neuronal firing rate on the same time scale, their relationship to coincident spiking in the millisecond range is unknown. Here, we present experimental evidence to reconcile the notions of synchrony at the level of spiking and at the mesoscopic scale. We demonstrate that only in time intervals of significant spike synchrony that cannot be explained on the basis of firing rates, coincident spikes are better phase locked to the LFP than predicted by the locking of the individual spikes. This effect is enhanced in periods of large LFP amplitudes. A quantitative model explains the LFP dynamics by the orchestrated spiking activity in neuronal groups that contribute the observed surplus synchrony. From the correlation analysis, we infer that neurons participate in different constellations but contribute only a fraction of their spikes to temporally precise spike configurations. This finding provides direct evidence for the hypothesized relation that precise spike synchrony constitutes a major temporally and spatially organized component of the LFP.
doi:10.1093/cercor/bhr040
PMCID: PMC3209854  PMID: 21508303
motor cortex; oscillation; population signals; synchrony

Results 1-9 (9)