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Frequency dependence of signal power and spatial reach of the local field potential

Szymon Łęski, Henrik Lindén, Tom Tetzlaff, Klas H. Pettersen, Gaute T. Einevoll

arXiv:1302.4559v1q-bio.NCq-bio.QM

TL;DR

The paper addresses how frequency-dependent LFP measurements relate to underlying neural activity, a link needed to interpret recordings from new microelectrodes. Using biophysical simulations with reconstructed morphologies and a simplified model, it finds that correlated synaptic inputs can boost low-frequency power and that low-frequency components may reach farther spatially than high-frequency components. The simplified model has reported deviations from full simulations, including for correlated populations.

  • Problem

    How LFP frequency dependence relates to underlying neural activity remains difficult to interpret, despite the need for a quantitative link as multisite recordings expand.

    Source: Author Summary · p. 1, §1 · p. 2

  • Method

    The study combines biophysical modeling with reconstructed neuronal morphologies and a frequency-resolved simplified model of population LFP.

    Source: Author Summary · p. 1, §3 · p. 13

  • Results

    Correlated synaptic inputs onto pyramidal-cell populations may boost low-frequency LFP power, whose generation region may be larger and whose signal may spread farther beyond the active population than high-frequency LFP.

    Source: Author Summary · p. 1, §3 · pp. 13, 15

  • Takeaways & Limitations

    Low-frequency correlations between electrodes a millimeter apart may arise from volume conduction, whose spatial range diminishes as LFP frequency increases.

    Source: §3 · p. 15

    The simplified model agrees qualitatively with simulations but overestimates low-frequency power in some comparisons and works better for larger than smaller correlated populations.

    Source: §2.4 · p. 8

Abstract

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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 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) that the LFP generated by a synaptically activated population spreads further outside the population edge due to volume conduction.

Author Summary

Synaptic-input correlations can boost low-frequency LFP components, which may be less local than high-frequency components both in their generation region and their spread beyond an active population.

Author Summary

  • Correlations in synaptic inputs onto pyramidal-cell populations may significantly boost the generated LFP’s low-frequency components.The study uses biophysical modeling with reconstructed neuronal morphologies.Source: Author Summary · p. 1
  • The low-frequency LFP may have a larger signal-generation region around the recording electrode than the high-frequency LFP.The passage describes this as one sense in which low-frequency components may be less local.Source: Author Summary · p. 1
  • Low-frequency LFP generated by a synaptically activated population may spread farther beyond its edge through volume conduction than high-frequency LFP.Source: Author Summary · p. 1

1 Introduction

The interpretation of LFPs remains difficult despite improved recording technology, and estimates of their spatial reach vary widely. This study examines frequency-dependent LFP power and locality using biophysical modeling and a frequency-resolved simplified model.

1 Introduction

  • Reported cortical LFP spatial-reach estimates range from a few hundred micrometers to several millimeters, with variation presumed to reflect network correlations and brain state.Source: §1 · p. 2
  • LFP interpretation is challenging because thousands of neurons contribute, while multisite recordings have renewed interest in its population-level information.The passage describes the biophysical link between neural activity and recorded potentials as well understood.Source: §1 · p. 2
  • High-frequency components are expected to be more local than low-frequency components because dendritic filtering reduces effective current dipoles as frequency increases.Source: §1 · p. 2
  • The study examines frequency-dependent LFP power and locality, including spatial reach around a contact and signal extension beyond an active population.The frequency-resolved simplified model is used to guide investigation of these questions.Source: §1 · p. 3
  • The frequency-resolved model incorporates faster single-neuron LFP decay at higher frequencies and frequency-dependent transfer of input correlations into correlations between cell contributions.It is reported to account well for results from comprehensive numerical investigations.Source: §1 · p. 3

2.1 Simulations

The simulations calculate LFP at the center of a disc-like population using reconstructed cell models and specified synaptic-input patterns. Inputs are designed to avoid imposing synaptic frequency filtering, so dendritic filtering and population summation shape the simulated frequency dependence.

2.1 Simulations

  • The simulations place 10,000 cells homogeneously on a 1,000 µm-radius disc and calculate LFP at the center, at soma level.The population signal is examined as contributions from increasingly distant neurons are included.Source: §2.1 · pp. 3–4, Figure 2 · p. 4
  • Three reconstructed morphologies are used: layer-3 and layer-5 pyramidal cells and layer-4 stellate cells, with different input-placement patterns for pyramidal cells.Pyramidal-cell inputs are apical, basal, or homogeneous; stellate-cell input is homogeneous.Source: §2.1 · p. 4
  • Synaptic currents use brief α-functions and Poissonian spike statistics, yielding white power spectra through 500 Hz so synapses impose no frequency filtering.The remaining frequency filtering comes from intrinsic dendritic effects and summation of single-neuron LFP contributions.Source: §2.1 · p. 4
  • The calculated extracellular potentials are called “LFPs” even at frequencies up to 500 Hz, which are often considered outside the LFP band.Source: §2.1 · p. 4

2.2 Simplified model of population LFP

The simplified frequency-resolved model represents each cell’s contribution by a distance- and frequency-dependent shape function and combines contributions according to their correlation. It distinguishes summing variances for uncorrelated cells from summing amplitudes for fully correlated cells.

2.2 Simplified model of population LFP

  • The model factorizes a cell’s LFP power into input-current power and a shape function describing distance- and frequency-dependent signal decay.The shape function is assumed to depend only on frequency and lateral distance, neglecting other sources of single-cell variation.Source: §2.2 · pp. 4–5
  • For uncorrelated input, the model adds single-cell LFP variances to obtain population power over a disc-shaped population.The expression uses planar cell density and integrates contributions over radial distance.Source: §2.2 · p. 5
  • For fully correlated single-cell LFPs, the model adds amplitudes rather than variances to calculate population power.Source: §2.2 · p. 5
  • In the frequency-domain formulation, pairwise correlation is represented by coherence, approximated by a population-averaged LFP coherence.The model relates root mean square amplitude to the power spectral density integrated from 0 Hz to half the sampling frequency.Source: §2.2 · p. 5

2.3 Numerical evaluation of ingredients of simplified model

Population LFP frequency dependence reflects both frequency-varying single-cell spatial decay and coherence between single-cell contributions. Combining these ingredients yields simplified-model predictions that closely match the example population simulation.

2.3.1 Frequency dependence of shape function

  • Higher-frequency single-cell LFP decays faster with distance, and the cutoff distance r∗ generally decreases as frequency increases.For layer-5 cells with basal input, the dipole-decay transition shifts from about 80 µm near 0 Hz to about 40 µm at high frequency.Source: §2.3.1 · pp. 5–6
  • Across the tested morphologies and input patterns, r∗ declines with frequency except for layer-5 cells with apical input, which show a minimum around 150 Hz.The exception reflects input positioned far above the soma, with frequency-dependent dipole shrinkage accompanied by an apical shift in mean dipole position.Source: §2.3.1 · p. 6, Figure 4 · p. 7

2.3.2 Frequency dependence of coherence

  • Population-averaged coherence is higher at low frequencies, consistent with low-frequency return currents producing more similarly oriented effective dipoles across cells.For the layer-5 basal-input example, high-frequency dipoles are expected to be more randomly oriented, while some low-frequency return current passes through the apical dendrite.Source: §2.3.2 · p. 7

2.3 Numerical evaluation of ingredients of simplified model

  • Combining frequency-dependent shape functions and coherence predicts the example population LFP power spectrum in excellent agreement with simulation.The model combines both ingredients in Equation 5.Source: §2.3 · p. 5, §2.3.2 · p. 7

2.3.2 Frequency dependence of coherence

  • Coherence falls with frequency for pyramidal cells receiving asymmetric input, most strongly for layer-5 basal input; filtering is nearly absent with homogeneous input.For homogeneous layer-4 stellate input, coherence is essentially zero, so input correlations do not translate into correlations between single-cell LFP contributions.Source: §2.3.2 · p. 7

2.4 Population LFP

In disc-like populations of layer-5 pyramidal cells with basal input, correlated synaptic inputs greatly increase low-frequency LFP power and make it grow more with population radius. The simplified model captures the qualitative pattern but has specific deviations.

2.4 Population LFP

  • With uncorrelated input, LFP power saturates around R ≃100–200 µm, making sources more than a few hundred micrometers away negligible.This pattern holds across all considered frequency bands.Source: §2.4 · p. 7, Figure 6 · p. 9
  • With cin = 1, low-frequency population LFP power is up to two orders of magnitude larger than in the corresponding uncorrelated case.For cin = 1, power near 0 Hz is an order of magnitude larger than power at 60 Hz.Source: §2.4 · p. 8, Figure 6 · p. 9
  • With correlated input, low-frequency power grows faster with population radius than high-frequency power, and total power does not appear to saturate as radius increases.The reported radius-dependent pattern is shown for correlated-input cases in Figure 6.Source: §2.4 · p. 8, Figure 6 · p. 9
  • The simplified model overestimates power near 0 Hz and at 60 Hz, despite qualitatively matching full simulations.Calculating coherence separately for each population radius significantly improves the predictions.Source: §2.4 · pp. 8–9

2.5 Frequency-dependence of spatial reach

Spatial reach varies with frequency and input correlation, especially for pyramidal cells receiving asymmetric synaptic input. Correlations can make low-frequency LFP components arise from a substantially larger region than high-frequency components, with effects depending on morphology and stimulation pattern.

2.5 Frequency-dependence of spatial reach

  • For asymmetric-input pyramidal populations, spatial reach grows with input correlation, especially at lower frequencies; homogeneous-input effects are weaker and stellate-cell reach is nearly invariant.The effects depend sensitively on cell morphology and synaptic stimulation pattern.Source: §2.5 · p. 9
  • With apical input, uncorrelated reach is already about 300–400 µm, and smaller input correlations are needed to saturate reach.The supplied passage notes that the maximum reach is the maximum possible in the setup.Source: §2.5 · pp. 9–10
  • Spatial reach is the radius contributing 95% of the center LFP's RMS amplitude relative to the largest modeled population, R = 1000 µm.Figure 7 plots full-simulation results as dots and simplified-model predictions as lines.Source: §2.5 · p. 9, Figure 7 · p. 10
  • For uncorrelated input, the simplified model gives spatial reach of approximately 1.85r∗ when α = 0.95 in the stated limit.The expression is derived from Equations 3 and 6 as rϵ approaches zero.Source: §2.5 · p. 10

2.6 Decay of extracellular potential outside the population

Correlated synaptic input can boost extracellular LFPs and extend their detectable signal beyond an active population, with the frequency pattern depending on input location. The simplified model captures much of this behavior but overestimates correlated signals far outside the population.

2.6 Decay of extracellular potential outside the population

  • For basal input, high correlations boost near-DC and 60-Hz signals by up to two orders of magnitude, while 500-Hz signals gain little; apical input boosts the three bands more uniformly.These effects are reported both inside and outside the population.Source: §2.6 · pp. 10–11, Figure 8 · p. 11
  • A correlated population 500 µm outside the electrode can produce a larger near-DC signal than an uncorrelated population surrounding the electrode.The passage reports a similar, smaller effect at 60 Hz and an even more pronounced effect for apical input.Source: §2.6 · p. 11
  • After normalization at the population center, LFP decay is more abrupt around the population edge for uncorrelated than correlated cases, particularly at low frequencies.The difference is more pronounced where population-averaged coherence is largest.Source: §2.6 · pp. 11–12, Figure 9 · p. 12
  • Detectability compares the active population's LFP RMS amplitude with background LFP RMS amplitude from uncorrelated, homogeneously distributed inputs.The assumed background is a model of ambient LFP noise, which can vary between experimental situations.Source: §2.6 · p. 12
  • With basal input at cin = 1, near-DC signal-to-noise ratio remains nearly 0.5 1000 µm beyond the edge, while the 125-Hz ratio reaches 0.5 only 200 µm beyond it.For uncorrelated input, the ratio falls to 0.5 about 100 µm outside the population and below 0.1 less than 500 µm outside.Source: §2.6 · p. 12
  • The simplified model agrees excellently with simulations for uncorrelated input but overestimates signal power far outside correlated populations.The model predicts r^-2 amplitude decay there, whereas simulations approach r^-3 decay because correlated vertical dipoles dominate.Source: §2.6 · p. 13
  • Including electrode-position dependence in population-averaged coherence, which decays as 1/X^2 for X ≫ R, corrects the model's far-field prediction.The modified theory includes the X-dependence of both G1 and coherence.Source: §2.6 · p. 13

3 Discussion

In simulations of LFPs, large frequency-dependent effects arise when pyramidal populations receive correlated, spatially asymmetric input. The results link these effects mainly to frequency-dependent coherence among neuronal contributions and show that low-frequency signals can reach farther than high-frequency signals.

3 Discussion

  • For basal input to layer-5 pyramidal cells, near-DC LFP power is an order of magnitude larger than power at 60 Hz, and low-frequency signals extend farther outside the active population.The modeled synaptic currents have white-noise power spectra.Source: §3 · p. 13
  • Most observed low-pass filtering is attributed to coherence among single-neuron LFP contributions, which translates synaptic-input correlations into neuronal-signal correlations more strongly at low frequencies.The simplified population-LFP model combines single-neuron distance decay, population geometry and density, and frequency-dependent coherence.Source: §3 · p. 13
  • At cin = 0.01, modeled reach can be about 800 µm near 0 Hz and 400 µm at 60 Hz; the 1-mm population limit underestimates reach for correlated populations.For uncorrelated input, reach is generally at most about 200 µm, except for apical input on large pyramidal cells.Source: §3 · p. 14
  • The model's frequency-dependent reach is sensitive to cell morphology and synaptic stimulation pattern, and does not reproduce equal spread reported in a macaque auditory-cortex study that used a different measurement approach.The cited study measured vertical spread using LFP amplitude at a given latency; this work modeled horizontal spread using long-period RMS amplitude and unknown input correlation.Source: §3 · p. 14
  • Low-frequency signals from asymmetric-input populations may extend a millimeter or more beyond the active population, so volume conduction may contribute to cross-electrode correlations.The volume-conduction contribution is expected to have a diminishing spatial range at higher frequencies.Source: §3 · pp. 14–15
  • Approximate LFP power-law exponents that vary with coherence suggest input coherence as a possible mechanism for experimentally observed spectral-exponent differences.The passage relates higher coherence in slow-wave sleep to larger α than in awake states, citing.Source: §3 · pp. 15–16
  • The simulations assume frequency-independent extracellular conductivity; any frequency-dependent conductivity would multiply the modeled LFP filtering by a factor proportional to 1/σcond(f).Source: §3 · pp. 16–17
  • The analysis uses linear dendritic RC circuits and current-based synapses; the forward-modeling scheme can also be used to investigate active conductances, whose adequacy for real neurons remains unknown.The linear setup decouples LFP frequencies in the frequency-resolved analysis.Source: §3 · p. 17

4 Methods

The methods combine detailed neuron models and population simulations to generate LFPs, then derive a simplified mean-field account using simulated cell shape functions and population coherence.

4.1 LFP simulations

  • The LFP simulations adapted an established scheme with shorter synaptic time constants to keep input-current power effectively flat below 500 Hz.The smaller time constant also required a smaller numerical time step.Source: §4.1 · p. 17

4.1.1 Cell models

  • Three passive compartmental cell models received 1,000 excitatory synapses each, driven by independent or shared Poisson spike trains and distributed across dendritic regions.The models were layer-3 and layer-5 pyramidal cells and layer-4 stellate cells; pyramidal-cell stimulation was apical, basal, or homogeneous.Source: §4.1.1 · p. 17

4.1.2 Calculation of LFP

  • Extracellular potentials were calculated with the line-source method in a homogeneous, isotropic, resistive, infinite medium, using an ideal point electrode at soma level.Population measurements sampled the center or lateral positions, and differently sized populations were formed by summing subsets of a 1-mm-radius population.Source: §4.1.2 · p. 18

4.1.3 Single-cell shape functions

  • Single-cell shape functions were estimated from LFP power spectra at multiple distances, averaging 100 simulations at each distance.Source: §4.1.3 · p. 18

4.1.4 Population simulations

  • Population simulations used 10,000 identical neurons distributed uniformly in a 1-mm-radius disc, with each cell randomly rotated around the vertical axis.Source: §4.1.4 · p. 18

4.1.5 Software

  • The simulations used NEURON with its Python interface, and LFPy to calculate extracellular fields.Source: §4.1.5 · p. 18

4.2 Derivation of the mean-field model

  • The mean-field derivation factorizes each cell’s LFP power using input-current PSD and a frequency-dependent shape function, then expresses population PSD using averaged pairwise coherence.It rewrites the pairwise sum using population averages.Source: §4.2 · pp. 18–19
  • For closed-form predictions, the model assumes power-law shape functions and converts sums to integrals; off-center predictions instead use simulated shape curves and numerical integration.Source: §4.2 · p. 19

4.3 Data analysis

The analysis estimates population-averaged LFP coherence across frequency and electrode position, defines spatial reach by a 95% amplitude criterion, and uses standard spectral and scientific-computing tools.

4.3.1 Population-averaged LFP coherence

  • Population-averaged LFP coherence was estimated using the technique of and evaluated at the population center or lateral positions.Source: §4.3.1 · p. 19
  • Finite-length signals can yield spurious negative coherence estimates; these indicate coherence too small to estimate reliably, so the figures plot |cΦ| in such cases.Source: §4.3.1 · p. 19

4.3.2 Frequency analysis

  • Power spectra used Welch’s average periodogram with Hanning windows of 32 or 128 ms, 50% overlap, and frequency bins from 0 to 500 Hz.Coherence was binned at the same frequencies.Source: §4.3.2 · p. 19

4.3.3 Spatial reach of LFP

  • Spatial reach was defined as the smallest population radius whose LFP amplitude exceeded 95% of the full-population amplitude at that frequency.Population radii ranged from 0 to 1,000 µm in 25-µm increments.Source: §4.3.3 · p. 19

4.3.4 Software

  • Data analysis used NumPy and SciPy with IPython; plots were made with Matplotlib.Source: §4.3.4 · p. 20
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