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

Abstract

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 ([Formula: see text]) value of about [Formula: see text]. Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs: the low-frequency ([Formula: see text]) 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.

Figure 1. Spatial reach of different frequency components of LFP for different levels of synaptic input correlations

. Color lines denote parts of the whole population (gray, radius = 1 mm) which contribute 95% of LFP amplitude at given frequency in the middle of the population, at the soma level. Results for layer-5 pyramidal cell with basal input.

Figure 2. Simulation setup.

A. Input spike trains are either generated independently for each cell (uncorrelated input), or chosen from a common pool (correlated input: every two cells share a fraction of inputs). B. Model cells (red: L3 pyramidal cell, green: L4 stellate cell, blue: L5 pyramidal cell) are placed with constant planar density on a disc of radius , in this example with the recording electrode at the population center. Electrode positions shown as black dots. C. The population LFP is a sum of contributions from cells at different distances . The dependence of the amplitude of the population LFP on the population radius serves to define the spatial reach (see text). The correlations between inputs give raise to correlations between single-cell LFP contributions. D. The synapses used in simulations yield a flat power spectrum of input current, but because of the frequency-dependence of single-cell shape functions and population-averaged coherence (see text), the resulting power spectrum of the population LFP is not flat (E). This LFP filtering effect will be present for any synapse type, such as the exponential synapse which in addition yields non-flat power spectrum of the input current (dashed curves in D, E).

Figure 3. Ingredients of the simplified LFP model for soma-level LFP for layer-5 cell with basal synaptic input.

A. Spatial decay in lateral direction for the squared single-cell shape functions for three different frequencies and . B. Single-cell LFP spectra for three different lateral distances from the soma (dotted vertical lines in A). C. Log-log plot of the squared near-DC () shape function (dots) approximated by a piecewise-linear function with cutoff distance (line; see Eq. 7). D. Frequency dependence of the cutoff distance . E. Population-averaged LFP coherence for different input correlation levels . Dots not connected with lines indicate that is plotted in place of spurious negative values (see Methods). F. Power spectra of the compound LFP (); dots correspond to simulation; lines correspond to predictions from simplified model, Eq. 6, based on and given in D and E, respectively.

Figure 4. Frequency dependence of the cutoff distance for soma-level LFP for all situations considered:

homogeneous (solid), apical (dashed) and basal synaptic input (dotted) applied to the layer-3 pyramidal cell (A), the layer-5 pyramidal cell (B), and the layer-4 stellate cell (C). Cell morphologies depicted in Figure 2B. Dots in A, B, C represent the actual frequency resolution, thin lines serve to guide the eye.

Figure 5. Frequency dependence of the population-averaged LFP coherence for soma-level LFP for all situations considered.

Dots represent the actual frequency resolution, thin lines serve to guide the eye. Dots not connected with lines indicate that is plotted, see Methods. A, B, C: population of layer-3 cells; D, E, F: population of layer-5 cells, G: population of layer-4 cells; A, D: apical synaptic input; B, E: basal synaptic input; C, F, G: homogeneous synaptic input.

Figure 6. Power spectral density of population LFP at the soma level as a function of frequency and the population radius.

Full simulation results (dots) and simplified model predictions (lines) for the LFP at the center of disc-like populations of layer-5 pyramidal cells receiving basal synaptic input. Three different input correlation levels are considered. A, B, C: PSD of population LFP for three population radii . D, E, F: dependence of power of three different frequency components on the population radius .

Figure 7. Spatial reach at soma level for different frequency components of LFP.

Spatial reach is defined as the radius of a subpopulation contributing 95% of the root mean square amplitude of LFP compared to the amplitude for . LFP is calculated at the soma level at the center of the population. Full simulation results plotted with dots; predictions from the simplified model (Equation 6) based on calculated values of and given in Figures 4 and 5, respectively, are shown with lines. A, B, C: population of layer-3 cells; D, E, F: population of layer-5 cells, G: population of layer-4 cells; A, D: apical synaptic input; B, E: basal synaptic input; C, F, G: homogeneous synaptic input.

Figure 8. Decay of extracellular potential at the soma level outside populations of layer-5 cells with asymmetric input.

Each of the panels shows full simulation results (dots) and predictions from simplified model, Equation 6 (lines) for one frequency band (0, 60, 500 Hz) and four input correlation levels. Horizontal dotted lines indicate ‘noise level’ (power of the signal generated by a population of uncorrelated cells with homogeneous input, see text). A, B, C: basal synaptic input. D, E, F: apical synaptic input.

Figure 9. Decay of extracellular potential at the soma level outside populations of layer-5 cells with asymmetric input.

Same as Figure 8, but with PSDs normalized to 1 at the population center, and the distance axis zoomed in to highlight the behavior around the edge of the population. A, B, C: basal synaptic input. D, E, F: apical synaptic input.

Figure 10. LFP signal power at the soma level as functions of frequency and distance from basally-activated pyramidal-cell populations.

Colormaps (A, C, E, G) show the power of extracellular signal of a population of layer-5 cells receiving basal synaptic input for four levels of input correlation as functions of frequency and distance from center of populations. Black solid and dotted lines denote signal to noise ratio of 0.5 and 0.1, respectively. B, D, F, H: power spectra of extracellular signal at different distances, lines: prediction from simplified model in Equation 6, dots: full simulation. Thin vertical dotted lines with dots in A, C, E, G denote the distances at which the power spectra are shown, that is, at the center (), population edge (), and two distances outside ( and ).

Figure 11. LFP signal power at the soma level as functions of frequency and distance from apically-activated pyramidal-cell populations.

Colormaps (A, C, E, G) show the power of extracellular signal of a population of layer-5 cells receiving apical synaptic input for four levels of input correlation as functions of frequency and distance from center of populations. Black solid and dotted lines denote signal to noise ratio of 0.5 and 0.1, respectively. B, D, F, H: power spectra of extracellular signal at different distances, lines: prediction from simplified model in Equation 6, dots: full simulation. Thin vertical dotted lines with dots in A, C, E, G denote the distances at which the power spectra are shown, that is, at the center (), population edge (), and two distances outside ( and ).

Figure 12. Population-averaged LFP coherence at the soma level as a function of distance from center of population of layer-5 pyramidal cells.

A, B, C: basal synaptic input, D, E, F: apical synaptic input. Dots not connected with lines indicate that is plotted in place of spurious negative values (see Methods). Dashed lines mark decay.

Figure 13. Depth-dependence of LFP power in the center of a population of layer-5 pyramidal cells.

PSD of the LFP for different correlation levels and different patterns of synaptic input. Population radius: . Values in each panel are normalized separately. A, B, C, D: apical synaptic input; E, F, G, H: basal synaptic input; I, J, K, L: homogeneous synaptic input. A, E, I: ; B, F, J: ; C, G, K: ; D, H, L: .

Figure 14. Simplified-model predictions of the depth-dependence of LFP power in a population of layer-5 pyramidal cells.

PSD of the LFP for different correlation levels and different patterns of synaptic input as predicted by the simplified model of the population LFP. Population radius: . Values in each panel are normalized separately. A, B, C, D: apical synaptic input; E, F, G, H: basal synaptic input; I, J, K, L: homogeneous synaptic input. A, E, I: ; B, F, J: ; C, G, K: ; D, H, L: .

Figure 15. Ingredients of the simplified model of the depth-dependence of LFP power.

Top row: squared shape functions for the lowest-frequency component (0 Hz) of the LFP generated by layer-5 cells with apical (A), basal (B) or homogeneous (C) synaptic input, at different recording depths. Bottom row: population-averaged LFP coherence , calculated at different depths in a maximally correlated () population of layer-5 cells with either apical (D), basal (E), or homogenous (F) distribution of synapses.