Fig 1.
Measurable signals should be available from neural simulations at different levels of abstraction.
Neural circuits, here represented by a putative cortical column (panel A), are studied at different levels of biological detail, depending on the scientific question (panel B). By using a forward model (panel C) one can calculate measurable signals (panel D) from neural activity simulated at different levels of abstraction. In general, calculations of such brain signals are only biophysically well-founded when using biophysically detailed cell models, while simplified representations of neurons will require “heuristic” approaches where it can be hard to estimate the accuracy of the resulting brain signal predictions.
Fig 2.
Illustration of the principle underlying the kernel method.
A: The postsynaptic perspective, where all incoming synaptic input to a postsynaptic cell is taken into account, and the time-dependent LFP contribution of the postsynaptic cell is calculated. The blue and red colors are used to illustrate positive and negative regions, respectively, of the LFP at a moment in time. The total population LFP V ( r , t ) is then the sum of all such single-cell contributions . This is the standard way of calculating LFP signals from neural simulations. B: The presynaptic perspective, where all outgoing synapses from a single cell are considered. For passive cells with static (no plasticity), current-based synapses, every action potential of a presynaptic neuron j will evoke the same postsynaptic currents, and hence, each action potential has a fixed LFP response
. By taking into account all postsynaptic targets, the single-cell kernel
can be calculated, and the single-cell LFP contribution can be found by convolving the single-cell kernel with the corresponding spike train of the presynaptic cell. The population LFP is again the sum of all single-cell contributions, and if this is done for all cells, and all external incoming synapses, the LFP calculated by these two approaches will be identical, under the assumptions listed above.
Fig 3.
Illustration of kernel method with toy model.
A: Two toy single-cell kernels (blue and orange), and the mean, that is, the population kernel (black). B: Raster plot of the two corresponding spike trains, with the same color code as in panel A. Each colored marker corresponds to a spike, and the individual spike trains are plotted at different heights along the y-axis. C: The population rate (average number of spikes per time bin, Δt = 0 . 1ms), that is, the mean firing rate from the spike trains in panel B. D: The gray line shows the ground truth toy LFP signal calculated as the sum of each single-cell contribution, which is again calculated by convolving the single-cell kernels with the corresponding spike trains. The black line shows the LFP calculated by convolving the population kernel with the population rate. The red line shows the difference between the ground truth LFP and the population kernel LFP.
Fig 4.
Error in population kernel predictions depend on kernel heterogeneity and spike-train correlations.
Each column shows 1000 single-cell kernels with different amplitude standard deviations (top), and different levels of spike-train correlations (middle). Spike trains with varying levels of correlations were generated through Multiple Interaction Processes (MIP) [42], controlled by the parameter f, where f = 0 corresponds to uncorrelated homogeneous Poisson processes, while f = 1 corresponds to fully correlated (identical) spike trains (see Methods). The mean firing rate is shown in black, and the standard deviation in gray. The toy LFP is calculated (bottom). Relative error Erel, quantified by the normalized standard deviation of the difference between the ground truth signal and the population kernel signal (see Methods), vanishes for identical kernels, regardless of correlation (first column). For variable kernels with some correlation, the kernel approach will result in some relative error (second column). For variable kernels and zero correlation, the relative error will be large (third column). For perfect correlation, the relative error vanishes regardless of kernel variability (fourth column).
Fig 5.
Parameter scan for simple toy-model LFP.
A: The LFP amplitude (quantified by its standard deviation across time) for different levels of amplitude variability in single-cell kernels, and different levels of correlations between spike trains. B: Simulated absolute error, quantified by the standard deviation of the difference between the ground truth signal and the population kernel signal. C: Simulated relative error, quantified by the standard deviation of the difference between the ground truth signal and the population kernel signal (panel B), normalized by the ground truth signal amplitude (panel A). D, E: Same as in panels B and C, but predicted from theory (equation (7)). F: Difference map between results from simulations (panel B) and theory (panel D), for the absolute error. G: Difference map between results from simulations (panel C) and theory (panel E), for the relative error. Correlated spike trains were generated using MIP processes (see Methods).
Fig 6.
Effect of neuron and synapse heterogeneity on the variability of single-cell LFP kernels.
A: A population of cortical pyramidal neurons (morphologies depicted in shades of light gray and soma locations as black dots) receives synaptic input from a presynaptic population. Each incoming axon forms, in total, connections with different postsynaptic neurons. The strength J of each synapse is randomly drawn from a lognormal distribution. The synaptic time constant
and the synaptic delay are drawn from normal distributions (graphs to the left). The vertical position of each synapse is drawn from the segment locations of the cells weighted by a normal distribution (green curve to the right). Some exemplary synapse positions are plotted on the postsynaptic population as green dots. Vertical soma positions are drawn from a capped normal distribution (black curve to the right). Horizontal soma positions are uniformly distributed on a disc within radiusRpop. The LFP response to an activation of all
synapses of a single incoming axon is calculated for different cortical depths (dark red dots). The EEG response outside the head, directly above the population, is calculated using a simple spherical head model. For each parameter configuration, we generate 100 single-cell kernels resulting from different random realizations of neuron and synapse parameters. Each of these kernels describes the postsynaptic LFP (EEG) response to the firing of a different presynaptic neuron. B–D: LFP and EEG responses for different synaptic target zones (B: apical; C: basal; D: uniform). Gray: single-cell kernels. Black: population kernel. The “basal input” case is used as the “default case” throughout this study. E: Mean (solid curves) and standard deviation (bands) of the maximum LFP deflection at different cortical depths for different synaptic target zones (see legend). See Methods for details on the model and parameter values.
Fig 7.
Example of LFP kernels, spike trains, and the resulting LFP signals.
A: The LFP kernels at different depths (see Fig 6A) with each single-cell kernel in gray and the population kernel in black. The kernels shown here are from the “default” case, corresponding to Fig 6C. B: Raster plot of uncorrelated spike trains (see Methods) with a firing rate of 10 s−1. Below the spikes, the population firing rate (constructed by summing all individual spike trains) is shown in black. C: The ground truth LFP signal (gray), the population kernel LFP signal (black), and the difference between them (red), at different depths. D: The relative error at different depths (see Methods), either observed from simulations (solid curve) or predicted from theory (dotted curve).
Fig 8.
Comparison of how different parameter configurations affect LFP amplitude and population kernel errors.
For uncorrelated Poisson input with a rate of 10 s−1 (see Methods), the figure shows the standard deviation of the LFP at different depths (column 1), and the absolute (column 2) and relative error (column 3) from using the population kernel, for different modifications of the original parameter set (“default”). Each row corresponds to varying a certain feature, see Methods and Table 1 for full description of different parameter configurations. A: Three different synaptic input regions, that is, apical dendrites, basal dendrites (“default”), or uniformly distributed over the whole cell. B: Three different numbers of postsynaptic targets Kout (outdegree), including half and double of the default value of 500. C: Three different levels of variability of synaptic parameters, including half and double the parameter values used for controlling the variability of the synaptic time constants, the synaptic delays, and the synaptic weights. D: Three different standard deviations of the normal distribution used when drawing synaptic locations, including half and double the default value of 100 μm. E: Three different radii, including half and double the default radius of 250μm.
Fig 9.
Generating different types of correlated spiking.
A, B: Spiking activity generated by Multiple Interaction Processes (MIP; [42]) with firing rate and correlation coefficients
(A) and 0.01 (B). C, D: Spiking activity generated by a recurrent network of point neurons [45] operating in the asynchronous irregular (“AI”; C) and in the slow synchronous irregular regime (“SI slow”; D). Top panels: Raster plots for 100 exemplary neurons. Bottom panels: Normalized spike-train auto- (black) and cross-covariance (gray) functions. The depicted curves represent population averaged correlations obtained from binned spike trains of an ensemble of 100 neurons, with an observation time of 15.5 s, and a binsize of 2−4 ms. See Methods for details on the spike-generation models and parameter values.
Fig 10.
Effect of spike-train statistics on population kernel errors.
A: Dependence of the LFP amplitude on the recording depth for various presynaptic spike-train ensembles generated by the MIP and by the Brunel network model (see legend and main text), for the kernels corresponding to the default case. MIP spike trains are characterized by the firing rate ν and the spike-train correlation coefficient . Brunel spike trains are obtained from the “AI” and the “SI slow” regime. Solid and dashed lines refer to the ground-truth signal and the LFP predicted by the kernel method. B: Same as panel A, but showing the absolute prediction error. Solid and dotted lines represent results obtained from simulations and theory, respectively. C: Same as panel B, but showing the relative prediction error. D: The maximum error across depth for MIP spike trains with different firing rates ν and correlations f. E: Same as panel D but showing the maximum relative error across depth.
Fig 11.
Summary of errors for different kernel parameters and different types of spiking activity.
A: Maximum ground truth LFP amplitude across depths for different combinations of kernel parameters (rows) and spiking activity (columns). B: Maximum simulated relative errors across depths, for different combinations of kernel parameters (rows) and spiking activity (columns). The rows and columns are sorted so the largest relative errors are in the bottom left, while the lowest relative errors are in the top right. C: The relative error as a function of the signal amplitude for the LFP signal (black dots), and for the EEG signal (gray dots) for all parameter combinations shown in panels A and B. Since the EEG signal intrinsically has a much lower signal amplitude, the LFP and EEG signals are normalized by the maximum observed signal amplitude seen in either of the two signals, so they are easier to visually compare. The dashed line is a visual guideline corresponding to a perfect inverse correlation.
Fig 12.
Illustration of the kernel approach applied to a rate model.
A: Stimulus induced switching dynamics of rate model described by equation (8), with Δ = 2, η = − 10, , and τ = 100ms [47]. The stimulus I(t) is a square pulse with an amplitude of 4, a delay of 1s, and a duration of 3s, resulting in switching dynamics similar to what was observed by Montbrió et al. [2015, Fig 2(a)] [46]. B: Population kernel for the “default” case (basal input) of the setup introduced in Sect 2.3 “sec:kernel_heterogeneity” (see Fig 6C). C: Transient behavior as observed in the population kernel LFP and EEG signals, calculated by convolving the population rate in panel A with the kernels in panel B. Before the convolution with the LFP/EEG kernel, the population rate is transformed from units of hertz to units of spikes/Δt, and scaled by the considered size of the presynaptic population which was in this case 10,000 [46].
Table 1.
Parameter combinations used for calculating the kernels, where the names of the columns correspond to the parameter combinations tested in Figs 8 and 11. Blank spaces indicate no change from the default values, and only parameters that are varied between simulations are included.