Hybrid Scheme for Modeling Local Field Potentials from Point-Neuron Networks

Abstract

With rapidly advancing multi-electrode recording technology, the local field potential (LFP) has again become a popular measure of neuronal activity in both research and clinical applications. Proper understanding of the LFP requires detailed mathematical modeling incorporating the anatomical and electrophysiological features of neurons near the recording electrode, as well as synaptic inputs from the entire network. Here we propose a hybrid modeling scheme combining efficient point-neuron network models with biophysical principles underlying LFP generation by real neurons. The LFP predictions rely on populations of network-equivalent multicompartment neuron models with layer-specific synaptic connectivity, can be used with an arbitrary number of point-neuron network populations, and allows for a full separation of simulated network dynamics and LFPs. We apply the scheme to a full-scale cortical network model for a ∼1 mm² patch of primary visual cortex, predict laminar LFPs for different network states, assess the relative LFP contribution from different laminar populations, and investigate effects of input correlations and neuron density on the LFP. The generic nature of the hybrid scheme and its public implementation in hybridLFPy form the basis for LFP predictions from other and larger point-neuron network models, as well as extensions of the current application with additional biological detail.

Keywords: cortical microcircuit; electrostatic forward modeling; extracellular potential; multicompartment neuron modeling; point-neuron network models.

Figure 1.

Overview of the hybrid LFP modeling scheme for a cortical microcircuit model. (A) Sketch of the point-neuron network representing a 1 mm² patch of early sensory cortex (adapted from Potjans and Diesmann 2014). The network consists of 8 populations of LIF neurons, representing excitatory (E) and inhibitory neurons (I) in cortical layers 2/3, 4, 5, and 6. External input is provided by a population of TC neurons and cortico-cortical afferents. The color coding of neuron populations is used consistently throughout this paper. Red arrows: excitatory connections. Blue arrows: inhibitory connections. See Tables 1–2, 5–6 for details on the network model. (B) Spontaneous ($t<900\mathrm{ms}$) and stimulus-evoked spiking activity (synchronous firing of TC neurons at time t = 900 ms, denoted by thin vertical line) generated by the point-neuron network model shown in panel A, sampled from all neurons in each population. Each dot represents the spike time of a particular neuron. (C) Populations of LFP-generating multicompartment model neurons with reconstructed, layer-, and cell-type specific morphologies. Cells are distributed within a cylinder spanning the cortex. Layer boundaries are marked by horizontal black lines (at depths z relative to cortex surface z = 0). Only one representative neuron for each population is shown (see Fig. 4 for a detailed overview of cell types and morphologies). Sketch of a laminar recording electrode (gray) with 16 contacts separated by $100\mathrm{\mu }\mathrm{m}$ (black dots). (D) Depth-resolved LFP traces predicted by the model (cf. Tables 3 and 4). Note that channel 1 is at the pial surface, so that channel 2 corresponds to a cortical depth of 100 μm and so forth.

Figure 2.

Cell types and morphologies of the multicompartment-neuron populations. The 8 cortical populations Y of size N_Y in the microcircuit network model are represented by 16 subpopulations of cell type y with detailed morphologies M_y (Binzegger et al. 2004, Izhikevich and Edelman 2008). Neuron reconstructions are obtained from cat visual cortex and cat somatosensory cortex (source: NeuroMorpho.org (Ascoli et al. 2007), Contreras et al. (1997), Mainen and Sejnowski (1996), Kisvárday and Eysel (1992), Stepanyants et al. (2008) cf. Table 7). Each morphology M_y is here shown in relation to the layer boundaries (horizontal lines). Colors distinguish between network populations as in Figure 1. The number of compartments ($n_{\mathrm{comp}}$), frequencies of occurrence (F_y), relative occurrence (F_yY), and cell count (N_y) are given for each cell type $y\in Y$.

Figure 3.

Example LFP responses from single-synapse activations of layer 4 neurons. (A) Illustration of the nontrivial relationship between apical synaptic input (red circle) onto a reconstructed morphology (black) of a pyramidal cell in layer 4 and the corresponding extracellular potential. The exponential synaptic input current $I_{i,j}\left(t\right)$ (upper inset) results in deflections in the extracellular potential $\varphi (r,t)$ here shown as time courses at 2 locations in proximity to the input site and the basal dendrites (green and blue circles, respectively; lower inset). The color-coded isolines show the magnitude of the scalar extracellular potential at t = 2 ms (vertical black line in insets) in the vicinity of the cell. (B) Same as in panel A, however, with the synaptic input current relocated to a basal dendrite, resulting in an extracellular potential with a different spatiotemporal signature less dependent on the geometry of the apical dendritic tree. At the location denoted by the blue circle, the extracellular potential changes sign with time due to interactions between signal propagation in the passive model neuron and volume conduction. (C) Same as panels B and C for a spiny stellate cell in layer 4 receiving an excitatory synaptic input on a basal dendrite.

Figure 4.

Constructing spatial synaptic connectivity for the cortical microcircuit model. (A) Illustration of pooling of presynaptic cell types. Presynaptic populations X in the point-neuron model (left box; here $X=$ L4E) consist of multiple cell types x (here $x\in \left\{\mathrm{p4,}\mathrm{ss4(L4),}\mathrm{ss4(L23)}\right\}$). The layer-specific number of synapses k_yXL (dash-dotted lines) formed between one cell of postsynaptic cell type y (right part of panel A: morphology projected onto cortical layers 1–6; here y = p5(L56)) and a presynaptic population X is given by the sum of all individual cell-type resolved synapse counts k_yxL (dotted or dash-dotted lines). (B) Bi-directional cell- and layer-specific pooling and dispersing of synapses between presynaptic and postsynaptic cell types. Postsynaptic populations Y (right box; here Y = L5E) in the point-neuron model consist of multiple cell types y (here $y\in \left\{\mathrm{p5(L56),}\mathrm{p5(L23)}\right\}$). A given presynaptic population X (left box; here X = L4E) containing cell types x (here $x\in \left\{\mathrm{p4,}\mathrm{ss4(L4),}\mathrm{ss4(L23)}\right\}$) forms cell-type and layer-specific connections within Y (black connection tree). For the number of synapses K_yXL between population X and cells of type y in layer L (right-most branching of connection tree) the synapse count K_YX between all cells in X and Y can be obtained by pooling all synapses onto cell types $y\in Y$ and input layers L. Conversely, for a given total number of synapses K_YX between all cells in X and Y, the number of synapses K_yXL onto a specific cell type y and layer L can, as described by Equation (9), be obtained by calculating the cell-type and layer specificity of connections ${\mathcal{T}}_{\mathit{yX}}$ and ${\mathcal{L}}_{\mathit{yXL}}$ (see Fig. 5) from anatomical data (Table 7).

Figure 5.

Connectivity of the cortical microcircuit model. (A) Connection probability C_YX between presynaptic population X and postsynaptic population Y of the cortical microcircuit model by Potjans and Diesmann (2014) given in Table 5. Zero values are shown as gray here and in subsequent panels. (B) Layer- and cell-type specific connectivity map C_yXL, where X, y, and L denote presynaptic populations, postsynaptic cell types, and the synapse location (layer), respectively. This map is computed from the connectivity of the point-neuron network (panel A), cell-type (panel C), and layer specificity (panel D) of connections. (C) Cell-type specificity ${\mathcal{T}}_{\mathit{yX}}$ of connections quantified as the fraction of synapses between presynaptic and postsynaptic populations X and Y formed with a specific postsynaptic cell type y. (D) Layer specificity ${\mathcal{L}}_{\mathit{yXL}}$ of connections denoting the fraction of synapses between population X and cell type y formed in a particular layer L. Both ${\mathcal{T}}_{\mathit{yX}}$ and ${\mathcal{L}}_{\mathit{yXL}}$ in panels C and D, respectively, are calculated from anatomical data (Binzegger et al. 2004, Izhikevich and Edelman 2008), cf. Table 8.

Figure 6.

Overview of output signals obtained from application of the hybrid scheme to a cortical microcircuit (spontaneous activity). Point-neuron network: (A) Spiking activity. Each dot represents the spike time of a point neuron (color coding as in Figure 1). (B) Population-averaged firing rates for each population. (C) Population-averaged somatic input currents (red: excitatory, blue: inhibitory, black lines: total). (D) Population-averaged somatic voltages. Averaged somatic input currents and voltages are obtained from 100 neurons in each population. Multicompartment model neurons: (E) Somas of excitatory (triangles) and inhibitory (stars) multicompartment cells and layer boundaries (gray/black ellipses). Illustration of a laminar electrode (gray) with 16 recording channels (black circles). (F) Depth-resolved CSD obtained from summed transmembrane currents in cylindrical volumes centered at each contact. (G) Depth-resolved LFP calculated at each electrode contact from transmembrane currents of all neurons in the column. Channel 1 is at pial surface, channel 2 at $100\mathrm{\mu }\mathrm{m}$ depth, etc.

Figure 7.

Network activity following transient activation of TC afferents. (A) Raster plot of spiking activity before and after δ-shaped thalamic stimulus presented at t = 900 ms (vertical black line in panels A, C, and D). (B) Population-averaged firing rate histogram for each population (color coding as in panel A). (C) Depth-resolved compound CSD of all populations (shown both in color and by the black traces). (D) Depth-resolved compound LFP (shown both in color and by the black traces) at each electrode channel as generated by all populations. Channel 1 is at pial surface, channel 2 at 100 μm depth, etc.

Figure 8.

Effect of network dynamics on LFP. Comparison of two different thalamic input scenarios and two different networks. Top: Reference network, spontaneous activity. Center: Reference network, oscillatory thalamic activation. Bottom: Original model by Potjans and Diesmann (2014), spontaneous activity. (A,F,K) Population-resolved spiking activity. (B,G,L) Population-averaged firing rate spectra. (C,H,M) Depth-resolved LFP. (D,I,N) LFP power spectra in layer 1 and at typical somatic depths of network populations. (E,J,O) LFP power spectra across all channels. Channel 1 is at pial surface, channel 2 at 100 μm depth, etc.

Figure 9.

Composition of CSD and LFP during spontaneous activity. (A) Representative morphologies of each population Y illustrating dendritic extent. (B) LFP (black traces) and CSD (color plot) produced by the superficial population L23E for spontaneous activity in the reference network. (C) Similar to panel B for population L6E (summing over contributions of $y\in \left\{\mathrm{p6(L4),}\mathrm{p6(L56)}\right\}$). (D) CSD variance as function of depth for each individual subpopulation (colored lines) and for the full compound signal (thick black line). (E) Same as in panel D, but for LFPs. Variances ${\sigma }^2<{10}^{-7}{\mathrm{mV}}^2$ not shown. (F) Compound LFP (black traces) and CSD (color plot) resulting from only excitatory input to the LFP-generating multicompartment model neurons. (G) Conversely, LFP (black traces) and CSD (color plot) resulting from only inhibitory input to the neurons. (H) Full compound LFP (black traces) and CSD (color plot) resulting from both excitatory and inhibitory synaptic currents. (I) Compound CSD variance as a function of depth with all synapses intact (thick black line), or having only excitatory (red) or inhibitory synapse input (blue). (J) Same as in panel I, but for the LFP signal.

Figure 10.

Decomposition of CSD and LFP into contributions due to excitatory and inhibitory inputs for thalamic activation. (A–E) Oscillatory thalamic activation (f = 15 Hz). (F–J) Transient thalamic activations at $t=900+n·1000\mathrm{ms}$ for $n\in \{0,1,2,3,4\}$. Same row-wise figure arrangement as in Figure 9F–J.

Figure 11.

Effect of single-cell LFP cross-correlations on compound-LFP power spectra during spontaneous activity (A,B) and for oscillatory thalamic input (C,D). A,C Compound-LFP power spectra $P_{\varphi }(\mathbf{r},f)$ (black traces) and compound spectra $P_{\varphi }^0(\mathbf{r},f)$ (red traces) obtained when omitting cross-correlations between single-cell LFPs (red traces; computed for 10% of the cells and multiplied by a factor 10) at recording channels corresponding approximately to the centers of layers 1, 2/3, 4, 5, and 6. B,D Depth and frequency-resolved ratio $P_{\varphi }(\mathbf{r},f)/P_{\varphi }^0(\mathbf{r},f)$ of LFP power spectra, cf. Equation (16).

Figure 12.

Prediction of LFPs from downsized networks. Top row: Spontaneous activity. Bottom row: Oscillatory thalamic activation. (A,E) Full-scale LFP traces $\varphi (\mathbf{r},t)$ (black) and low-density predictors ${\varphi }^{\gamma \xi }(\mathbf{r},t)$ (red) obtained from a fraction $\gamma =0.1$ of neurons in all populations and upscaling by a factor $\xi ={\gamma }^{-1/2}$. (B,F) Correlation coefficients (gray bars) between full-scale LFP and low-density predictor shown in panels A and E, respectively. The dashed lines denote 1% significance levels obtained after computing the chance correlation coefficients for 1000 trials. (C,G) Power spectra $P_{\varphi }(\mathbf{r},f)$ and $P_{{\varphi }^{\gamma \xi }}(\mathbf{r},f)$ of full-scale LFPs (black) and low-density predictors with $\gamma =0.1$ and $\xi ={\gamma }^{-1/2}$ (red) or $\xi ={\gamma }^{-1}$ (gray). (D,H) Ratio $P_{\varphi }(\mathbf{r},f)/P_{{\varphi }^{\gamma \xi }}(\mathbf{r},f)$ between power spectra of full-scale LFP and low-density predictor with $\gamma =0.1$ and $\xi ={\gamma }^{-1/2}$ (cf. Eq. 17).

Figure 13.

Linear prediction of LFPs from population firing rates. (A) LFP responses ${\overline{H}}_X(\mathbf{r},\tau )$ (kernels) to simultaneous firing of all neurons in a single presynaptic population X (see subpanel titles) at time τ = 0 ms, normalized by size N_X of the presynaptic population (red/blue: responses to firing of excitatory/inhibitory presynaptic populations). (B) Spike-trigger-averaged LFPs (stLFP) triggered on spikes of L5E neurons during spontaneous activity (left) and oscillatory thalamic network activation (right), averaged across all L5E spikes (T = 5000 ms simulation time). (C,F) LFP traces of the full model (black) compared with predictions (red) obtained from superposition of linear convolutions of population firing rates ${\nu }_X$ with LFP kernels ${\overline{H}}_X(\mathbf{r},\tau )$ shown in panel A. (D,G) Correlation coefficients (gray bars) between LFPs and population-rate predictors shown in panels C and F. The dashed lines denote 1% significance levels obtained after computing the chance correlation coefficients for 1000 trials. (E,H) Power spectra of LFPs (black) and the population-rate predictors (red) for different recording channels. Panels C–E and F–H show results for spontaneous activity and oscillatory thalamic activation, respectively.