Fig 1.
Interpretability of multivariate SFC through GPLA.
(A) Schematic for the concept of biophysical interpretability. A biophysical model allows to make predictions about some observable quantity derived from neural data, that we call Quantity of Interest (QoI). The QoI is biophysically interpretable whenever its variations can be explained by changes in some property of the model. (B) (Top-left) A coupling matrix is estimated from electrophysiology data by gathering complex SFC estimates of all spike-LFP pairs in a rectangular matrix. Coefficients (Cnm) contain information similar to complex-valued PLV up to a scaling factor: the magnitude indicates the strength of coupling, and the angle reflects the average timing of the spike occurrence within the period of the corresponding LFP oscillation. (Bottom-left) The coupling matrix can be approximated using its largest singular value and the corresponding singular vectors. Singular vectors represent the dominant LFP (blue array) and spiking patterns (red array) and the singular value (d1), called generalized Phase Locking Value (gPLV), characterizes the spike-field coupling strength for the phenomenon under study and the chosen frequency. The magnitude of each vector entry indicates a relative coupling of the corresponding unit/channel, and the phase indicates the relative timing with respect to other units/channels. By convention, the phase of the LFP vector coefficients’ average is set to zero, such that the phase of the spike vector average reflects the overall phase shift of the spike pattern with respect to the LFP pattern. (Top-right) A biophysical model accounts for the underlying network connectivity and dynamics, as well as the measurement process that leads to the collected data. This leads to a theoretical account of multivariate spike-field coupling. (Bottom-right) Model reduction entails simplifying assumption, leading to a low-rank description of the coupling in the model based on the key mechanistic parameters. This description is compared to the left-hand side low-rank decomposition, obtained from experimental data, to infer parameters and interpret the data. All clip art in this figure was designed and drawn by authors M.B. and S.S..
Table 1.
Summary of normalization by spike count and whitening application in all figures.
Fig 2.
Illustration of GPLA on simple simulations.
(A) Normalized amplitude of LFP-like oscillatory signals. (B) gPLVs for different models demonstrated in C-F (C-F) Various scenarios of spike-LFP coupling. Left: schematic representation of the modulating LFP oscillation (circle), and 6 representative neuron-like-units (indicated by the triangles). The color of each connecting line indicates the locking phase (see bottom colorbar for color code). Center: LFP-like signals within the window specified by the blue box in A and spikes are represented by overlaid red vertical lines. Right: resulting spike vector is represented in the third column. (C) Spiking activity globally synchronized to the trough of the LFP oscillation. (D) Sequential discharge of spikes coupled to the LFP. (E) Three clusters of neurons discharge at different phases of the LFP oscillation (a similar model was also used in Fig 3). (F) Spiking activity uncoupled to LFP oscillation (independent homogeneous spike trains). Also see Table A in S1 Appendix for a methodological summary.
Fig 3.
Comparison of GPLA and uni-variate spike-field coupling.
(A) Normalized amplitude of LFP-like transient oscillatory signal with additive Gaussian white noise (used in the first simulation). (B) LFP-like signal and overlaid spike raster (colored vertical lines—colors indicate each population of units with common locking phase) within the window specified by the blue box in (A). (C) Spike vector coefficients in the complex plane (colors correspond to B). Each dot represents one coefficient of the spike vector corresponding to a single neuron (note that within each cluster, dots are overlapping as they are similarly coupled). (D) Complex PLVs represented in the complex plane. Angles indicate the locking phase and the radius of the PLV. The gray point clouds indicate the PLV of multiple simulations and larger black dots indicate the average values. (E) Performance comparison (in percentage of simulations with significant coupling) of PLV, pooled PLV (pPLV) and gPLV, for three individual neurons. (F) Example oscillation, original (blue trace) and noisy (black trace) used in the second simulation. (G) Example coupling matrix related to simulation with a large amount of noise (σ = 5) (H-I) Comparison of GPLA-based and PLA-based estimation of PLVs for (H) different number of trials and (I) different levels of firing rate. Signal-to-Noise Ratio (SNR) is defined as the ratio of coupling strength (PLV) to estimation error (the difference between estimated PLV and the ground truth). Also see Table A in S1 Appendix for a methodological summary.
Fig 4.
Statistical analysis of GPLA with a theoretical significance test.
(A) LFPs are synthesized by mixing several oscillatory components (Ok(t)). The vertical red line evidences the phase shift between them. (B) Two exemplary spike trains (each from one of the coupled populations) and the corresponding LFPs. In the LFP trace on the top, the oscillatory component with the highest frequency is dominant while the bottom one is dominated by the lowest frequency component. (C) An exemplary coupling matrix for a simulation with two coupled populations. (D-E) Theoretical Marchenko-Pastur distribution (red lines) and empirical distribution (gray bars) for (D) simulation without coupling and (E) with coupling between multivariate spikes and LFP (F) Performance of GPLA for the detection of coupling between spike trains and LFPs for different strength of coupling (y-axis) and different number of spiking units/LFP channels. (G) Type I error for different numbers of spiking units/LFP channels (x-axis), quantified as the percentage of simulations wherein a significant coupling between spike trains and LFPs is detected in absence of ground truth coupling. The horizontal green line indicates the %5 threshold. (H) Mean-squared-error of GPLA-based estimation of the number of populations coupled to LFP for varying coupling strengths (y-axis) and numbers of coupled populations (x-axis). See also Table A for a methodological summary.
Fig 5.
Generative model of spike-LFP coupling.
(A) A two-population neural field model of neural dynamics. Vk, λk and sk indicate respectively somatic membrane potential, firing rate and post-synaptic current for Excitatory (k = E) and Inhibitory (k = I) populations. η indicates the exogenous input to the circuit. Arrows indicate the causal dependence between variables of the model. (B) Schematic representation of the model’s connectivity: local inhibition and long range excitation, together with the driving by exogenous synaptic currents. (C) Schematic representation of the contribution of postsynaptic currents to the electric field, affected by the spatial distribution of synapses over the dendritic tree and the geometry of pyramidal cells. From left to right: Schematic representation of pyramidal neurons, electric field, electrode (gray bar), contribution of each current (EPSC, IPSC and exogenous current, leak current is also contributing to LFP but is not shown) to the LFP profile along the electrode’s axis (D) Simple microcircuit structure leading to a temporal ordering of the local activities of different kinds LFP → excitation → inhibition (E) Simple microcircuit structure leading to a temporal ordering of activities of the same kind across space: the location receiving stronger exogenous input leads other locations, such that amplitude gradient leads to phase gradients.
Fig 6.
GPLA of hippocampal SWRs generated by a biophysical model of [78].
(A) Hippocampal multi-compartment model. Top: Canonical circuits of CA1 and CA3. Bottom: Schematic of the whole model (blue, excitatory connections; red, inhibitory. (B) From top to bottom: Example broad band CA1 LFP trace, band-pass filtered trace of the CA1 LFP in ripple band (80–180 Hz), and population firing rate of CA1 neurons. (C) CA1 gPLVs. Triangles indicate the significance assessed based on empirical (blue triangles, p<0.05) and theoretical (red triangles) tests. (D) LFP vectors for GPLA of CA1 (blue and red curves are overlapping), superimposed to ground truth dipolar LFP profile passively generated by the two compartment models of the pyramidal cell population. The right-hand side schematic illustrates the vertical dimensions of one cell’s compartments. (E) Spike vector coefficients for CA1 in several frequency bands (left: pyramidal cells, right: interneurons). (F) Average phase lag between LFP and spike vectors across frequencies for: outcome GPLA on hippocampal SWRs, theoretical analysis of Mass2D (without and with feedforward inhibition) and MassAlpha neural mass models. Dashed green line indicate MassAlpha filtered over the frequency bands used for GPLA. (G) Difference between phases of E and I populations based on GPLA the MassAlpha neural mass model filtered in the same bands (IPSP was used as LFP proxy). (H) Spike vector resulting from GPLA jointly applied to CA1 and CA3 in the gamma band (20–40 Hz). Related Supplementary Figures: S1 Fig, Use of EPSP as LFP proxy; S2 Fig, Joint GPLA of CA3 and CA1 activities; S3 Fig, Joint GPLA of CA3 and CA1 activities.
Fig 7.
Neural field simulation using EPSP as LFP proxy.
(A) Simulation with weak recurrent inhibition. Example time course at center location for exogenous input (top), E- and I- populations rates (bottom). (B) Same as A for strong recurrent inhibition. (C) gPLV as a function of frequency for both models. (D) Phase of spike vector coefficients as a function of their modulus for the frequency band yielding maximum gPLV for both models (each dot one coefficient, and the continuous lines are plotted based on linear regression). (E) Shift between averaged phase of spike vector and averaged phase of LFP vector, as a function of frequency. (F) Schematic of the spike vector’s phase gradient in the two models according to Eq 8. X-axis is the distance from center and y-axis is the connectivity strength. Line color indicates the phase according to the colorbar on the right. (G) Resulting GPLA in 3 frequency bands (indicated on the left) for weak recurrent inhibition (model schematized in A). (H) Same as G for strong recurrent inhibition (model schematized in B). In both G and H, color of pixel code the values of spike/LFP vector coefficients, with colorbar on top of H. Colors are represented in HSV mode, in which a complex number (reiϕ) is represented by hue and brightness of a pixel. Hue of a pixel indicates the phase (ϕ) and the brightness of a pixel indicates the magnitude (r). Related supplementary Figures: S5 Fig, Phase-modulus relation dependency on level of inhibition; S6 Fig, GPLA using IPSP as LFP proxy.
Table 2.
List of 2D neural field model parameters.
Fig 8.
Application to electrophysiological recordings in non-human primate PFC.
(A) Location of the Utah array, anterior to the arcuate sulcus (red line) and inferior to the principal sulcus (green line). (B) Broadband trace of the recorded LFP (from the recording channels indicated in C). (C) Utah array spatial map identifying channel IDs shown in B. (D) Spike rasters for all recorded neurons. (E-F) Example spike trains (red bars) and filtered LFP (black traces) in the frequency ranges (E) 3–5 Hz and (F) 15–30 Hz. (G) gPLV values. Triangles indicate the significance assessed based on surrogate (blue triangles) and analytical test (red triangles) tests. (H) Phase of spike vector coefficients as a function of its modulus for the frequencies indicated in the legend (one dot per coefficient, continuous lines indicate linear regression). (I-K) LFP and spike vectors for frequency (I) 3–5 Hz, (J) 5–15 Hz, and (K) 15–30 Hz. The first column depicts the LFP (blue dots) and spike (red dots) in the complex plane. The second column depicts the fitted von Mises distribution to phase of LFP and spike vectors. Third and forth columns respectively represent the spatial distribution of phase of LFP and spike vectors values on the array (see C). White pixels in the third column (LFP vector) indicate the recording channels that were not used in the recording and in the fourth column (spike vector), white pixels indicate the recording channels with insufficient number of spikes (multiunit activity with a minimum of 5 Hz firing). In the last two columns, colors are represented in HSV mode, in which a complex number (reiϕ) is represented by hue and brightness of a pixel. The hue of a pixel indicates the phase (ϕ) and the brightness of a pixel indicates the modulus (r). The colorbar is depicted on the right. Related supplementary Figure: S7 Fig, Analysis of PFC Utah array data.