Table 1.
Identified cell counts from single-unit recordings by brain area and spatial phase-coding subtype.
Fig 1.
An example LS neuron with spatially correlated rate and phase.
Recording data for a 2.2-h session in an 80-cm diameter arena. Sinusoids aligned to phase axes indicate theta waves, with peaks oriented to the top (horizontal axes) or left (vertical axes). (A) Spike-trajectory plot. Red dots: animal position at time of spike; gray line: trajectory. Inset: spike train autocorrelogram (top) and spike theta-phase distribution (bottom). (B+C) Spatial maps computed with an adaptive Gaussian kernel (Methods). (B) Firing ratemap. (C) Spike mean-phase map (left) and phase-vector map (right) with normalized MVL indicated by color saturation (color wheel; Methods). (D) Example 1-s traces of hippocampal LFP theta waves and spikes during periods of low (top) or high (bottom) firing rate. Highlights show theta cycles. (E) Example 15-s trajectory segment (line) showing bursts (circles) emitted as the rat traveled from a low-rate region to a high-rate location (blue-to-red bursts) and back to a low-rate region (red-to-green bursts; left). Likewise, plotted against firing rate, burst phase first advanced (blue-to-red bursts) and then delayed (red-to-green bursts; right). Left background: ratemap from (B). (F) Mean-phase (C) distributions (grayscale) conditioned on mean rate (B). Red lines: circular-linear regressions; multiple theta cycles shown (y-axis) for clarity.
Fig 2.
Phaser cells encode space with positive or negative phase shifts.
(A) Selection of phase-coding recordings based on spatial phase information (Iphase, x-axis), total phase shift (y-axis), and rate-phase coupling strength (circle diameter). Phaser cell recordings were divided into ‘negative’/‘positive’ subtypes according to the bottom-right/top-right regions selected by the criteria. Circles: significant Iphase recordings; contours: kernel density-estimate of non-significant recordings; red hatch lines: region excluded by the first two phaser cell criteria (see numbered listing of criteria above in Results). (B) Spatial uncertainty is related to the magnitude of phase shift for negative and positive phaser cell recordings. (C) Spatial distributions of mean resultant vector length (MVL) across phase maps (mean ± 90% empirical c.i.). (D+E) Pair-wise comparisons of early vs. late (<1 h) session activity (D) or between days (E). Within-cell spatial correlations were higher (left) and absolute changes in total phase shift were lower (right) than baseline comparisons between cells. Histograms: normalized by pair count, bin size from the Freedman-Diaconis rule. (F+G) Spatial comparison of MVL (x-axis) and within-session change in the phase code (y-axis) at every location in the phase map. (F) Example LS cell from Fig 1. Inset: mean-phase maps (top) and whole-session MVL (bottom; black, 0; white, maximum MVL). (G) Average density of all negative (left) and positive (right) phaser cell recordings.
Fig 3.
Mechanisms and temporal organization of the phaser cell code.
Our thesis is that phaser cell activity is distinct from hippocampal phase precession and encodes spatial isocontours, not specific locations. (A) Schematic of symmetric rate-phase coupling (cf. Fig 1E, right) that deflects in one direction and then retraces in the opposite direction as the animal moves through a high-activity region. Inset from Fig 1E for illustration. (B) Mean rate-phase relationship across normalized traversals of 1,071 place fields from Souza & Tort (2017) [43]. Arrow: unidirectionality of phase precession. (C) Schematic of ramp-depolarization model with symmetric inputs, as is the case prior to learning [8]. Sinusoid: theta inhibition; green line: depolarizing input. (D) Schematic of a spatial phase code modeled on the LS cell in Fig 1 in which theta phase (left) maps to an isocontour level of underlying spatial inputs reflected by mean firing rate (right). (E+F) Negative and positive phaser cell recordings were segregated by theta phase. Multiple theta cycles shown for clarity. (E) Rate-phase regressions across normalized mean firing-rates. Line width: thin, |r| < 1/3; medium, 1/3 ≤ |r| < 2/3; thick, |r| > 2/3. (F) Distributions of typical spike theta-phases computed as spatial averages. Histograms: positive composited over negative; lines: density estimates using a circular π/4 bandwidth Gaussian kernel. Panel (B) was adapted from figure 5B of Souza & Tort (2017) [43] as permitted by the CC-BY 4.0 International License (creativecommons.org/licenses/by/4.0/).
Fig 4.
Example phaser cells illustrate the diversity of spatial phase codes.
For example recordings of negative (A) and positive (B) phaser cells, we show the ratemap (top), phase-vector map (middle), and conditional spike-phase distribution with rate-phase regression lines (bottom, as in Fig 1F). Maximal firing rates (top rows, color bar axes) were consistent with the moderate range of phaser cell firing rates (S2 Fig, panel A, left). Negative phaser cells demonstrated visibly stronger spatial modulation and rate-phase coupling compared to positive phaser cells, consistent with analyses of spatial uncertainty (Fig 2B), phase reliability (Fig 2C), and location-specific phase-code stability (Fig 2G).
Fig 5.
Space–trajectory GLM reproduced allocentric spatial modulation.
(A) Actual firing ratemaps (top) and LQW-SD-predicted ratemaps (bottom) for the negative phaser cell examples in Fig 4A. Reconstructions were built from spike-count predictions in each 3 × 3 grid section (Methods). White lines: grid section boundaries; arrows: normalized GLM directional (D) weights; Strength: DSI; Homogeneity: DHI. (B+C) GLM spike-count predictions for phaser cells were driven by allocentric spatial variables. The GLM coefficients (B) and maximal contributions (C; Eq (2)) from the spatial (L, Q, W) and trajectory-based (S, D) variables for phaser and nonphaser cells are shown in 95% box-and-whisker plots with outliers (× markers). For phaser cells, the purely allocentric, second-order spatial predictors (L and Q) dominated the GLM.
Table 2.
Parameters for dynamical theta-bursting neuron models.
Table 3.
Input and conductance parameters for model phaser units.
Fig 6.
Dynamical models of theta-bursting negative and positive phasers.
A model theta-burster (blue, ‘Negative’) with inhibitory theta and excitatory external input (green) provided feedforward inhibition to another theta-burster (orange, ‘Positive’). (A-C) A 20-s simulation. (A) A triangle-wave input (top) produced spiking (Low1, Low2) and bursting (High) in the negative phaser (middle) and a complementary pattern in the positive phaser (bottom). (B) Expanded intervals from the highlights in (A). Sinusoid: the reference theta wave of the simulation. (C) Negative vs. positive phaser spike phase across external input levels. Lines: circular-linear input-phase regressions. (D+E) Rate-phase coupling for the negative (D) and positive (E) phasers. A 1-hr simulation of 10-s to 62-s triangle-wave cycles sampled mean firing rates and mean spike phases for 512 input-level bins. Grayscale: conditional phase distributions; red line: circular-linear rate-phase regressions.
Fig 7.
Demonstration of a 1D phaser network and target cell learning.
(A+B) We defined a set of 64 place and 64 notch tuning functions as 1D spatial inputs on the range [0, 1] (Methods). (A) Spatial inputs (top, ‘Place’-driven network; bottom, ‘Notch’-driven network) drive 128 pairs of negative (blue circles) and positive (orange circles) phasers. Inputs excite the negative phasers which suppress the positive phasers (Fig 6A). Phasers at position 0.5 are highlighted. (B) A 1-hr simulation sampled spike phase for a 1-min triangle-wave trajectory traversing the space. For the highlighted phasers in (A), joint space-phase distributions of spike timing (left) show the phaser inputs to a downstream target neuron (right). From top to bottom (input/phaser network layer): place/positive, place/negative, notch/negative, and notch/positive. (C-G) Supervised competitive learning over presynaptic phaser inputs trained a ‘target burster’ model (B, right) to follow a spatial phase code. (C) Supervised phase code for training with two modes: theta trough on the left (position 0), theta peak on the right (position 1). Black: desired activity modes; white: untrained. Inset: prior to training, the target burster randomly drifted in phase due to a stochastic input current (Eq (11); S7 Fig, panel D). (D) Competitive kWTA weights (Table 4; Methods) for connections from each of the four input/phaser network layers in (A) to the target burster. (E) Total weighted phaser network input to the target burster. (F+G) 1-hr simulations of a 1-min triangle-wave trajectory spanning the range [0, 1]. Target burster output (burst phase) is shown without (F) and with (G) intrinsic noise (σ; Table 4). Arrows: phase trajectories for rightward (F, lower arrow) or leftward (F, upper arrow) movement; gray rectangles: supervised phase code from (C); red highlight: region with minimal phaser input based on panels (D) and (E). Multiple theta cycles are shown (y-axis) for clarity.
Table 4.
Input, noise, and learning parameters for target models.
Fig 8.
Realistic open-field phaser entrainment of path integration networks.
Simulations of 1,000 pairs of negative and positive phasers with generative 2D open-field spatial inputs (S10 Fig, panel A) entrained target cells (A-D) and networks (E-H). (A) Bursting phase map of a target neuron without phaser input. (B) Two supervised 2D phase codes with different phase offsets that emulate oscillatory path integration in the 45° direction. (C) 2D space-phase distributions of total kWTA-weighted phaser input to the target neuron (Table 4). (D) Phase maps of the target burster with phaser input. (E-H) Bayesian decoding of position from burst phase (Eq (13); Methods) of three collections of 64 target neurons representing path integration networks. (E) Supervised phase codes for each unit in the target networks. (F) Decoded sequences for an example 6-s trajectory for each target network (64 target units) or the combination of all three networks (192 target units). Heatmap: composited sequential posteriors; magenta line/circles: sequential MAP position estimates; blue line: actual trajectory. (G+H) Path-integration error-correction performance was quantified by decoding a benchmark 60-s trajectory from network activity and 100 bootstrapped unit samples of network activity. Plus symbols: network performance; curves: bootstrap mean; error bars: bootstrap s.e.m. (G) or 95% c.i. (H). (G) Decoded position error according to the number of decoded units. (H) The timescale of error-correction was measured as the HWHM of temporal auto-correlations of decoding error (Methods).