Position-theta-phase model of hippocampal place cell activity applied to quantification of running speed modulation of firing rate

Abstract

Spiking activity of place cells in the hippocampus encodes the animal's position as it moves through an environment. Within a cell's place field, both the firing rate and the phase of spiking in the local theta oscillation contain spatial information. We propose a position-theta-phase (PTP) model that captures the simultaneous expression of the firing-rate code and theta-phase code in place cell spiking. This model parametrically characterizes place fields to compare across cells, time, and conditions; generates realistic place cell simulation data; and conceptualizes a framework for principled hypothesis testing to identify additional features of place cell activity. We use the PTP model to assess the effect of running speed in place cell data recorded from rats running on linear tracks. For the majority of place fields, we do not find evidence for speed modulation of the firing rate. For a small subset of place fields, we find firing rates significantly increase or decrease with speed. We use the PTP model to compare candidate mechanisms of speed modulation in significantly modulated fields and determine that speed acts as a gain control on the magnitude of firing rate. Our model provides a tool that connects rigorous analysis with a computational framework for understanding place cell activity.

Keywords: firing rate variability; phase precession; spatial navigation.

Fig. 1.

Parametric model of place cell activity. (A) Schematic of model: firing rate at each theta phase and position predicted by the PTP model (heatmap). The firing rates are derived by multiplying the spatial tuning curve (below x axis) with the theta-phase tuning curve (beside y axis). The theta-phase tuning curve shifts as the preferred phase (*) decreases as a function of position (Lower Left), capturing phase precession. Parameters for this schematic derived from fitting model to real place field data. (B) Model equations: Spatial input function is a Gaussian function of position with 3 parameters: amplitude $A_x$, width ${\sigma }_x$, and center $x_0$. Theta modulation function is a Von Mises function (approximately circular Gaussian) of theta phase normalized to height 1 with one parameter: $k_{\theta }$. Theta modulation is centered on a preferred phase in the precession function, which changes linearly with position according to slope $m_{\theta }$ and intercept $b_{\theta }$.

Fig. 2.

Simulating place cell data. (A) Single-trial simulation. (i) Simulated theta-phase–position trajectory imposed on place field schematic (as in Fig. 1A) for trial 1 (blue), and trial 2 (purple) at the same speed with a different initial theta phase at the start of the place field [not to be confused with onset of phase precession, i.e., the preferred phase at position 0 (2)]. (ii) Simulated firing rates for model fit to place field in C computed for trajectories in trial 1 and 2 in i. Spiking for each trial (below) simulated via Poisson process. (B) Summary visualization of simulated data: Spiking was simulated for 42 trials with varying speeds and initial theta phases, using the best-fit PTP model from real place cell in C. (i) Mean firing rate vs. position, averaged over trials. (ii) Theta phase vs. position for each spike. (C) Summary visualization of real place cell data in example place field. (i and ii) Same as in B.

Fig. 3.

Speed-dependent variability can cause spurious correlations with running speed. (A) Alignment between spatial input and theta modulation can affect expected firing rate in place fields. Position and theta phase are simulated for 2 trials with identical running speed, differing only in the initial theta phase at the entrance of the place field. The resulting model equations and predicted firing rates are shown: (i) The spatial input function (black) is identical for the 2 trials, while the phase modulation functions are shifted according to the initial theta phase for trials 1 (blue) and 2 (green). (ii) Predicted firing rate vs. position for trials 1 and 2 computed by multiplying the spatial input and phase modulation functions in i. (iii) Mean firing rates for trials 1 and 2, averaged over position. Error bars correspond to SE (SEM) predicted from Poisson variance of spiking. (B) Mean firing rate across place field simulated as a function of initial theta phase and running speed. Variability caused by initial theta phase increases at higher speeds. (C) Firing rate vs. position in the fast (red), medium (yellow), and slow (blue) sets of trials in 3 simulated experiments. In each experiment, 30 trials were simulated with randomized speeds and initial theta phases. Mean firing rate was computed as a function of position for each set of trials. Conditions were identical for each simulated experiment with no explicit speed dependence in the model; however, apparent speed modulation appeared by chance, both negatively (experiment 1) and positively (experiment 3).

Fig. 4.

Real pyramidal cells show heterogeneous distribution of speed correlations. (A) Speed dependence of example place field with ostensible negative speed modulation: (i) Trial vs. position of each spike, trials ordered by mean running speed in the place field. (ii) Firing rate vs. position for fastest (red), middle (yellow), and slowest (blue) thirds of trials. (iii) Mean firing rate across place field vs. speed, each point representing one trial. r values throughout indicate Kendall rank correlation coefficient (P = 0.025). (B) Same as A for ostensibly unmodulated example place field (P = 0.13). (C) Same as A for ostensibly positively speed-modulated place field (P = 8.8e-04). (D) Distribution of speed–firing-rate correlations across all place fields in dataset.

Fig. 5.

Speed modulation is statistically significant in a small number of place fields. (A) Mean firing rate vs. speed for real example place field (same as Fig. 4C), for real experiment (black) and simulated experiment (green). Simulation performed using model fit to this place field and conditions identical to real experiment. Kendall correlation coefficient for simulated experiment is r = 0.27 (P = 0.032). (B) Null distribution of speed–firing-rate correlations (green) computed from 20,000 simulations of the experiment in A, each using the empirically measured position and theta phase from the original experiment. Empirical correlation in black. For this example field, we find evidence for speed modulation beyond what can be explained by the PTP model (P = 0.001). (C) One-tailed P values computed from null distribution for all place fields in the dataset. Two tests are shown, one for positive speed–firing-rate relationships (red) and one for negative speed–firing-rate relationships (blue). Values range from 0 to 1, corresponding to the proportion of simulated experiments with correlation values larger (for negative test) or smaller (for positive test) than that of the real experiment. Significant speed modulation is defined as P < 0.05. (D) Proportion of place fields for which there was statistical significance for negative or positive speed modulation, and the proportion for which there was no evidence for speed modulation.

Fig. 6.

Speed model comparison: variants of basic model that include speed dependence. (A) Gain control model: (i) Hypothetical relationship between amplitude parameter and speed. (ii) Spatial input function as it varies with speed. (iii) Phase modulation function (stationary with respect to speed). (iv) Hypothetical relationship between firing rate and speed for gain control model. (B) Phase modulation model: same as A, except phase selectivity varies with speed instead of amplitude. (C) Dual modulation model: same as A, except both amplitude and phase selectivity vary with speed. (D) Proportion of place fields in each speed modulation category best fit by each model variant. Positively and negatively modulated fields are by majority best fit with a gain control model, while unmodulated fields are mostly best fit with the original PTP model.