Conversion of a phase- to a rate-coded position signal by a three-stage model of theta cells, grid cells, and place cells

Abstract

As a rat navigates through a familiar environment, its position in space is encoded by firing rates of place cells and grid cells. Oscillatory interference models propose that this positional firing rate code is derived from a phase code, which stores the rat's position as a pattern of phase angles between velocity-modulated theta oscillations. Here we describe a three-stage network model, which formalizes the computational steps that are necessary for converting phase-coded position signals (represented by theta oscillations) into rate-coded position signals (represented by grid cells and place cells). The first stage of the model proposes that the phase-coded position signal is stored and updated by a bank of ring attractors, like those that have previously been hypothesized to perform angular path integration in the head-direction cell system. We show analytically how ring attractors can serve as central pattern generators for producing velocity-modulated theta oscillations, and we propose that such ring attractors may reside in subcortical areas where hippocampal theta rhythm is known to originate. In the second stage of the model, grid fields are formed by oscillatory interference between theta cells residing in different (but not the same) ring attractors. The model's third stage assumes that hippocampal neurons generate Gaussian place fields by computing weighted sums of inputs from a basis set of many grid fields. Here we show that under this assumption, the spatial frequency spectrum of the Gaussian place field defines the vertex spacings of grid cells that must provide input to the place cell. This analysis generates a testable prediction that grid cells with large vertex spacings should send projections to the entire hippocampus, whereas grid cells with smaller vertex spacings may project more selectively to the dorsal hippocampus, where place fields are smallest.

Figure 1. Producing grid fields by beat interference between theta oscillators

As a rat runs across a linear track at a constant speed of 30 cm/s (top), a pair of detuned theta oscillators with frequencies of 7 Hz (red) and 8 Hz (green) interfere with each other to produce a 1 Hz beat oscillation (magenta) riding on a 7.5 Hz carrier (blue). If the blue trace is assumed to be a place or grid cell’s membrane potential, then the cell will fire when its membrane potential exceeds a spike threshold (dashed line). Spikes come in bursts (black lines) at the 7.5 Hz carrier frequency, and undergo phase precession against the 7 Hz theta oscillator (bottom graph).

Figure 2. Ring attractor model of the rat head direction circuit

A) A stationary activity bump when the head is still. B) A shifting bump when the rat is turning. C) Reciprocal weights in the ring attractor. D) Direction and speed of an oscillating bump depend upon the weighting coefficient, γ (t), for the asymmetric weight component (Equation 9).

Figure 3. A three-stage model of theta cells, grid cells, and place cells

The model’s first stage is a bank of ring attractors consisting of theta cells; λ₀ denotes a “reference ring” oscillating at a frequency that does not depend on velocity; λ_1,…,λ₁₂ are “theta rings” oscillating at frequencies modulated by velocity with differing gains. Each grid cell in the second stage receives a shared input from theta cell θ_0,0 in the reference ring, and an unshared input from a cell θ_{i, j} in one of the theta rings. The place cell in the third stage sums weighted inputs from grid cells to produce a Gaussian place field.

Figure 4. Phase precession by grid cells

A) The interference waveform for grid cell G_1,0 (top), with the timing of its “spikes” plotted against the phase of its own carrier signal (bottom). B) The oscillation generated by theta cell θ_0,0 (top), with the timing of G_1,0 ’s “spikes” plotted against the phase of θ_0,0 (bottom). C) The oscillation generated by theta cell θ_{i, j} (top), with the timing of G_1,0 ’s “spikes” plotted against the phase of θ_{i, j} (bottom). D) The interference waveform for grid cell G_4,0 (top), with the timing of G_1,0 ’s “spikes” plotted against the phase of G_4,0 ’s carrier signal (bottom).

Figure 5. Spatial frequency content of Gaussian place fields

Left column shows simulated place cell membrane potentials (y-axis) as a rat runs across the length of a 10 m linear track (x-axis). Right column shows the absolute value of the normalized weight (y-axis) at each spatial frequency (x-axis, logarithmic scale) in the fourier spectra of the Gaussian place fields at left. In the weight vector plots (A–C, right panels), each of the 72 bars represents the weight of one of the 72 grid cells, with filled and empty bars representing positively and negatively weighted inputs, respectively (the DC weight, w_k, is omitted from the plot for scale reasons). A) Membrane potentials and weight vectors of two simulated place cells with large (red) and small (blue) place fields formed by summing input from all 72 grid cells; the location (x-axis) and theta phase (y-axis) of individual place cell spikes are plotted above the membrane potential traces to illustrate phase precession. B) Same as ‘A’ except only 36 grid cells with high spatial frequencies were summed. C) Same as ‘A’ except only 36 grid cells with low spatial frequencies were summed. D) Target Gaussian place fields for the simulations in ‘A–C’ (left) and their corresponding spatial frequency spectra (right).