An oscillatory interference model of grid cell firing

Abstract

We expand upon our proposal that the oscillatory interference mechanism proposed for the phase precession effect in place cells underlies the grid-like firing pattern of dorsomedial entorhinal grid cells (O'Keefe and Burgess (2005) Hippocampus 15:853-866). The original one-dimensional interference model is generalized to an appropriate two-dimensional mechanism. Specifically, dendritic subunits of layer II medial entorhinal stellate cells provide multiple linear interference patterns along different directions, with their product determining the firing of the cell. Connection of appropriate speed- and direction-dependent inputs onto dendritic subunits could result from an unsupervised learning rule which maximizes postsynaptic firing (e.g. competitive learning). These inputs cause the intrinsic oscillation of subunit membrane potential to increase above theta frequency by an amount proportional to the animal's speed of running in the "preferred" direction. The phase difference between this oscillation and a somatic input at theta-frequency essentially integrates velocity so that the interference of the two oscillations reflects distance traveled in the preferred direction. The overall grid pattern is maintained in environmental location by phase reset of the grid cell by place cells receiving sensory input from the environment, and environmental boundaries in particular. We also outline possible variations on the basic model, including the generation of grid-like firing via the interaction of multiple cells rather than via multiple dendritic subunits. Predictions of the interference model are given for the frequency composition of EEG power spectra and temporal autocorrelograms of grid cell firing as functions of the speed and direction of running and the novelty of the environment.

Figure 1

A: Dual oscillator interference model of phase precession, showing the sum of an oscillatory somatic input (v_s) at 10Hz, and an oscillatory dendritic input at 11.5 Hz (v_d). That is, v_s+ v_d, where v_s=a_scos(w_s t), v_d = a_dcos(w_d t+φ_d), with w_s =10×2π, w_d=11.5×2π, a_s = a_d = 1, φ_d = 0. The sum of the two oscillations is an interference pattern comprising a high frequency “carrier” oscillation (frequency 10.75Hz) modulated by a low frequency “envelope” (frequency 0.75Hz; rectified amplitude varies at 1.5Hz). B: Schematic showing a cell whose firing rate is the rectified sum of both inputs (Θ is the Heaviside function). The top row of A represents the phase at which peaks of the interference pattern occur – i.e. peaks of the overall membrane potential when summing the dendritic and somatic inputs and thus likely times for the firing of an action potential.

Figure 2

Directional interference patterns (see equation 5), showing the positive part of a single directional interference pattern (A, rightward preferred direction), and the product of two (B), three (C) or (6) such patterns oriented at multiples of 60° to each other. i) Pattern generated by straight runs at 30cm/s from the bottom left hand corner to each point in a 78×78cm square. ii) Pattern generated by averaging the values generated at each locations during 10 minutes of a rat's actual trajectory while foraging for randomly scattered food in a 78cm cylinder (white spaces indicate unvisited locations). iii) As ii) but shown with 5cm boxcar smoothing for better comparison with experimental data. All oscillations are set to be in phase (φ_i = 0) at the initial position (i: bottom left corner; ii: start of actual trajectory – indicated by an arrow in Figure 5). The plots show f(x(t)) = Θ(Π_i=1ⁿ cos(w_i t+φ_i) + cos(w_s t)), for n=1 (A), 2 (B), 3 (C) and 4 (D), with w_i = w_s + βscos(ø - ø_i), where s is running speed, spatial scaling factor β = 0.05×2π rad/cm (i.e., 0.05 cycles/cm), preferred directions: ø₁ = 0° (i.e. rightwards), ø₂ = 60°, ø₃ = 120°, ø₄ = 180°, ø₅ = 240°, ø₆ = 310°. Θ is the Heaviside function. All plots are auto-scaled so that red is the maximum value and blue is zero.

Figure 3

Simulation of 'grid cell' firing as the product of three linear interference patterns with different combinations of preferred directions (see equation 5). Simulated cells with regular grid firing patterns achieve high firing rates more often than those with irregular patterns, so long as collinear patterns are excluded. A) The most frequently high-firing grid cell (firing at 90% of maximum firing rate, 8Hz, 1310 times in the 28125 locations sampled by a rat in 10mins, same trajectory as Figure 2). B) The median frequency high-firing cell is shown in the middle (433 times). C) The least often high-firing cell on the right (228 times). Cell firing rate was simulated as the product of the firing envelopes of the three inputs to each cell, to facilitate speed and reliability. All unique combinations of preferred directions (ø₁, ø₂, ø₃) selected from (0°, 10°, .. ,350°) such that all angles differ by at least 20° were simulated. These firing rate maps are shown with 5cm boxcar smoothing for better comparison with experimental data.

Figure 4

Schematic of the interference model of grid cell firing. Left: Grid cell (pale blue) receiving input from three dendritic subunits (red, blue, green). Middle: Spikes, Somatic theta input (black) and dendritic membrane potential oscillations (MPOs, red, blue, green). The rat is shown (upper right) running perpendicular to the preferred direction of one of the subunits (blue), so the MPO of this subunit oscillates at theta frequency. Since the rat is running approximately along the preferred directions of the other two subunits (red and green, i.e. within 30° of their preferred directions), these MPOs oscillate above theta frequency. Spikes are shown at the times of the peaks of the product of the 3 interference patterns each MPO makes with the somatic theta input. The locations of the peaks of the envelopes of the 3 interference patterns are shown on the environment in the corresponding colors (red, green and blue stripes, lower right). The locations of grid cell firing are shown in pale blue and occur in the environment wherever the 3 envelopes all peak together (upper right).

Figure 5

Schematic of the association of grid to environment via phase reset of grid cells by place cells. Left: Diagram showing anatomical connection from mEC grid cell (pale blue) with three dendritec subunits (green, blue, red) to hippocampal place cell (gold) and feedback from place cell onto the dendrites of the grid cell. Center: the maximal firing of the place cell occurs in phase with theta (above, dashed line), and resets dendritic membrane potentials to be in phase with theta (below), Right: the path of the rat in the open field and the place cell's firing field (gold, above). In a familiar environment connections from the place cell to the grid cell are developed due to their coincident firing fields (right: above and middle). These connections enable maximal firing of the place cell to reset the phases of the grid cell's dendritic membrane potential oscillations (MPOs) to be in phase with theta – forcing the grid to stay locked to the place field at that location, by ensuring that the envelopes of the three dendritic linear interference patterns coincide at the location of the place field (below right). Sensory input from the environment (especially boundaries), via lateral entorhinal cortex (lEC), keeps the place field locked to the environment. For convenience, only one place cell and one grid cell are shown – in practice we would expect multiple grid cells (with firing at the place field) to project to the place cell, and multiple place cells (with coincident place fields) to project to the grid cell. See Figure 4 for details of the grid cell model.

Figure 6

Effect of error in speed s and heading ø on simulated grid cell firing, and correction by phase resetting (same trajectory and simulated grid cell as Figure 2Cii). A) No error. B) Error in the estimate of current direction (addition of random variable from Normal distribution N(0, 10°)) and distance traveled (multiplication by 1+δ, where δ is drawn from N(0, 0.1)) for each time step (1/48s). C) Similar error to B, but with phases reset to φ₁ = φ₂ = φ₃ = 0 whenever the animal visits a single location within 78cm cylinder (i.e. within 2cm of arrow: reset 84 times in 10 mins). These firing rate maps are shown with 5cm boxcar smoothing for better comparison with experimental data.