Models of place and grid cell firing and theta rhythmicity

Abstract

Neuronal firing in the hippocampal formation (HF) of freely moving rodents shows striking examples of spatialorganization in the form of place, directional, boundary vector and grid cells. The firing of place and grid cells shows an intriguing form of temporal organization known as 'theta phase precession'. We review the mechanisms underlying theta phase precession of place cell firing, ranging from membrane potential oscillations to recurrent connectivity, and the relevant intra-cellular and extra-cellular data. We then consider the use of these models to explain the spatial structure of grid cell firing, and review the relevant intra-cellular and extra-cellular data. Finally, we consider the likely interaction between place cells, grid cells and boundary vector cells in estimating self-location as a compromise between path-integration and environmental information.

Figure 1

Theta-phase precession of place cell firing. (a) As a rat runs on a track, a place cell in the hippocampus fires as the animal passes through a specific region (the “place field,” b). This firing rate code for location is also a temporal code (c): spikes (ticks) are fired at successively earlier phases of the theta rhythm of the local field potential (blue trace), referred to as “theta-phase precession.” The theta phase of firing correlates with the distance traveled through the place field (d) better than with other variables, such as time spent in the place field or the place cell’s instantaneous firing rate. Firing phase precesses from late to early phases of theta as the animal runs through the start, middle and end of the place field, irrespective of whether spikes are fired at a low or high rate on that run (e; mean and s.e.m. over n=34 cells). Adapted from [5]. (f) Schematic representation of the data of Harvey et al. (2009) showing local field potential theta (blue) intracellular membrane potential (MPO, black) and spikes (ticks) from a place cell as a mouse runs from left to right through the place field in a virtual arena. Note that the intracellular MPO increases in amplitude and in frequency relative to the LFP theta when the mouse is in the place field, and that the spikes are aligned to the intracellular MPO. The firing rate peaks in the center of the place field, while the membrane potential shows a more asymmetric ramp-like depolarization.

Figure 2

Oscillatory models of phase precession. (a) Dual Oscillator model. The somatic MPO (black) is the sum of two oscillatory inputs having different frequencies in the place field: a somatic input proportional to LFP theta (sin(2πf_bt), blue), and a dendritic input (sin(2πf_at), red) whose frequency (f_a) increases above LFP theta frequency (f_b) in the place field (proportionally to running speed, see Figure 3c). The somatic MPO frequency exceeds LFP theta and its amplitude peaks in the middle of the field. Spikes occur at the peaks of the somatic MPO (ticks; f_b=8Hz, f_a=9Hz in place field). (b) Dual Oscillator model with shunt inhibition. The somatic input can be a shunt (i.e. modulatory) input. Here somatic input ([3+sin(2πf_bt)]/2, blue) multiplies dendritic input (sin(2πf_at), red) to give the somatic MPO (black). (c-e) Somato-Dendritic Interference. The somatic MPO (black) is the sum of two theta-frequency inputs: a somatic input (sin(2πft), blue), and a dendritic input (Dsin(2πft+δ), red) with amplitude D, which increases through the place field, and phase advance δ. The MPO can show significant phase precession, but has a decreasing amplitude in the place field: MPO=Rsin(2πft+φ), where R²=1+D²+2Dcos(δ) and φ=arctan{sin(δ)/[cos(δ)+1/D]}. (c) MPO theta-phase (φ) and amplitude (R) for different values of dendritic phase advance (δ). The dendritic amplitude D increases like 2t as the animal runs through the place field from t=0 to 1.0s, f=8Hz. (d) Example somatic input (blue), dendritic input (red) and MPO (black) for δ=162°. (e) Somato-Dendritic interference with divisive inhibition: dendritic input D[1+sin(2πft)] is divided by somatic input [1.5+sin(2πft)], producing a MPO with low plateaus early in the place field and higher sharper peaks later in the place field. The increasing dendritic amplitude D delays the peak of the flat low early MPOs more than the sharper higher late MPOs, producing phase precession. (f) Depolarizing Ramp model. Somatic input is proportional to LFP theta (blue), dendritic input is a ramp-like depolarization in the place field (red). The resulting somatic MPO (black) crosses a firing threshold (green) successively earlier on each cycle of LFP theta, producing successively earlier phases (and greater numbers) of spikes.

Figure 3

Multiple oscillator model of grid cell firing. (a) Example of the spatial distribution of spikes from a grid cell (red dots) as a rat forages in a 1m² box (path shown in black). Adapted from [11]. (b) Example of theta-phase precession in a grid cell from medial entorhinal cortex (layer II) as a rat runs left-to-right through two of its firing fields on a linear track. Black dots (above) show theta-phase of spikes, red dots (below) show spatial position on the track. Adapted from [12]. (c) A “velocity-controlled oscillator” (VCO) extends the dual oscillator model (see Figure 2a) to two dimensions: the active oscillation has a frequency (f_a) that varies relative to the baseline frequency (f_b) like: f_a(t)=f_b(t)+βs(t)cos(ø(t)-ø_d), where s(t) is running speed, ø(t) is running direction and ø_d is that VCO’s preferred running direction. The dependence of the active frequency on running direction and speed is shown in middle and right panels respectively (the preferred direction is left-to-right). These dependencies cause the phase of the active oscillation relative to baseline to reflect displacement along the preferred direction (see mid top panel of d). In the dual oscillator model: f_a(t)=f_b(t)+βs(t). (d) Multiple VCOs with different preferred directions combine with each other, and the somatic baseline input, to produce grid firing. Grid scale depends on the constant β as: G=2/√3β. Adapted from [25;48], see also [26].

Figure 4

Environmental inputs to place cells, and interactions with grid cells. (a-c) “Boundary vector cells” (BVCs) as the environmental inputs to place cells. (a) An example BVC tuned to respond to a boundary nearby to the East (above), and one tuned to respond at a longer distance (below). (b) A place cell’s firing can be modeled as the thresholded sum of its BVC inputs. Firing rate maps are shown for four environments (small square, circle, square rotated relative to distal cues, large square) for two different BVCs (above). The firing rate maps for the place cell are shown below. (c) A place cell’s firing patterns in several different shaped environments (top left) can be used to infer the BVCs driving its firing (top right). The inferred model can then predict how the place cell will fire in new environmental configurations (bottom right), usually producing a reasonable qualitative fit when the place cell’s firing is recorded in these new configurations (bottom left). Adapted from [16]. (d) Three example boundary vector cells recorded in the subiculum. Each column shows the receptive field (above), and firing rate map in a series of environments (below). Adapted from [17]. (e) Schematic proposed interaction between place cells (PC), BVCs and grid cells (GC). BVCs anchor place fields to the environment and place cells anchor grid cell firing fields to the environment (via phase-resetting of velocity controlled oscillators (VCOs) in the multiple oscillator model). In return, grid cells provide a path integrative input to place cells. Adapted from [24].