Computation by oscillations: implications of experimental data for theoretical models of grid cells

Abstract

Recordings in awake, behaving animals demonstrate that cells in medial entorhinal cortex (mEC) show "grid cell" firing activity when a rat explores an open environment. Intracellular recording in slices from different positions along the dorsal to ventral axis show differences in intrinsic properties such as subthreshold membrane potential oscillations (MPO), resonant frequency, and the presence of the hyperpolarization-activated cation current (h-current). The differences in intrinsic properties correlate with differences in grid cell spatial scale along the dorsal-ventral axis of mEC. Two sets of computational models have been proposed to explain the grid cell firing phenomena: oscillatory interference models and attractor-dynamic models. Both types of computational models are briefly reviewed, and cellular experimental evidence is interpreted and presented in the context of both models. The oscillatory interference model has variations that include an additive model and a multiplicative model. Experimental data on the voltage-dependence of oscillations presented here support the additive model. The additive model also simulates data from ventral neurons showing large spacing between grid firing fields within the limits of observed MPO frequencies. The interactions of h-current with synaptic modification suggest that the difference in intrinsic properties could also contribute to differences in grid cell properties due to attractor dynamics along the dorsal to ventral axis of mEC. Mechanisms of oscillatory interference and attractor dynamics may make complementary contributions to the properties of grid cell firing in entorhinal cortex.

Figure 1

Frequency of subthreshold oscillations across a population of 137 cells in medial entorhinal cortex. A. The mean subthreshold oscillation frequency in dorsal cells shows a significantly larger increase in frequency with voltage compared to a smaller slope of increase in ventral cells. B-C. The frequency of cells decreases systematically from dorsal to ventral at approximate membrane potentials of −50 mV (B) and −45 mV (C).

Figure 2

Oscillation frequency decreases systematically from dorsal to ventral. A-B. The average subthreshold oscillation frequency is plotted for each anatomical bin at a membrane potential of approximately −50 mV (A), −45 mV (B) and −55 mV (B). C. Simulations of the oscillatory model using the average frequency from each anatomical bin shown in part A. The field size and spacing of the grid decreases as the frequency of the subthreshold membrane oscillation decreases. Left: Trajectory plotted in gray and spikes in black. Right: Smoothed plot of the firing rate data shown on the left.

Figure 3

Effect of steady current injection on spiking frequency of stellate cells. A. Firing frequency of a single stellate cell at successively higher current amplitude steps 2.5 seconds long in duration. B. Frequency of spiking (spikes per second) relative to the normalized injected current amplitude for 15 stellate cells. C. Frequency of spiking relative to the normalized injected current shown for 8 individual cells along the dorsal-ventral axis. Legend lists the distance of each cell from the dorsal surface (mm). D. Example of a single stellate cell's spiking frequency relative to normalized injected current.

Figure 4

Changes in the voltage-dependence of oscillations corresponds to changing the B parameter. A. Simulation of the oscillatory model with a virtual rat running on an 18 m linear track. The B value was set to .004. B. Simulations of the oscillatory model on an 18 m linear track with the B value set to 0.0015, resulting in larger field size and spacing of the grid firing fields. C. Variations in the B parameter result in different field size and spacing of grid firing fields. Large B values result in smaller field size and spacing while smaller B values result in larger field size and spacing. D-E. Frequency changes with voltage associated with the additive model (D) and multiplicative model (E).

Figure 5

Variance of oscillation period along the dorsal to ventral axis. . A. Histograms of the distribution of oscillation period in four dorsal (left) and four ventral (right) cells. B. The oscillation period increases from dorsal to ventral mEC. C. The standard deviation of the oscillation period increases from dorsal to ventral mEC. D. The range of the oscillation period increases from dorsal to ventral mEC. E. Example of the Weber law. The standard deviation of the oscillation period is linearly related to the average interval length.

Figure 6

A. Effects of noise in the oscillatory model. As the noise levels in the model are increased, the firing of the simulated grid cell becomes less consistent and loses its grid pattern of spatial periodicity. B. Layer V cells show higher frequency membrane oscillations in dorsal compared to ventral portions of medial entorhinal cortex at a membrane potential of −45 mV and −40 mV. C. Qualitative differences between m-current (layer V) and h-current (layer II) dependent subthreshold oscillations. Two different layer V cells (left; top 4.2 mm from dorsal surface, bottom 4.7 mm from the dorsal surface) show higher amplitude oscillations compared to two different layer II cells (right; top 4.4 mm from dorsal surface, bottom 4.6 mm from dorsal surface).