Fig. 4.

Spatial resolution, as quantified by the least mean squared error (LMSE) between the animal's position and the position decoded from the stochastic spikes of place cells with idealized Gaussian place fields (Eq. 3). A: the spatial resolution grows linearly with the area of the represented region if each place cell has exactly one place field, whether the place field size is constant (Standard Rep.) or grows with the environmental size (Scaled Rep.). In contrast, the LMSE is constant for any area given the Poisson distribution of place fields used for the megamap (Flexible Rep.). The apparent advantage of the single-peaked codes over the flexible representation when A < 1/λ is an artifact, since many cells in the flexible representation are silent in these small regions. The maximum likelihood estimates (MLEs; Eq. 5) approximate the LMSE (black; Eq. 4), as expected since the MLEs are unbiased estimators. (The mean error along either dimension is negligible: −0.0080 ± 0.0076 cm for the single-peaked place code and 0.0006 ± 0.0063 cm for the megamap.) B: the standard deviation of the Gaussian place fields (Eq. 3) grows with the environmental area only for the scaled representation. C: the number of co-active place cells at any single position is constant for the scaled and flexible representations but decreases for the standard representation, since the number of cells (N) is constant. For all simulations presented, N = 22,500, T = 250 ms, λ = −ln(0.8) m⁻², and a = 15 Hz (see materials and methods for more details).