Orange (blue) curves are the results of simulations in (a)periodic networks. Successively darker shades (of orange or blue) represent simulations with successively higher neural variability (

,

,

, and 1, respectively). Identical colors across panels represent simulations with identical network parameters. Velocity inputs are zero everywhere, and network size is

, except where stated otherwise. (A) Phase drift and (B) angular drift of the periodic (orange, CV = 1) and aperiodic (blue,

) networks. In (A), the drift in cm corresponds to a measured drift in neurons by assuming the same gain factor as in the simulations with a trajectory, as in Figure 5. (C) The summed square 2-d drift in position estimation as a function of elapsed time, for two different values of CV, in the absence of velocity inputs. The squared drift (small open circles) can be fit to straight lines (dashed) over 25 seconds (for longer times the traces deviate from the linear fit due to the finite time of the simulation), indicating that the process is diffusive. The slope of the line yields the diffusion constant

for phase (translational) drift of the population pattern, in units of neurons

^{2}/s. The same fitting procedure applied to the squared angular drift as a function of time yields the angular diffusion constant

. (D) Diffusion constants measured as in (C), for networks of varying size and CV. The diffusion constant is approximately linear in CV

^{2}, and in the number of neurons

. To demonstrate the linearity in

, the plots show

multiplied by

, upon which the data for

and

approximately collapse onto a single curve. (E) An estimate of the time over which a periodic spiking network (with the same parameters as the corresponding points in (C) and (D)) can maintain a coherent grid cell response, plotted as a function of N, for two values of neural stochasticity. The estimate is based on taking the diffusion relationship

, and solving for the time when the average displacement

is 10 pixels, about half the population period, and estimating the diffusion constants from (D) to be

*ND*≃2500 neurons

^{2}/s. The coherence time scales like

, where

is the period of the population pattern. (F) Rotational diffusivity,

, in an aperiodic network of size 128×128 also increases linearly with CV

^{2}. The diffusion constant was measured from simulations lasting 20 minutes.

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