(A) Pattern formation in the neural population: Left, schematic depiction of the outgoing weights of a neuron in the network. All neurons have the same connectivity pattern, and the width of the inhibitory surround is parameterized in our model by (see Methods). Center, circularly symmetric center-surround connectivity, with sufficiently strong local inhibitory flanks, produces a regular triangular lattice population pattern in the neural sheet through spontaneous destabilization of the uniform mode (Turing instability). Right, the pattern period depends on the width of the inhibitory surround. (B) The velocity shift mechanism by which velocity inputs drive pattern flow: Each neuron in the sheet is assigned a preferred angle (color coded), which means two things. First, the outgoing weight profile, instead of being centered exactly on the originating neuron, is shifted by a small amount along the preferred angle in the neural sheet (left). Each patch in the neural sheet contains neurons with all preferred angles. Second, the direction preference means that the neuron receives input from head direction cells tuned to the corresponding angle (center and right). (C) Snapshots of the population activity, when the networks (periodic boundaries, above; aperiodic boundaries, below) are driven by a constant velocity input in the rightward direction. In the periodic network, as the pattern flows, it wraps around the opposite edge. In the aperiodic network, as the pattern flows, blobs move away from the left edge and new ones spontaneously form through the same dynamics that govern pattern formation. (Boundaries are considered in more detail in the paper and in later figures.) The green lines represent an electrode at a fixed location in the neural sheet, and the circle above them represents the activity state of the targeted neuron (gray = inactive, yellow = active). Network parameters are as in Figure 2A–C and Figure 2D–F.

Simulation of network response, with velocity inputs corresponding to a rat's recorded trajectory in a 2 m circular enclosure [50]. The boundary conditions in the neural sheet are periodic in (A–C) and aperiodic in (D–F). (A,D) Instantaneous activity within the neural sheet (color represents the firing rate: black corresponds to vanishing rate). The red curve in (D) represents the fading profile of inputs to the network. (B,E) Grid cell response: average firing rate of a single neuron (located at the electrode tip in (A,D)), as a function of the rat's position within the enclosure. (C,F) Velocity integration in the network: Top: Actual distance of the rat from a fixed reference point (black), compared to the network's integrated position estimate, obtained by tracking the flow of the pattern in the population response (blue). The reference point is at the left-bottom corner of the square in which the circular enclosure is inscribed. Middle: Accumulated distance between the integrated position estimate and the actual position. Bottom: Orientation of the three main axes in the population response during the trajectory. Note that there is no rotation in the periodic network, and little rotation in the aperiodic one.

(A–C) Same simulation as in Figure 2D–F, but with a sharper input profile (red curve above B). The SN pattern has no periodicity (A), the integration error is large (thick line in (C), upper plot; note the different scale compared to Figure 2E, whose error is represented by the thin line), and the population response rotates frequently ((C), lower plot). (D1–D3) Network velocity response as a function of different input profiles: Input profile decay is least abrupt in (D1), more abrupt in (D2), and most abrupt in (D3) ( for (D1), (D2), and (D3), respectively; network size is 128 neurons per side() for all). (D4) The input profile at the boundaries is identical to D2 (), but the network is larger (256 neurons per side or ). (D2) corresponds to the parameters in (A–C), and (D1) corresponds to the parameters in Figure 2D–F.

(A) Periodic network manifold: Points within the trough represent stable states of the network that will persist in the absence of perturbing inputs. If the network is placed at a state outside the trough, it will rapidly decay to a state within the trough. Points in the trough consist of continuous translations of the population-level pattern. Rotations, stretches, or other local or global deformations of the pattern lie outside the trough. Rat velocity inputs drive transitions between points in the trough (red arrow). (B) Aperiodic network manifold: all rotations of a stable population pattern are energetically equivalent, and so form a continuous attractor manifold. Translations are not equivalent (rippled energy functional). Rat velocity inputs, when large enough to overcome the ripple, drive translations of the population pattern; however, the flat rotational mode means that the network can also rotate.

(A) SN response in a stochastic spiking periodic network. The parameters and input velocity trajectory are as in Figure 2A–C, except that spiking is simulated explicitly and the spikes are generated by an inhomogeneous Poisson process. (B) SN response in a stochastic spiking aperiodic network. The parameters are as in Figure 2D–F, except that spiking is simulated explicitly and the spikes are generated by a point process with a CV of (see Methods). Each red dot represents a spike.

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 CV², 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 neurons²/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 CV². The diffusion constant was measured from simulations lasting 20 minutes.

Green lines represent the same fixed electrode locations in the neural population, across all plots. (A) Left: Single-neuron response. Right: Input head direction/velocity tuning curves, and an instantaneous snapshot of the underlying population response, which together produced the SN response on the left. (B) The SN grid (left) expands along one direction when the amplitude of the head direction/velocity inputs for that direction is lowered relative to other directions (right, first panel), while the population patterns remain unchanged. Alternatively, the same SN expansion could have been produced by keeping the amplitude of the head direction/velocity inputs fixed, if the population patterns were stretched (right, second panel). The latter scenario is inconsistent with the attractor hypothesis, because deformations of the pattern are not part of the attractor manifold. In the former (continuous attractor) scenario, the phase relationships between neurons is preserved despite the SN expansion; in the second, phase relationships must change. (C) The SN grid (left) rotates if the head direction/velocity inputs to the network are rotated, while the population remains unchanged. The same rotation could have been produced by rotating the population pattern, but keeping the head direction/velocity inputs intact. The latter possibility is inconsistent with the attractor hypothesis. Again, the former (continuous attractor) scenario can be distinguished from the latter by whether phase relationships between neurons in the population are preserved. (SN plots and the left column of population responses were produced from a simulation with network parameters as in Figure 2D–F, and by appropriately scaling or rotating the velocity/head direction inputs. Right population plots are hypothetical.)