Fig 1.
The temporal adaptation kernel K.
(A) Impulse response of the filter (Eq 2). A positive peak with amplitude K(0) = 1/τS − μ/τL ≈ 3.4 spikes/s is followed by a slow negative response. Note that the kernel is small for t > τmax, i.e., |K(t)| < 0.01|K(0)| for t > τmax, with τmax = 5τL = 0.8 s. (B) Frequency response of the filter. The dashed vertical line indicates the filter’s resonance frequency kres = 1.23 s−1. Parameter values: τS = 0.1 s, τL = 0.16 s, μ = 1.06. The integral of the filter is 1 − μ = −0.06.
Fig 2.
Example virtual-rat trajectories.
Colored lines denote example virtual-rat trajectories obtained by integrating Eq 11 starting at the center of the gray disk. Filled dots indicate the position of the virtual rat at time τmax = 5τL = 0.8 s. Note that the trajectories are smooth within time stretches shorter than τmax. Parameter values: v = 0.25 m/s, θσ = 0.7. The disk radius is vτmax = 20 cm.
Fig 3.
Input correlation function C for spatially-regular inputs.
The function is circularly symmetric, i.e., it depends only on the distance |r − r′| between the receptive-field centers r and r′ (Eq 54). In the attraction domain (red shaded area) the correlation is positive and the synaptic weights grow in the same direction. In the repulsion domain (blue shaded area) the correlation is negative and the synaptic weights grow in opposite directions. Parameter values: σ = 6.25 cm, rav = 0.4 s−1, τS = 0.1 s, τL = 0.16 s, μ = 1.06, Wtot = 1 s, L = 1 m, v = 0.25 m/s.
Fig 4.
Impact of the adaptation kernel on grid-pattern formation.
(A1-A2) Critical spatial frequency kmax (A1) and largest eigenvalue λmax (A2) as a function of the kernel integral 1 − μ and the long kernel time constant τL. The short time constant is τS = 0.1 s. The black lines are iso-levels (see annotated values). Regions enclosed by two adjacent iso-lines are colored uniformly (darker colors denote larger values). Within the black region in A1 we obtain λmax ≤ 0 s−1 (see white region in A2). Within the black region in A2 we obtain kmax = 0 m−1 (see white region in A1). The dashed horizontal line indicates zero-integral kernels. The star denotes the parameter values τS = 0.1 s, τL = 0.16 s, μ = 1.06 of the kernel in Fig 1. (B1-B2) Same as in A but varying the short kernel time constant τS. The long time constant is τL = 0.16 s. The eigenvalue spectrum is estimated from Eq 32. Further parameter values: σ = 6.25 cm, rav = 0.4 s−1, Wtot = 1 s, ρ = 900 m−2, L = 1 m, v = 0.25 m/s, a = 1.1 s−1.
Fig 5.
Grid-pattern formation with spatially-regular inputs.
(A) Eigenvalue spectrum λ(k) of the averaged weight dynamics (Eq 32). The black solid line shows the continuous spectrum in the limit of infinite-size environments; the red dots show the discrete eigenvalues for a square arena of side length L = 1 m with periodic boundaries. The horizontal dashed line separates positive and negative eigenvalues. The vertical gray line indicates the critical spatial frequency kmax = 3 m−1. The eigenvalue at frequency k = 0 is not shown. Parameter values: τS = 0.1 s, τL = 0.16 s, σ = 6.25 cm. (B) Time-resolved distribution of N = 900 synaptic weights updated according to the STDP rule in Eqs 3–6. Red triangles indicate the time points shown in C. Inset: fraction of weights close to the lower saturation bound (wi < 5 ⋅ 10−3). (C) Top row: evolution of the synaptic weights over time. Weights are sorted according to the two-dimensional position of the corresponding input receptive-field centers. Note that each panel has a different color scale (maximum weight at the bottom-left corner, see B for distributions). Bottom row: Fourier amplitude of the synaptic weights at the top row. The red circle indicates the frequency kmax = 3 m−1 of the largest eigenvalue (see panel A). (D) Time evolution of weights' Fourier amplitudes for wave vectors k at the critical frequency |k| = kmax. Wave vector angles (color coded) are relative to the largest mode at the end of the simulation (t = 106 s). The black triangles indicate time points in C. (E) Gridness score of the weight pattern over time. The gridness score quantifies the degree of triangular periodicity. See Sec Numerical simulations for further details and parameter values.
Fig 6.
(A) Median gridness scores of the input synaptic weights for 40 random weight initializations and different learning-rate values, i.e., η = (2, 3, 5, 10) ⋅ 10−5. The weight development is simulated with the detailed spiking model with spatially-regular inputs and constant virtual-rat speed (see also Fig 5). (B) Median gridness scores of the input synaptic weights simulated with constant (black line) and variable (green line) virtual-rat speeds for 40 random weight initializations. Variable running speeds are obtained by sampling from an Ornstein-Uhlenbeck process with long-term mean m/s, volatility σv = 0.1 m ⋅ s−1.5 and mean-reversion speed θv = 10 s−1. The inset shows the distribution of running speeds (mean: 0.25 m/s std: 0.02 m/s). Note that the long-term mean
of the process equals the speed v in constant-speed simulations. See Sec Numerical simulations for further details and additional parameter values.
Fig 7.
Spatial scale of the grid patterns.
Example grid patterns obtained with different adaptation kernels K (Eq 2, top row) and different input tuning curves (Eq 9, left-most column). For each choice of the functions K and
, the synaptic weights (left) and their corresponding Fourier spectra (right) at the end of the simulation are shown (t = 106 s). The synaptic-weight maps have different color scales (maximal values at the bottom-left corner). The red circles indicate the spatial frequency kmax of the weight patterns. Synaptic weights were obtained by simulating the average weight dynamics in Eq 16. Note that we used a larger enclosure (L = 2 m) as compared to the one in Figs 5 and 6 (L = 1 m). See Sec Numerical simulations for further details and parameter values.
Fig 8.
Geometric properties of the grid patterns.
(A) Distribution of grid spatial phases (A1) and grid orientations (A2) for patterns at frequency kmax = 3 m−1 in an arena of side-length L = 2 m (σ = 6.25 cm, τL = 0.16 s; see also Fig 7, bottom-left panel). Distributions were obtained from the average weight dynamics in Eq 16 for 200 random initializations of the synaptic weights (t = 106 s). Only patterns with gridness scores larger than 0.5 were considered (197/200). Panel A3 shows example weight patterns for the two most common orientations in A2 (maximal values at the bottom-left corner). (B) Same as in A but for patterns at spatial frequency kmax = 2 m−1 in an arena of side-length L = 2 m (σ = 6.25 cm, τL = 0.35 s; see also Fig 7, bottom-right panel). A fraction of 182/200 grids had a gridness score larger than 0.5. See Sec Numerical simulations for further details and parameter values.
Fig 9.
Grid-pattern formation with spatially-irregular inputs.
(A) Four examples of irregular input firing-rate maps (top row) and the corresponding Fourier spectra (bottom row). The maximal firing rate (spikes/s) is reported at the bottom-left corner. The red circles indicate the spatial frequency kmax = 3 m−1. (B) Four examples of output firing-rate maps (top row) and the corresponding Fourier spectra (bottom row). The gridness score is reported at the bottom-right corner. Output firing-rate maps were estimated from the average weight dynamics in Eq 16 (t = 106 s) for four different realizations of the spatial inputs. (C-F) Distribution of gridness scores (C), grid spatial frequencies (D), grid spatial phases (E), and grid orientations (F) for 100 random realizations of the spatial inputs. The red vertical line in C indicates the mean score (0.77). See Sec Numerical simulations for further details and parameter values.
Fig 10.
Scale factor Φ and largest eigenvalue λmax for spatially-irregular inputs.
(A) The scale factor Φ for M > 1 superimposed fields (Eq 70). The black dots are obtained by estimating the power spectrum at frequency |k| = 1 m−1 for 3600 input realizations. The red line is the theoretical curve in Eq 82. (B) The largest eigenvalue λmax as a function of the number of superimposed fields M. The black dots are obtained by computing the eigenvalues of the correlation matrix Cij − aδij for N = 3600 inputs, where δij is the Kronecker delta (Eq 21). The red line is obtained from Eqs 71 and 82. Note that, according to Eq 71, the largest eigenvalue is always at the critical frequency kmax = 3 m−1 for any value of M. Parameter values as in Fig 9 (see Sec Numerical simulations).
Fig 11.
Grid scale with after-spike hyperpolarizing potentials.
The critical spatial frequency kmax is plotted as a function of the output-kernel integral −μout and the output-kernel time constant τout (Eqs 31 and 32 with K = Keq). The black lines are iso-levels (see annotated values). Regions enclosed by two adjacent iso-lines are colored uniformly (darker colors denote larger values). The input-kernel time constant is τin = 5 ms. Similar results are obtained with different values of τin < τout. Parameter values: σ = 6.25 cm, v = 0.25 m/s, L = 1 m. rav = 0.4 s−1.
Table 1.
Model parameters.
Table 2.
Default parameter values for the numerical simulations.
See also Table 1 for short descriptions of the parameters. TL: top-left, TR: top-right, BL: bottom-left, BR: bottom-right. Note that in Fig 6A the learning rate η is varied from 2 ⋅ 10−5 to 10 ⋅ 10−5 and that in Fig 6B the virtual-rat running speed is sampled from an Ornstein-Uhlenbeck process with long-term mean .