Input correlation function <i>C</i> for spatially-regular inputs.

<p>The function is circularly symmetric, i.e., it depends only on the distance |<b>r</b> − <b>r</b>′| between the receptive-field centers <b>r</b> and <b>r</b>′ (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e118" target="_blank">Eq 54</a>). 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: <i>σ</i> = 6.25 cm, <i>r</i>_av = 0.4 s⁻¹, <i>τ</i>_S = 0.1 s, <i>τ</i>_L = 0.16 s, <i>μ</i> = 1.06, <i>W</i>_tot = 1 s, <i>L</i> = 1 m, <i>v</i> = 0.25 m/s.</p

Grid-pattern formation with spatially-regular inputs.

<p>(A) Eigenvalue spectrum λ(<i>k</i>) of the averaged weight dynamics (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e070" target="_blank">Eq 32</a>). 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 <i>L</i> = 1 m with periodic boundaries. The horizontal dashed line separates positive and negative eigenvalues. The vertical gray line indicates the critical spatial frequency <i>k</i>_max = 3 m⁻¹. The eigenvalue at frequency <i>k</i> = 0 is not shown. Parameter values: <i>τ</i>_S = 0.1 s, <i>τ</i>_L = 0.16 s, <i>σ</i> = 6.25 cm. (B) Time-resolved distribution of <i>N</i> = 900 synaptic weights updated according to the STDP rule in Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e009" target="_blank">3</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e014" target="_blank">6</a>. Red triangles indicate the time points shown in C. Inset: fraction of weights close to the lower saturation bound (<i>w_i</i> < 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 <i>k</i>_max = 3 m⁻¹ of the largest eigenvalue (see panel A). (D) Time evolution of weights' Fourier amplitudes for wave vectors <b>k</b> at the critical frequency |<b>k</b>| = <i>k</i>_max. Wave vector angles (color coded) are relative to the largest mode at the end of the simulation (<i>t</i> = 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.</p

Geometric properties of the grid patterns.

<p>(A) Distribution of grid spatial phases (A1) and grid orientations (A2) for patterns at frequency <i>k</i>_max = 3 m⁻¹ in an arena of side-length <i>L</i> = 2 m (<i>σ</i> = 6.25 cm, <i>τ</i>_L = 0.16 s; see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g007" target="_blank">Fig 7</a>, bottom-left panel). Distributions were obtained from the average weight dynamics in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e036" target="_blank">Eq 16</a> for 200 random initializations of the synaptic weights (<i>t</i> = 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 <i>k</i>_max = 2 m⁻¹ in an arena of side-length <i>L</i> = 2 m (<i>σ</i> = 6.25 cm, <i>τ</i>_L = 0.35 s; see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g007" target="_blank">Fig 7</a>, 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.</p

Spatial scale of the grid patterns.

<p>Example grid patterns obtained with different adaptation kernels <i>K</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e007" target="_blank">Eq 2</a>, top row) and different input tuning curves (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e021" target="_blank">Eq 9</a>, left-most column). For each choice of the functions <i>K</i> and , the synaptic weights (left) and their corresponding Fourier spectra (right) at the end of the simulation are shown (<i>t</i> = 10⁶ s). The synaptic-weight maps have different color scales (maximal values at the bottom-left corner). The red circles indicate the spatial frequency <i>k</i>_max of the weight patterns. Synaptic weights were obtained by simulating the average weight dynamics in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e036" target="_blank">Eq 16</a>. Note that we used a larger enclosure (<i>L</i> = 2 m) as compared to the one in Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g005" target="_blank">5</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g006" target="_blank">6</a> (<i>L</i> = 1 m). See Sec Numerical simulations for further details and parameter values.</p

Grid scale with after-spike hyperpolarizing potentials.

<p>The critical spatial frequency <i>k</i>_max is plotted as a function of the output-kernel integral −<i>μ</i>_out and the output-kernel time constant <i>τ</i>_out (Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e069" target="_blank">31</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e070" target="_blank">32</a> with <i>K</i> = <i>K</i>^eq). 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 <i>τ</i>_in = 5 ms. Similar results are obtained with different values of <i>τ</i>_in < <i>τ</i>_out. Parameter values: <i>σ</i> = 6.25 cm, <i>v</i> = 0.25 m/s, <i>L</i> = 1 m. <i>r</i>_av = 0.4 s⁻¹.</p

Scale factor Φ and largest eigenvalue λ_max for spatially-irregular inputs.

<p>(A) The scale factor Φ for <i>M</i> > 1 superimposed fields (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e152" target="_blank">Eq 70</a>). The black dots are obtained by estimating the power spectrum at frequency |<b>k</b>| = 1 m⁻¹ for 3600 input realizations. The red line is the theoretical curve in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e169" target="_blank">Eq 82</a>. (B) The largest eigenvalue λ_max as a function of the number of superimposed fields <i>M</i>. The black dots are obtained by computing the eigenvalues of the correlation matrix <i>C_ij</i> − <i>aδ_ij</i> for <i>N</i> = 3600 inputs, where <i>δ_ij</i> is the Kronecker delta (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e048" target="_blank">Eq 21</a>). The red line is obtained from Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e156" target="_blank">71</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e169" target="_blank">82</a>. Note that, according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e156" target="_blank">Eq 71</a>, the largest eigenvalue is always at the critical frequency <i>k</i>_max = 3 m⁻¹ for any value of <i>M</i>. Parameter values as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g009" target="_blank">Fig 9</a> (see Sec Numerical simulations).</p

Grid-pattern formation with spatially-irregular inputs.

<p>(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 <i>k</i>_max = 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e036" target="_blank">Eq 16</a> (<i>t</i> = 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.</p

Model parameters.

<p>Model parameters.</p

Impact of the adaptation kernel on grid-pattern formation.

<p>(A1-A2) Critical spatial frequency <i>k</i>_max (A1) and largest eigenvalue λ_max (A2) as a function of the kernel integral 1 − <i>μ</i> and the long kernel time constant <i>τ</i>_L. The short time constant is <i>τ</i>_S = 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 λ_max ≤ 0 s⁻¹ (see white region in A2). Within the black region in A2 we obtain <i>k</i>_max = 0 m⁻¹ (see white region in A1). The dashed horizontal line indicates zero-integral kernels. The star denotes the parameter values <i>τ</i>_S = 0.1 s, <i>τ</i>_L = 0.16 s, <i>μ</i> = 1.06 of the kernel in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g001" target="_blank">Fig 1</a>. (B1-B2) Same as in A but varying the short kernel time constant <i>τ</i>_S. The long time constant is <i>τ</i>_L = 0.16 s. The eigenvalue spectrum is estimated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.e070" target="_blank">Eq 32</a>. Further parameter values: <i>σ</i> = 6.25 cm, <i>r</i>_av = 0.4 s⁻¹, <i>W</i>_tot = 1 s, <i>ρ</i> = 900 m⁻², <i>L</i> = 1 m, <i>v</i> = 0.25 m/s, <i>a</i> = 1.1 s⁻¹.</p

Time scales of learning.

<p>(A) Median gridness scores of the input synaptic weights for 40 random weight initializations and different learning-rate values, i.e., <i>η</i> = (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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005782#pcbi.1005782.g005" target="_blank">Fig 5</a>). (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 <i>σ</i>_<i>v</i> = 0.1 m ⋅ s^−1.5 and mean-reversion speed <i>θ</i>_<i>v</i> = 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 <i>v</i> in constant-speed simulations. See Sec Numerical simulations for further details and additional parameter values.</p