Interactive Demo | Shaoyang Cui

Grid Cell Demo 03: From Grid Cells to fMRI Hexadirectional Signal

Thu, 14 May 2026 00:00:00 +0000

This demo now focuses only on voxel-level fMRI outputs and decoding.

Core intuition:

Within one voxel, many cells with different spatial phases can reduce position-locked map contrast after averaging.
But if local cells share a similar grid orientation $\phi$, the six-fold directional term can survive at population level.
After HRF convolution, we can still decode hexadirectional structure from BOLD.

Decoding protocol in this page:

Use first half of time points as training data.
Fit $\cos(6\theta)$ and $\sin(6\theta)$ GLM terms to estimate population $\hat\phi$.
Use second half as test data.
Build test regressor $\cos(6(\theta-\hat\phi))$ and estimate test $\beta_{\text{hex}}$.
Compare recovered $\hat\phi$ against ground-truth $\phi$ (modulo 60 degrees).
Visualize aligned vs misaligned bins and directional preference on a 360-degree polar plot.

You can customize:

voxel cell count
initial phase distribution
HRF kernel
signal/noise settings
trajectory length

and inspect how recovery quality changes.

Voxel-level fMRI simulation + decoding. We generate synthetic Right EC BOLD with a forward HRF, then decode hexadirectional structure with a GLM that uses a potentially different decode HRF on regressors.

cells per voxel 220 phase distribution grid orientation phi (ground truth) 18 deg noise level 0.18 trajectory samples 800 runs (CV) 4 forward HRF (generation) decode HRF (GLM design)

Forward HRF Kernel

Decode HRF Kernel

phi true / phi estimated0.0 deg / 0.0 deg

phi error (mod 60)0.00 deg

beta_hex train0.000

beta_hex test0.000

aligned - misaligned (test)0.000

cv setupleave-one-run-out

BOLD and Decoded Predictor

Aligned vs Misaligned Bins (Test)

Cell Orientation: Ground Truth vs Decoded

Folded 60 deg Profile

Grid Cell Demo 02: Three-Wave Interference and Spatial Autocorrelogram

Wed, 13 May 2026 00:00:00 +0000

This page strips away the rat trajectory and keeps only the theoretical three-wave construction.

The question here is:

if three direction-tuned waves interfere in space, what firing map does that imply?
once that firing map is built, what does its spatial autocorrelogram look like?

The demo below now exposes a broader family of deformations. You can adjust:

beta and the three k_i scales for anisotropic wavelength changes
theta_1, theta_2, theta_3 for angle distortions
A_1, A_2, A_3 for amplitude imbalance
stretch x, stretch y, and shear for global affine deformation
coord warp and phase warp for position-dependent distortion
amp modulation for spatially varying amplitude bias
baseline and threshold for excitability / threshold effects

The left panel shows the theoretical firing pattern itself. The right panel shows the spatial autocorrelogram computed from that map. I also display a simple gridness estimate, so the demo can be used not only qualitatively but also as a first quantitative probe of how each deformation changes hexagonal order.

This demo turns the hypotheses in the note into explicit parameters. You can deform wave lengths, wave angles, amplitudes, affine geometry, nonlinear phase warp, and thresholding, then inspect both the firing map and its spatial autocorrelogram.

beta base 15.5 k1 scale 1.00 k2 scale 1.00 k3 scale 1.00 theta_1 0 deg theta_2 60 deg theta_3 120 deg A1 1.00 A2 1.00 A3 1.00 stretch x 1.00 stretch y 1.00 shear 0.00 coord warp 0.00 phase warp 0.00 amp modulation 0.00 baseline b 0.00 threshold 0.80

Gridness 0.000

Ring Radius 0

R60 / R30 0.000 / 0.000

Theoretical Firing Pattern

The firing map after three-wave interference, amplitude modulation, affine distortion, coordinate warp, phase warp, baseline shift, and thresholding.

Spatial Autocorrelogram

The normalized spatial autocorrelogram. Gridness is computed from rotational correlations on the first ring around the center peak.

Grid Cell Demo 01: Rat Walk Demo in a 2 x 2 Arena

Tue, 12 May 2026 00:00:00 +0000

This page is the first step from static geometry toward an actual moving-animal simulation.

The setup is intentionally simple:

the arena is a fixed $2\times 2$ square
the rat is controlled by W, A, S, D
the right side shows four moving waves: theta, wave 1, wave 2, wave 3
the middle panel accumulates the cell’s firing as the rat moves

The heatmap keeps the full firing history until you reset it, so the spatial pattern emerges directly from the trajectory itself rather than from a pre-drawn lattice.

Firing threshold 0.95

Rat in 2 x 2 Arena

While you drag, the rat moves exactly with the pointer inside the square world, and that movement updates the phase evolution of the directional waves.

Grid Cell Firing

Pale circles show the theoretical firing sites predicted by the same four-wave score and the same threshold. The brighter heat layer records the actual online firing accumulated from the rat's trajectory.

Oscillatory Waves

These four traces show the moving theta wave together with the three direction-tuned waves.