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 Notes 04: From fMRI to Grid-Cell-Like Coding

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

This note is about a practical question I kept encountering:

How do we move from noisy fMRI in the right entorhinal cortex (Right EC) to a defensible claim of grid-cell-like coding?

The short answer is: we do not look for literal hexagonal “spots” in BOLD images. We test whether Right EC activity is modulated by movement direction with a 60-degree periodicity.

If movement direction aligns with a latent grid orientation, BOLD is higher; if it is offset by 30 degrees, BOLD is lower. This is the classic hexadirectional modulation logic.

1) Start from trajectory, not static location

Suppose the participant moves in a virtual environment and we track position:

$$ (x(t), y(t)) $$

Instantaneous movement direction is

$$ \theta(t)=\mathrm{atan2}(y(t+\Delta t)-y(t), x(t+\Delta t)-x(t)) $$

In practice, low-speed or stationary periods are excluded. The directional model is about motion direction during navigation, not occupancy.

2) Estimate grid orientation in Right EC

The standard modeling idea is that many grid cells in a local EC population can share a common orientation (while differing in spatial phase). At voxel level, phase-specific structure can average out, but orientation-locked directional modulation can remain.

For each time point, construct:

$$ \cos(6\theta(t)),\ \sin(6\theta(t)) $$

The factor 6 encodes six-fold rotational symmetry (60-degree periodicity).

Fit a GLM on training data (for example, N-1 runs):

$$ \mathrm{BOLD}_{EC}(t)=\beta_{\cos}\cos(6\theta(t))+\beta_{\sin}\sin(6\theta(t))+\mathrm{nuisance\ regressors} $$

Then recover the putative grid orientation:

$$ \phi=\frac{1}{6}\mathrm{atan2}(\beta_{\sin},\beta_{\cos}) $$

Because of 60-degree periodicity, $\phi$ is defined modulo 60 degrees.

3) Test in independent data (critical)

The key methodological constraint is avoiding circularity: do not estimate $\phi$ and test $\phi$ on the same data.

Use cross-validation (for example leave-one-run-out): estimate $\phi$ on training runs, test on held-out run, rotate folds, then average.

In test data, build:

$$ \cos(6(\theta(t)-\phi)) $$

Then fit:

$$ \mathrm{BOLD}_{RightEC}(t)=\beta_{\mathrm{hex}}\cos(6(\theta(t)-\phi))+\mathrm{nuisance\ regressors} $$

If $\beta_{\mathrm{hex}} > 0$, Right EC is stronger when $\theta$ aligns with $\phi + k\cdot60^\circ$ than when it is shifted toward $\phi + 30^\circ + k\cdot60^\circ$.

This is the aligned > misaligned signature.

4) Intuitive visualization

A useful check is directional binning:

aligned bins: $\phi \pm 15^\circ$ and every 60-degree repetition
misaligned bins: $\phi+30^\circ \pm 15^\circ$ and every 60-degree repetition

Plot Right EC beta by bins. A consistent aligned > misaligned gap supports the model.

5) Group-level significance

A common inference flow:

Estimate one $\beta_{\mathrm{hex}}$ per subject.
Run one-sample t-test or permutation test at group level.
Restrict inference to Right EC ROI with proper correction (for example small-volume correction / FWE).
Verify effect is significantly above zero.

More conservative pipelines also do voxel-wise inference in EC with permutation + TFCE + FWE correction.

6) Necessary control analyses

A 6-fold effect alone is not enough. I would treat these controls as mandatory:

Symmetry controls: test 4-, 5-, 7-, 8-fold models. The effect should be specific to 6-fold.
Behavioral/perceptual confounds: control speed, head direction, turning angle, visual flow, button/motor effects, and head motion artifacts.
Direction sampling balance: ensure movement directions are not heavily skewed by task geometry.
Stability/coherence checks:
- similar $\phi$ across runs
- non-random orientation structure within Right EC (for example Rayleigh-type checks).

7) What conclusion is valid

A careful claim is:

Right EC BOLD shows significant six-fold directional modulation relative to an estimated grid orientation, replicated out-of-sample and stronger than non-6-fold control symmetries. This supports a grid-cell-like population code in Right EC.

What I should avoid saying:

fMRI proves single-neuron grid cells in Right EC.

fMRI supports population-level grid-like representation, not direct single-cell firing-field proof.

One-line takeaway

Right EC grid-like signal in fMRI means: estimate hidden orientation $\phi$ on independent data, then show BOLD is higher for $\theta=\phi+k\cdot60^\circ$ than for $\theta=\phi+30^\circ+k\cdot60^\circ$, with 6-fold specificity and confounds controlled.

References

Doeller CF, Barry C, Burgess N. Evidence for grid cells in a human memory network. Nature. 2010.
Doeller Lab copy of the paper PDF: Evidence for grid cells in a human memory network.
Nature Communications (2024): Grid-like entorhinal representation of an abstract value space during prospective decision making.
Null/constraint evidence in passive navigation context: Failure to detect entorhinal grid-like signals in a passive virtual navigation paradigm.

FMRI | Shaoyang Cui