Grid-Cell-Notes | 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 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.

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 Notes 03: Environmental Geometry Deforms Grid Patterns

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

This note is a literature checkpoint rather than a derivation. The question is simple:

If grid cells really implement a clean internal spatial metric, then what happens when the animal explores an enclosure whose geometry is strongly polarized or deformed?

The short answer from experiment is: the grid is not perfectly rigid. In sufficiently non-regular environments, grid firing patterns can become stretched, less hexagonal, locally non-uniform, and history-dependent.

The Phenomenon

The clearest empirical message from the literature is that environmental geometry can reshape grid firing in at least three related ways:

Global distortion: the lattice can become less hexagonal and more elliptical.
Local distortion: different parts of the same enclosure can carry different local grid structure.
Deformation-specific remapping of metric: compression or expansion of the enclosure can change grid spacing and apparent field positions, often in a boundary-dependent way.

So the naive statement

$$ \text{grid cell firing fields are a universal, geometry-invariant hexagonal metric} $$

is too strong.

Classic Trapezoid Result

One of the strongest demonstrations is the trapezoid experiment of Krupic et al. (2015), who compared firing in square and trapezoidal environments.

Their main qualitative findings were:

in squares, grid patterns retained stronger hexagonal regularity
in trapezoids, the pattern became more elliptical and less homogeneous
the distortion was not only global; the narrow and wide sides of the trapezoid showed different local structure

Biological data from Figure 3 of Krupic et al. (2015), cropped from the paper PDF. The same cells show cleaner hexagonal structure in the square and visibly distorted, stretched patterns in the trapezoid.

Biological data from Figure 4 of Krupic et al. (2015), cropped from the paper PDF. The left and right sides of the trapezoid do not carry identical local grid structure: the pattern rotates, stretches, and changes field size across the enclosure.

For my own purposes, this is the most important observation: the distortion is not just a single uniform affine warp of an otherwise perfect lattice. The same enclosure can induce different local grid structure in different subregions.

Beyond Trapezoids: Environmental Deformation

Later work pushed this farther by asking what happens when familiar enclosures are compressed, stretched, or otherwise deformed.

In Keinath, Epstein, and Balasubramanian (2018), the authors argued that environmental deformations produce systematic shifts in the grid metric that are better understood as boundary-tethered, history-dependent distortions than as a simple global rescaling. In other words, the apparent grid can depend on how recent boundary encounters anchor the phase pattern.

This is useful because it says the deformation effect is not merely a static shape issue. It depends on how self-motion integration interacts with environmental boundaries over time.

Related evidence from Munn et al. (2020) showed that entorhinal velocity-related signals themselves reflect environmental geometry. That result matters because it suggests the distortion may enter not only at the level of the final firing map, but also upstream in the velocity coding and path-integration machinery.

What I Take From These Papers

For this notebook, the current takeaway is:

A grid pattern should not be treated as an always-perfect Euclidean ruler.
Strongly polarized geometry can bias orientation, spacing, ellipticity, and local homogeneity.
A narrow rectangle or a trapezoid should not be expected to preserve the same lattice visible in a square.
If I build demos later, it will be worth distinguishing:
- a purely internal oscillatory/interference mechanism
- boundary-driven anchoring or correction
- velocity-code distortion induced by enclosure geometry

So the next conceptual question is no longer only “how do multiple waves form a hexagonal pattern?” It is also:

$$ \text{how does environmental geometry bend or constrain that pattern?} $$

References

Krupic J, Bauza M, Burton S, Barry C, O’Keefe J. Grid cell symmetry is shaped by environmental geometry. Nature. 2015.
Keinath AT, Epstein RA, Balasubramanian V. Environmental deformations dynamically shift the grid cell spatial metric. eLife. 2018.
Munn RGK, Rikhye RV, Sreenivasan S, et al. Entorhinal velocity signals reflect environmental geometry. Nature Neuroscience. 2020.

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.

Grid Cell Notes 01: Oscillatory Interference Along a Preferred Direction

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

This column is meant to be a running notebook. I do not want it to be limited to prose. For grid cells, intuition often becomes much clearer when formulas and visual structure are placed side by side, and even clearer when some parameters can be manipulated directly.

This first note focuses on the simplest case of the oscillatory-interference picture: the animal moves along the preferred direction of one wave, that wave changes frequency relative to a background pacemaker, and the drifting phase difference generates a fixed firing spacing.

Symbol setup

To keep the notation fixed, I will use:

$\theta_i$: the preferred direction of wave $i$
$\Phi_0$: the phase of the background pacemaker in the brain
$\Phi_i$: the phase of wave $i$
$\omega_0$: the baseline angular frequency of the pacemaker
$\beta$: the gain that converts motion into frequency shift
$v(t)$: the animal’s instantaneous speed

In this note, I only consider the case where the animal is moving exactly along the preferred direction of wave $i$. Under that assumption, the frequency of the wave changes according to

$$ \omega_i(t)=\omega_0+\beta v(t). $$

So the directional dependence is absorbed into the assumption itself: I am already on the preferred axis of this wave.

Why focus on relative phase

Before writing down formulas, the key intuition is this: in an oscillatory-interference view of grid cells, what matters is not an isolated wave by itself, but how one oscillation lines up against another.

Here the background pacemaker provides one oscillatory reference, and the direction-tuned wave provides another. If their peaks keep overlapping, the interference is strong. If they drift apart, the interference weakens.

So if we assume that a grid cell tends to fire when wave peaks align strongly enough, then the central object to track is not the absolute phase of either oscillation alone, but their relative phase.

That is why the quantity

$$ \Phi_i - \Phi_0 $$

becomes the natural variable. Once motion changes the frequency of the direction-tuned wave, this relative phase starts to drift. Every time the two waves come back into peak alignment, the interference becomes strong again, and that repeated re-alignment is what sets a spatial firing period.

Phase drift relative to the pacemaker

The key quantity is not just the absolute phase of a wave, but the phase difference relative to the background pacemaker:

$$ \Delta \Phi_i(t)=\Phi_i(t)-\Phi_0(t). $$

Its time derivative is

$$ \frac{d}{dt}\Delta \Phi_i(t)=\omega_i(t)-\omega_0. $$

Therefore,

$$ \frac{d}{dt}\Delta \Phi_i(t)=\beta\, v(t). $$

So your basic intuition is correct: once movement changes the frequency of wave $i$, the phase difference $\Phi_i-\Phi_0$ is no longer fixed and starts drifting over time.

If the speed is constant, this becomes

$$ \Delta \Phi_i(t)=\Delta \Phi_i(0)+\beta vt. $$

Re-alignment: first think with constant speed

The next question is: when do the two waves meet again at their peaks?

The clean way to say it is not “when $\Delta\Phi_i(t)=2\pi$”, because the initial relative phase may not be zero. What matters is that the change in relative phase reaches another full cycle:

$$ \Delta\Phi_i(t)-\Delta\Phi_i(0)=2\pi. $$

Under the constant-speed assumption,

$$ \Delta \Phi_i(t)=\Delta \Phi_i(0)+\beta vt, $$

so the re-alignment condition becomes

$$ \beta vt = 2\pi. $$

From this, the time between successive peak re-alignments is

$$ T_i = \frac{2\pi}{\beta v}. $$

This is the first way to think about the problem:

Assume $v$ is constant.
Solve for the time it takes the relative phase to accumulate another $2\pi$.
Convert that time interval into a traveled distance.

In one such period, the animal moves

$$ d=vT_i=v\frac{2\pi}{\beta v}=\frac{2\pi}{\beta}. $$

So under constant speed, the firing spacing is already a constant determined only by $\beta$.

Re-alignment: now remove the constant-speed assumption

The more interesting question is whether this result depends on $v$ being constant.

It turns out that it does not.

Instead of solving for a fixed period $T_i$, we can directly track accumulated phase drift:

$$ \frac{d}{dt}\Delta\Phi_i(t)=\beta v(t). $$

Integrating from time $0$ to time $t$ gives

$$ \Delta\Phi_i(t)-\Delta\Phi_i(0)=\int_0^t \beta v(\tau)\, d\tau. $$

If we ask for the next firing event, we impose the same re-alignment condition:

$$ \Delta\Phi_i(t)-\Delta\Phi_i(0)=2\pi. $$

So we obtain

$$ \int_0^t \beta v(\tau)\, d\tau = 2\pi. $$

Pulling $\beta$ out,

$$ \beta \int_0^t v(\tau)\, d\tau = 2\pi. $$

But

$$ \int_0^t v(\tau)\, d\tau $$

is exactly the distance traveled along this direction during that interval. Therefore the firing distance satisfies

$$ d=\int_0^t v(\tau)\, d\tau=\frac{2\pi}{\beta}. $$

This is the more robust formulation:

it does not require constant speed
it only requires that movement stays along the preferred direction
it shows directly that the relevant spatial spacing is fixed by $\beta$

So the realignment story can be understood in two equivalent ways:

Constant-speed view: solve for the re-alignment period first, then multiply by speed.
Integral view: accumulate phase drift until it reaches $2\pi$, and read off the distance directly.

The second view is the stronger one, because the conclusion survives even when $v(t)$ changes over time.

Interpretation

The logic can now be summarized very compactly:

The background pacemaker oscillates at $\omega_0$.
Motion along the preferred direction changes the wave frequency to $\omega_0+\beta v$.
That creates a phase drift rate of $\beta v(t)$ relative to the pacemaker.
Each time the accumulated phase drift reaches another $2\pi$, the peaks re-align.
The traveled distance associated with that re-alignment is $$ d=\frac{2\pi}{\beta}. $$

So one wave already gives a stripe-like periodic firing structure with a spacing fixed by $\beta$. The next conceptual step is then to ask how several direction-tuned waves, each with its own preferred direction, can combine to form a 2D grid-like firing pattern.

Interactive sketch

The demo for this note is intentionally restricted to the one-wave case. It keeps only one preferred direction and assumes the movement direction is parallel to it, matching the assumptions of the derivation above.

This demo keeps only one wave, and assumes the movement direction is parallel to that wave's preferred direction.

beta 0.140 theta_1 0 deg

Wave

A single oscillatory component along the preferred direction.

Grid Cell Firing

With only one wave, the firing pattern is a stripe-like grating rather than a full 2D hexagonal grid.

Later, I want to add a more faithful demo: control a moving rat in a 2D scene, update several direction-tuned waves according to velocity-dependent phase drift, and then simulate the resulting grid-cell firing pattern over the trajectory.

Why this is only a starting point

This picture is still incomplete, even in the aligned-direction case.

Real grid-cell theory quickly brings in questions such as:

How should the pacemaker itself be modeled?
How exactly do several direction-tuned oscillators interact?
Why are multiple modules needed?
How does path integration update phase over time?
What dynamical mechanism stabilizes the lattice?
How does a population of grid cells support decoding?

Those are the questions I want this column to gradually move toward. For now, the point is narrower: make the one-direction phase-drift logic precise before moving into a full spatial simulation.

Grid Cell Notes 02: Multiple Waves and Velocity Projection

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

In the first note, I only kept the simplest aligned case: one wave, one preferred direction, and motion parallel to that direction. That already gives a clean stripe-like firing periodicity with spacing

$$ d=\frac{2\pi}{\beta}. $$

Now I want to move one step closer to the full grid-cell picture by introducing multiple waves.

Fixed movement direction, multiple preferred directions

I now assume:

the animal moves at constant speed $v$
the movement direction is fixed
wave $i$ has preferred direction $\theta_i$
$\Phi_0$ is still the background pacemaker phase

The projection of velocity onto wave $i$ is

$$ v\cos\theta_i, $$

where $\theta_i$ is the angular difference between the fixed movement direction and the preferred direction of wave $i$.

So the frequency of wave $i$ becomes

$$ \omega_i=\omega_0+\beta v\cos\theta_i. $$

The relative phase drift satisfies

$$ \frac{d}{dt}\Delta\Phi_i=\omega_i-\omega_0=\beta v\cos\theta_i. $$

Therefore, the re-alignment period of wave $i$ is

$$ T_i=\frac{2\pi}{\beta v\cos\theta_i}. $$

If I care about the distance measured along wave $i$’s own preferred direction, then during one re-alignment period the effective displacement is the projected velocity multiplied by the period:

$$ d_i=(v\cos\theta_i)T_i=\frac{2\pi}{\beta}. $$

So the key point is:

the time between two re-alignments depends on $\cos\theta_i$
but the distance measured along the preferred direction of that wave is still the same constant $2\pi/\beta$

The projection factor changes how quickly phase accumulates in time, not the preferred-direction spacing set by $\beta$.

One wave

With only one preferred direction, the picture is still the stripe case from Note 01. The wave repeatedly re-aligns with the pacemaker after every projected distance

$$ \frac{2\pi}{\beta}, $$

so the firing pattern is a family of parallel bands orthogonal to that wave vector.

Two waves

Now add a second preferred direction. Each wave has its own projected phase drift,

$$ \frac{d}{dt}\Delta\Phi_1=\beta v\cos\theta_1,\qquad \frac{d}{dt}\Delta\Phi_2=\beta v\cos\theta_2. $$

Each wave alone would define its own stripe family. When both are present, the firing pattern is strongest where the two stripe families cross. So the geometry is no longer a single banded pattern; it becomes a lattice of pairwise crossings.

The main role of $\theta_1$ and $\theta_2$ is to control the angle between those two stripe families, and therefore the shape and spacing of the crossing structure.

Three waves

Adding a third preferred direction gives a third stripe family. Now the most salient firing locations are the points where all three families come into alignment at once.

This is the step that moves the picture from simple stripe crossings toward the familiar grid-like arrangement. In the symmetric case, when the preferred directions are separated in a balanced way, the triple intersections become distributed across space in a regular pattern.

So in this note the logic is:

each wave contributes a stripe family
each stripe family is controlled by its projected phase drift
pairwise crossings already create localized structure
three-wave crossings push that structure toward a grid-cell-like firing pattern

A geometric superposition picture

A simple geometric way to write the multi-wave field is

$$ w_i(x)=\cos(k_i^\top x+\phi_i), $$

where the orientation of $k_i$ is tied to $\theta_i$.

Then the combined field is

$$ s(x)=\frac{1}{N}\sum_{i=1}^{N} w_i(x), $$

and the firing map is produced through a readout

$$ g(x)=f(s(x)). $$

This is still a simplified spatial picture, but it is a useful bridge between the oscillatory phase-drift intuition and the 2D firing pattern we want to understand.

Interactive sketch

The demo below keeps the model at this geometric-superposition level. You can introduce one wave, then two, then three, while adjusting the key parameters:

$\beta$
$\theta_1$
$\theta_2$
$\theta_3$

The left panel shows the active waves and their superposition in a 1D slice. The right panel shows each wave’s stripe family in its own color, together with highlighted intersections.

With only wave 1 active, the model produces a stripe-like periodic structure set by beta and theta_1.

beta 0.140 wave grids

Wave 1

theta_1 0 deg

Wave 2

theta_2 60 deg

Wave 3

theta_3 120 deg

Wave

A 1D spatial slice showing the active waves and their superposition under the current beta and theta_i values.

Grid Cell Firing

Each active wave contributes one colored stripe family. Bright highlights mark where the currently introduced stripe families intersect.

What this note adds

Compared with Note 01, the conceptual change is small but important:

Motion is no longer assumed to be aligned with every wave.
Each wave sees only the projected component $v\cos\theta_i$.
Different waves therefore drift against the pacemaker at different rates in time.
Each wave still carries the same preferred-direction spacing $2\pi/\beta$.
Their combined interference can move from stripes to crossings and eventually to grid-like firing.

The next step after this note is to stop treating the pattern as static geometry and instead directly simulate phase accumulation while a rat moves through space.