plan.form

1 · The Neural Field — Wilson-Cowan & Bifurcation

The simulation runs two coupled ordinary differential equations per spatial point on a 512 × 512 grid, updated 8 times per rendered frame via WebGL2 ping-pong render targets:

τ_E · ∂E/∂t = −E + σ( w_EE · K_exc⊛E  −  w_EI · I  +  I_ext · (1−α) )
τ_I · ∂I/∂t = −I + σ( w_IE · E  −  w_II · I )

σ(x) = 1 / (1 + exp(−5 · (x − 0.28)))   [sigmoid activation]
⊛ = spatial convolution with lateral connectivity kernel

The spatial convolution uses two ring samplers in GLSL: a 12-point ring (30° spacing, 6-fold symmetry) at radius 6 px for short-range excitation, and a 16-point ring at radius 20 px for long-range inhibition. Their difference approximates the Mexican-hat lateral connectivity kernel — the anatomical reality that excitatory neurons form local clusters while inhibitory interneurons project further. The 12-point ring's 6-fold symmetry is deliberate: an 8-point ring would introduce 4-fold bias and spuriously favour square-lattice planforms over the hexagonal class V1 prefers.

Strictly speaking, this is a reduced formulation: the inhibitory equation uses only local E (no spatial kernel on the E → I coupling), placing the system between the full Wilson-Cowan model and the Amari (1977) single- population neural field. The simplification is standard, preserves the Turing-instability behaviour, and keeps the GPU cost low enough for real-time integration at 60 fps.

The parameter α represents the DMT effect. As α rises from 0 to 1, two things happen simultaneously: external input is suppressed by factor (1−α), and lateral excitatory coupling w_EE rises from 0.45 to 1.75. Below α ≈ 0.42, every spatial point converges to the same resting activity (E* ≈ 0.35). Above this threshold, the homogeneous state loses stability.

This is a Turing instability (Turing, 1952). The critical wavevector k* — the spatial frequency that first goes unstable — is determined by the ratio of excitatory to inhibitory kernel radii. For r_exc = 6 px and r_inh = 20 px, this selects a characteristic wavelength of approximately 52 px, producing the hexagonal planform. The same mathematics that spaces the spots on a leopard's coat spaces the nodes of the DMT pattern.

The external input I_ext includes a persistent high-frequency edge signal derived from the image texture. This seeds planform nodes at image texture boundaries — ensuring the pattern that emerges is topographically anchored to the input, not arbitrary.

2 · Form Constants — Klüver to Bressloff

In 1928, psychologist Heinrich Klüver systematically documented the geometric patterns that appear universally across mescaline experiences, sensory deprivation, hypnagogia, and migraine aura. He identified four fundamental classes: gratings and lattices, cobwebs, tunnels and funnels, and spirals. He called these form constants and proposed they reflect the intrinsic organisation of the visual system.

Fifty years later, Ermentrout and Cowan (1979) provided the mathematical explanation. They showed that the visual cortex, modelled as a neural field with lateral connectivity, undergoes symmetry-breaking instabilities that produce exactly these four pattern classes. The patterns are not content imposed on the cortex — they are the cortex's own structural resonances, made visible when external suppression is removed.

Bressloff et al. (2001) extended this analysis with full consideration of V1's functional architecture — orientation preference columns, ocular dominance, the anisotropy of long-range horizontal connections — and showed that the symmetry group of V1 connectivity constrains the emergent planforms to a small set of geometric forms. These correspond precisely to Klüver's form constants. The hexagonal planform, which this simulation produces, is predicted for uniform, isotropic inputs under elevated cortical excitability — the gain increase classical hallucinogens are thought to induce via 5HT_2A-mediated action on V1.

3 · Cortical Geometry — Retino-Cortical Map & Hyperbolic Space

The visual field does not map linearly to V1 surface area. The central 10° of vision maps to approximately half of V1's surface, despite covering only a small fraction of the total visual field (Horton & Hoyt, 1991). This magnification follows a logarithmic relationship — the retino-cortical transform — described precisely by the complex logarithm:

w = log(z)    where z = x + iy is position in the visual field

In polar coordinates (r, θ):
  cortical x = log(r)
  cortical y = θ

Implemented in GLSL:
  cx = (log(r) + 3.22) / 3.57
  cy = atan(y, x) / (2π) + 0.5

Radial patterns in visual space become vertical stripes in cortical coordinates. Concentric rings become horizontal stripes. The planform — which lives in cortical coordinates — appears as a radially symmetric mandala when mapped back through the inverse transform. The bilateral mirror applied before the log-polar transform reflects V1's biological symmetry: each hemisphere processes the contralateral visual field, so any cortical pattern has a symmetric counterpart. The "double creature" forms arise from this symmetry operation.

Andres Gomez-Emilsson (2016) of the Qualia Research Institute proposed that DMT transforms the underlying geometry of phenomenal space from Euclidean to hyperbolic. In hyperbolic space, the circumference of a circle grows exponentially with radius rather than linearly — providing exponentially more "room" per unit radius. This accommodates the information overload of DMT-amplified lateral connectivity, and may explain the reported quality of "more real than real" — more information is encoded per perceived unit of space than is possible in Euclidean geometry. Note: the QRI essay is a speculative philosophical proposal, not a peer-reviewed publication. It is used here as conceptual inspiration for the Poincaré disk transform, not as empirical evidence.

The Poincaré disk model maps the entire infinite hyperbolic plane into a unit disc. Implemented here via:

r_texture = tanh( atanh(r) / (1 + t · 1.2) )

where r = normalised screen radius, t = curvature parameter [0,1]

Combined with four-fold colour averaging (sampling all four mirror positions and averaging), this produces the disc-shaped mandala characteristic of peak psychedelic visual geometry. The centre expands without apparent limit; the boundary recedes forever.

During the final transition, vertex displacement is recentred around the image's mid-luminance point, causing dark areas (the "floor" between creature forms) to recede and bright areas (the creature surfaces) to protrude further. The bilateral symmetry of the planform activates the fusiform face area (Kanwisher et al., 1997) — which responds to the structural features of a face (bilateral symmetry, nodal eye positions, axial organisation) independent of whether the input is literally a face. Higher cortex completes the pattern. This may be a neural correlate of what Terence McKenna described as machine elves: deterministic pattern completion operating on a supercritical neural field. Whether that accounts for the full phenomenology — the agency, the narrative, the emotional weight of entity encounters — is an open question the model does not resolve.