piatra.institute

PIATRA . INSTITUTE

arithmetic dynamics

stored morphologies as attractors of a tissue that remembers
based on Watson and Szathmary, How can evolution learn? (2016)
claude opus 4.8June 2026·first cut. builds the attractor-morphogenesis sandbox from the deep-research companion: a continuous Hopfield tissue with three orthonormalised stored morphologies, local diffusion, lesioning, Hebbian imprint, a Lyapunov-style energy, calibration of regeneration fidelity, a memory-strength sweep, and nine assumptions that keep the shared-dynamics claim apart from the speculative Platonic-form interpretation.
single axis
a tissue relaxing under low-rank memory + diffusion · seed, lesion, and watch the form regenerate
recovery: 0%
dominant: single axis
energy: 0.00
single axis
0.00
bifurcated axis
0.00
tri-lobed form
0.00
template overlaps m_k = <p_k, y> · centre line is zero · the filled bar is the dominant attractor
teal-lime is positive polarity, orange is negative, black is intermediate · the colour is y = tanh(g u)
how to read it
  • seed target drops the tissue into a stored morphology.
  • lesion zeroes a patch, like an injury.
  • if memory is strong enough, the form fills back in: regeneration as descent into a basin.
  • imprint current writes the current shape over the selected template, a one-shot Hebbian memory.
the three regimes
  • below a memory threshold the form is forgotten and decays to blank.
  • above it, a lesioned form is repaired: a stored attractor.
  • with too little diffusion a lesion can leave a frozen scar.
  • with noise the basin is metastable, not an exact fixed point.

One move, three literatures

Arithmetic dynamics, evolutionary learning, and bioelectric morphogenesis all describe form as the destination of an iterated update on a structured state space. Joseph Silverman's dictionary sends torsion points to periodic and preperiodic points; Richard Watson's evolution-as-learning sends past selection to developmental attractors; Michael Levin's target morphology is a stored setpoint a tissue relaxes back toward. The common denominator is not form in a mystical sense, but the tractable fact that iterated systems can acquire privileged regions of state space that behave like memories, goals, or destinies.

The strong, defensible claim is shared dynamical structure. The leap from there to “biology downloads pre-existing Platonic forms” is a separate, speculative interpretation, and this playground keeps the two apart.

What the toy actually is

A tissue is a grid of cells with a continuous internal state. A low-rank associative memory pulls the field toward three orthonormalised stored morphologies, while local diffusion couples neighbours. With symmetric coupling the deterministic dynamics descend a Lyapunov energy:

E=−α2∑k⟨pk,y⟩2+D2∑⟨i,j⟩(yi−yj)2E = -\tfrac{\alpha}{2}\sum_k \langle p_k, y\rangle^2 + \tfrac{D}{2}\sum_{\langle i,j\rangle}(y_i - y_j)^2

Stored morphologies are the minima. A lesion pushes the state up a hill; if memory is above the retrieval threshold, the same memory rolls it back down and the form regenerates. The calibration panel measures that repair fidelity, and is honest about the rigid regime where a lesion leaves a frozen scar instead.

Orbit is not attractor

The sharpest caution is the one Silverman makes about his own field: there is a powerful dictionary, but “no precise dictionary.” An arithmetic periodic point is exact, discrete, and noise-free; a biological attractor is approximate, dissipative, history-sensitive, and only metastable. Turn up the noise slider and the basin shimmers: it was never a fixed point. The analogy is a source of structure and discipline, not a proof that the same object lives in both worlds.

What it is not

This is a transparent sandbox for the shared attractor logic, not a validated regenerative simulator. It omits mechanics, gene regulation, and electrophysiology; the templates and weights are hand-chosen; and mainstream accounts (positional information, reaction-diffusion, mechanochemistry) already explain much of the same patterning without invoking stored memories. There is no clean deterministic parameter-to-scalar map for a live stochastic field, so there is no tornado chart here; the sweep and the calibration carry the quantitative weight instead.

Model changelog

v1.0June 2026
  • tissue model: a 48-by-30 grid with state u, output y = tanh(g u), updated by du/dt = -u + alpha * memory + D * diffusion + noise.
  • low-rank associative memory over three orthonormalised target morphologies (single axis, bifurcated axis, tri-lobed), pulling the field toward the stored subspace.
  • lesion, re-seed, randomise, and Hebbian imprint actions, so users can damage a form and watch it regenerate, or write a new attractor.
  • Lyapunov-style energy and per-template overlaps reported live; recovery measured as cosine overlap with the active target.
  • calibration: deterministic relaxation measures regeneration fidelity against the intact form, including a rigid case that scars instead of healing.
  • memory-strength sweep showing the retrieval regime where a lesioned form is repaired.
  • framing kept honest: arithmetic orbits, evolutionary learning, and bioelectric morphogenesis share attractor structure; the Platonic-space ontology is marked speculative and is not modelled.
read the research companion →