foundationalcomputationneuroscience

The Spiking Prototype: a Nambu-Goto-Enforced Substrate and What It Does (Not) Achieve

v0.1 — a 3-D LIF/ALIF blob with node-placed functional plasticity, two slow homeostats, Dale's-law E/I balance, and a global surface-minimisation watcher; the honest baseline the local-rule programme must beat

Danyiel Colin2026-06 last modified Jun 20, 2026

Criticality and Nambu-Goto surface minimisation are emergent global properties: neither may be enforced from outside at runtime without making the result circular. This paper presents v0.1, a deliberately honest prototype that enforces the geometry anyway — a global watcher slides the free interior of a 3-D embedding down the gradient of the physical-network surface action S = Σ π w L and rewires by proximity — so there is a working substrate and a measured baseline. The atomic unit is a leaky integrate-and-fire neuron with spike-frequency adaptation; its functional plasticity lives on the node (a learnable membrane time constant and threshold), not on O(E) edges, and its synapses are scalar weights under local STDP behind a three-factor interface. Excitability is regulated by two slow, literature-grounded homeostats — Triesch intrinsic plasticity (threshold) and Turrigiano synaptic scaling (weight) — over a Dale's-law excitatory/inhibitory balance. We justify each choice and then test the substrate honestly. Across 36 runs (sizes, seeds, and ablation arms) the watcher minimises surface every pass; the population is alive and rate-regulated; and the branching ratio σ_MR sits sub-critical at ≈0.32, tight across seeds — the expected negative under globally-enforced geometry. The load-bearing finding is a clean discriminator: ablate the homeostat and σ_MR collapses to ≈1.0 (synchrony), so excitability regulation, not the geometry, is what keeps the substrate measurable. v0.1's job is to establish the substrate and prove its central property is not an artefact of its regulator — and to make the case that the enforced geometry must instead be *earned* by a local rule, which is the next paper.

spiking-neural-networkLIFcriticalitynambu-gotohomeostasisSTDPself-organizationnode-migrationPSI-E1substrate

Introduction

Purpose of this section. State the reframe, the honest scope of v0.1, and the falsifiable question it sets up.

  • The reframe. Criticality and Nambu-Goto surface minimisation are emergent global properties of a population of locally-acting units. Enforcing either at runtime — a global servo that pins mean activity, or a watcher that solves the surface PDE and moves nodes — makes "it emerged" circular: the consequence has been hardcoded as the cause.
  • What v0.1 is. A deliberately honest prototype that does enforce the geometry (a global watcher), so there is a working substrate and a measured baseline. The watcher is named as a shortcut, not a result.
  • What v0.1 is for. To (a) justify a minimal spiking substrate component by component, (b) characterise honestly what enforced-NG buys — expected: not criticality — and (c) state the question the next paper answers: can a purely local rule earn the surface minimisation the watcher fakes?
  • Predecessor. SORN (self-organising recurrent networks; Lazar–Pipa–Triesch) is the closest ancestor — local plasticity + homeostasis producing structured dynamics — which we extend with an explicit 3-D physical geometry.

The architecture, and why each piece

Purpose of this section. Justify the atomic unit and every component — the spine of the paper. Each choice is defended against its alternative.

  • The atomic unit — LIF/ALIF spiking, not a continuous map. Why spiking over the earlier continuous coupled-map-lattice with KAN edges: O(N) state vs O(E·(G+k)) splines, biologically faithful, async-portable (the same unit runs clock-driven now and event-driven later), and — decisively — spikes make avalanche/criticality measurement native, which a dense continuous field does not (its branching-ratio estimator degenerates). Spike-frequency adaptation (ALIF) gives a node-local slow timescale.[3][4]
  • Functional plasticity on the node, not the edge. A learnable membrane time constant (parametric-LIF) and threshold per neuron — O(N) — instead of a learnable function per synapse — O(E). The synapse is ~a scalar gain; the nonlinearity lives in the cell.
  • Scalar synapses + STDP behind a three-factor interface. Local, online, and structured so a top-down learning signal (e-prop) drops in unchanged at scale — the sync→async bridge.[5]
  • Two slow homeostats — both legs. Triesch intrinsic plasticity slides the threshold toward a firing-rate set-point (regulates rate); Turrigiano synaptic scaling multiplicatively rescales incoming excitatory weights (regulates magnitude, bounds STDP runaway). Each reads only the cell's own activity — never a population aggregate.[6][7]
  • Dale's-law E/I balance. A fixed inhibitory subset desynchronises the recurrent net; without it the population synchronises and presents manufactured criticality (σ_MR→1). The fast lateral divisive-normalisation leg is validated but kept off the default path.[8][9]
  • 3-D embedding + the global NG watcher. Each cell carries a position in ℝ³; the watcher slides the free interior down the gradient of the physical-network surface action S = Σ_e π w L_e and rewires by proximity, so geometry feeds back onto the dynamics. This is the enforced shortcut — stated plainly as the thing 0002 replaces.[10]

What we test, and why

Purpose of this section. Lay out the probe suite and the ablation matrix, and justify each measurement — what question it answers and what would falsify it.

  • NG-is-happening witness. Confirm the watcher actually minimises surface: the per-pass descent ΔS ≤ 0 (a backtracking line search guarantees it at every scale). The honest nuance: the proximity rewire densifies the graph, so total S can rise even as each node descends — the witness is the descent, not total S.
  • Criticality battery. The central question — does criticality emerge? Gauges: the Wilting–Priesemann branching ratio σ_MR (subsampling-robust), avalanche size/duration power-law exponents and the crackling relation, Lempel-Ziv complexity (rate-controlled), and the 1/f aperiodic exponent. Expectation: sub-critical under globally-enforced geometry — the honest negative, and the baseline 0002 must beat.[11][12]
  • Perturbation response. Poke one neuron and measure how the perturbation propagates against an identical twin (spike-difference spread, divergence trend, perturbational complexity / PCI-style LZc) — does the substrate support non-trivial propagation?[13]
  • Ablations as discriminators — how we attribute behaviour to components. Each arm removes one axis: NG-off (geometry frozen — is the geometry doing anything?), homeostat-off (excitability regulation — the load-bearing test), STDP-off (synapses frozen), scaling-off (does Turrigiano bound runaway?), lateral-on (the fast divisive-normalisation leg). The contrast between arms, not any single number, is what isolates a mechanism's role.

Results

Purpose of this section. Report the measured baseline — 36 runs (sizes 4³ and 5³, seeds 0–2, six ablation arms) — and read the honest negatives and the one load-bearing positive. All numbers are from the certified sweep (verdict.json: prototype_works, ng_minimisation, discriminator_present all true).

The substrate is alive and the watcher minimises surface. Every NG-on arm descends surface on every pass (frac_passes_descending = 1.0); the migration removes surface scaling with the free interior (≈0.4 at side 4, ≈2.4–4.0 at side 5); the population holds a regulated firing rate (~0.06/step).

Criticality is sub-critical — the expected negative. The branching ratio on the full operating arm is tight across seeds and sizes:

size seed 0 seed 1 seed 2
4³ (N=64) 0.234 0.299 0.308
5³ (N=125) 0.322 0.322 0.326

σ_MR ≈ 0.32 (median), well below 1 — under globally-enforced geometry the substrate does not self-organise to criticality. A critical, well-posed σ_MR≈1 would have been the surprise.

The load-bearing result — the homeostat, not the geometry, keeps it measurable. Ablating the slow excitability axis collapses σ_MR to ≈1.0 (median 1.0000 vs the full arm's 0.32) — synchronised degeneracy, not criticality. This is the cleanest discriminator in the matrix and the figure below.

Perturbations propagate. A single poke spreads through the population with perturbational complexity (LZc of the spike-difference raster) of 0.95–0.99 across all full-arm runs — non-trivial propagation despite sub-critical branching.

A real scale effect, reported as data. At size 5³ the control arms (homeostat-off, NG-off, STDP-off) fall below the avalanche-applicability floor — they lack a leg the full configuration has, so they degrade as designed. This is recorded (arms_not_applicable), not hidden by the gate.

Interactive chart · click to open
Branching ratio σ_MR by ablation arm (36 runs). The full arm sits sub-critical (~0.32); homeostat-off collapses to ~1.0 (synchrony) — excitability regulation, not the enforced geometry, is what keeps the substrate in a measurable regime. click to expand

A baseline, not the thing

Purpose of this section. Close by stating plainly why this substrate is a baseline, and hand off the falsifiable question to the next paper.

  • The circularity, named. The watcher solves the surface problem globally and moves the nodes there. So "the substrate minimises its wiring surface" is true by construction, not by emergence — we hand-solved the PDE and called it a property. v0.1 is honest about this: it is the control, not the claim.
  • What the baseline establishes. A justified spiking substrate; a measured, reproducible operating point (σ_MR≈0.32, sub-critical); and a clean attribution — the homeostat is load-bearing, the enforced geometry is not what makes the dynamics measurable.
  • The question for 0002. Can a purely local update rule — each cell reading only its own state and its immediate neighbours, never the global action — be distilled such that, run with the watcher off, the wiring surface still descends? And does the same frozen rule hold across orders of magnitude in N (a law, not a fit)? That is the contribution of the next paper.
  • Deferred. The tissue comparison and the holographic area-law test need a second timescale and the scaled async substrate; they belong to v0.2/v0.3, not here.

References

  1. [1]Lazar, A., Pipa, G., & Triesch, J. (2009). SORN: a self-organizing recurrent neural network. Front. Comput. Neurosci., 3, 23.
  2. [2]Del Papa, B., Priesemann, V., & Triesch, J. (2017). Criticality meets learning. PLOS ONE, 12(5), e0178683.
  3. [3]Abbott, L. F. (1999). Lapicque's introduction of the integrate-and-fire model neuron (1907). Brain Res. Bull., 50(5–6), 303–304.
  4. [4]Fang, W., et al. (2021). Incorporating learnable membrane time constant (PLIF). ICCV.
  5. [5]Bi, G., & Poo, M. (1998). Synaptic modifications in cultured hippocampal neurons (STDP). J. Neurosci., 18(24), 10464–10472.
  6. [6]Turrigiano, G. G. (2008). The self-tuning neuron: synaptic scaling. Cell, 135(3), 422–435.
  7. [7]Triesch, J. (2005). A gradient rule for the plasticity of a neuron's intrinsic excitability. ICANN.
  8. [8]Carandini, M., & Heeger, D. J. (2012). Normalization as a canonical neural computation. Nat. Rev. Neurosci., 13(1), 51–62.
  9. [9]van Vreeswijk, C., & Sompolinsky, H. (1996). Chaos in neuronal networks with balanced E/I activity. Science, 274(5293), 1724–1726.
  10. [10]Meng, X., Piazza, F., Both, J., Barzel, B., & Barabási, A.-L. (2026). Surface optimisation governs the local design of physical networks. Nature, 649. doi:10.1038/s41586-025-09784-4.
  11. [11]Wilting, J., & Priesemann, V. (2018). Inferring collective dynamical states from widely unobserved systems. Nat. Commun., 9, 2325.
  12. [12]Beggs, J. M., & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. J. Neurosci., 23(35), 11167–11177.
  13. [13]Casali, A. G., et al. (2013). A theoretically based index of consciousness (PCI). Sci. Transl. Med., 5(198), 198ra105.
  14. [14]Touboul, J., & Destexhe, A. (2017). Power-law statistics and universal scaling in the absence of criticality. Phys. Rev. E, 95(1), 012413.
  15. [15]Bonachela, J. A., & Muñoz, M. A. (2009). Self-organization without conservation: true or just apparent scale-invariance? J. Stat. Mech., P09009.