Atlas Geometric Structure Methodology v1.1

The Geometric Structure of Synaptic Plasticity

LTP/LTD in hippocampal CA1 synapses is a statistical manifold equipped with the Fisher information metric, with a stratified boundary defined by the BCM modification threshold. This identification was confirmed by M4 lateral convergence across three independent research traditions.

Primary Object

Statistical Manifold

Formally Proven

Metric

Fisher Information Matrix

Formally Proven

Boundary

BCM Threshold Stratification

Formally Proven

M4 Lateral Convergence — 3 Independent Traditions

Tradition 1

Information geometry of learning rules — Amari (1998), Kreutzer et al. (2022)

Tradition 2

Energetics of synaptic plasticity — Karbowski (2021)

Tradition 3

Geometry of weight distributions — Pogodin et al. (2024)

Geometric Object Identification

The Atlas Geometric Structure Methodology was applied to synaptic plasticity (LTP/LTD in hippocampal CA1 synapses). Phase 0 (M2 Variable Lens Rotation) rotated all 15 Atlas variables onto the subdomain. Seven variables voted for a statistical manifold, four for a graph, two for a Riemannian manifold, one for a stratified space, and one for a symplectic manifold. One variable (σ — entropy) produced no valid signal, constituting an M10 blind spot.

Object Type	Votes	Variables
Statistical manifold	7	Φ, κ, ρ, J, D, A, L
Graph	4	S, Λ, C, T
Riemannian manifold	2	κ, G
Stratified space	1	B
Symplectic manifold	1	H
Unknown (M10 blind spot)	1	σ

Physical Substrate

The physical substrate is the two-step phosphorylation/dephosphorylation cycle of AMPA receptors (AMPARs) in hippocampal dendritic spines. This cycle, regulated by kinases (PKA, CaMKII) and phosphatases (PP1, PP2A), controls the number of AMPARs in the postsynaptic density and thereby determines synaptic conductance. The substrate exhibits bistability — two stable states corresponding to potentiated and depressed synaptic weights.

\frac{dv}{dt} = \frac{h r(1 - \alpha r) - (v - v_0)}{\tau_w} + \frac{\sqrt{2D}}{\tau_w} \eta(t)

Stochastic BCM synaptic current — Karbowski (2021), J Comput Neurosci 49:71–106

Methodology Summary

Phase 0 (M2): 15 Atlas variables rotated onto synaptic plasticity — 7 votes for statistical manifold

Phase 1 (M1): 8-question decision tree — continuous manifold with stratified boundary confirmed

Phase 1 (M4): Lateral convergence — 3 independent traditions confirm statistical manifold

Phase 1 (M5): Exclusions — symplectic, pure graph, non-commutative, topos excluded

Phase 2: Substrate identified — AMPAR phosphorylation dynamics, Fokker-Planck equation

Phase 3: 10 anatomy layers mapped — Layers 0, 1, 5, 6, 9 fully/formally mapped; Layers 7 is a gap

Phase 4: 15 domain faces — 2 formally proven, 1 structural isomorphism, 12 partial views

Phase 5: 14 mathematical structures — 6 formally proven, 5 structural isomorphisms, 1 conjectured

Phase 6: 9 derivations — 5 formally proven, 3 structural isomorphisms, 1 conjectured

Phase 7: 6 cross-domain theorems, 6 open problems, epistemic audit of 8 claims