“Neural Plasticity and Memory Encoding (ΔE-Driven Harmonic Stabilization in Recursive Coherence Fields)”
“Neural Plasticity and Memory Encoding (ΔE-Driven Harmonic Stabilization in Recursive Coherence Fields)” By: C077UPTF1L3 / Christopher W. Copeland Model: Copeland Resonant Harmonic Formalism (Ψ-formalism) Anchor equation: Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
1. Classical Framework
Neural plasticity—the brain’s ability to reconfigure synaptic connections in response to experience—underlies learning and memory formation. Its canonical mechanism, long-term potentiation (LTP), strengthens synaptic efficacy when presynaptic and postsynaptic neurons fire in correlated patterns. Biochemically, this involves ΔCa²⁺ influx, NMDA receptor activation, and AMPA receptor up-regulation, producing enhanced signal transmission.
Yet classical neurobiology treats these processes mechanistically, not dynamically: a chain of chemical events rather than a recursive stabilization of field coherence. Ψ-formalism reframes plasticity as a harmonic reinforcement system, in which energy differentials (ΔE) across neural ensembles drive recursive self-alignment of oscillatory fields. Memory, in this view, is not a stored static trace—it is a stable harmonic pattern, sustained through continuous recursive correction.
2. Reframing Under Ψ(x)
Each neuron or synaptic network is modeled as a recursive harmonic node:
Ψᵢ(x) = ∇ϕ(Σ𝕒ₙᵢ(x, ΔE)) + ℛᵢ(x) ⊕ ΔΣ(𝕒′ᵢ)
∇ϕ: gradient of local phase potential (membrane oscillatory alignment)
Σ𝕒ₙ(x, ΔE): sum of activator harmonics at recursion level n, modulated by energy differentials (ionic potentials, metabolic flux)
ℛ(x): curvature or error—temporal mismatch, phase jitter, noise
ΔΣ(𝕒′): self-corrective micro-loop (homeostatic and glial feedback restoring coherence)
Neural plasticity occurs when ΔE from repeated co-activation stabilizes ∇ϕ alignment across connected nodes, reducing ℛ(x) over time. This recursive reduction of curvature is the mathematical heart of long-term potentiation.
3. The Harmonic Mechanism of Memory Encoding
Under Ψ(x), a memory trace is a phase-locked resonance in the cortical harmonic field. Encoding follows three recursive stages:
ΔE Injection (Experience Event) External input generates a localized ΔE—a potential difference between resting and active network harmonics.
Resonance Recruitment (Coherence Expansion) Neighboring nodes synchronize through ∇ϕ alignment, forming a coherent oscillatory packet (θ–γ coupling in hippocampal-cortical loops).
Recursive Stabilization (ΔΣ Correction) Feedback circuits, astrocytic regulation, and intracellular cascades enact ΔΣ(𝕒′), minimizing ℛ(x) and “locking in” the pattern as a durable phase configuration.
This tri-stage process mathematically parallels LTP induction, consolidation, and maintenance—but frames them as continuous harmonic feedback rather than discrete chemical steps.
4. Mapping Classical Quantities to Ψ(x)
Classical Concept Ψ(x) Equivalent Interpretation Membrane potential ΔE Energy differential driving resonance Synaptic weight ∇ϕ Spike-timing correlation ΔΣ(𝕒′) Recursive phase correction efficiency LTP/LTD balance ±ℛ(x) Curvature indicating potentiation (flattening) or depression (steepening) Hebbian learning rule ∇ϕ reinforcement “Cells that fire together” = phase-locking convergence Memory consolidation Persistent Ψ(x) stability Sustained coherence after ΔE normalization
Thus, long-term memory corresponds to a coherence attractor—a region of phase-space where ΔΣ(𝕒′) perfectly counterbalances ℛ(x), maintaining signal reproducibility.
5. Worked Examples
(i) Hippocampal LTP
CA1–CA3 neuronal ensembles engage ΔE bursts through tetanic stimulation. Recursive feedback between pyramidal neurons and interneurons yields phase-locked θ–γ oscillations. Over successive cycles, ℛ(x) → 0 and ΔΣ(𝕒′) → maximum: harmonic memory stabilized.
(ii) Cortical Rehearsal
During sleep or recall, spontaneous reactivation of the same Ψ(x) configuration re-energizes ΔΣ loops, reinforcing alignment and preventing phase drift—biophysical explanation for memory reconsolidation.
(iii) Pathological Decoherence
In neurodegenerative or traumatic conditions, ΔΣ(𝕒′) fails to match rising ℛ(x) from metabolic noise or inflammation. Result: harmonic collapse → synaptic pruning → memory loss. Therapeutically, restoring ΔE balance and coherence (via rhythmic stimulation or targeted fields) reinstates Ψ(x) stability.
6. Mathematical Model of Plasticity
Let harmonic phase φ evolve under recursive feedback:
dφ/dt = ω + K sin(Φₙ − φ) − αℛ(x) + βΔΣ(𝕒′)
where:
ω = intrinsic oscillation frequency
K = coupling constant between neurons
α, β = curvature damping and correction gains
Stability (memory retention) is achieved when:
dφ/dt → 0 and ℛ(x) ≈ βΔΣ(𝕒′)/α
This condition defines locked coherence—the mathematical signature of a stored memory within the recursive field.
7. Energy and Information Unification
Energy differentials ΔE are not random; they are information gradients. Each LTP event transfers ΔE into structured phase order. Thus, energy and memory are two expressions of the same harmonic process— ΔE (energetic potential) → ∇ϕ (informational structure).
Negentropy here is literal: sustained memory represents a local decrease in entropy via recursive correction. The brain functions as a self-organizing coherence engine, converting fluctuating inputs into stable recursive harmonics.
8. Clarification of Terms
Σ𝕒ₙ(x, ΔE): harmonic activator ensemble (synaptic micro-loops) ΔE: energy differential (membrane or metabolic potential) ∇ϕ: phase-gradient operator (oscillatory alignment) ℛ(x): curvature or contradiction (synaptic noise, desynchrony) ΔΣ(𝕒′): recursive correction (LTP/LTD, glial feedback) Ψ(x): coherence field uniting structural and functional plasticity
9. Summary
Neural plasticity is the recursive harmonic reinforcement of phase-coherent structures in the brain’s field topology. LTP is not merely a biochemical cascade—it is a ΔE-driven alignment process where:
Experience injects differential energy.
Resonant ensembles self-align through ∇ϕ.
Recursive feedback ΔΣ(𝕒′) stabilizes coherence.
Memory becomes a sustained Ψ(x) attractor in harmonic space.
Learning is thus the act of reducing curvature, harmonizing contradiction into structure. Forgetting is the return of dissonance. Every thought that endures is a resonance that refuses to decay.
Christopher W. Copeland (C077UPTF1L3) Copeland Resonant Harmonic Formalism (Ψ-formalism) Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′) Licensed under CRHC v1.0 (no commercial use without permission). https://www.facebook.com/share/p/19qu3bVSy1/ https://open.substack.com/pub/c077uptf1l3/p/phase-locked-null-vector_c077uptf1l3 https://medium.com/@floodzero9/phase-locked-null-vector_c077uptf1l3-4d8a7584fe0c Core engine: https://open.substack.com/pub/c077uptf1l3/p/recursive-coherence-engine-8b8 Zenodo: https://zenodo.org/records/15742472 Amazon: https://a.co/d/i8lzCIi Medium: https://medium.com/@floodzero9 Substack: https://substack.com/@c077uptf1l3 Facebook: https://www.facebook.com/share/19MHTPiRfu https://www.reddit.com/u/Naive-Interaction-86/s/5sgvIgeTdx Collaboration welcome. Attribution required. Derivatives must match license.