Part 3: The Membrane-Field Transducer
The mechanism of action. How we use frequency-selective electromagnetic fields to lower the activation energy for regeneration.
TASK 4: The Membrane–Field Interaction Core
Version: 1.1 (Updated with Validation)
Status: DEFINED
Date: 2026-01-16
1. The Transducer Model
The primary mechanism for coupling external control fields (u) to the biological state (SH) is the Cell Membrane. It acts as a nonlinear rectifier and integrator of electromagnetic energy.
We model a tissue patch as a distributed network of these equivalent circuits:
1.1 The Circuit Element
Each cell membrane is modeled as:
Zmem(omega) = Rleak parallel frac{1}{jomega Cmem} parallel Zactive(Vmem)
- Cmem: The lipid bilayer capacitance (≈ 1 μ F/cm2). Stores the polarization energy.
- Rleak: The passive ionic leakage (Ohmic).
- Zactive: The voltage-gated ion channels (Nonlinear, Time-variant). Modeled as variable resistors RNa(V, t), RK(V, t).
- Ipump: The ATP-driven metabolic current source (e.g., Na+/K+ ATPase).
2. Field Coupling Physics
How does an external Electric Field E{ext} create a change in Transmembrane Potential Δ V?
2.1 The Schwan Equation (Low Frequency)
For a spherical cell of radius R in a field E, the induced potential is:
Δ Vmem = 1.5 R E cos(theta) frac{1}{1 + jomega τ}
Where:
* τ = R Cmem (ρint + ρext/2) is the relaxation time constant.
* Implication: Larger cells couple more strongly to the field.
2.2 The Frequency Windows
The coupling efficiency depends strictly on frequency (f):
* DC (0 Hz): Screened by the Debye layer. No penetration into cytoplasm. Moves ions in ECM only (σdc).
* Low Frequency (10 Hz – 10 kHz): Maximum coupling to Vmem. The membrane acts as an insulator, dropping the entire field potential across the bilayer. This is the Control Window.
* RF (> 1 MHz): Capacitive short-circuit. The field passes through the cell, heating the cytoplasm (Dielectric Heating). Useful for Tnoise but not for Vmem control.
3. Safe Field Envelopes
To ensure the control u(t) remains in the Permissive Window and does not cause damage (exit the safety envelope), we enforce:
| Parameter | Limit | Physical Reason | Failure Mode |
|---|---|---|---|
| Induced Potential | Δ Vmem < 100 mV | Electroporation Threshold | Pore formation, Lysis |
| Electric Field | E < 100 V/m (in tissue) | Dielectric Breakdown | Tissue burn, necrosis |
| Current Density | J < 10 mA/cm2 | Electrochemical Limit | pH shifts, gas evolution |
| Temperature Rise | Δ T < 1^circ C | Metabolic Stress | Protein denaturation |
4. The Nonlinear Rectification Effect
Why does an AC field produce a biological effect?
Because the membrane conductance G(V) is nonlinear:
langle I rangle = frac{1}{T} int0T G(Vrest + Vac sin omega t) (Vrest + Vac sin omega t) dt neq 0
Even a zero-mean AC field produces a non-zero mean change in ionic concentration due to the rectification of voltage-gated channels. This allows us to “pump” the state vector SH using purely oscillatory fields, avoiding electrode corrosion.
5. Summary
- The Knob: We control Eext(omega, t).
- The Lever: The cell membrane converts E{ext} into Δ V.} and Δ [Ion]int
- The Effect: This shifts the resting potential and ionic gradients, pushing the system state SH out of the Scar Attractor and towards the Regeneration Attractor.
6. Heterogeneity & Geometry (Validation Update)
6.1 Tissue Heterogeneity
The Schwan equation assumes a uniform sphere. Real tissue is a heterogeneous mix of:
* Fibroblasts: Small, spindle-shaped (R ≈ 10 μ m).
* Macrophages: Large, amoeboid (R ≈ 20 μ m).
* Nerve Axons: Very long cylinders (τ gg τ_{sphere).
Impact: The field E does not induce a uniform Δ Vmem. Large cells (Macrophages/Nerves) are “louder” listeners to the field than small fibroblasts. The control field must be tuned to target specific populations via frequency selection (omega τ ≈ 1).
6.2 Electrode Geometry
The field distribution E(x,y,z) depends critically on electrode placement:
* Parallel Plate: Uniform E. Best for experimental control.
* Coaxial: E propto 1/r. High gradient near center.
* Patch: Fringing fields. Shallow penetration depth (delta ≈ dseparation).
Requirement: The simulator (Task 6) must include a Finite Element Model (FEM) of the electrode geometry to calculate the local field experienced by the tissue voxel.