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 ($S_H$) 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:
$$ Z_{mem}(\omega) = R_{leak} \parallel \frac{1}{j\omega C_{mem}} \parallel Z_{active}(V_{mem}) $$
- $C_{mem}$: The lipid bilayer capacitance ($\approx 1 \mu F/cm^2$). Stores the polarization energy.
- $R_{leak}$: The passive ionic leakage (Ohmic).
- $Z_{active}$: The voltage-gated ion channels (Nonlinear, Time-variant). Modeled as variable resistors $R_{Na}(V, t), R_{K}(V, t)$.
- $I_{pump}$: The ATP-driven metabolic current source (e.g., Na+/K+ ATPase).
2. Field Coupling Physics
How does an external Electric Field $\vec{E}{ext}$ create a change in Transmembrane Potential $\Delta V$?
2.1 The Schwan Equation (Low Frequency)
For a spherical cell of radius $R$ in a field $E$, the induced potential is:
$$ \Delta V_{mem} = 1.5 R E \cos(\theta) \frac{1}{1 + j\omega \tau} $$
Where:
* $\tau = R C_{mem} (\rho_{int} + \rho_{ext}/2)$ 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 ($\sigma_{dc}$).
* Low Frequency ($10 Hz – 10 kHz$): Maximum coupling to $V_{mem}$. 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 $T_{noise}$ but not for $V_{mem}$ 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 | $\Delta V_{mem} < 100 mV$ | Electroporation Threshold | Pore formation, Lysis |
| Electric Field | $E < 100 V/m$ (in tissue) | Dielectric Breakdown | Tissue burn, necrosis |
| Current Density | $J < 10 mA/cm^2$ | Electrochemical Limit | pH shifts, gas evolution |
| Temperature Rise | $\Delta T < 1^\circ C$ | 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:
$$ \langle I \rangle = \frac{1}{T} \int_0^T G(V_{rest} + V_{ac} \sin \omega t) (V_{rest} + V_{ac} \sin \omega t) dt \neq 0 $$
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 $S_H$ using purely oscillatory fields, avoiding electrode corrosion.
5. Summary
- The Knob: We control $\vec{E}_{ext}(\omega, t)$.
- The Lever: The cell membrane converts $\vec{E}{ext}$ into $\Delta V$.}$ and $\Delta [Ion]_{int
- The Effect: This shifts the resting potential and ionic gradients, pushing the system state $S_H$ 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 ($R \approx 10 \mu m$).
* Macrophages: Large, amoeboid ($R \approx 20 \mu m$).
* Nerve Axons: Very long cylinders ($\tau \gg \tau_{sphere}$).
Impact: The field $E$ does not induce a uniform $\Delta V_{mem}$. 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 ($\omega \tau \approx 1$).
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 ($\delta \approx d_{separation}$).
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.