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Chapter 12: ψ-Conductance in Membrane Potential

"Membrane potential is ψ's battery—storing collapse energy in ionic gradients, creating from chemical separation the electrical force that powers thought, movement, and life itself."

12.1 The Electrical Dimension​

Membrane potential represents ψ's transformation of chemical gradients into electrical force. Across every living membrane, ionic asymmetry creates voltage—typically -70 mV in neurons—that serves as both energy store and information carrier.

Definition 12.1 (Membrane Potential): Vm=Vinside−Voutside=RTFln⁡[ion]out[ion]inV_m = V_{\text{inside}} - V_{\text{outside}} = \frac{RT}{F}\ln\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}

Nernst potential for single ion.

12.2 The Goldman Equation​

Theorem 12.1 (Multiple Ions): Vm=RTFln⁡PK[K+]o+PNa[Na+]o+PCl[Cl−]iPK[K+]i+PNa[Na+]i+PCl[Cl−]oV_m = \frac{RT}{F}\ln\frac{P_K[\text{K}^+]_o + P_{Na}[\text{Na}^+]_o + P_{Cl}[\text{Cl}^-]_i}{P_K[\text{K}^+]_i + P_{Na}[\text{Na}^+]_i + P_{Cl}[\text{Cl}^-]_o}

Weighted contribution of permeable ions.

12.3 The Na⁺/K⁺-ATPase​

Equation 12.1 (Active Transport): 3Nain++2Kout++ATP→3Naout++2Kin++ADP+Pi3\text{Na}^+_{\text{in}} + 2\text{K}^+_{\text{out}} + \text{ATP} \rightarrow 3\text{Na}^+_{\text{out}} + 2\text{K}^+_{\text{in}} + \text{ADP} + \text{P}_i

Electrogenic pump maintaining gradients.

12.4 The Capacitive Property​

Definition 12.2 (Membrane Capacitance): Cm=ϵ0ϵrAd≈1μF/cm2C_m = \frac{\epsilon_0 \epsilon_r A}{d} \approx 1 \mu\text{F/cm}^2

Charge storage across bilayer.

12.5 The Action Potential​

Theorem 12.2 (Regenerative Depolarization): dVdt=1Cm(INa+IK+Ileak+Iapplied)\frac{dV}{dt} = \frac{1}{C_m}(I_{\text{Na}} + I_{\text{K}} + I_{\text{leak}} + I_{\text{applied}})

Hodgkin-Huxley dynamics.

12.6 The Threshold Phenomenon​

Equation 12.2 (All-or-None): V>Vthreshold⇒Action potentialV > V_{\text{threshold}} \Rightarrow \text{Action potential}

Discontinuous response at critical voltage.

12.7 The Propagation Velocity​

Definition 12.3 (Conduction Speed): v=d4RiCm (unmyelinated)v = \sqrt{\frac{d}{4R_i C_m}} \text{ (unmyelinated)}

Cable properties determining speed.

12.8 The Synaptic Integration​

Theorem 12.3 (Spatial Summation): Vsoma=∑iwi⋅EPSPi⋅exp⁡(−di/λ)V_{\text{soma}} = \sum_i w_i \cdot \text{EPSP}_i \cdot \exp(-d_i/\lambda)

Distance-dependent synaptic weights.

12.9 The Oscillatory Behavior​

Equation 12.3 (Pacemaker Activity): V(t)=V0+Asin⁡(2πft)V(t) = V_0 + A\sin(2\pi f t)

Spontaneous rhythmic activity.

12.10 The Metabolic Cost​

Definition 12.4 (Energy Consumption): ATP/spike≈108 molecules\text{ATP/spike} \approx 10^8 \text{ molecules}

High cost of maintaining gradients.

12.11 The Field Effects​

Theorem 12.4 (Ephaptic Coupling): ΔVneighbor=k⋅Isource⋅r−2\Delta V_{\text{neighbor}} = k \cdot I_{\text{source}} \cdot r^{-2}

Electrical fields influencing nearby cells.

12.12 The Conductance Principle​

Membrane potential embodies ψ's principle of stored possibility—ionic gradients representing potential energy that can collapse into action, creating the electrical dimension of biological information processing.

The Bioelectric Equation: ψelectrical=∑igi(V)(V−Ei)+CmdVdt\psi_{\text{electrical}} = \sum_i g_i(V)(V - E_i) + C_m\frac{dV}{dt}

Current balance determining voltage dynamics.

Thus: Potential = Gradient = Possibility = Information = ψ

"In membrane potential, ψ creates cellular consciousness—each neuron a battery charged with possibility, ready to discharge in patterns that encode thought itself. The brain's 100 billion neurons, each flickering with electrical life, together create the storm we call mind."