Comprehensive Neurobiology Notes: Electrochemical Gradients, Channel Dynamics, and Action Potentials

Fundamentals of Graphing and System Modeling

• Variable Assignment on Coordinate Axes:

  • The horizontal axis (x-axis) conventionally represents the independent variable, which progresses on its own (most commonly time, $t$).

  • The vertical axis (y-axis) represents the dependent variable, which changes as a function of the independent variable.

• Data Point Interpolation and Plotting Dynamics:

  • Connecting discrete data points with straight lines vs. smooth curves depends on the underlying physical process. Straight line connections should not be drawn by default without evaluating the system's dynamics.

  • When data points represent discrete experimental observations without intermediate measurements, interpolating curves assumes unknown behavior.

• The Principle of Causality (Causal Universe):

  • In a causal physical system, effects cannot precede their causes ($t_{\text{effect}} \ge t_{\text{cause}}$).

  • On time-series plots, if a discrete event or perturbation occurs at time $t = b$, no deflection, response curve, or change in the dependent variable can appear prior to $t = b$.

The Leaky Tank Model and Electrical Analogs

• Hydrostatic Pressure and Fluid Mechanics:

  • Hydrostatic pressure in a fluid container is generated by molecular collisions against the container walls and increases proportionally with fluid depth.

  • Diving Physiology & Pressure Equalization:

  • Descending underwater (in a pool or ocean) to depths below approximately $10\,\text{ft}$ increases external hydrostatic pressure against the tympanic membrane (eardrum).

  • Without active pressure compensation, the unbalanced external pressure forces the eardrums inward, causing severe pain and risking membrane rupture.

  • Valsalva Technique: Divers plug their nose and exhale gently to force air through the eustachian tubes into the middle ear. This equalizes internal ear pressure with external hydrostatic pressure and must be performed repeatedly during descent.

• Step-by-Step Current Dynamics in a Fluid Tank:

  • Interval $a \rightarrow b$: The fluid tank has no open holes/leaks. Water flow current $I = 0$.

  • State $b$: A single leak opens at the bottom of the full tank. Water current jumps rapidly to a baseline value ($I = 1$). The current remains relatively constant assuming the tank volume is large relative to the leak rate.

  • State $c$: A second identical leak opens (2 total leaks). Water current doubles instantly ($I = 2$).

  • Pressure Increase: Doubling the water level (doubling hydrostatic pressure) doubles the total current output across both open leaks from $2$ to $4$.

  • Partial Occlusion: Plugging one leak reduces the total current by half (from $4$ back to $2$).

  • Separation of Time Scales: Relieving fluid pressure by draining tank volume is a slow, macroscopic process relative to the near-instantaneous change in flow current measured immediately across open holes.

• Biophysical Equivalences (Fluid to Electrical):

  • Hydrostatic Pressure $\equiv$ Electrical Voltage ($V$) (Electrical potential difference or driving force).

  • Fluid Flow Rate / Current $\equiv$ Electrical Current ($I$) (Flow of charged particles).

  • Tank Leaks / Holes $\equiv$ Ion Channels / Membrane Conductances ($G$).

  • Pressure Gradient (Internal vs. External) $\equiv$ Electrochemical Driving Force.

  • Key Distinction: Voltage can be positive or negative relative to ground, whereas absolute pressure cannot be negative (in a vacuum, atmospheric pressure can only push outward).

Questions & Discussion: Plotting and Homework Standards

• Question: When plotting discrete state transitions, should lines be drawn vertically or curved?

• Answer: If the transition time scale is instantaneous relative to the plotted time axis, a vertical step function is appropriate. If the physical process has finite transition kinetics, a smooth curve following the onset time is drawn. Deflections preceding event onset are strictly incorrect.

• Question: How does total current scale with open channel count?

• Answer: For non-interacting, identical ion channels, total ionic current $I$ is directly proportional to the number of open channels $N$ multiplied by single-channel current $i$:   $I = N \times i$   If zero channels are open, total current is strictly $0$.

Electrochemical Gradients and the Nernst Potential

• Physical Nature of Electrical Current:

  • In metallic conductors, current is carried by mobile electrons ($e^-$).

  • In biological aqueous solutions, current is carried by mobile charged dissolved ions ($K^+$, $Na^+$, $Cl^-$, $Ca^{2+}$).

  • Directional Convention: Standard electrical current is defined as the net flow of positive charge. The movement of a positive ion in direction $+x$ produces an identical electrical current to a negative ion moving in direction $-x$.

• The Nernst Equation:

  • Calculates the equilibrium potential (reversal potential) for a specific ion species across a membrane:     $E_{\text{ion}} = \frac{RT}{zF} \ln\left(\frac{[\text{Ion}]_{\text{out}}}{[\text{Ion}]_{\text{in}}}\right)$

  • Gas Constant ($R$): Represents the kinetic thermal energy of particle motion ($8.314\,\text{J\,mol}^{-1}\text{K}^{-1}$).

  • Absolute Temperature ($T$): Expressed in Kelvin ($\text{K}$). Human core body temperature is $37^\circ\text{C} = 310\,\text{K}$.

  • Environmental variations: Bird body temperature $\approx 40^\circ\text{C}$ ($313\,\text{K}$); Antarctic sea stars operate near ocean water temperatures down to $-5^\circ\text{C}$ ($268\,\text{K}$) without freezing due to high salinity.

  • Valence ($z$): Electrical charge of the ion ($+1$ for $K^+$ and $Na^+$; $+2$ for $Ca^{2+}$).

  • Faraday's Constant ($F$): Charge per mole of electrons ($\approx 96,485\,\text{C\,mol}^{-1}).\n - **Logarithmic Concentration Ratio**: The magnitude and sign of E_{\text{ion}} depend strictly on the ratio of extracellular to intracellular ion concentration.\n\n- **Quantitative Calculation for Potassium (K^+)**:\n - Given concentrations: [K^+]_{\text{out}} = 20\,\text{mM}$and$[K^+]_{\text{in}} = 400\,\text{mM} (squid giant axon values):\n    \frac{[K^+]{\text{out}}}{[K^+]{\text{in}}} = \frac{20\,\text{mM}}{400\,\text{mM}} = 0.05\n    \ln(0.05) \approx -2.9957\n    \frac{RT}{zF} = \frac{(8.314\,\text{J\,mol}^{-1}\text{K}^{-1})(310\,\text{K})}{(1)(96485\,\text{C\,mol}^{-1})} \approx 0.0267\,\text{V} = 26.7\,\text{mV}\n    E_K = 0.0267\,\text{V} \times (-2.9957) \approx -0.07998\,\text{V} = -79.98\,\text{mV} \approx -80\,\text{mV}\n\n- **Conservation of Intracellular Concentration**:\n - A mole is an extremely large quantity (6.022 \times 10^{23}\text{ particles}). The physical number of ions moving across a cell membrane to shift membrane potential is infinitesimally small relative to total bulk concentration.\n - Intracellular and extracellular ion concentrations remain essentially constant during action potentials.\n - Extreme nutritional salt deprivation can collapse extracellular Na^+ gradients, severely impairing neural signaling. Ruminants (cows) utilize environmental salt licks to maintain systemic ionic balance.\n\n\n# Membrane Dynamics and Resting Membrane Potential\n\n- **Definitions and Terminology**:\n - **Equilibrium Potential / Reversal Potential (E_{\text{ion}}$)**: The specific membrane potential at which electrical driving force perfectly balances chemical concentration driving force, resulting in net zero ionic flux ($I_{\text{ion}} = 0).\n - **Resting Membrane Potential (V_m$)**: The baseline steady-state electrical potential across a cell membrane when no active signaling occurs (typically$-65\,\text{mV} in neurons).\n\n- **Driving Force Mechanics**:\n - Electrochemical driving force is defined as (V_m - E_{\text{ion}}).\n - At V_m = -65\,\text{mV}$, potassium ($E_K = -80\,\text{mV}$) is **not** at equilibrium. The outward concentration force exceeds the inward electrical force; opening$K^+$channels causes$K^+ to flow **out** of the cell.\n - If V_m$is experimentally pulled to$-80\,\text{mV}$, net$K^+$flow ceases ($I_K = 0).\n - If V_m$is hyperpolarized to$-90\,\text{mV}$, electrical attraction dominates, pulling$K^+ **into** the cell against its concentration gradient.\n - Sodium (E_{Na} \approx +35\,\text{mV}$to$+50\,\text{mV}$) experiences aligned chemical and electrical driving forces at$-65\,\text{mV}$, causing rapid inward$Na^+$flow ($Na^+$influx) whenever$Na^+ channels open.\n\n- **Ionic Basis of the Resting Potential**:\n - **Glial Cells**: Possess exclusively passive K^+$leak channels. Consequently, glial membrane potential sits exactly at$E_K = -80\,\text{mV}.\n - **Neurons**: Possess passive leak channels for both K^+$and$Na^+.\n - **Correction of Common Misconception**: The primary reason the resting neuron sits at -65\,\text{mV}$is **not** pump electrogenicity ($3\,Na^+$out /$2\,K^+$in), but because the resting membrane has a **far higher permeability to$K^+$than to$Na^+$** ($P_K \gg P_{Na}).\n - At -65\,\text{mV}$, a small inward$Na^+$leak current through a few channels balances a moderate outward$K^+ efflux through many channels, establishing steady state.\n - Dynamically changing the relative permeability ratio (P_{Na} / P_K$) allows the cell to set its membrane potential to any arbitrary voltage bounded between$E_K = -80\,\text{mV}$and$E_{Na} = +35\,\text{mV}.\n\n\n# Equivalent Circuit Model, Capacitance, and Myelin\n\n- **Electrical Circuit Model of the Cell Membrane**:\n - **Batteries**: Represented by Nernst equilibrium potentials (E_K, E_{Na}, E_{Cl}) generated by ionic concentration gradients.\n - **Conductors / Resistors**: Represented by membrane ion channels. Conductance (G$) is the reciprocal of resistance ($R = 1/G). Parallel channel conductances add directly:\n    G_{\text{total}} = G_1 + G_2 + \dots + G_n\n - **Capacitor (C)**: The lipid bilayer acts as a dielectric insulator separating inner and outer charged solutions, storing electrostatic charge.\n\n- **Capacitance Dynamics**:\n - Membrane capacitance slows down voltage transitions, causing applied current pulses to produce rounded, exponential charging curves rather than instantaneous step changes.\n\n- **Myelination and Multiple Sclerosis**:\n - Wrapping multiple layers of glial plasma membrane around an axon increases the physical separation between intracellular and extracellular conducting fluids.\n - **Functional Consequences**: Myelination drastically decreases membrane capacitance (C_m$) and increases membrane resistance ($R_m), eliminating charge storage delay and facilitating rapid saltatory conduction.\n - **Multiple Sclerosis (MS)**: An autoimmune disorder directed against central myelin sheaths. Demyelination restores high membrane capacitance and low resistance, causing conduction slowdown, action potential failure, and severe motor/cognitive impairment.\n\n\n# Voltage-Gated Channels and Action Potential Generation\n\n- **Structural Biochemistry of Voltage-Gated Na^+ Channels**:\n - Composed of transmembrane protein subunits containing hydrophobic alpha-helices embedded in the lipid bilayer.\n - **Voltage-Sensing Domain**: Contains a specialized alpha-helix (S4 helix) lined with positively charged amino acid residues.\n - **Mechanical Gating Mechanism**:\n - At negative resting potential (-65\,\text{mV}), the internal negative field pulls the positively charged S4 helix downward, keeping the channel gate closed.\n - Membrane depolarization reduces internal negativity; electrostatic repulsion forces the positively charged helix upward, driving a conformational change that opens the pore.\n\n- **Stochastic Single-Channel Kinetics**:\n - Individual channel opening and closing transitions are probabilistic, driven by thermal collision energy.\n - Depolarization increases the **probability** (P_{\text{open}}) that a given channel is open and extends its average open duration.\n\n- **Action Potential Spike Kinetics**:\n - Total duration of a typical neuronal action potential is approximately 5\,\text{ms}.\n - **Hodgkin-Huxley Positive Feedback Loop**:\n 1. Initial membrane depolarization reaches threshold.\n 2. Voltage-gated Na^+$channels open ($P_{Na} increases).\n 3. Inward Na^+ current flows down electrochemical gradient.\n 4. Membrane potential depolarizes further.\n 5. Additional voltage-gated Na^+ channels open.\n - **Termination Threshold**: The positive feedback loop self-limits near E_{Na} \approx +35\,\text{mV}$, as the net electrochemical driving force on$Na^+ approaches zero.\n\n\n# Voltage Clamp Analysis and Neuropharmacology\n\n- **Squid Giant Axon Model System**:\n - The giant axon of the marine squid genus *Loligo* measures up to 1\,\text{mm} in diameter, allowing insertion of axial internal electrodes for voltage clamp electrophysiology.\n\n- **Voltage Clamp Technique**:\n - An electronic feedback circuit measures V_m$, compares it with a user-defined command voltage, and continuously injects holding current to keep$V_m clamped at a constant level.\n - Eliminates capacitive current delays (I_c$) after an initial brief charging transient, allowing direct measurement of individual ionic currents ($I_{\text{ion}}).\n\n- **Pharmacological Channel Blockers (Neurotoxins)**:\n - **Tetrodotoxin (TTX)**:\n - Isolated from the pufferfish (order Tetraodontiformes; toxic liver/organs), blue-ringed octopus, and certain newt species.\n - Mechanism: Highly specific antagonist of **voltage-gated Na^+$channels** ($Na_v$); does not block passive$Na^+ leak channels.\n - Physiological Effect: Completely abolishes action potentials, causing motor paralysis, sensory blockage, and fatal respiratory failure via paralysis of the diaphragm and intercostal muscles.\n - Voltage Clamp Utility: Applying TTX isolates outward voltage-gated K^+ currents.\n - **Tetraethylammonium (TEA)**:\n - Mechanism: Highly specific blocker of **voltage-gated K^+$channels** ($K_v).\n - Voltage Clamp Utility: Applying TEA isolates inward voltage-gated Na^+ currents.\n - **Experimental Separation**: Co-application and independent application of TTX and TEA allow complete mathematical separation of transient inward Na^+$currents and sustained outward$K^+$$ currents during action potential analysis.