• Ion flow is governed by:

  • Membrane permeability.

  • Concentration gradients.

  • Electrical driving forces.

• Membrane potential arises from:

  • Selective ion channels.

  • Active transport mechanisms.

• Equilibrium potential ( Nernst ) defines the voltage at which ion movement is balanced.

Capacitance

• A capacitor stores charge by separating it across an insulating layer.

• Cell membrane acts as a capacitor with intracellular and extracellular fluid as conductors.

• Stored charge :

  $$Q=C\ast V$$

  • $Q$$Q$ = charge (coulombs)

  • $C$$C$ = capacitance (farads, F)

  • $V$$V$ = voltage (volts)

• The capacitive current is:

  $$I_c=\frac{dQ}{dt}=C\frac{dV}{dt}$$

• Faster voltage change $\Rightarrow$$\Rightarrow$ more current is needed to charge the capacitor.

Time constant ( $\tau$$\tau$ )

• Determines how fast the voltage changes in response to current.

• Represents the time to reach 63% of final voltage after a step current

Steps to Calculate Electrical Properties

1. Measure the time constant ($\tau$$\tau$) – time to reach 63% of steady state.

2. Measure steady-state voltage response to injected current.

   • Use $G=\mathrm{\Delta }I/\mathrm{\Delta }V$$G = \Delta I / \Delta V$ to find conductance.

   • Resistance: $R=1/G$$R = 1/G$.

3. Calculate capacitance using:

   $$C=\frac{\tau }{R}$$

4. Determine cell area:

   $$\text{Cell Area}=\frac{C}{1\mu \text{F}/{\text{cm}}^2}$$

5. Find specific membrane resistance:

   $$R_m=R\ast (\text{Cell Area})$$

• Cell membrane behaves as an RC circuit, affecting signal transmission speed.

• Larger capacitance slows down voltage changes.

• Time constant ($\tau$$\tau$) determines how quickly the membrane potential responds.

• Membrane resistance and capacitance influence action potential dynamics.

Voltage-Gated Channel Structure

• S4 segment:

  • Contains positively charged amino acids.

  • Functions as the voltage sensor.

• Ion channels exhibit high selectivity through steric and electrostatic mechanisms.

• Ion movement replaces hydration shell interactions with channel protein interactions.

• Voltage-gated channels rely on charged amino acids to detect voltage changes.

• Inward rectifiers regulate resting membrane potential by blocking outward K⁺ flow at positive voltages.

• Electrostatic forces, steric fit, and hydration energy govern ion flow through channels.

Membrane as an Electrical Circuit

• Capacitor: lipid bilayer stores charge.

• Resistor: ion channels regulate current.

• Battery: Nernst potential represents ion concentration gradient.

Equivalent Circuits & Passive Electrical Properties

1. Membrane as an RC Circuit

   • Capacitance ($C_m$$C_m$): lipid bilayer stores charge.

   • Resistance ($R_m$$R_m$): ion channels in parallel.

   • Ohm’s Law: $\mathrm{\Delta }V=I\times R$$\Delta V = I \times R$.

2. Input Resistance

   $$R_{\mathrm{in}}=\frac{\mathrm{\Delta }V}{I}.$$

   • Higher $R_{\mathrm{in}}$$R_{\mathrm{in}}$ = bigger voltage change for a given current.

3. Time Constant

   $$\tau =R_m{C}_m.$$

   • Governs how quickly membrane voltage responds to current steps. Numerically, the time to reach ~63% of the final voltage after a step.

4. Length Constant

   $$\lambda =\sqrt{\frac{R_m}{R_i}}$$

   • Determines how far a passive potential spreads along a cable-like structure (dendrite/axon) before decaying significantly.

• Electrophysiology allows precise measurement of membrane currents and potentials.

• Voltage-clamp and patch-clamp techniques revolutionized understanding of ion channels.

• Membrane properties can be modeled as electrical circuits.

• Time constant ($\tau$$\tau$) affects how fast neurons respond to stimuli.

• Single-channel and whole-cell recordings provide insights into ion channel function.

Current-Voltage ( I / V ) Plots

• $y=(m\ast x)+(b)$$y = ( m * x ) + ( b )$

• x-intercept = reversal potential for ion

  • $0=(excel-m\ast ¿x?)+(excel-b)$$0 = ( excel-m * ¿x? ) + ( excel-b )$

• slope = conductance

  • $g=\Delta I/\Delta V$$g = ΔI / ΔV$

  • conductance ( g ) = flow of ions through some separation channel

• linear slope portion = can assume Ohm's Law = constant conductance

• positive slope portion = leak channels , normal voltage gated K+ and Na+ channels

• negative slope portion = inward rectifier currents , inactivation time

1. Chord Conductance

   $$g=\frac{I}{V_m-E_{\mathrm{ion}}}.$$

   • Evaluate at various voltages to map how channels open with depolarization.

2. Separating Na/K Currents

   • Under voltage clamp, set $g_{\mathrm{Na}}=0$$g_{\mathrm{Na}}=0$ to isolate $I_K$$I_K$, or $g_{\mathrm{K}}=0$$g_{\mathrm{K}}=0$ to isolate $I_{\mathrm{Na}}$$I_{\mathrm{Na}}$.

   • Examine each current’s activation, inactivation, reversal potential, etc.

Conductance-Voltage ( G / V ) Plots

• Derived from I-V data using :

• Normalized conductance shows sigmoidal activation curve.

• we just need Gmax

• then divide in half ( Gmax / 2 )

• we then want the corresponding voltage point on the curve , we call this ( V1/2 )

• V1/2 is the voltage at which 50% of the channels are open

• if V1/2 is more negative , the channel activates at lower voltages ( probably sodium )

• if V1/2 is more positive , the channel requires a stronger depolarization to activate ( probably potassium )

Single-Channel Events & Stochastic Gating

1. Microscopic Currents

   • Individual channels open in discrete, square-step events.

   • The macroscopic current is the sum of many channels.

2. Na${}^{+}$$^+$ vs. K${}^{+}$$^+$ Channels

   • Na${}^{+}$$^+$: open quickly upon depolarization, then inactivate.

   • K${}^{+}$$^+$: slower to open but often remain open longer with minimal inactivation (in classical models).

3. Voltage Dependence

   • Higher depolarization usually increases open probability (especially for K${}^{+}$$^+$ channels).

4. Open-Time / Closed-Time

   • By repeated single-channel recordings, you measure average open times, closed times, gating rate constants, etc.

Studying Na⁺ Channel Activation

• Prepulses at negative voltages relieve inactivation.

Studying Na⁺ Channel Inactivation

• Inactivation protocol:

  • Prepulse at various voltages to control inactivation.

  • Test pulse applies the same voltage to assess remaining available channels.

• Inactivation range:

  • Na⁺ channels inactivate between -80 mV and -50 mV.

• Observations:

  • More negative prepulse → More channels available to open.

  • Less negative prepulse → More channels remain inactivated.

Consequences of Na⁺ Channel Inactivation

1. Transient Na⁺ currents:

   • Even with sustained depolarization, Na⁺ channels inactivate.

2. Refractory period:

   • Prevents immediate reactivation of Na⁺ channels after an action potential.

3. Unidirectional action potential propagation:

   • Ensures nerve impulses travel in one direction.

4. Prolonged depolarization leads to paralysis:

   • Na⁺ channel inactivation prevents further excitation.

• Voltage-gated Na⁺ channels exhibit rapid activation and inactivation.

• I-V and G-V plots help analyze activation kinetics.

• Inactivation prevents sustained Na⁺ influx, crucial for action potentials.

• Refractory periods and unidirectional propagation depend on Na⁺ inactivation.

Influence of Ion Concentrations

• Lower $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ → smaller AP amplitude.

• Higher $[{\mathrm{K}}^{+}{]}_{\mathrm{out}}$$[\mathrm{K}^+]_{\mathrm{out}}$ → more depolarized rest, smaller amplitude or fails to fire.

Voltage Clamp Protocols (Activation/Inactivation)

(Relevant to Guides #6, #7)

1. Voltage-Step Protocols

   • Condition at a certain potential (e.g. -90 mV), then step to a test potential (e.g. -20 mV).

   • Measure current amplitude to see how many channels activate or remain inactivated.

2. Midpoint of Activation

   • The $V_m$$V_m$ where $g$$g$ is half of $g_{max}$$g_{\max}$. Determined from $g$$g$–$V$$V$ curves.

3. Midpoint of Inactivation

   • The conditioning $V_m$$V_m$ that inactivates half the channels.

   • Found by applying various prepulses, then stepping to a fixed test potential.

4. Na vs. K

   • Na${}^{+}$$^+$ channels typically show both strong activation and inactivation; K${}^{+}$$^+$ channels require stronger depolarization, often show little inactivation.

Skeletal Muscle Weakness & Inexcitability

1. Shifts in Resting $V_m$$V_m$

   • Disease can produce abnormal channel gating, changing $V_m$$V_m$ from e.g. -70 to -90 mV and making it harder to fire or causing partial channel inactivation.

2. Input Resistance & Time Constant

   • If $R_{\mathrm{in}}$$R_{\mathrm{in}}$ or $C_m$$C_m$ are altered, the fiber’s response changes. Could need bigger/longer stimuli to reach threshold.

3. Pathophysiological Relevance

   • Small disruptions in channel function or ion gradients can cause muscle weakness or periodic paralysis.

Electrical Propagation in Axons

1. Passive Spread (Cable Properties)

   • Voltage decays with distance if no regenerative channels are involved.

   • Larger diameter reduces $R_i$$R_i$, bigger $\lambda$$\lambda$. Lower $C_m$$C_m$ or $g_{\mathrm{leak}}$$g_{\mathrm{leak}}$ can help signals travel faster/farther.

2. Unmyelinated Axons

   • Conduction velocities can be a few to ~20 m/s (like in squid giant axon).

   • Speed often scales with axon diameter.

3. Myelinated Axons

   • Myelin drastically lowers $C_m$$C_m$ and raises $R_m$$R_m$ in internodes → saltatory conduction (jump node-to-node).

   • Can reach 100+ m/s in large vertebrate axons.

4. Experimental Observations

   • Changing diameter, $C_m$$C_m$, or $g_{\mathrm{leak}}$$g_{\mathrm{leak}}$ changes conduction speed.

   • More “wraps” (myelin) → less internodal capacitance → faster conduction.

Passive Axon Properties

• Voltage spread along an axon without ion channels is insufficient for signaling.

• Enhancement mechanisms:

  • Voltage-gated ion channels: amplify and regenerate the signal.

  • Myelin: insulates the axon and speeds up conduction.

Squid Giant Axon: An Evolutionary Solution

• Diameter ~500 μm → increases conduction speed.

• Unmyelinated, but large size compensates for slow conduction.

• If all human neurons were this large, they would take up too much space.

Comparing Myelinated vs. Unmyelinated Axons

Myelin decreases capacitance and increases resistance:

• Decreasing capacitance → Less charge needed to depolarize the next node.

• Increasing resistance → Reduces current loss between nodes.

• Voltage-gated Na⁺ and K⁺ channels enable action potential propagation.

• Myelin dramatically increases conduction velocity while saving space.

• Saltatory conduction allows for efficient and rapid neural signaling.

• The balance between conduction speed, space, and metabolic cost drives axonal adaptations.

Calcium Action Potentials

1. Ca${}^{2+}$$^{2+}$ Channels

   • Open upon depolarization but often inactivate slower, leading to longer-duration APs (plateau-like phases).

2. Replacing Na${}^{+}$$^+$ with Ca${}^{2+}$$^{2+}$

   • Produces slower, longer spikes if Ca${}^{2+}$$^{2+}$ is the main inward current. Peak amplitude might be smaller, but duration is often larger.

3. Cardiac-Like APs

   • Characterized by a plateau due to sustained Ca${}^{2+}$$^{2+}$ current, slower repolarization.

   • Increasing $g_{\mathrm{Ca}}$$g_{\mathrm{Ca}}$ → prolongs plateau.

   • Increasing $g_{\mathrm{Na}}$$g_{\mathrm{Na}}$ → faster upstroke, higher peak.

4. Role of K${}^{+}$$^+$ Channels

   • Eventually repolarize the membrane, but if Ca${}^{2+}$$^{2+}$ channels remain open, the net outward current is delayed, prolonging the AP.

Properties of Voltage-Activated Calcium Channels

1. Subunit Structure :

   • main ${\alpha }_1$$\alpha_1$ subunit

     • has 4 domains

     • the main pore-forming subunit that determines ion selectivity, conductance, and gating properties

   • $\gamma$$\gamma$ :

     • Four-transmembrane-spanning protein, initially found in skeletal muscle calcium channels but also present in the brain (e.g., stargazin).

   • $\beta$$\beta$ :

     • Cytoplasmic protein that binds to the α1 subunit’s intracellular I-II linker, helping with trafficking to the plasma membrane and modulation

   • ${\alpha }_2$$\alpha_2$ :

     • Derived from a single gene and proteolytically cleaved into an extracellular α2 portion and a membrane-spanning δ portion, which stabilizes the channel at the plasma membrane.

   

2. Gate / Voltage Sensor :

   • S4 segment :

     • Contains positively charged amino acids , arginine and lysine , every 3rd or 4th residue

     • unctions as the voltage sensor

   • Depolarization causes deformation in P-loops , creating a pore

3. Selectivity Filter for Calcium $\left(C{a}^{+2}\right)$$\left(\ Ca^{+2} \ \right)$ :

   • Calcium selectivity is determined by the pore loop ( P-loop ) glutamates :

     • Glutamate residues ( E ) are conserved in each of the four domains ( I-IV ) of the α1 subunit

     • These residues form a high-affinity binding site for Ca2+ ,

       • ensuring preferential permeability over other ions

   

   

4. How would you classify or identify different subtypes of calcium channels ?

   • Voltage-gated calcium channels are classified into :

     • High-Voltage Activated ( HVA )

     • Low-Voltage Activated ( LVA ) ( T-type ) channels

   • These subtypes can be distinguished using :

     • Pharmacological blockers ( e.g., ω-Agatoxin IVA for P/Q-type channels )

     • Genetic expression ( different subtypes are present in different tissues )

     • Electrophysiology ( LVA T-type channels activate at lower voltages than HVA channels )

   Channel Type	Clone (CaV)	Function	Blockers
   L-type	CaV1.1 - CaV1.4	Skeletal, cardiac muscle contraction, secretion	DHP (nifedipine), verapamil
   P/Q-type	CaV2.1	Neurotransmitter release (central)	ω-Agatoxin IVA
   N-type	CaV2.2	Neurotransmitter release (peripheral & central)	ω-Conotoxin GVIA
   R-type	CaV2.3	Uncertain	SNX 482
   T-type	CaV3.1 - Cav3.3	Cell excitability	Mibefradil, ethosuximide (controversial)

• L-type blockers:

  • Dihydropyridines (DHPs): Nifedipine, Amlodipine.

  • Non-DHPs: Verapamil (cardiac selective).

• P/Q-type blockers:

  • ω-Agatoxin IVA (funnel web spider toxin).

• N-type blockers:

  • ω-Conotoxin GVIA (cone shell mollusk toxin).

  • Used for neuropathic pain management.

• T-type blockers:

  • Mibefradil, ethosuximide (antiepileptics).

${\alpha }_1$$\alpha_1$ Subunits

1. Q: Explain why the resting $V_m$$V_m$ is usually closer to $E_{\mathrm{K}}$$E_{\mathrm{K}}$ than $E_{\mathrm{Na}}$$E_{\mathrm{Na}}$ in typical neurons. A: The cell is more permeable to $K^{+}$$K^+$ at rest (many K${}^{+}$$^+$ leak channels), so $V_m$$V_m$ lies near $E_{\mathrm{K}}$$E_{\mathrm{K}}$.

2. Q: Sketch/describe an RC charging curve and label $\tau$$\tau$. What if you double $C_m$$C_m$? A: An RC curve rises toward its final voltage, reaching 63% of the total change at $t=\tau$$t=\tau$. Doubling $C_m$$C_m$ doubles $\tau$$\tau$, slowing the voltage response.

3. Q: On an $I$$I$–$V$$V$ plot, identify the slope and reversal potential. A: The slope is conductance ($g$$g$). Where $I=0$$I=0$ is the reversal potential ($E_{\mathrm{rev}}$$E_{\mathrm{rev}}$), usually near $E_{\mathrm{ion}}$$E_{\mathrm{ion}}$ if one ion dominates.

4. Q: How does changing $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ affect AP shape/amplitude? A: Higher $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ → larger driving force for Na${}^{+}$$^+$ → bigger, faster upstroke and more positive peak. Lower $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ does the opposite.

5. Q: Absolute vs. relative refractory periods in terms of Na${}^{+}$$^+$, K${}^{+}$$^+$ channel states? A: Absolute: most Na${}^{+}$$^+$ channels are inactivated → no new AP possible. Relative: some Na${}^{+}$$^+$ channels have recovered but K${}^{+}$$^+$ conductance is high → requires stronger stimulus.

6. Q: Why might a muscle fiber at $-90\text{ mV}$$-90\text{ mV}$ fail to fire if channels become inactivated? A: If resting $V_m$$V_m$ is shifted or channels are in a state with fewer available Na${}^{+}$$^+$ channels, normal stimuli might not reach threshold.

7. Q: Raising $[{\mathrm{K}}^{+}{]}_{\mathrm{out}}$$[\mathrm{K}^+]_{\mathrm{out}}$ from 4.5 mM to 25 mM: effect on resting potential? A: Depolarizes it (less negative) because $E_{\mathrm{K}}$$E_{\mathrm{K}}$ becomes more positive, so $V_m$$V_m$ shifts closer to that new $E_{\mathrm{K}}$$E_{\mathrm{K}}$.

8. Q: A short current pulse doesn’t reach threshold, but a longer one does. Why? A: The membrane (RC circuit) needs enough time to charge up to threshold. A short pulse may end before hitting threshold.

9. Q: Multiple myelin wraps: effect on conduction velocity?

   A: Myelin greatly reduces $C_m$$C_m$ (and increases $R_m$$R_m$) in internodes, so fewer charges are needed to depolarize each segment. Conduction is much faster.