High Level Overview

• holding cell more negative forces sodium channels to complete cycle faster

  • more of them moving from inactivated to closed state

• Passive ( no voltage gated channels , seen in dendrites ) :

  • voltage decays with distance if no regenerative channels are involved

• Dynamic ( voltage gated channels present , seen in unmyelinated axons ) :

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

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

Reverse Calculate The Intercellular Concentration ( denominator ) from Some Random Voltage

• lets say they tell you $V_m=-80mV$$V_m = -80\ mV$ and $[K{]}_{\text{outside}}=4mM$$[\ K\ ]_{\text{outside}} = 4\ mM$

• what must $[K{]}_{\text{inside}}$$[\ K\ ]_{\text{inside}}$ be then ?

Example Cell , Finding Conductance Ratio and gTotal

• pick random values for :

  • $E_K=-90mV$$E_K = -90\ mV$

  • $E_{Na}=+55mV$$E_{Na} = +55\ mV$

  • Resting Membrane Potential = $-55mV$$-55\ mV$

• At steady state , you set the sum of all currents equal to zero

  $$0=(g_{Na}\ast {\text{DF}}_{Na})+(g_K\ast D{F}_K)$$

• we are solving for ratio $\frac{g_{Na}}{g_K}$$\frac{g_{Na}}{g_{K}}$

  • and we expect it to be near $0.2$$0.2$

  • because we are at resting membrane potential ( current is zero )

    • so sodium channels are not very conductive ( small g )

    • potassium channels are 5 times more conductive than sodium at resting membrane potential

  • for this example :

    $$\frac{I_{Na}}{I_K}=\frac{0mA}{0mA}=\frac{g_{Na}\ast 110mV}{g_K\ast 35mV}$$

• we could do the algebra again , but a shortcut is just to remember to flip them from the way you would naturally write it like above :

• we can also pick some random current value for $I_{Na}$$I_{Na}$ and $I_K$$I_K$ at RMP.

  • we expect current at RMP to be zero

    • so $I_{Na}$$I_{Na}$ and $I_K$$I_K$ must be equal and opposite

    • let $I_{Na}=-30mA$$I_{Na} = -30\ mA$ = inward current

    • let $I_K=+30mA$$I_{K} = +30\ mA$ = outward current

• we then project out from the $I_{Na}$$I_{Na}$ and $I_K$$I_K$ values to their Nernst values

• then we can project outward for potassium , and downward for sodium ,

  • allowing you to draw the orange line in the middle

• the orange line in the middle = total current

  • the slope of it = total conductance

Capacitance ( C )

• adding myelin to an axon increases the diameter

  • $C\frac{1}{\propto }\text{diameter}$$C\ \frac{1}{\propto}\ \text{diameter}$

  • so this decreases the capacitance

• decreasing the capacitance , decreases the time constant

  • $\tau \propto C$$\tau \propto C$

  • decreasing the time constant = charging and discharging times are faster for the cell

Time Constant ( τ )

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

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

  • scaled by $V_{\text{max}}\ast (1-\frac{1}{e})$$V_{\text{max}}* \left(\ 1 - \frac{1}{e} \ \right)$

• Also represents the time to reach 63% of discharging voltage after a step current

  • you can also think of this as 37% of the initial value

    • scaled by $V_{\text{max}}\ast \frac{1}{e}$$V_{\text{max}}* \frac{1}{e}$

• this image is for current $I$$I$ , but its for visualization of the 63% and 37%

Length Constant ( λ )

• tells us how far voltage spreads in an axon cylinder shape

• $\lambda \propto R$$\lambda \propto R$ ,

  • so larger length constance = larger resistance

    • faster conduction

    • $V\propto R$$V \propto R$

• scale by $V_{max}\ast \frac{1}{e}$$V_{max} * \frac{1}{e}$

Chord Conductance ( g )

• $V$$V$ is the membrane voltage at which you measured the current

• $I$$I$ is the measured current at voltage $V$$V$

• $E_{\text{rev}}$$E_{\text{rev}}$ is the reversal potential ( voltage where net current is zero )

• $I_{\text{rev}}$$I_{\text{rev}}$ is the measured current at $E_{\text{rev}}$$E_{\text{rev}}$ ( often zero if $E_{\text{rev}}$$E_{\text{rev}}$ is truly the point where the I–V curve crosses zero )

• it is the chord conductance = the slope of the line connecting $(E_{\text{rev}},I_{\text{rev}})$$(\ E_{\text{rev}}\ ,\ I_{\text{rev}}\ )$ to $(V,I(V))$$(\ V\ ,\ I(\ V\ )\ )$ on the I–V plot

• 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.

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 is then the inverse

     • $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_s$$R_s$ ):

   $$R_m=\frac{R_s}{\text{Cell Area}}$$
   $$R_s=R_m\ast \text{Cell Area}$$

• 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

• Also note from the $R_m=\frac{R_s}{\text{Cell Area}}$$R_m = \frac{R_s}{\text{Cell Area}}$ version , that

  • $R_m\frac{1}{\propto }\text{Cell Area}$$R_m\ \frac{1}{\propto}\ \text{Cell Area}$

  • smaller cells have higher resistance ,

    • what does this mean for charge/discharge times , the $\tau$$\tau$ ?

      • higher resistance is higher $\tau$$\tau$

    • what does this mean for length constant ?

      • its proportional to $R$$R$ in general , and specifically the ratio $\frac{R_m}{R_i}$$\frac{R_m}{R_i}$

        • ( square root is just a special scaling , doesn't impact proportionality statement )

        • so $\lambda \propto R$$\lambda \propto R$

      • therefore a larger $\lambda$$\lambda$ as well

• you could use :

  $$\text{Resistance}=\frac{\rho \ast \text{Length}}{\text{Area}}$$

• and still $R\frac{1}{\propto }\text{Area}$$R\ \frac{1}{\propto}\ \text{Area}$

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.

• these larger axons have a very low intracellular resistance ( $r_i$$r_i$ )

• this increases the length constant by a lot

  • ( decreasing denominator ➡️ increases result )

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

Voltage-Gated Channel Structure

• S4 segment:

  • Contains positively charged amino acids.

  • Functions as the voltage sensor.

Membrane as an Electrical Circuit

• Capacitor: lipid bilayer stores charge.

• Resistor: ion channels regulate current.

• Battery: Nernst potential represents ion concentration gradient.

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

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 9

   • 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.

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

  • this resets all the sodium channels to closed state , where they are ready to open as soon as possible

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.

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