Neuronal Computation, Membrane Potential, and Electrochemical Gradients

Fundamentals of Neuronal Computation and Membrane Physics

• Neuronal Computation Overview:

  • Neurons operate similarly to computers by receiving inputs, processing those inputs, and generating an output at the appropriate time.
  • Inputs: Synapses serve as the primary input structures to neurons.
  • Outputs: Neurons generate electrical outputs (such as action potentials) based on internal computational integration.

• Voltage and Potential Difference:

  • Voltage Definition: The work required to separate electrical charges. It represents a potential difference between two distinct points.
  • Neuronal Context: Voltage across a neuronal membrane represents the difference in electrical charge between the inside of the cell and the outside of the cell.
  • Scale of Voltage: Neuronal voltages are extremely small and measured in millivolts ($mV$).
    • Milli prefix: $10^{-3}$ (i.e., $1\text{ mV} = 10^{-3}\text{ V}$).

• Analogy to Chemical Batteries:

  • Battery Mechanism: A physical battery functions via a chemical separation where one solution donates electrons and another accepts electrons across an isolating barrier.
  • Electromotive Force: A physical force acts on the separated electrons, but they cannot travel without a conducting pathway.
  • Circuit Flow: When a conducting path is established, electrons flow through the circuit, creating electrical current.
  • Neuronal Electrical Current: Neurons do not use free electrons to carry current. Instead, electrical charges are carried by mobile charged ions moving across the membrane under environmental forces.

• The Cell Membrane as an Insulator:

  • Structure: Neuronal membranes consist of a lipid bilayer (phospholipid bilayer).
  • Molecular Alignment: Composed of hydrophilic (polar) heads facing aqueous environments and hydrophobic tails facing inward, forming a tight barrier.
  • Physical Function: In physical terms, the lipid bilayer acts as an electrical insulator.
  • Permeability Properties: The lipid bilayer prevents the passive passage of electrical charges, charged ions, and polar molecules, effectively isolating the intracellular fluid from the extracellular fluid.

• Aqueous Cellular Environments and Ion Dissociation:

  • Intracellular Environment: The inside of all cells consists of an aqueous solution containing dissolved salts, proteins, and organic compounds.
  • Extracellular Environment: Neurons are bathed externally in cerebrospinal fluid (CSF), which is an aqueous, salt-rich fluid.
  • Salt Dissociation: When inorganic salts dissolve in aqueous solutions, they dissociate completely into their constituent ionic components.
    • Example: Potassium chloride ($KCl$) in solution dissociates into potassium cations ($K^+$) and chloride anions ($Cl^-$).

Membrane Potential, Ion Channels, and Equilibrium

• Necessity of Ion Channels:

  • Because pure lipid bilayers are impermeable to charged ions, a completely unbroken membrane prevents current flow or movement of charge.
  • To create electrically active neurons, specialized transmembrane proteins called ion channels are embedded within the lipid bilayer to provide pathways for ion movement.

• Equilibrium vs. Physiological Viability:

  • Hypothetical Unregulated System: If a membrane containing open, non-selective ion channels is placed between two ionic solutions, ions diffuse freely until reaching equilibrium.
  • Biological Definition of Equilibrium: Thermodynamic equilibrium in the nervous system means there is no net electrical gradient or voltage difference, resulting in a complete cessation of electrical activity.
  • Consequence: Equilibrium in the nervous system is equivalent to cell death. Neurons must actively maintain non-equilibrium state conditions to remain functional.

• Selective Permeability:

  • Neurons establish a resting membrane potential by utilizing selectively permeable membranes that allow specific ions to pass while blocking others.
  • Mechanism of Charge Separation: If a membrane is permeable only to potassium ($K^+$) and not to chloride ($Cl^-$), $K^+$ moves down its concentration gradient toward the outside, leaving an excess of uncompensated $Cl^-$ on the inside.
  • Generation of Voltage: This asymmetric movement creates a charge separation across the membrane, establishing a measurable electrical voltage.

• Intracellular Anions:

  • The interior of neurons contains non-permeable organic molecules, including proteins, lipids, and internal organelles, which collectively carry a net negative charge.
  • These non-permeable molecules are designated collectively as organic anions ($A^-$).

• Baseline Resting Potential:

  • Due to selective ion permeability and intracellular organic anions, the intracellular space of a neuron at rest is negative relative to the extracellular space.
  • Standard Values: Typical resting membrane potentials ($V_m$) range between $-55\text{ mV}$ and $-70\text{ mV}$ (frequently cited at $-65\text{ mV}$ or $-70\text{ mV}$).

Electrochemical Gradients and the Nernst Equation

• Dual Forces Acting on Ions:

  • 1. Concentration Gradient (Chemical Force):
    • Ions passively move from regions of high concentration to regions of low concentration.
    • $K^+$ concentration is significantly higher inside the cell than outside, generating a concentration force driving $K^+$ outward.
  • 2. Electrical Gradient (Electrostatic Force):
    • Opposite charges attract; like charges repel.
    • Because the inside of the cell is negatively charged (e.g., $-70\text{ mV}$), positively charged cations like $K^+$ and $Na^+$ experience an electrostatic force pulling them inward.

• Equilibrium Potential ($E_{\text{ion}}$):

  • Definition: The specific membrane potential at which the electrical gradient pulling an ion across the membrane exactly balances the concentration gradient pushing it in the opposite direction.
  • Dynamic Equilibrium: At $E_{\text{ion}}$, individual ions continue to cross the membrane via open channels, but the inward current precisely equals the outward current, resulting in zero net flux ($\text{Net Current} = 0$).
  • Every distinct ion species ($K^+$, $Na^+$, $Cl^-$, $Ca^{2+}$) possesses its own unique equilibrium potential based on its concentration ratio and electrical valence.

• The Nernst Equation:

  • Function: Used to calculate the equilibrium potential ($E_{\text{ion}}$) for a single ion species that is selectively permeable across a membrane.
  • General Theoretical Form:

$E_{\text{ion}} = \frac{RT}{zF} \times \text{ln}\frac{[\text{ion}]_{\text{outside}}}{[\text{ion}]_{\text{inside}}}$

*   **Variables Defined:**
    *   $R$ = Universal Gas Constant.
    *   $T$ = Absolute Temperature in Kelvin ($K$). Calculated at mammalian body temperature: $37^\text{o}\text{C} + 273 = 310\text{ K}$.
    *   $z$ = Valence (electrical charge) of the specific ion ($+1$ for $K^+$ and $Na^+$, $-1$ for $Cl^-$, $+2$ for $Ca^{2+}$).
    *   $F$ = Faraday's Constant.
    *   $\text{ln}$ = Natural logarithm.

• Simplified Form at Body Temperature ($37^\text{o}\text{C}$):
  • Converting natural logarithm to base-10 logarithm ($\text{ln}(x) = 2.3 \times \text{log}_{10}(x)$), incorporating constants ($R$, $F$), using body temperature ($37^\text{o}\text{C}$), and scaling units to millivolts yields:

$E_{\text{ion}} = \frac{61.54}{z} \times \text{log}_{10}\frac{[\text{ion}]_{\text{outside}}}{[\text{ion}]_{\text{inside}}}$

*   *Note:* The pre-factor is frequently approximated as $61.54\text{ mV}$ or $62\text{ mV}$.

• Equilibrium Potential Values for Specific Ions:
  • Potassium ($K^+$):
    • Typical Concentrations: $[K^+]_{\text{outside}} = 5\text{ mM}$, $[K^+]_{\text{inside}} = 100\text{ mM}$ (ratio of $1:20$).
    • Valence ($z$): $+1$
    • Calculated Value: $E_K \text{ is approximately } -80\text{ mV} \text{ to } -97\text{ mV}$.
  • Sodium ($Na^+$):
    • Typical Concentrations: $[Na^+]_{\text{outside}} = 150\text{ mM}$, $[Na^+]_{\text{inside}} = 15\text{ mM}$ (ratio of $10:1$).
    • Valence ($z$): $+1$
    • Calculated Value: $E_{Na} = \frac{61.54}{+1} \times \text{log}_{10}(10) = +61.54\text{ mV} \text{ (approximately } +62\text{ mV})$.
  • Chloride ($Cl^-$):
    • Valence ($z$): $-1$
    • Calculated Value: $E_{Cl} \text{ sits near resting membrane potential, approximately } -65\text{ mV}$.

Driving Force and Directionality of Ion Flow

• Fundamental Rule of Ion Movement:

  • Ions always move across the membrane through open conductances in a direction that pulls the current membrane voltage ($V_m$) toward that ion's specific equilibrium potential ($E_{\text{ion}}$).

• Electrochemical Driving Force Definition:

  • Driving Force Formula:

$\text{Driving Force} = V_m - E_{\text{ion}}$

*   **Sign Conventions and Directional Rules:**
    *   **Positive Driving Force ($V_m - E_{\text{ion}} > 0$):** Indicates that the cell membrane must become more negative.
        *   Cations ($+$ charge) flow OUT of the cell.
        *   Anions ($-$ charge) flow INTO the cell.
    *   **Negative Driving Force ($V_m - E_{\text{ion}} < 0$):** Indicates that the cell membrane must become more positive.
        *   Cations ($+$ charge) flow INTO the cell.
        *   Anions ($-$ charge) flow OUT of the cell.
    *   **Zero Driving Force ($V_m - E_{\text{ion}} = 0$):** The system is at equilibrium for that ion; no net current flows.

• Directional Scenarios for Potassium ($K^+$) ($E_K = -97\text{ mV}$):

  • Scenario A ($V_m = -120\text{ mV}$):
    • $\text{Driving Force} = -120\text{ mV} - (-97\text{ mV}) = -23\text{ mV}$ (Negative driving force).
    • Cell must become more positive.
    • $K^+$ flows INTO the cell (moving with its electrical gradient, but against its concentration gradient).
  • Scenario B ($V_m = -97\text{ mV}$):
    • $\text{Driving Force} = -97\text{ mV} - (-97\text{ mV}) = 0\text{ mV}$.
    • Zero net current; $K^+$ moves inward and outward at equal rates.
  • Scenario C ($V_m = -60\text{ mV}$):
    • $\text{Driving Force} = -60\text{ mV} - (-97\text{ mV}) = +37\text{ mV}$ (Positive driving force).
    • Cell must become more negative.
    • $K^+$ flows OUT of the cell (moving with its concentration gradient, but against its electrical gradient).
  • Scenario D ($V_m = +20\text{ mV}$):
    • $\text{Driving Force} = +20\text{ mV} - (-97\text{ mV}) = +117\text{ mV}$ (Large positive driving force).
    • Cell must become more negative.
    • $K^+$ flows OUT of the cell (moving with both its concentration gradient and its electrical gradient).

• Behavior of Sodium ($Na^+$) Flow:

  • Because $E_{Na} \text{ is } +62\text{ mV}$ and real neurons rarely achieve a membrane potential above $+62\text{ mV}$, sodium has a persistent, strong negative driving force under virtually all physiological conditions.
  • Opening of sodium channels almost exclusively results in an influx of $Na^+$ into the cell, driving $V_m$ in a positive direction (depolarization).

• Behavior of Chloride ($Cl^-$) Flow:

  • Because $E_{Cl} \text{ is } -65\text{ mV}$ (nearly identical to standard resting membrane potential):
    • If $V_m$ is slightly more negative than $-65\text{ mV}$, $Cl^-$ flows OUT to make the cell more positive.
    • If $V_m$ is slightly more positive than $-65\text{ mV}$, $Cl^-$ flows IN to make the cell more negative.
  • $Cl^-$ play a crucial role in mediating inhibitory synaptic transmission by stabilizing membrane potential near $-65\text{ mV}$.

Active Transport Mechanisms and Ion Pumps

• Passive vs. Active Transport:

  • Ion Channels (Passive): Allow ions to flow down established electrochemical gradients without consuming metabolic energy.
  • Ion Pumps (Active): Specialized membrane-spanning proteins that expend metabolic energy (typically via ATP hydrolysis) to transport ions against their electrochemical gradients, maintaining concentration differences.

• Sodium-Potassium ATPase ($Na^+/K^+$ ATPase):

  • Mechanism: Uses energy derived from burning one ATP molecule to actively pump three sodium ions ($3\text{ Na}^+$) OUT of the cell and two potassium ions ($2\text{ K}^+$) INTO the cell.

$\text{1 ATP} \rightarrow 3\text{ Na}^+\text{ (out)} + 2\text{ K}^+\text{ (in)}$

*   **Energy Consumption:**
    *   Consumes approximately $50\text{\%}$ of all energy utilized by the brain.
    *   Consumes approximately $\frac{1}{3}$ ($33.3\text{\%}$) of total whole-body energy expenditure at rest.

• Chloride Transporters:

  • $KCC2$ (Potassium-Chloride Cotransporter 2):
    • Function: Chloride exporter (pumps $Cl^-$ out of the cell).
    • Mechanism: Utilizes ATP and co-transports $K^+$ out alongside $Cl^-$ to extrude chloride.
  • $NKCC1$ (Sodium-Potassium-Chloride Cotransporter 1):
    • Function: Chloride importer (pumps $Cl^-$ into the cell).
    • Mechanism: Secondary active transport that does not directly burn ATP; uses the steep inward $Na^+$ gradient as an energy source to drive $Cl^-$ and $K^+$ into the cell.
  • Spatial and Functional Heterogeneity: The relative ratio of $KCC2$ and $NKCC1$ expression determines internal $Cl^-$ concentration within specific cell types or subcellular regions (e.g., dendrites vs. cell body/soma).

• Calcium ($Ca^{2+}$) Regulation and Transporters:

  • Intracellular Free Calcium Levels: Extracellular calcium is far higher than intracellular calcium ($[Ca^{2+}]_{\text{inside}} \text{ is held near } 0.0002\text{ mM}$, virtually zero free floating intracellular calcium).
  • Signaling Role: Unbound intracellular calcium triggers intense biochemical signaling cascades. Extremely low resting baseline levels permit massive signal amplification upon channel opening.
  • Primary Calcium Pumps:
    • 1. Plasma Membrane Calcium ATPase (PMCA): Directly hydrolyzes ATP to actively pump $Ca^{2+}$ out across the plasma membrane.
    • 2. Sodium-Calcium Exchanger (NCX): Secondary active transporter that uses the potential energy of the inward $Na^+$ gradient to extrude $Ca^{2+}$ out of the cell while allowing $Na^+$ to flow inside.

The Goldman-Hodgkin-Katz (GHK) Equation and Resting Potential

• Leak Channels and Relative Permeabilities:
  • Resting membrane potential is governed by passive "leak" channels that remain continuously open.
  • The membrane is simultaneously permeable to multiple ions, but at vastly different rates.
  • Resting Permeability Ratio:

$P_K : P_{Na} : P_{Cl} = 1 : 0.05 : 0.45$

*   Because potassium permeability ($P_K$) is by far the highest, resting membrane potential sits relatively close to $E_K$, but is pulled slightly more positive due to minor resting permeabilities to $Na^+$ and $Cl^-$.

• Goldman-Hodgkin-Katz (GHK) Voltage Equation:
  • Function: Calculates the overall resting membrane potential ($V_m$) when the membrane is simultaneously permeable to multiple distinct ionic species.
  • Theoretical Equation:

$V_m = \frac{RT}{F} \times \text{ln}\frac{P_K[K^+]_{\text{out}} + P_{Na}[Na^+]_{\text{out}} + P_{Cl}[Cl^-]_{\text{in}}}{P_K[K^+]_{\text{in}} + P_{Na}[Na^+]_{\text{in}} + P_{Cl}[Cl^-]_{\text{out}}}$

*   *Note on Valence:* Because chloride is a monovalent anion ($z = -1$), its concentration terms are inverted relative to cations (intracellular concentration is placed in the numerator, extracellular concentration in the denominator).

• Simplified GHK Equation at Body Temperature ($37^\text{o}\text{C}$):

$V_m = 61.54 \times \text{log}_{10}\frac{P_K[K^+]_{\text{out}} + P_{Na}[Na^+]_{\text{out}} + P_{Cl}[Cl^-]_{\text{in}}}{P_K[K^+]_{\text{in}} + P_{Na}[Na^+]_{\text{in}} + P_{Cl}[Cl^-]_{\text{out}}}$

• Calculated Resting Voltage:

  • Plugging standard physiological concentration values and relative permeabilities ($1 : 0.05 : 0.45$) into the GHK equation yields an exact calculated resting potential of approximately $-68.35\text{ mV}$.

• Summary of Resting Potential Determination:

  • 1. Semipermeable lipid bilayer insulates charge.
  • 2. Active ion pumps (e.g., $Na^+/K^+$ ATPase) establish steep concentration gradients.
  • 3. Continuous presence of resting leak channels creates high selective permeability for $K^+$ relative to $Na^+$ and $Cl^-$.
  • 4. Membrane rests slightly more positive than $E_K$ (at $-68\text{ mV}$ to $-70\text{ mV}$).

• Experimental/Thought Scenario: Extreme Hyperkalemia:

  • Hypothetical Condition: Increasing extracellular potassium ($[K^+]_{\text{outside}}$) by 100-fold.
  • Effect on Concentration Ratio: The ratio $\frac{[K^+]_{\text{outside}}}{[K^+]_{\text{inside}}}$ increases significantly, causing the logarithmic term in the Nernst equation to shift drastically.
  • Effect on $E_K$: The reversal potential for potassium ($E_K$) shifts above $+100\text{ mV}$ into positive values.
  • Effect on Neuron: The resting membrane potential ($V_m$) depolarizes severely, eliminating the negative baseline required for electrical excitability.

Questions & Discussion

• Question: Are electrical and concentration gradients always opposite to each other?

  • Response: No, they are not always opposite. Their relative directions depend entirely on the valence of the ion, the precise concentration gradient between the inside and outside of the cell, and the current membrane voltage ($V_m$) relative to the ion's equilibrium potential. In certain instances (e.g., potassium at $V_m = +20\text{ mV}$), both concentration and electrical gradients act in the exact same direction.

• Question: Is the voltage inside of the cell always changing?

  • Response: Yes, the intracellular potential is continuously fluctuating. Neurons receive constant, dynamic synaptic inputs that alter ion conductances. At rest, neurons sit near $-55\text{ mV}$ to $-70\text{ mV}$, but during active signals such as action potentials, current flow across the membrane temporarily drives the voltage into positive values.