Sections 2.4–2.8

2.4 Channels and transporters move solutes passively or actively across neuronal membranes

The lipid bilayer of the plasma membrane and membranes of intracellular vesicles are impermeable to most ions and water-soluble molecules; specialized transport proteins are required to move solutes across.
- The lipid bilayer consists of phospholipids with hydrophilic heads and hydrophobic tails, forming a densely packed, impenetrable barrier for hydrophilic molecules.
Two broad classes of membrane transport proteins:
- Channels: provide an aqueous pore that spans the lipid bilayer; solutes pass directly through this water-filled pore when the channel is open. These channels often have a selectivity filter (group of amino acids) making them selective for specific ions.
  - Leak channels: Always open, not gated, and continuously follow concentration and electrical gradients. Typical leak channels, such as potassium channels, are selective for particular ions.
  - Gated channels: Can be open or closed, responding to various signals like membrane potential, ligands, or neurotransmitters.
- Transporters: have two gates that open/close alternately, moving solutes from one side to the other; do not create an open continuous pore.
Passive vs active transport:
- Passive transport: solutes move down their electrochemical gradients through channels or transporters, without external energy input. Often faster through channels than transporters.
- During passive flux, ions follow the electrochemical gradient, driving the membrane potential towards the equilibrium potential (also known as Nernst or reversal potential) for that specific ion.
- Active transport: solutes are moved against their electrochemical gradients, requiring external energy (e.g., ATP hydrolysis, light, or coupling to another gradient).
Energy sources for active transport:
- Chemical energy from ATP hydrolysis via ATPases (most common).
- Light-driven pumps (energy from photon absorption).
- Coupled (cot)ransporters: energy from transporting a second solute down its gradient is used to move another solute up its gradient; subtypes include:
- Symporters: two solutes move in the same direction.
- Antiporters (exchangers): two solutes move in opposite directions.
Figures referenced:
- Figure 2-8A: channels mediate passive transport with an aqueous pore.
- Figure 2-8B: transporters mediate passage via sequential gate openings and energy use.
- Figure 2-9: electrochemical gradients combine chemical and electrical gradients to drive movement.
Electrochemical gradients:
- For uncharged solutes, movement follows the chemical gradient (high to low concentration).
- For ions, movement creates an electrical gradient; the electrochemical gradient is the combination of chemical and electrical gradients and governs direction/magnitude of net movement.
- Thought Experiment: Imagine a semi-permeable membrane only allowing K+ ions. If K+ ions are 10x more concentrated on one side, they will initially flow down their chemical gradient, leaving behind uncompensated negative charges. This separation of charges across the membrane generates a measurable membrane potential. Eventually, the electrical force opposing K+ movement will balance the chemical force, reaching an electrochemical equilibrium where net flux is zero.
Summary relationships:
- Channels: rapid flux when open; selective to specific ions.
- Transporters: slower, require conformational changes; may couple to other gradients or energy sources.
- Net transport under a given condition is often directed (one-way) due to gradients and membrane properties.

2.5 Neurons are electrically polarized at rest because of ion gradients across the plasma membrane and differential ion permeability

The primary goal for this section is to understand the nature of electrical signaling in individual neurons. Neurons communicate through electrical signals (e.g., the 86 billion neurons in the brain generate ~150 Watts of electricity).
Electrical signals rely on a membrane potential: a voltage difference across the plasma membrane, measurable by intracellular recording with a microelectrode.
Resting membrane potential ( $V_m$ ) in neurons typically ranges from about $-50 ext{ mV}$ to $-80 ext{ mV}$ , depending on cell type. Generally, the inside of a neuron is negative relative to the outside (e.g., -60 to -70 mV).
- Nomenclature:
  - Depolarization: The membrane potential becomes less negative.
  - Hyperpolarization: The membrane potential becomes more negative than the resting potential.
Causes of electrical polarization:
- Unequal ion concentrations inside vs. outside the cell.
- Differential membrane permeability to ions (more permeable to K+ than Na+ or Cl– at rest).
An ion is an atom or molecule with a net electrical charge (total number of electrons is unequal to its total number of protons). Cations are positively charged, Anions are negatively charged. Electrostatic force pulls opposite charges together. An electric current is the flow of electric charge.
Typical ionic gradients (model neuron values):
- Key ions involved: Sodium ( $Na^+$ ), Potassium ( $K^+$ ), Calcium ( $Ca^{2+}$ ), Chloride ( $Cl^-$ ), and organic anions (mostly large, non-permeable, concentrated intracellularly).
- $[K^+]i ext{ (e.g., ~120 mM)} ext{ is much higher than } [K^+]o ext{ (e.g., ~4 mM)}$
- $[Na^+]o ext{ (e.g., ~150 mM)} ext{ is much higher than } [Na^+]i ext{ (e.g., ~15 mM)}$
- $[Cl^-]{o} ext{ (e.g., ~120 mM value in transcript; 150 mM text) } ext{ is much higher than } [Cl^-]{i} ext{ (e.g., ~5 mM value in transcript; 10-30 mM text) }$ (intracellular Cl– is relatively low in many neurons)
- $[Ca^{2+}]o ext{ (e.g., ~1.2 mM)} ext{ is much higher than } [Ca^{2+}]i ext{ (very little at rest)}$
Na+/K+ ATPase (Na+-K+ pump):
- Uses energy from ATP hydrolysis to pump 3 Na+ out and 2 K+ in per cycle, maintaining Na+ and K+ gradients and thus the resting potential by actively transporting ions against their electrochemical gradients.
- This pump is very slow (100-10,000 times slower than ion channels).
- Estimated to account for about one third or more of neuronal energy use, underscoring its importance. If this pump is turned off, the ion gradients would eventually dissipate, and the resting potential would diminish to 0 mV, leading to neuronal dysfunction.
Ion cotransporters maintaining the Cl− gradient:
- K+-Cl− cotransporter moves K+ and Cl− together, helping maintain intracellular Cl− levels.
Equilibrium potentials (Nernst potentials) for key ions (at room temperature; example values from text):
- The Nernst equation calculates the equilibrium potential for a single ion. If a cell membrane were exclusively permeable to one ion, its membrane potential would be equal to that ion's equilibrium potential.
- $Ei = rac{RT}{zF} ext{ln} \left(\frac{[\text{ion}]o}{[\text{ion}]_i}\right)$
- R is the gas constant, T is the absolute temperature, z is the valence of the ion (e.g., +1 for K+, +1 for Na+, -1 for Cl-, +2 for Ca2+), and F is the Faraday constant.
- Simplification at room temperature (25°C): The term $rac{RT}{F}$ simplifies to approximately 25 mV. By converting the natural logarithm $ext{ln}$ to a base-10 logarithm $ext{log}_{10}$ (multiplying by 2.3), the equation can be approximated as:
- $Ei = rac{58}{z} \text{log}{10} \left(\frac{[\text{ion}]o}{[\text{ion}]i}\right) ext{ at 25°C}$
- This simplification is common, but the full Nernst equation should be used if temperature is not 25°C.
- Examples (using concentrations from lecture transcript):
 - $EK \approx -87 ext{ mV}$ (using $[K^+]o=4 ext{ mM}$ , $[K^+]_i=120 ext{ mM}$ , $z=+1$ )
 - If $[K^+]o$ doubles to 8 mM, $EK$ would depolarize to approximately -68 mV.
 - $E{Na} \approx +58 ext{ mV}$ (using $[Na^+]o=150 ext{ mM}$ , $[Na^+]_i=15 ext{ mM}$ , $z=+1$ )
 - $E{Cl} \approx -80 ext{ mV}$ (using $[Cl^-]o=120 ext{ mM}$ , $[Cl^-]_i=5 ext{ mM}$ , $z=-1$ - Calculated by instruction in the lecture, note the reversal of outside/inside for negative ions in GHK below)
Resting potential is typically between EK and ENa due to multiple ions contributing with their respective permeabilities; Cl− often has a potential near EK, so its driving force is small at rest.
Driving force for an ion i: $DFi = Vm - E_i$
When multiple ions contribute, the Goldman–Hodgkin–Katz (GHK) framework is used to compute Vm based on permeabilities (P) and ion concentrations. In neurons, the resting potential is determined by a mix of potassium, sodium, and chloride ions, as calcium channels are typically not open at rest ( $P_{Ca} = 0$ ).
Goldman–Hodgkin–Katz (GHK) framework (summary):
- Vm is given by the weighted average of equilibrium potentials, weighted by the relative permeability ($P$) of the membrane to each ion:
- $Vm = rac{RT}{F} \text{ln} \left(\frac{PK[K^+]o + P{Na}[Na^+]o + P{Cl}[Cl^-]i}{PK[K^+]i + P{Na}[Na^+]i + P{Cl}[Cl^-]_o}\right)$
- Note: For chloride ( $Cl^-$ ), the inside and outside concentrations are inverted in the numerator and denominator compared to cations, to account for its negative valence.
- Using typical relative permeabilities ( $PK ext{ approx. 1, } P{Na} ext{ approx. 0.04, } P{Cl} ext{ approx. 0.1}$ ) and ion concentrations, the calculated $Vm ext{ is approximately -67 mV}$ . If sodium permeability were to significantly increase, the neuron's membrane potential would depolarize towards $E_{Na}$ (+58 mV).
Permeability vs conductance:
- Permeability reflects intrinsic ability of the membrane to pass a given ion (depends on open channel numbers). It acts as a proxy for how many channels are in the membrane that are permeable to a given ion.
- Conductance depends on permeability and the presence/concentration of ions; both terms are used to describe ease of ion flow.
- In context, Vm is not equal to any single ion’s equilibrium potential due to multiple active conductances.
Anatomical Underpinnings of Membrane Potential Change:
- Pyramidal Cell Example:
  - Dendrites and Soma (Blue part): The receptive component of the neuron, where it receives synaptic inputs (connections with other neurons).
  - Axon Initial Segment and Axon (Red part): The transmission and effector component, where information is integrated, and the decision to fire an action potential is made.
  - The fundamental function of a neuron is the transformation from synaptic inputs into action potential output.
- For simplicity, in current discussions, neurons are often modeled as little balls (single compartments), ignoring their complex dendritic arbors.

2.6 The neuronal plasma membrane can be described in terms of electrical circuits

The membrane behaves like an electrical circuit with resistive and capacitive elements:
- Lipid bilayer acts as a capacitor (membrane capacitance, $C_m$ ) because it stores charge but blocks direct current. A capacitor consists of two parallel conductors separated by an insulator and acts as a charge-storing device.
- Membrane as a resistor network due to ion channels; a given ion path behaves like a resistor with a battery equal to its equilibrium potential in series with the membrane potential. Ion channels can be modeled as resistors ( $R_I$ ) and ion gradients as batteries (equivalent to the equilibrium potential for that ion).
- These elements (capacitor, resistor, battery) are connected to the membrane's electrical potential (Vm) and an input current generator (IT), which represents synaptic input delivering current into the cell.
Ohm’s law relationships (basic circuit):
- For a resistor: $V = IR$ or equivalently $I = gV$ with $g = 1/R$ (conductance). Ohm's Law describes the relationship between voltage (volts, V), current (amperes, A), and resistance (ohms, $ext{M}\Omega$ ) or conductance (siemens, S).
Key circuit concepts:
- Series: currents are the same; resistances add: $R{total} = R1 + R2$ ; $g{total} ( = 1/R_{total})$ is not simply additive in series.
- Parallel: voltages are the same; currents add: $I{total} = I1 + I2$ ; $rac{1}{R{total}} = rac{1}{R1} + rac{1}{R2}$ (for resistors); conductances add: $g{total} = g1 + g_2$ . For capacitors in parallel, the capacitances sum up.
The neuronal membrane as a circuit:
- The lipid bilayer is an insulator with effectively infinite resistance, but ion channels create parallel current paths that act as resistors with their own batteries (eq. potentials) in the circuit.
- The membrane as a capacitor (with surface area proportional capacitance, typically $ext{~1 }\mu ext{F/cm}^2$ ).
Simple battery + resistor model (left in Figure 2-13): illustrates Ohm’s law and series/parallel combinations.
RC circuit: battery + capacitor; when the switch is closed, the capacitor charges toward the battery voltage with time constant $au = RC$ . If a resistor and capacitor are in series, the current and voltage across each component evolve over time when energized.
In real neurons, membranes are modeled as a parallel RC circuit because many ion channels (resistive paths) are in parallel across the membrane while the membrane itself stores charge (capacitance).

2.7 Electrical circuit models can be used to analyze ion flow across the glial and neuronal membranes

Simplified model (only K+ permeability): two parallel paths:
- Membrane capacitance path ( $C_m$ )
- K+ conducting path (with conductance $gK$ ) and a K+ equilibrium potential $EK$ in series with the membrane potential Vm.
- At rest, net current is zero, so $Vm = EK$ in this simplified case.
More realistic neuron model: four parallel paths (K+, Cl−, Na+, plus capacitance):
- Equations for Vm based on multiple parallel paths:
- $I = IK + I{Cl} + I{Na} + I{Capacitor}$
- In steady rest, $I{Capacitor} = 0 \Rightarrow IK + I{Cl} + I{Na} = 0$
- Vm is given by:
- $Vm = rac{gK EK + g{Cl} E{Cl} + g{Na} E{Na}}{gK + g{Cl} + g{Na}}$
This formulation mirrors the Goldman–Hodgkin–Katz (GHK) framework and provides a practical way to relate conductances and equilibrium potentials to the resting membrane potential.
Distinctions:
- Conductance ( $g$ ) is the functional measure of ion flow given the membrane’s ionic environment; permeability is an intrinsic property related to the number of open channels.
- Vm reflects a weighted balance among multiple ion conductances and their driving forces, not the single equilibrium potential of any one ion.
Ion currents through each pathway:
- $IK = gK (Vm - EK)$
- $I{Cl} = g{Cl} (Vm - E{Cl})$
- $I{Na} = g{Na} (Vm - E{Na})$
- These currents contribute to the overall membrane current and are voltage-dependent because $gK, g{Na}$ change with membrane potential (and time).
Practical takeaway:
- The resting Vm is a composite result of multiple ion conductances and their driving forces, not the single equilibrium potential of any one ion.

2.8 Passive electrical properties of neurons: electrical signals evolve over time and decay over distance

Two key passive properties govern how signals evolve:
- Time constant $au_m$ (temporal dynamics): the rate at which voltage changes in response to current inputs.
- Length (space) constant $\lambda$ (spatial decay): how far electrical signals propagate along a neuron before decaying.
Experimental setup (idealized neuron fiber): inject current at one site and record at multiple distances along the fiber (Figure 2-16A).
- Observation (Square Current Pulse): When a constant square current pulse is injected into a neuron, the membrane potential change does not follow a square response. Instead, it rises exponentially to a steady state (depolarization for positive current) and decays exponentially back to resting potential when the current is turned off.
- This exponential rise is because the membrane capacitance must first be charged. Once charged, current flows primarily through membrane resistors (ion channels), causing the membrane potential to change. The exponential decay happens as the capacitor discharges.
Circuit model for a cable-like fiber (Fig. 2-16C): each membrane segment is a parallel RC circuit with:
- Membrane resistance per unit area (Rm) and membrane capacitance (Cm).
- Resting potential represented as a battery Er.
- Internal (axial) resistance Ri linking segments.
Key relationships:
- For a parallel RC circuit driven by a transient current, the membrane potential change follows an exponential approach with a time constant $au = Rm Cm$ .
- The smaller the time constant, the faster the membrane potential responds to current inputs. A faster tau means the cell can respond more quickly to synaptic input.
- Practical Determination of RC Circuit Parameters from Experiment:
 1. Input Resistance ( $R{in}$ ): Measured by taking the steady-state voltage change ( $\Delta V{m,ss}$ ) in response to a known injected current ( $I{in}$ ). Calculated as $R{in} = \Delta V{m,ss} / I{in}$ (e.g., 30 mV / 100 pA = 300 M $\Omega$ ).
 2. Time Constant ( $\tau$ ): Measured as the time it takes for the voltage response to reach 63% of its maximum/steady-state value. (e.g., 20 milliseconds).
 3. Membrane Capacitance ( $Cm$ ): Calculated using the relationship $au = R{in} Cm$ , so $Cm = \tau / R_{in}$ .
Significance of $C_m$ and $au$ :
- Membrane capacitance ( $C_m$ ) is a convenient measure of cell size or membrane area. This is because the total membrane capacitance of a cell, composed of many parallel RC circuits, is approximately proportional to the surface area of the cell (assuming uniform capacitance per unit area, as membrane thickness does not change).
 - A bigger cell has a higher $C_m$ and a lower input resistance (if ion channel density is constant).
 - A smaller cell has a lower $C_m$ and a higher input resistance (if ion channel density is constant).
- The time constant ( $\tau$ ) significantly affects how fast a neuron can respond to current injections or synaptic inputs. Cells with a smaller tau respond faster.
 - If ion channel density increases (more channels), $Rm$ decreases, and thus $au$ decreases (assuming constant cell size, so $Cm$ is constant). This leads to a smaller voltage response (less depolarization) and a faster rise time.
Distance-dependent decay (length constant, $\lambda$ ):
- The axial spread of the membrane potential change is governed by the axial resistance Ri and membrane resistance Rm, with diameter d influencing Ri.
- Observation: When recording at increasing distances from the current injection site along a fiber, the maximum amplitude of the membrane potential change decreases, and the rise to the peak is slower.
- This occurs because injected current Not only escapes through the membrane but also flows along the fiber through its axial resistance ( $R_I$ ). As current flows, a portion continuously leaks out through the membrane, causing less and less current to reach more distant segments.
- The membrane potential at distance x decays approximately as an exponential: $V(x) = V_0 e^{-x/\lambda}$ where $\lambda$ increases with higher membrane resistance (Rm) or larger fiber diameter (which increases cross-sectional area and reduces Ri).
- Longer length constants allow signals to travel farther before attenuating. It is the distance from the initial depolarization site where the membrane potential change has dropped to 37% (or by two-thirds).
- Factors influencing $\lambda$ :
 - Diameter (d): $\lambda$ is directly proportional to diameter (thicker structure = longer $\lambda$ ).
 - Membrane Resistance (Rm): Higher membrane resistance (less leaky membrane, fewer open channels) = longer $\lambda$ .
 - Axial Resistance (Ri): Lower axial resistance (less intracellular resistance to current flow) = longer $\lambda$ . The formula is $\lambda = \sqrt{\frac{Rm}{Ri}}$ (general relation).
Practical implications:
- Larger diameter fibers (e.g., squid giant axon, with $\lambda$ ~5 mm) support longer distance signaling.
- Myelination increases distance over which signals propagate by effectively increasing length constant and reducing leak.
- Dendrites, being very thin, have a very short length constant and a high time constant.
Box 2-2 (parallel RC dynamics) highlights the differential equations governing a parallel RC circuit fed by a current source, deriving:
- $I = IR + IC$ and detailed forms leading to:
- $V(t) = I R_m (1 - e^{-t/\tau})$
- $I_R(t) = I (1 - e^{-t/\tau})$
- $I_C(t) = I e^{-t/\tau}$
- with the time constant $\tau = Rm Cm$ and the interpretation of one, two, or three time constants corresponding to 63%, 86%, and 95% charging, respectively.
Cable property analogy:
- The parallel RC circuit models the membrane; the axial resistances model the internal resistance along the fiber.
- The product $Rm Cm$ sets the temporal integration window for synaptic inputs and propagating voltage signals.
Active signaling is needed to propagate signals over long distances (e.g., from spinal cord to toe) because passive decay would attenuate signals over short distances alone.
- Observed neuronal electrical activity includes small fluctuations (noise) and occasional large, transient depolarizations called action potentials (APs).
Experimental setup using an axon (Figure 2-18, top): apply step current pulses via a stimulating electrode and record at the injection site (electrode a).
Observations with increasing current:
- Hyperpolarizing pulses (negative current) cause negative changes in membrane potential proportional to current magnitude.
- Small depolarizing pulses (positive current) produce proportional depolarization at electrode a.
- At a certain threshold or higher, a much larger, transient depolarization occurs and is not proportional to input—this is an action potential.
- Action potentials of the same magnitude can be elicited with successive stimuli once threshold is reached, demonstrating the all-or-none nature.
Conceptual takeaway:
- Voltage-gated channels (primarily Na+ and K+ in many neurons) underlie the generation and propagation of action potentials, providing an active mechanism to overcome passive decay.
- Once initiated, the action potential propagates along the neuron, enabling long-distance signaling to effector targets (muscle, glands, other neurons).

Connections to broader principles and real-world relevance

Energy efficiency and neuronal function:
- The Na+/K+ ATPase is a major energy consumer in neurons, linking cellular energetics to electrical signaling accuracy and stability.
Electrical circuits as a unifying framework:
- Modeling membranes as RC circuits and membranes with parallel ion conductances provides intuitive and quantitative insight into how neurons integrate inputs over time and space.
Relevance to neural computation:
- Time constants and length constants determine how synaptic inputs are integrated across dendritic trees and how signals degrade over distance, shaping neural coding and network dynamics.
Practical implications:
- Myelination and axon diameter are evolutionary strategies to optimize signal conduction speed and distance.
- Understanding passive vs active properties informs how neurons respond to stimulation, how synaptic inputs are integrated, and how disorders affecting ion channels (channelopathies) can disrupt signaling.

Key equations and constants (summary)

Nernst equation for ion i:
- $Ei = \frac{RT}{zF} \text{ln} \left(\frac{[\text{ion}]o}{[\text{ion}]_i}\right)$
- Simplified at 25°C: $Ei = \frac{58}{z} \text{log}{10} \left(\frac{[\text{ion}]o}{[\text{ion}]i}\right)$
Resting membrane potential (Goldman–Hodgkin–Katz form, simplified to conductance-weighted):
- $Vm = \frac{PK[K^+]o + P{Na}[Na^+]o + P{Cl}[Cl^-]i}{PK[K^+]i + P{Na}[Na^+]i + P{Cl}[Cl^-]_o} \text{ (full GHK, weighted by permeability)}$
- $Vm = \frac{gK EK + g{Cl} E{Cl} + g{Na} E{Na}}{gK + g{Cl} + g{Na}} \text{ (conductance-weighted, derived from GHK)}$
Ion currents (for each ion):
- $IK = gK (Vm - EK)$
- $I{Cl} = g{Cl} (Vm - E{Cl})$
- $I{Na} = g{Na} (Vm - E{Na})$
Simple RC membrane model (time response):
- $V(t) = I R_m (1 - e^{-t/\tau})$
- $I_R(t) = I (1 - e^{-t/\tau})$
- $I_C(t) = I e^{-t/\tau}$
- with $\tau = Rm Cm$
Length (space) constant and exponential decay along a fiber:
- $V(x) = V_0 e^{-x/\lambda}$
- $\lambda = \sqrt{\frac{Rm}{Ri}}$ (general relation; larger diameter and higher membrane resistance increase $\lambda$ )
Cable-like relationships and current distribution along passive fibers:
- Parallels between the time constant and space constant govern how signals spread and integrate over time and distance.