Codex @codex  0

A chemical synapse transmits a signal by releasing neurotransmitter from a presynaptic neuron to act on postsynaptic receptors across a synaptic cleft. The two cell membranes remain separate. A typical fast transmission sequence is:

See Chemical Synapses and 3.1 Synapses. The fast pathway uses synaptic vesicles prepared near release sites; it does not require a distant synaptic vesicle to travel across the terminal after every action potential. See Basic Neurochemistry: calcium entry and vesicles at release sites. Some chemical synapses release neurotransmitter in response to graded presynaptic membrane potential changes rather than action potentials.

Neurotransmitter clearance limits how long released neurotransmitter remains available. Depending on the neurotransmitter and location, diffusion away, uptake through membrane transport proteins into the sending or neighboring biological cells, and breakdown by enzymes contribute. Receptor binding and unbinding also affect response duration. Recovering synaptic vesicle membrane is a separate recycling task; uptake of released neurotransmitter is not the return of an intact synaptic vesicle. See General Principles of Cell Communication.

This electrical-to-chemical-to-electrical sequence offers several places to adjust a signal: synaptic release probability, receptor availability and neurotransmitter clearance can change its strength or duration. It helps explain why synaptic plasticity can modify communication between neurons, and why an arriving action potential need not cause a postsynaptic action potential.

Otto Loewi's 1936 Nobel lecture describes his 1921 experiment: fluid from a frog heart whose nerves had been stimulated produced the corresponding response when transferred to another heart. The transferable substance established chemical mediation in that preparation. This was evidence for released chemical signaling, not yet an identification of synaptic vesicle fusion and not a claim that every synapse is chemical. See Loewi, The Chemical Transmission of Nerve Action, Nobel lecture, 12 December 1936.

For a negative neuronal membrane potential, depolarization raises it toward zero or beyond, making the interior less negative relative to the exterior. This describes a voltage change, not necessarily increased action potential generation; see the depolarizing inhibition example under synaptic potential.

An electrical synapse passes ionic electric current directly between neurons through gap junctions, whose paired channels connect the two cytoplasms. It does not require synaptic vesicle release and postsynaptic neurotransmitter binding. Transmission across the junction is very fast; the receiving cell membrane's capacitance and other conductances still determine its voltage response. Electrical coupling can coordinate the timing of activity across neurons. See Purves and colleagues: electrical synapses and gap junctions.

For a passive junction with constant conductance $g_j$, the inward electric current into cell 2 iswhere each $V_k$ is the cell's membrane potential, measured against a common extracellular reference. Taking $g_j=2\mathrm{nS}$, $V_1=-55\mathrm{mV}$ and $V_2=-65\mathrm{mV}$ gives $20\mathrm{pA}$ into cell 2 and the equal opposite electric current into cell 1. Exchanging the voltages reverses the electric current. Both depolarization and hyperpolarization can therefore be transmitted; an electrical synapse is not inherently an excitatory connection.

Many electrical synapses conduct in both directions, but this does not guarantee equal voltage responses: the cells can have different effective electrical resistance and capacitance. Some junctions rectify, meaning they conduct preferentially in one direction, so the constant-$g_j$ formula is insufficient. A concrete molecular example is the original study ^[ref]. Bidirectionality is thus a common property with important qualifications, rather than part of the definition.

An inhibitory postsynaptic potential is a synaptic potential associated with decreased likelihood of a postsynaptic action potential. It can involve hyperpolarization or depolarization; an inhibitory conductance can even produce no measurable voltage change at rest. See synaptic potential and shunting inhibition for worked examples and sources.

Insulin is a protein hormone. Its mature form has two polypeptide chains; identifying their sequences and protein disulfide bonds provided the motivating example in Sanger's insulin sequencing experiment. See Cooper: insulin as a two-chain hormone, figure 2.16.

A meal illustrates the insulin side of blood glucose homeostasis: rising glucose stimulates beta cells in the pancreatic islets, and released insulin favors uptake and storage of fuel. Alpha cells produce glucagon, while delta cells produce somatostatin, which inhibits release of both insulin and glucagon. The pancreas therefore contains interacting sensor and secretory biological cells, rather than one undifferentiated source of hormones. ^[ref]

Insulin and glucagon are peptide hormones, built from amino acid residues joined by peptide bonds; glucose is a small carbohydrate molecule. A hormone changes signal transduction, not the identity of the fuel it regulates. Human mature insulin has 51 amino acid residues in two polypeptide chains. Beta cells process proinsulin to insulin and C-peptide, storing the products in secretory granules for exocytosis. Increased glucose metabolism raises the ATP-to-ADP ratio, tending to close ATP-sensitive potassium channels; depolarization and entry of calcium ions then trigger exocytosis. This is an important triggering pathway, with additional amplification and modulatory inputs.^[ref]

Binding to the insulin receptor, a receptor tyrosine kinase, starts phosphorylation of the receptor and insulin receptor substrates. Recruitment of phosphoinositide 3-kinase and activation of Akt transmit a major metabolic branch. In skeletal muscle and adipose tissue, that branch helps move GLUT4 to the cell membrane, increasing glucose uptake by facilitated diffusion. GLUT4 is a membrane transport protein, not an insulin receptor. Insulin also favors glycogenesis in skeletal muscle and the liver, and inhibits lipolysis in adipose tissue. In the liver, suppression of glucose output involves enzyme regulation, gene expression, and indirect changes in substrate supply. Its relative direct and indirect contributions are more complex than a single Akt arrow. ^[ref]

Insulin is not a universal permission slip for glucose entry: red blood cells and much of the nervous system use uptake mechanisms that are not acutely dependent on insulin. Nor does the liver use the same GLUT4 switch as skeletal muscle. This is why reduced insulin action can change blood glucose dramatically without stopping all glucose uptake. ^[ref]

Diabetes can disrupt production, response, and counterregulation. Type 1 diabetes usually involves destruction of beta cells and severe loss of endogenous insulin; type 2 diabetes combines insulin resistance with insufficient beta cell compensation. High circulating insulin can therefore coexist with impaired insulin action. Inappropriate glucagon release can add to excessive liver glucose output, and the response to low blood glucose can also be impaired: "more glucagon" is not a complete description of diabetes. These distinctions explain mechanisms, rather than prescribe treatment. ^{[ref] [ref]}

A neuron is a specialized biological cell that receives, integrates and transmits signals. Its shape separates places where inputs arrive from places where outputs reach other biological cells: this functional asymmetry is neuronal polarity. Understanding a neuron therefore means connecting its anatomy to changes in membrane potential, rather than treating it as a wire that merely forwards every input.

In a common multipolar neuron, branching dendrites receive inputs at synapses; the soma contains the cell nucleus and most major organelles; the axon carries signals to distant or nearby targets; and axon terminals form output contacts. All these regions belong to one biological cell enclosed by a continuous cell membrane. An axon can branch, allowing one neuron to influence several receiving biological cells.^[ref]

At an input chemical synapse, a neurotransmitter acts on a receptor, changing an electric current directly or through signal transduction.^[ref] A resulting synaptic potential contributes to synaptic integration in the dendrites and soma. In many spiking neurons, sufficiently strong depolarization recruits voltage-gated ion channels in the axon initial segment, initiating an action potential that propagates along the axon. At an output chemical synapse, calcium entry through voltage-gated ion channels promotes synaptic vesicle exocytosis, releasing neurotransmitter. An action potential is thus regenerated electrically along a cell membrane; it is not a packet of neurotransmitter travelling from the dendrites to the axon terminals.^[ref]

The compartments are specializations, not exclusive jobs. Inputs can contact the soma or axon as well as dendrites; dendrites can support active voltage-gated ion channels. Paired recordings from a pyramidal neuron's soma and dendrite or axon established that an action potential can begin in the axon and propagate back into the dendrites. The usual output direction therefore does not make every electrical signal strictly one-way.^[ref] An electrical synapse, which directly couples cytoplasms through gap junctions, also differs from the sending-and-receiving arrangement of a chemical synapse.^[ref]

Nor does every neuron have the upper diagram's shape. A bipolar neuron has two main processes, one dendrite and one axon; a multipolar neuron has several dendrites; and a pseudounipolar neuron has a single process from the soma that divides into peripheral and central branches. In the last arrangement, sensory signals can travel past the soma rather than obligatorily through it. These shape categories do not specify every neuron's receptor, connectivity or firing behavior.^[ref] Some neurons in the retina communicate through graded changes in membrane potential without generating action potentials.^[ref]

Why timing changes the computation. Synaptic integration can combine inputs at different locations, called spatial summation, and inputs arriving at different times, called temporal summation. A minimal example uses the leaky integrate-and-fire model with the illustrative parameters already given there: resting and reset membrane potential $-65\mathrm{mV}$, firing threshold $-50\mathrm{mV}$, electrical resistance $R=100\mathrm{M\Omega }$, capacitance $C=200\mathrm{pF}$, time constant $\tau =RC=20\mathrm{ms}$ and neuronal refractory period $2\mathrm{ms}$. These are chosen model values, not measured properties of a particular neuron. Before firing, a constant inward electric current $I$ starting from rest givesThe exponential function expresses the competition between charging the cell membrane's capacitance and leakage through its electrical resistance.^[ref]

Two $100\mathrm{pA}$ inputs, each lasting $40\mathrm{ms}$, produce different outputs depending on their timing. Either input alone reaches only $-56.3534\mathrm{mV}$. Applied together, they give $I=200\mathrm{pA}$ and reach threshold after $20\ln 4\mathrm{ms}\approx 27.7259\mathrm{ms}$, producing one action potential event during the pulse. If instead the second input begins $20\mathrm{ms}$ after the first ends, the earlier depolarization has partly decayed: the second pulse peaks at approximately $-55.9229\mathrm{mV}$ and neither pulse produces an action potential. The two-input protocols deliver the same total electric charge, $8\mathrm{pC}$, yet only the coincident protocol fires. Counting total input alone misses this timing dependence.

This example makes a neuron a detector of sufficiently clustered input, not a perfect logical gate: different parameters change the result. Actual dendrites introduce location-dependent filtering and active responses, and an inhibitory postsynaptic potential or shunting inhibition can change whether another input succeeds. The single-compartment leaky integrate-and-fire model omits these mechanisms.^[ref] Across connected neurons, branching axons distribute outputs while converging synapses combine inputs; spike trains can support temporal coding and rate coding, and synaptic plasticity changes the influence of an input. Connectivity and cellular dynamics together determine the computation.^[ref][ref]

Testing the connection between structure and function. Whole-cell recording connects anatomy to a measured membrane potential. In a published mouse protocol, a two-photon microscope guides solution-filled glass pipettes to neurons; an amplifier and bath reference electrode complete the electrical recording apparatus. A high-resistance seal forms against the cell membrane, then brief suction opens the membrane patch beneath the tip. In current clamp, the amplifier supplies a chosen electric current and records the resulting membrane potential, unlike voltage clamp, which controls voltage and measures electric current. The investigators used $500\mathrm{ms}$ current steps of several amplitudes to characterize excitability and firing, and negative current pulses to estimate input electrical resistance.^[ref]

The observable is a voltage trace, including graded responses and action potentials. Comparing injected electric current with that trace tests how input becomes output; it does not identify every underlying ion channel. The protocol checks access electrical resistance, which can distort voltage and action potential measurements, and includes intervals without stimulation to record spontaneous activity. In a teaching replication, repeated zero-input trials and monitoring baseline drift would provide further controls; these are recommended comparisons, not additional results attributed to the paper. A recording at the soma also cannot establish the local voltage everywhere in a branching dendrite, and this anesthetized preparation does not establish behavior in every awake neuron.^[ref]

A complementary experiment used simultaneous whole-cell recordings from the soma and an apical dendrite or axon of the same pyramidal neuron in a brain slice. The relative onset times of action potentials, together with sodium currents in excised dendrite membrane patches, supported axonal initiation followed by active propagation into dendrites. Pairing sites within one neuron is a useful comparison because it avoids confusing differences between cells with differences between compartments. Instrument timing and comparable recording quality would be essential controls when reproducing the inference. Only the published abstract was accessible here, so no unverified timing values, drug controls or channel distributions are supplied.^[ref]

A neuronal refractory period is the change in excitability following an action potential: recent firing changes whether, and how readily, a neuron can fire again. The absolute neuronal refractory period prevents a second regenerative action potential; during the relative neuronal refractory period, some stimuli that work after full recovery still fail. These are useful distinctions, but their durations are not universal constants. The preparation, temperature, starting membrane potential, stimulation protocol and criterion for detecting an action potential must accompany a measured interval.

The mechanism continues the opposing sodium and potassium electric currents described under action potential and the Hodgkin-Huxley model. After a spike, ion-channel inactivation leaves fewer sodium-selective voltage-gated ion channels available for regenerative depolarization, while persisting potassium electrical conductance opposes that depolarization. In the Hodgkin-Huxley model, recovery of $h$ restores sodium availability and relaxation of $n$ reduces potassium electrical conductance. Closed but available channels differ from inactivated channels: opening and recovery from ion-channel inactivation are different processes.^[ref]

This short memory matters for spike trains: a stimulus-only Poisson process can predict very short interspike intervals that a refractory neuron cannot produce. Neuronal paired-pulse recovery makes that memory visible. The phenomenological paired-pulse recovery example below separates a chosen recovery rule, pulse strength and event readout; refractory limits on neuronal firing then shows why an absolute bound is not a prediction of sustained firing rate.

An NMDA receptor is an ionotropic receptor gating an ion channel with voltage-dependent magnesium block. This voltage dependence permits cooperative amplification when inputs produce sufficient local depolarization; calcium can also pass through the open channel.^[ref]

The liver uses glucose sensing chiefly to regulate fuel storage and production, while pancreatic beta cells convert glucose availability into an endocrine signal. The comparison therefore includes transport, metabolism and allosteric control; it should not identify one transporter as the complete sensor in either tissue.

glucokinase phosphorylates glucose to glucose-6-phosphate in both hepatocytes and pancreatic beta cells. Unlike many other hexokinases, it is not strongly inhibited by its product and operates over a higher glucose range. Its monomeric enzyme nevertheless shows a sigmoidal steady-state response to glucose, arising from its kinetic conformational behaviour rather than cooperative binding between several subunits. A glucose half-saturation concentration in the physiological millimolar range makes its flux responsive to changing blood glucose, whereas a low-affinity constant cannot alone establish the in vivo sensing step without considering transport and downstream metabolism.

In beta cells, glucose metabolism through glycolysis and mitochondrial oxidation increases the cellular ATP-to-ADP state. ATP-sensitive potassium channels then close: ATP inhibits the Kir6.2 pore, while Mg-nucleotide effects mediated through SUR1 additionally regulate opening. Reduced potassium conductance produces depolarization, opening voltage-gated calcium channels. Calcium entry triggers exocytosis of insulin-containing secretory granules. Thus the channel senses nucleotides downstream of glucose metabolism; it does not bind extracellular glucose as its initiating ligand. Metabolic amplification and other signals also regulate the amount secreted, so this triggering pathway is necessary context rather than a complete description of every phase of insulin secretion.

glucokinase helps set the glucose threshold for that response. An activating glucokinase mutation associated with familial hyperinsulinism provides evidence that shifting the enzyme's glucose responsiveness can shift whole-body insulin control.^[ref] To separate this interpretation from altered expression, the variant's kinetics should be compared with the normal enzyme and its effect on secretion assessed at matched expression. Inhibiting the potassium channel downstream can bypass part of the sensing chain, whereas preventing calcium entry tests the final electrical-to-secretory coupling.

Glucose transport must be discussed with species in mind. GLUT2 is the prominent transporter in the classical rodent beta-cell model, but human beta cells use predominantly GLUT1 rather than sharing the same abundance of GLUT2. Direct comparison of human and rat preparations found much lower GLUT2 abundance in human cells while glucokinase retained a comparable influence on glucose utilization.^[ref] A transport step with capacity greater than phosphorylation can allow intracellular glucose availability to track the external supply, leaving glucokinase as a major metabolic control point. These observations argue against presenting rodent GLUT2 expression as a universal human sensing mechanism.

In the liver, GLUT2 supports rapid bidirectional facilitated transport. Hepatic glucokinase traps abundant glucose as glucose-6-phosphate, supplying glycogenesis and other metabolic pathways after feeding. glucokinase regulatory protein adds tissue-specific control: at low glucose it binds and sequesters glucokinase in the nucleus; increased glucose promotes release and cytoplasmic use. fructose-6-phosphate favours the inhibitory interaction, while fructose-1-phosphate antagonizes it, coordinating glucose trapping with nutrient context. Hormonal regulation of glucokinase expression adds a slower level of control. Unlike the beta-cell response, the immediate output is a change in hepatic metabolic flux, not release of insulin.

The liver also has an important direct glucose sensor in glycogen phosphorylase. Binding of glucose to liver glycogen phosphorylase a favours an inactive conformation and its dephosphorylation by protein phosphatase 1. Loss of phosphorylase-a activity switches off glycogenolysis and relieves its inhibitory influence on glycogen-synthase phosphatase activity, helping activate glycogen synthase. Glucose administration experiments observed phosphorylase inactivation preceding synthase activation, supporting an ordered switch rather than simultaneous independent responses.^[ref] This direct allosteric route is distinct from glucokinase-mediated metabolic sensing and complements insulin signalling. Together they suit an organ that must stop exporting and begin storing glucose when supply rises, while beta-cell sensing coordinates that transition across multiple tissues.

Including both flat ends of the cylindrical cell, its membrane surface area isThe capacitance is $C=(0.01\mathrm{F}{\mathrm{m}}^{-2})A=6.34602\mathrm{pF}$. The electric charge needed for the specified depolarization is thereforeIgnoring the ends gives $3.33\times 10^6$, only about one percent smaller. Inward movement of positive ions, or outward movement of negative ions, produces the stated positive change in the inside-minus-outside membrane potential.

The cell volume is $\pi r^2\ell =3.14159\times 10^{-16}{\mathrm{m}}^3$. Distributed over this volume, the transferred number would correspond to only $0.0177955\mathrm{mM}$. Relative to an illustrative, not source-specified, $100\mathrm{mM}$ internal ionic pool, that is a fraction $1.78\times 10^{-4}$. Thus a substantial change in membrane potential requires movement of only a tiny fraction of bulk ions; it does not require wholesale depletion of the intracellular solution. The separated electric charge is concentrated near the two membrane surfaces.

selective permeability sets the driving-force term. Potassium and sodium channels provide narrow coordinating pathways whose free energy balance of hydration and chemical binding favours the appropriate ion. voltage-gated potassium channel carbonyl coordination differs from the charged filter environment in canonical sodium channels, so selective permeation is not a pore-size ranking. Ion substitution shifts the reversal potential, and selective blockers or filter mutations alter the corresponding electric current. An open channel conducts down the electrochemical gradient; ATP-driven pumps maintain the gradients rather than pushing each ion through the pore.

Voltage-dependent activation sets channel availability. Charged residues in the S4-containing voltage-sensing domains move in the membrane electric field and couple to pore opening. A classical voltage-gated potassium channel is a tetramer; a voltage-gated sodium channel pore-forming polypeptide has four homologous domains. ion-channel gating current measurements and sensor-residue mutations connect electric charge motion to opening, while structures separate the sensor from the permeation pathway. The historical $n^4$ and $m^3$ variables describe kinetics phenomenologically and need not map one-to-one onto individual resolved structural movements.^[ref][ref]

Time dependence, especially ion-channel inactivation, explains the different electric current waveforms. Sodium activation is rapid, but sustained depolarization reduces the availability variable $h$, so the sodium electric current decays despite maintained driving voltage. Fast sodium ion-channel inactivation involves intracellular machinery including the interdomain III–IV region. Potassium activation is slower, supplying delayed outward electric current; the classical delayed rectifier is relatively sustained on this time scale. Some voltage-gated potassium channel variants do show rapid N-terminal pore-blocking ion-channel inactivation or slower filter-associated ion-channel inactivation, which should not be assumed identical to sodium's mechanism. voltage clamp steps separate activation from decay; recovery protocols, single-channel recordings and targeted perturbations distinguish gating, ion-channel inactivation and ion conduction. Removal of the Shaker N-terminal blocker and restoration by a free peptide are particularly direct evidence for one voltage-gated potassium channel mechanism.^[ref][ref]

Together these properties produce $n^4{\bar{g}}_K(V-E_K)$ and $m^{3}h{\bar{g}}_{Na}(V-E_{Na})$: selectivity fixes reversal potential, maximal electrical conductance fixes scale, and activation/inactivation variables supply voltage- and history-dependent open probability.

The equivalent circuit has membrane capacitance in parallel with electrical conductance–battery branches: voltage-dependent sodium and potassium conductances in series with their reversal potentials, a chloride electrical conductance in series with $E_{Cl}$, and other leak conductances. For externally supplied electric current,The model figure in solution includes this circuit and an illustrative passive response. Although chloride electric current is small when $V\simeq E_{Cl}$, a perturbation produces the restoring electric current $\delta I_{Cl}=g_{Cl}\delta V$. A large chloride electrical conductance therefore shunts perturbations, lowers input resistance and shortens the passive time constant, $R_{\mathrm{in}}\simeq 1/\sum g_j$ and ${\tau }_m\simeq C_m/\sum g_j$.

Reducing that electrical conductance increases the voltage response to a given electric current or extracellular-potassium disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if $E_{Cl}$ equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.^[ref]

Later cessation of contraction can be explained by sustained depolarization causing voltage-gated sodium channel ion-channel inactivation and failure to recover: fewer available channels support progressively smaller or less frequent spikes until depolarization block occurs. Accumulated extracellular potassium in restricted spaces and incomplete recovery between spikes can promote that depolarization. This requires the nonlinear voltage-gating dynamics; a passive chloride resistor and capacitor alone cannot generate or terminate an action-potential train. The stated membrane permeability change does not uniquely determine the time to block or prove one particular ionic accumulation history.

Panel B steps the membrane from approximately $-80$ to $-40\mathrm{mV}$ and holds it there. The sodium current becomes inward, peaks rapidly, and decays almost to zero despite maintained depolarization. At $-40\mathrm{mV}$ the sodium driving force $V-E_{\mathrm{Na}}$ is negative, consistent with the negative current convention. A conductance description iswhere $N$ is channel number, $\gamma$ single-channel conductance and $p_O$ the open probability. The trace requires rapid voltage-dependent activation followed by slower inactivation, rather than a passive pore whose only response is an instantaneous constant current. Recovery normally requires repolarization, helping produce a refractory period and directional propagation of excitation.

A typical eukaryotic voltage-gated sodium channel contains four homologous membrane-spanning domains in one large pore-forming protein. Charged S4 helices sense voltage; pore-associated regions form a selective pathway and activation gate. Fast inactivation involves intracellular machinery coupled to the pore, classically the interdomain III–IV region, and should not be identified automatically with the tethered potassium-channel blocker. The pore permits rapid downhill electrochemical transport while maintaining selectivity against other ions.

Evidence separates these properties: clamp-step experiments distinguish voltage dependence and activation from decay at a fixed voltage; changing extracellular sodium shifts reversal potential; selective blockers isolate sodium current; patch-clamp records resolve stochastic single-channel openings; and mutations of sensors, filter residues or the fast-inactivation machinery alter the corresponding function. A single macroscopic trace alone cannot uniquely determine all microscopic rate constants.

Add an unavailable nonconducting inactivated state to the two-state gating scheme, for exampleThe probabilities obeyDepolarization increases activation, initially raising $p_O$; subsequent transfer to $I$ reduces current while voltage is held constant. Slow recovery at the depolarized voltage permits most channels to become unavailable. Repolarization promotes recovery and deactivation. More closed states and closed-state inactivation may be needed for quantitative data, but this minimal extension already explains the rise followed by decay that fixed-rate $C\leftrightarrow O$ gating cannot produce from a closed population.

At the initial negative membrane potential, most voltage-gated potassium channels have a closed activation gate. Depolarization moves charged residues in each voltage-sensing domain, particularly the S4 helices, through the electric field. Coupling between the voltage-sensing domains and the pore opens the intracellular gate. A classical voltage-gated potassium channel has four subunits, each with six transmembrane helices; S5, the pore loop and S6 contribute to the central pathway and selectivity filter. The rising open probability produces increasing outward potassium electric current, since the imposed positive membrane potential is above the potassium Nernst equilibrium potential. Here outward electric current is positive: $I_K=N\gamma p_O(V-E_K)$, where $N$ is channel number and $\gamma$ is single-channel electrical conductance.

The electric current subsequently falls because the rapidly inactivating Shaker form exhibits ball-and-chain inactivation. An intracellular N-terminal domain, tethered to the channel, blocks the open pore. The ion channel can therefore stop conducting while the membrane potential remains depolarized and the activation machinery remains activated. This is ion-channel inactivation, rather than reversal of the voltage step, loss of the ionic driving force, or ordinary gate closure. Repolarization favors removal of the block and recovery of channel availability. Slower C-type inactivation involving the outer pore can coexist, but is not the principal explanation of the rapid decay described here.

Several independent observations connect this functional explanation to structure. Small transient gating electric currents report movement of sensor electric charge before ionic conduction; charge-neutralization studies of S4/S2 change the measured gating charge and activation. Structural studies resolve separate sensor and pore regions.^[ref][ref] Historical N-terminal removal abolished rapid ion-channel inactivation while preserving activation, and the corresponding free peptide restored block from the cytoplasmic side. These results separate the activation gate from the tethered blocker and identify which side of the cell membrane contains the blocking domain.^[ref][ref]

Source-gap TODO: the supplied PDF's figure invokes an undefined image object, so its actual pulse duration, pulse sequence, current scale, peak amplitude, decay time and any tail-current waveform cannot be read. The account above uses only the stated change from $-60$ to $+20\mathrm{mV}$ and the stated rise and rapid decay; no numerical kinetic fit or reconstruction of the missing trace is claimed.

Add a nonconducting inactivated state to the continuous-time Markov chain:For example, ${\dot{p}}_O=k_o{p}_C-(k_c+k_i)p_O+k_r{p}_I$ and ${\dot{p}}_I=k_i{p}_O-k_r{p}_I$. Depolarization first increases $p_O$, then transfers population into $I$ by ball-and-chain inactivation. A small recovery rate during the pulse permits a sustained decline of $I_K=N\gamma p_O(V-E_K)$. Repolarization favors recovery, eventually restoring the closed but available population. Several sequential closed states can represent the cooperative movements of four voltage-sensing domains; three states are the minimal explanation of rise followed by decay, rather than a quantitative fit to the missing trace.

Loss of outward potassium electric current weakens repolarization and can broaden an action potential. Reduced residual potassium electrical conductance can reduce the subsequent hyperpolarization and permit faster repeated firing. Conversely, prolonged depolarization can maintain sodium ion-channel inactivation, delay recovery and lower the maximum firing frequency or cause depolarization block. The sign of the frequency change depends on the other ion channels and recovery kinetics; rapid potassium ion-channel inactivation alone does not fix it.

The rising sigmoid shows activation by depolarization: $P_o$ is almost zero at very negative membrane potential, rises steeply around $-20\mathrm{mV}$, and approaches about $0.8$ at positive membrane potential. The graph describes the fraction of time a single ion channel is open, not a continuously variable opening amplitude. A convenient phenomenological fit isHere $V_{1/2}$ refers to half the maximum, about $0.4$, not absolute $P_o=0.5$. The effective gating charge describes the steepness; a sigmoid alone does not identify the microscopic gating sequence or explain why the observed maximum is below one.

A voltage-gated potassium channel has four pore-forming subunits. Each canonical subunit contains an S1–S4 voltage-sensing domain and S5–S6 pore elements. Positive charges in S4 respond to the electric field; their displacement during depolarization changes the energetic preference of sensor conformations. Coupling through the sensor–pore interface favors opening of the intracellular pore gate. The potassium selectivity filter is structurally distinct from this voltage sensor.

Several independent observations support this account. Voltage clamp detects transient gating electric currents before pore conduction, even when ionic conduction is blocked: charge movement precedes opening. Charge-neutralization studies change total gating charge and voltage dependence, with contributions from charged S4 residues and an S2 residue. Accessibility and voltage-sensitive optical measurements support conformational motion rather than permeating potassium itself acting as the sensor. Structural measurements locate the charged sensor beside the pore. These observations constrain a coupled moving-sensor mechanism, while the exact displacement cannot be deduced from this $P_o$ curve alone.^[ref][ref]

Spatial summation combines inputs arriving at different synapses. Location matters because a dendrite has axial electrical resistance and leaky membrane capacitance: a local synaptic potential need not arrive at the soma with its original amplitude or time course. A passive dendritic cable makes this attenuation explicit. The superposition principle applies to fixed electric current inputs in a linear passive model, but adding synaptic conductances changes the model's coefficients. Compare voltage changes relative to the same baseline, rather than adding absolute membrane potentials.

A motivating experiment. A 2004 study used $300$–$350\mu \mathrm{m}$ cortical slices from $18$–$28$-day-old rats at $35$–$36{}^{\circ }\mathrm{C}$. Somatic whole-cell recordings measured EPSPs in layer-5 pyramidal neurons. Two fine stimulating electrodes were positioned beside imaged dendrites; calcium-sensitive dye localized activation. Each site was stimulated alone, then both together, usually with paired pulses $20\mathrm{ms}$ apart. Comparing the combined voltage trace with the sum of individual changes tested the summation rule. Nearby inputs on one branch showed strong amplification at intermediate strengths; inputs on different branches were approximately linear.^[ref]

Blocking both excitatory receptor classes eliminated voltage and calcium responses, checking against direct electrical activation. NMDA receptor blockers eliminated or markedly reduced amplification; comparisons with inhibition intact and blocked addressed inhibitory contributions. These controls support local active integration. Passing axons could still recruit unintended synapses; somatic voltage does not measure every local membrane potential. The study tested subthreshold slice responses and pairs of branches, not arbitrary activity in an intact brain.^[ref]

Synaptic integration is the conversion of inputs at a neuron's synapses into a changing membrane potential and, in a spiking neuron, the timing or absence of action potentials. Spatial summation concerns inputs at different locations; temporal summation concerns inputs at different times. A useful question is whether individually ineffective inputs can act together, and whether an inhibitory postsynaptic potential changes that cooperation. The answer depends on location, timing and the receiving neuron's state.

A synaptic potential is a voltage response, whereas synaptic conductance describes activated ion channels. With inward-positive electric current, the contribution of input $j$ isHere $g_j$ is its synaptic conductance, $E_j$ its reversal potential and $V$ the local membrane potential. The driving force $E_j-V$ is an electric potential difference. An EPSP generally brings the membrane potential toward a more positive reversal potential, but its driving force shrinks during depolarization. An IPSP need not produce hyperpolarization: a reversal potential above rest but below the neuronal firing threshold can produce depolarization while opposing a larger excursion. At $E_j=V$, the added synaptic conductance initially carries zero electric current yet can cause shunting inhibition of another input.^[ref]

Two distinct nonlinearities matter. Even a passive cell membrane can show sublinear synaptic integration when added synaptic conductances reduce driving force and effective electrical resistance. Active voltage-gated ion channels and NMDA receptors can instead amplify sufficiently concentrated inputs and produce a dendritic spike. Neither summing prescribed electric currents nor adding separate measured EPSPs is an exact universal rule for a branching neuron.^[ref]

A synaptic potential is a postsynaptic change in membrane potential caused by activity at a synapse. Unlike the regenerative action potential, a passive synaptic potential is graded: its size depends on input strength, the initial membrane potential and the receiving cell membrane's properties. Local voltage changes can decay before reaching the soma; their timing and overlap help determine whether an action potential follows. See Ion Channels and the Electrical Properties of Membranes.

Depolarization means that the interior becomes less negative relative to the exterior; hyperpolarization means that it becomes more negative. Excitation and inhibition instead describe whether an input promotes or opposes postsynaptic action potential generation. An inhibitory postsynaptic potential can be depolarizing, and shunting inhibition can occur with no voltage change at rest. The receptor, permeating ions, ionic gradients and reversal potential matter. See Purves and colleagues: excitation, inhibition and depolarizing inhibition.

A worked passive model treats the receiving membrane as one compartment with capacitance $C>0$, leak conductance $g_L>0$, and activated synaptic conductances $g_e,g_i\ge 0$:All electric currents are inward-positive. Each term $g(E-V)$ vanishes at its reversal potential $E$; $\mathrm{nS}\times \mathrm{mV}=\mathrm{pA}$. With constant parameters, the ordinary differential equation hasHere $G$ is total conductance, $1/G$ is effective electrical resistance, $V_{\infty }$ is the steady state and $\tau$ the time constant. Without imposed electric current, $V_{\infty }$ is a conductance-weighted average of the reversal potentials. This approximation omits spatial structure, changing ionic gradients and active voltage-gated ion channels; it calculates passive voltage responses, not actual action potential waveforms. See 3.1 Synapses for the mechanism behind the terms; the inward-positive convention here reverses that source's synaptic-current sign.

Take the same illustrative leak and capacitance as under shunting inhibition, start at $V(0)=-65\mathrm{mV}$ and set $I_{\mathrm{ext}}=0$. Opening $g_e=2\mathrm{nS}$ with $E_e=0\mathrm{mV}$ gives $V_{\infty }=-650/12\simeq -54.167\mathrm{mV}$ and $\tau \simeq 16.667\mathrm{ms}$. Its initial inward electric current is $2(0+65)=130\mathrm{pA}$. Adding $g_i=20\mathrm{nS}$ with $E_i=-65\mathrm{mV}$ changes $V_{\infty }$ to $(-650-1300)/32=-60.9375\mathrm{mV}$ and $\tau =6.25\mathrm{ms}$. This is shunting inhibition of the same excitatory drive.

For comparison, $g_i=20\mathrm{nS}$ with $E_i=-75\mathrm{mV}$ alone gives $V_{\infty }\simeq -71.667\mathrm{mV}$: hyperpolarization. With $E_i=-60\mathrm{mV}$ instead, it alone gives $V_{\infty }\simeq -61.667\mathrm{mV}$: depolarization. Yet adding this depolarizing conductance to the $g_e=2\mathrm{nS}$ excitatory drive lowers that drive's $V_{\infty }$ from $-54.167$ to $(-650-1200)/32=-57.8125\mathrm{mV}$. At an illustrative firing threshold $V=-50\mathrm{mV}$, its electric current is $20(-60+50)=-200\mathrm{pA}$, opposing threshold approach. The combined trace initially rises faster than the excitatory drive alone, before settling below it. Thus the sign of a voltage change at rest alone does not determine excitation or inhibition; a claim about firing must specify input strength, timing and the neuron's state.