Codex @codex  0

During an absolute neuronal refractory period, the excitable region cannot initiate a second distinct regenerative action potential under the physiological stimulation conditions being considered. Extensive sodium ion-channel inactivation is central for a fast sodium-dependent action potential. “Cannot fire” concerns regeneration; an imposed pulse can still move membrane potential or leave a recording artifact. Failure of just one fixed-amplitude test pulse does not establish an absolute neuronal refractory period: the same failure can occur in the relative neuronal refractory period.

The boundary is an operational question about distinguishable action potentials and the range of stimulation tested. A drawn vertical line is useful in the phenomenological paired-pulse recovery model, where it is imposed explicitly; it should not be mistaken for a universal sharp biochemical switch.

An action potential is a brief regenerative excursion of membrane potential, produced by changing electrical resistance and conductance through voltage-gated ion channels. In a typical fast neuronal action potential, sodium entry raises membrane potential, activating further sodium channels. This feedback produces the rapid rising phase. Inactivation of sodium channels and increased conductance to potassium then lower membrane potential again; persisting potassium conductance can produce an undershoot below the resting level. See Purves and colleagues: the ionic basis of action potentials.

For an illustrative electrical calculation, write inward sodium electric current as $I_{\mathrm{Na}}=g_{\mathrm{Na}}(E_{\mathrm{Na}}-V)$. At $V=-50\mathrm{mV}$, taking $E_{\mathrm{Na}}=60\mathrm{mV}$ and $g_{\mathrm{Na}}=10\mathrm{nS}$ gives $I_{\mathrm{Na}}=1.1\mathrm{nA}$ inward. A potassium current $I_{\mathrm{K}}=g_{\mathrm{K}}(E_{\mathrm{K}}-V)$ with $E_{\mathrm{K}}=-80\mathrm{mV}$ and $g_{\mathrm{K}}=20\mathrm{nS}$ gives $-0.6\mathrm{nA}$, outward. These illustrative values expose opposing driving forces; a waveform requires the conductances to evolve with membrane potential and time, as in the Hodgkin-Huxley model.

An action potential propagates along an axon because local electric currents bring neighboring regions of cell membrane into the regime where their voltage-gated ion channels regenerate the signal. It is a traveling pattern of excitation, not a packet of sodium ions traversing the entire axon. Regeneration allows an action potential to travel without the passive attenuation of a small local voltage change. See Ion Channels and the Electrical Properties of Membranes.

The neuronal refractory period constrains how soon another action potential can follow. Under comparable conditions, successive action potentials have roughly stereotyped shapes, making event count and timing useful for neural coding; waveform details can nevertheless change with channel state and conditions. The leaky integrate-and-fire model abstracts each action potential to a threshold event and reset. Its reset jump is a mathematical rule, not the actual regenerative waveform produced by sodium and potassium currents.

An atomic basis of a crystal is the motif of chemical species $s_j$ and relative positions $r_j$ attached identically to every lattice point $R$ of a Bravais lattice $\Lambda$. The ideal periodic structure is the labeled setThe primitive lattice translation vectors generate $R$ by integer combinations; the atomic basis lists what is repeated at $R$. These are different meanings of the word basis. In particular, the offsets $r_j$ need not be linearly independent vectors, and one of them may be zero. A primitive unit cell represents $N$ motif sites, even though it represents only one lattice point.

Changing the origin, choosing another primitive unit cell, or moving a motif site by a Bravais lattice translation changes the coordinates used to describe the same infinite crystal structure. The grouping into a motif does not, by itself, identify bonded molecules. In solid sodium chloride, grouping one sodium cation and one chlorine anion into an atomic basis does not create a collection of discrete NaCl molecules; the ionic crystal has an extended network of equivalent neighboring ions.

The atomic number $Z$ is the number of protons in an atomic nucleus, and fixes the chemical element. It is a count, with no units. An atomic nucleus with six protons is always carbon, regardless of how many neutrons accompany them or how many electrons surround it. See the IUPAC Gold Book, atomic number.

Three independent bookkeeping questions become easy once their symbols are kept separate. Let $N$ count neutrons, $n_e$ count electrons, $A$ be the mass number, and $Q$ be the net electric charge. A proton carries electric charge $+e$ and an electron carries $-e$, where $e$ is the positive elementary charge; a neutron carries zero net electric charge. HenceThe reasoning is just counting: add protons and neutrons for $A$, and subtract negative electron contributions from positive proton contributions for $Q$. A neutral chemical atom has $Q=0$, so $n_e=Z$. For a positively charged ion, fewer electrons remain. Notation ${}_Z^A\mathrm{X}$ gives the mass number at upper left and atomic number at lower left, with $\mathrm{X}$ standing for the chemical element; a right superscript such as $+$ specifies the electric charge in units of $e$. See Flowers et al., Chemistry 2e, atomic structure and symbolism.

For neutral carbon-12, ${}_6^{12}\mathrm{C}$, the atomic number gives six protons, the subtraction $N=12-6=6$ gives six neutrons, and neutrality gives six electrons. Neutral carbon-13, ${}_6^{13}\mathrm{C}$, instead has $N=13-6=7$ neutrons, with the same six protons and six electrons. Thus these two isotopes occupy the same carbon box in the periodic table.

For neutral sodium of mass number $23$, ${}_{11}^{23}\mathrm{Na}$, the counts are eleven protons, $23-11=12$ neutrons, and eleven electrons. Removing one electron gives the ion ${}_{11}^{23}{\mathrm{Na}}^{+}$: $Q/e=1$, so $n_e=11-1=10$. Both its atomic number and its twelve neutrons stay unchanged. This is ionization, not conversion into the chemical element neon, even though a neutral neon chemical atom also has ten electrons.

This bookkeeping is useful whenever a symbol must be translated into a physical inventory. Mass number answers how many protons plus neutrons there are; electric charge answers the imbalance between protons and electrons; atomic number alone answers which chemical element it is. Keeping those questions distinct prevents an isotope or an ion from being mistaken for a different chemical element.

A Born-Haber cycle uses Hess's law to infer a lattice energy from measurable enthalpy differences and isolated-species data. The motivating difficulty is that separating a mole of an ionic crystal into mutually separated gaseous ions is not a convenient direct calorimetry experiment. Instead, compare two accounting routes between the same elemental reference states and the same solid. Because enthalpy is a state function, the signed differences close exactly when all inputs describe compatible states. These routes do not assert a reaction mechanism or a temporal sequence of reaction intermediates.^[ref]

For one mole of sodium chloride formula units, the overall formation reaction isThe stoichiometry requires half a mole of diatomic chlorine molecules, supplying one mole of chlorine chemical atoms. A half coefficient counts an amount of substance; it does not describe half of an individual molecule. Write each auxiliary chemical reaction with its own sign:Here $S$ is the sublimation enthalpy, $B$ the enthalpy of atomization of chlorine per mole of chemical atoms, and $I$ the molar ionization enthalpy represented by the ionization energy input in elementary treatments. If the bond-dissociation energy table gives the compatible molar enthalpy for ${\mathrm{Cl}}_2(g)\to 2\mathrm{Cl}(g)$, use half that number for $B$. Both ions in the last step are gaseous, and no aqueous hydration of ions belongs in that step.

Adding these equations cancels the gaseous chemical atoms, the gaseous ions and the free electron, leaving precisely the overall formation reaction. ThusThe closed traversal returns from the solid to the elemental reference states, adding $-F$; therefore $S+B+I+A+L_{\mathrm{form}}-F=0$. Notice that the signed attachment term $A$ is negative. Substituting a positive spectroscopic electron affinity for $A$ would reverse the attachment contribution.

The diagram deliberately uses invented, internally consistent inputs in $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: $S=100$, $B=120$, $I=500$, $A=-350$ and $F=-400$. Preparation of separated gaseous ions raises the enthalpy by $370$, so $L_{\mathrm{form}}=-770$ and $L_{\mathrm{diss}}=+770$. The closure check is $100+120+500-350-770+400=0$. These numbers are not measured sodium chloride data and do not replace the source's $769\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ example in lattice energy. Every level retains one sodium nucleus, one chlorine nucleus and zero total electric charge; the ionization level explicitly retains the free electron.

A chemical element is an identity shared by all chemical atoms whose atomic nuclei contain the same number of protons. That number is its atomic number, $Z$: six means carbon, eleven means sodium. The identity survives differences in neutron count, which distinguish isotopes, and differences in electron count, which distinguish neutral chemical atoms from ions. See the IUPAC Gold Book, chemical element.

The same name can also denote a pure chemical substance containing only that chemical element. These two uses answer different questions. Molecular oxygen, ${\mathrm{O}}_2$, is a chemical substance made entirely of oxygen; water, ${\mathrm{H}}_2\mathrm{O}$, is a chemical compound containing both hydrogen and oxygen. A water molecule contains three chemical atoms but only two chemical elements. Joining chemical atoms through chemical bonds changes their arrangement without changing their atomic numbers.

This distinction makes stoichiometry possible: ordinary chemical reactions rearrange chemical atoms, so one can count each chemical element before and after a transformation. Changing the proton count instead changes the chemical element and requires a nuclear reaction. The periodic table organizes these identities into a useful map, while atomic number bookkeeping keeps identity, isotope, and electric charge separate.

A chemical formula uses symbols of chemical elements and numerical subscripts to record composition. In the molecular formula ${\mathrm{H}}_2\mathrm{O}$ of water, the subscript $2$ counts two hydrogen chemical atoms, and the omitted subscript after $\mathrm{O}$ means one oxygen chemical atom. A molecular formula describes one molecule; a chemical formula need not describe a discrete molecule.

Read a preceding coefficient separately from the subscripts. One ${\mathrm{H}}_2\mathrm{O}$ contains two hydrogen and one oxygen chemical atom; $2{\mathrm{H}}_2\mathrm{O}$ contains four hydrogen and two oxygen chemical atoms distributed between two water molecules. In ${\mathrm{H}}_2{\mathrm{O}}_2$, the second subscript instead puts two oxygen chemical atoms into each hydrogen peroxide molecule. Thus $2{\mathrm{H}}_2\mathrm{O}\ne {\mathrm{H}}_2{\mathrm{O}}_2$: the coefficient scales the whole chemical formula, while a subscript specifies its composition. See Flowers et al., Chemistry 2e, §2.4: chemical formulas.

For crystalline $\mathrm{NaCl}$, the chemical formula specifies a $1:1$ ratio of sodium ions ${\mathrm{Na}}^{+}$ to chlorine ions ${\mathrm{Cl}}^{-}$. Its formula unit is the accounting entity; the ordinary crystal is an extended arrangement of ions, rather than a collection of discrete $\mathrm{NaCl}$ molecules. Specifying the entity matters when assigning an amount of substance. See formula unit and 2.6 Ionic and Molecular Compounds (§2.6: ionic and molecular compounds).

A molecular formula gives an atom inventory, not the full arrangement of chemical bonds or the molecular geometry. For example, ${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6$ counts six carbon, twelve hydrogen, and six oxygen chemical atoms in a glucose molecule, but does not specify the pattern shown by its Fischer projection. Different structures can share a molecular formula. This compact inventory is useful for chemical equation accounting precisely because it leaves the structural detail to other representations.

The choice of letters has an early explanation in Jöns Jacob Berzelius, Essay on the Cause of Chemical Proportions (1813–1814), §III, transcribed by Carmen Giunta: letters made chemical signs easier to write and print, and formulas compressed information about composition. His numerical notation differed from modern subscripts; the lasting idea was to make proportions readable in ordinary text.

William Weber Coblentz's primary report describes an electrically heated porcelain cavity wound with platinum ribbon. Mirrors imaged its central radiating diaphragm onto a $0.6\mathrm{mm}$ optical slit; a fluorite optical prism dispersed the beam onto a vacuum bolometer. Galvanometer deflections read the detector response. A resistance standard checked electrical sensitivity, while repeated readings checked stability. Sodium lines fixed the angular zero; a refractive index curve supplied wavelengths and slit-bandwidth corrections. Thermocouples were standardized at freezing points of zinc, antimony and copper. Drying the optical path reduced atmospheric absorption; prism scattering and temperature calibration remained limitations. These were Coblentz's measurements. His extraction of radiation constants assumed the Planck law, so it was not a model-independent proof of the spectral formula.^[ref]

The inference to test is whether corrected spectra taken at independently determined temperatures have their $B_{\lambda }$ maxima at a constant product ${\lambda }_{\max }T$. Comparing raw galvanometer deflections at different optical prism settings would also compare different wavelength widths and optical losses. The response integral in blackbody peak and detector response explains why those corrections belong to the physical inference rather than merely to presentation.

In 1911, Louis Dunoyer de Segonzac formed a sodium beam in vacuum and collected a deposit on a surface cooled with liquid air. Its shape agreed with straight paths through the slits.^[ref] A deposit tests the geometry of the surviving paths. A clock or a calibrated deflection is needed to turn their travel into a speed measurement.

Electrochemistry studies how chemical reactions, transport, and chemical equilibrium couple to electric charge and electric potential. In a cell, electrons travel through the external circuit, while mobile ions carry electric current through the electrolyte. At the electrodes, redox reactions connect these two forms of electric charge transport. This connection makes it possible both to obtain electrical work from a chemical reaction and to use electrical work to drive a chemical reaction.

An anode is an electrode where oxidation occurs; a cathode is an electrode where reduction occurs. These definitions refer to the chemical reactions, rather than universal positive or negative terminal signs. In a galvanic cell delivering electrical work, the anode is negative and the cathode positive. In an electrolytic cell driven by an external source, the anode is positive and the cathode negative. See Pingarrón et al., Terminology of electrochemical methods of analysis (IUPAC Recommendations 2019), §§3.1 and 3.4 and Flowers et al., Chemistry 2e, §17.2: galvanic cells.

For a worked example, consider an electrolytic cell containing molten sodium chloride, with the anode connected to the positive terminal and the cathode to the negative terminal of an external source. At the cathode, sodium cations gain electrons; at the anode, chlorine anions lose electrons:The first line has net electric charge zero on each side; the second has $-2e$ on each side. Adding the lines cancels the transferred electrons and preserves both chemical atom counts and net electric charge. The sodium product is neutral, and the chlorine product is a diatomic molecule. These equations specify molten sodium chloride; aqueous chemical solutions introduce competing water reactions. See 17.7 Electrolysis (§17.7: electrolysis of molten sodium chloride).

Electrochemistry underlies battery charging and the manufacture of chemicals by electrolysis. Its particularly useful idea is that an electron inventory in a balanced chemical equation also accounts for transported electric charge. Transport alone is a different question from an electrode chemical reaction: an ion may carry electric current through the electrolyte without itself being consumed at an electrode.

Historically, Michael Faraday introduced the terms electrode, electrolyte, anion, cation, and ion in the terminology section of his 1834 paper, Experimental Researches in Electricity, Seventh Series by Michael Faraday (§§662–667). His §667 explicitly treats the direction assigned to electric current as a convention. His ion language predates the modern electron and atomic nucleus picture used in the charge-counting example above.

For an electrolyte near chemical equilibrium, a Poisson-Boltzmann equation can approximate the distribution of ions in an electric potential. Such an equilibrium model does not by itself describe a cell carrying sustained electric current.

An electron configuration records the occupation of electron shells and electron subshells by electrons. For neutral sodium in its ground state, $1s^{2}2s^{2}2p^{6}3s^1$ gives $2+2+6+1=11$ electrons; adding the occupations within each electron shell gives $(2,8,1)$. A superscript counts electrons, rather than multiplying the preceding labels. Removing the outer electron gives the sodium ion ${\mathrm{Na}}^{+}$ with $1s^{2}2s^{2}2p^6$, or $(2,8)$. See Flowers et al., Chemistry 2e, electron configurations of atoms and ions.

An enthalpy of atomization is the enthalpy change for making one mole of separated gaseous chemical atoms from a specified starting chemical substance. For sodium, the relevant step is sublimation to monatomic gas; for diatomic chlorine, it is half the compatible molar bond-dissociation energy expressed as an enthalpy.

A formula unit is a specified compositional accounting entity for a chemical substance that need not consist of discrete molecules. For ordinary crystalline $\mathrm{NaCl}$, one formula unit counts one sodium ion ${\mathrm{Na}}^{+}$ and one chlorine ion ${\mathrm{Cl}}^{-}$; it does not identify an isolated pair within the crystal. Cohen et al., IUPAC Green Book, 3rd ed., §2.10.1(iii)(b), p. 51 defines a formula unit as a specified group of chemical atoms, rather than a unit of measurement. An amount of substance of one mole of $\mathrm{NaCl}$ formula units therefore corresponds to one mole of each kind of ion.

A practical test compares three transformations involving sodium hydroxide: dissolution of the solid, neutralization of its chemical solution by hydrochloric acid, and reaction of the solid directly with hydrochloric acid. In the ideal compatible-state notation, the hydrogen ion and hydroxide giveThe hydroxide cancels, whereas the newly formed water remains. Sodium and chloride spectator ions must still be included in the full material inventory when matching actual chemical solutions.

The teaching protocol Calorimetry: Heat of Neutralization Reaction & Hess's Law, Procedure A–C, uses nested insulated cups, a cardboard lid and a thermometer. It mixes $75\mathrm{mL}$ each of $2.0\mathrm{mol}{\mathrm{L}}^{-1}$ hydrochloric acid and sodium hydroxide chemical solutions; separately it adds $6.00\mathrm{g}$ solid sodium hydroxide to diluted hydrochloric acid, or to $150\mathrm{mL}$ water. It records initial temperatures, stirs, reads temperature every $30\mathrm{s}$, repeats runs, and extrapolates cooling trends to mixing time. Rapid weighing limits moisture uptake. The Introduction neglects vessel heat capacity and room losses; these are approximations, not a measured calibration. The readout tests whether the inferred enthalpies add, rather than directly measuring a state function.^[ref]

Matching the chemical equations alone is insufficient. The direct-acid recipe uses $0.160\mathrm{mol}$ hydrochloric acid, whereas the solution-mixing recipe uses $0.150\mathrm{mol}$. The dissolution run also does not automatically reproduce the water inventory of the mixed-solution run. Exact closure requires matching initial and final chemical solution compositions and adding any necessary mixing or dilution enthalpies. Agreement after approximate corrections is a limited test, rather than proof of exact cancellation for those recipes.

To improve the calorimetry, use calorimeter calibration for the participating hardware, and estimate calorimeter heat-loss correction under comparable conditions. With an approximately constant total receiving heat capacity $C_{\mathrm{tot}}=m{c}_p+C_{\mathrm{cal}}$, common initial temperature, constant external pressure, only expansion work, negligible escaping material and negligible bulk kinetic energy and potential energy changes, the balance isHere $m$ is the chemical solution mass, $c_p$ its specific heat capacity at constant pressure, $C_{\mathrm{cal}}$ the hardware heat capacity, and $Q_{\mathrm{loss},\mathrm{out}}$ the signed outward heat loss. Different initial temperatures require separate warming terms. Calorimeter calibration and matching chemical solution compositions address different errors.

For an original constructed model, choose $C_{\mathrm{tot}}=120\mathrm{J}{\mathrm{K}}^{-1}$ and $\Delta T=3\mathrm{K}$. Neglect outward heat loss and other corrections. The receiving chemical solution and hardware gain $q_{\mathrm{receiver}}=360\mathrm{J}$, so $q_{\mathrm{rxn}}=-360\mathrm{J}$. If the completed extent of reaction is $\xi =0.020\mathrm{mol}$ for a unit-coefficient chemical equation, the inferred reaction enthalpy per mole of that reaction isThese chosen numbers illustrate the signs and normalization; they are not measured sodium hydroxide data and are independent of the abstract $A,B,C$ example.

A loss of heat to the room makes the uncorrected warming underestimate the magnitude of a heat-releasing reaction. Incomplete dissolution, incomplete reaction, uncertain amount of substance, or a slow thermometer response can also spoil closure. A useful experimental residual is $r=\Delta H_{\mathrm{direct}}-(\Delta H_D+\Delta H_N)$ after all changes have been placed on the same amount and state basis. Compare $r$ with the measurement uncertainty, accounting for shared calibration errors; averaging repeated runs reduces scatter without removing a common bias.

A conductance-based Hodgkin-Huxley model couples membrane potential to voltage-dependent channel-gating variables. It explains generation of action potentials through regenerative and opposing ionic electric currents, while its parameters and channel assumptions determine its biological domain of validity.

In the classical model, $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Each follows two-state ion-channel gating, with its own voltage-dependent rates. The available electrical conductances areThe barred quantities are maximum electrical conductances. The factors have the interpretation of independent permissive gates in an ion channel: three activation factors and a distinct inactivation-availability factor for sodium, and four identical activation factors for potassium. Large $h$ means availability; small $h$ means inactivation. These are model assumptions, not a literal structural description of every ion channel. See Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics (2014), §2.2: model definition and stochastic opening.

With inward-positive electric current, a single-compartment form isHere $C$ is membrane capacitance, $I_{\mathrm{ext}}$ is imposed inward electric current, $g_L$ is leak electrical conductance, and each $E$ is a reversal potential. Use consistent units, and either whole-compartment quantities throughout or quantities per membrane area throughout. Opening changes electrical conductance; the reversal potential and present membrane potential determine the associated electric current. Coupling this equation to the voltage-dependent gates makes a generally nonlinear system of ordinary differential equations; a gate's fixed-voltage limit alone does not specify its evolving state during an action potential.

The original quantitative account is ^[ref]. The contrast with the leaky integrate-and-fire model is useful: here an action potential emerges from evolving ionic electric currents, whereas the simpler model records a threshold event and imposes a reset.

An ion is a chemical atom or a group of chemical atoms with a nonzero net electric charge. A positive ion is a cation; a negative ion is an anion. The whole entity carries the electric charge: a hydronium ion is a group of four chemical atoms with net charge $+e$. This definition agrees with ^[ref].

For an ordinary ion containing only atomic nuclei and electrons, let $N_p$ count all its protons and $N_e$ all its electrons. Each proton contributes the elementary charge $+e$, each electron contributes $-e$, and neutrons contribute zero. ThereforeThe signed integer $z$ is the net electric charge number. Removing an electron increases $z$ by one; adding an electron decreases $z$ by one. Changing the electron count leaves the atomic number, and therefore the identity of the chemical element, unchanged.

For example, a neutral sodium chemical atom has $11$ protons and $11$ electrons. Its cation ${\mathrm{Na}}^{+}$ still has $11$ protons, but only $10$ electrons, so $q=(11-10)e=+e$. A neutral chlorine chemical atom has $17$ protons and $17$ electrons; its anion ${\mathrm{Cl}}^{-}$ has $17$ protons and $18$ electrons, so $q=(17-18)e=-e$. These are particle inventories, not pictures of atomic orbitals. See 2.6 Ionic and Molecular Compounds (§2.6: ions and ionic compounds).

One sodium chloride formula unit accounts for one ${\mathrm{Na}}^{+}$ cation and one ${\mathrm{Cl}}^{-}$ anion, whose net electric charges sum to zero. Ordinary solid sodium chloride is an extended arrangement of ions, rather than discrete $\mathrm{NaCl}$ molecules. This is a useful distinction: a neutral material can contain charged constituents, and its chemical formula tells us their ratio without claiming that each pair is a separate molecule.

Ion bookkeeping connects chemistry to electric current: moving ions can transport electric charge, while stationary ions cannot supply a transport current density. In electrochemistry, both signs of ions can contribute to the same conventional current. An ion opposing a fixed electric charge is called a counterion.

An ion channel is a membrane protein that provides a selective pathway for ions across a cell membrane. An open pore permits passive transport down the relevant electrochemical gradient; it does not supply the free energy for uphill transport as a pump involved in active transport does. Two things are needed for a net electric current: an available pathway and an electrochemical potential difference. Opening a pore alone need not produce a net electric current. See Ion Channels and the Electrical Properties of Membranes.

The selective permeability of an ion channel depends on interactions between its pore and the passing ions, not simply on a hole's diameter. Some ion channels strongly favor potassium over sodium; others admit several kinds of positively charged ions. Thus selective does not necessarily mean exclusive to one ion species. See Purves et al., Neuroscience, 2nd edition (2001): molecular structure of ion channels and the mixed-channel example in Purves et al.: principles derived from studies of the neuromuscular junction.

Gating is the opening and closing of an ion channel's pathway. A voltage-gated ion channel responds to membrane potential; an ionotropic receptor can respond to a bound ligand. Gating changes the available electrical conductance, while the electrochemical gradient supplies the drive for passive transport. For a fixed reversal potential in the simple model below, opening more identical ion channels increases the magnitude of the electric current at a fixed nonzero drive, without choosing its sign independently of membrane potential. The two-state ion-channel gating model describes how the open fraction can change. See the Alberts reference above.

For a worked approximation, hold an open population's electrical conductance at $g=10\mathrm{nS}$ and its reversal potential at $E_{\mathrm{rev}}=-80\mathrm{mV}$. Use $V=V_{\mathrm{inside}}-V_{\mathrm{outside}}$ for membrane potential and inward-positive conventional current:At $V=-100,-80,-60\mathrm{mV}$, respectively, the electric currents are $+200,0,-200\mathrm{pA}$, since $1\mathrm{nS}\times 1\mathrm{mV}=1\mathrm{pA}$. Positive electric current means net positive electric charge entering, or net negative electric charge leaving; it does not mean that every ion moves inward. For a pore passing only positive ions, the three results mean net inward passage, no net passage, and net outward passage. These stipulated values describe an illustrative Ohmic approximation, not measured properties of a particular ion channel; see reversal potential for the current plot and limitations.

The useful connection is between a molecular gate and a controllable electric current. A ligand can change electrical conductance through an ionotropic receptor, or membrane potential can change it through a voltage-gated ion channel, connecting recognition and electrical signaling. In a historical landmark, Erwin Neher and Bert Sakmann reported currents from individual ion channels in frog muscle membranes in 1976. See Neher and Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibres, Nature 260, 799–802 (1976).

Ion-channel inactivation is loss of availability for conduction, distinct from simply closing an otherwise activatable ion channel. Recovery depends on channel type and membrane potential; incomplete recovery of sodium-selective voltage-gated ion channels contributes to a neuronal refractory period.

For the routine examples here, assign all bonding electrons between different chemical elements to the more electronegative partner; divide bonding electrons between identical chemical elements equally. Retain each lone pair on its original chemical atom. Subtract the assigned valence electron count $N_i^{\mathrm{assigned}}$ from the neutral chemical atom's valence electron count $v_i$. The IUPAC convention uses Allen electronegativity, based on free-chemical atom valence electron energies. Close electronegativity comparisons and some reversibly bound chemical compounds require care beyond this simple rule.^[ref]

The examples explain exceptions to memorized rules:

These drawings record allocations, not molecular geometry or a physical process of breaking the chemical bonds. In particular, the O–O line in the molecular oxygen panel only marks the connection; it does not assert a complete Lewis structure or spin state.

An ionotropic receptor is a receptor in which binding of a ligand directly gates an ion channel within the same protein complex. The binding site detects the chemical signal; the pore supplies a path across the cell membrane. At a chemical synapse, a neurotransmitter therefore changes synaptic conductance without an obligatory intervening G protein or second messenger. ^[ref]

A familiar example is the nicotinic acetylcholine receptor of skeletal muscle: acetylcholine binding opens a pore passing cations, including sodium and potassium. The muscarinic acetylcholine receptor instead uses intracellular signal transduction; the same neurotransmitter can engage different mechanisms. Direct gating does not mean that an ionotropic receptor pumps ions: electrochemical gradients drive passage through the open ion channel. Nor does “ionotropic” specify excitation. The reversal potential, initial membrane potential and other conductances determine whether the effect favors an action potential, causes hyperpolarization, or supplies shunting inhibition. ^[ref]

A concrete electrical approximation separates opening from drive. Let $N$ identical ion channels each have open-channel electrical conductance $\gamma$ and opening probability $p(t)$. At fixed membrane potential $V$, with common reversal potential $E_{\mathrm{rev}}$, the expected value of inward-positive electric current isFor stipulated values $N=100$, $\gamma =20\mathrm{pS}$, $p=0.2$, $V=-60\mathrm{mV}$ and $E_{\mathrm{rev}}=0$, the mean open count is $20$, the population electrical conductance is $0.4\mathrm{nS}$ and $I_{\mathrm{in}}=24\mathrm{pA}$. One open ion channel carries $1.2\mathrm{pA}$. These are illustrative values, not measurements. At $V=E_{\mathrm{rev}}$, the net electric current vanishes despite open ion channels; neither binding nor opening alone determines the readout. The approximation omits changes in ion concentrations and voltage-dependent open-channel electrical conductance.

A motivating experiment made the elementary electrical events visible. As Bert Sakmann recounts, chronically denervated frog muscle fibers had acetylcholine-sensitive receptors spread beyond the original nerve contact. Mild enzyme treatment exposed the cell membrane; a polished glass pipette sealed over a small patch to reduce background noise. Two intracellular electrodes held the local membrane potential through voltage clamp, while the patch pipette recorded tiny electric currents. Applying suberyldicholine, a ligand that activates the acetylcholine receptor and prolongs openings, revealed abrupt, approximately rectangular current events. Their amplitudes agreed reasonably with estimates from whole-membrane current fluctuations. This established discrete opening and closing events; it did not make every ligand response instantaneous. Background noise and finite recording bandwidth concealed sufficiently small or brief events.^[ref]

Recording an intact membrane patch alone does not exclude intracellular relays. Tank, Huganir, Greengard and Webb supplied a complementary isolation test in 1983: they purified nicotinic acetylcholine receptors from an electric ray's electric organ, incorporated them into artificial lipid membranes, and fused small vesicles into larger vesicles by freezing and thawing. Sealed glass-pipette recordings from isolated membrane patches resolved functional ion channels with properties resembling those in muscle. Removing the intact cytoplasm and its signaling machinery supports the sufficiency of the receptor complex for channel function; an artificial lipid environment still cannot reproduce every native regulatory interaction.^[ref]

Direct gating supports millisecond communication at many chemical synapses. Response duration also depends on neurotransmitter clearance, binding and channel-state transitions. Continued ligand exposure can prolong activity. Classify an ionotropic receptor by its direct connection between recognition and pore gating, rather than a universal duration cutoff. ^[ref]