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Biology studies living systems: how biological cells maintain themselves, how organisms develop and reproduce, and how evolution generates diversity. Its scales connect molecules, genes, biological cells, organisms and the interactions studied by ecology. These are complementary viewpoints: a protein's chemistry can explain a cellular mechanism, while evolution explains why inherited variants of that mechanism persist. See Clark, Douglas and Choi, Biology 2e: the science of biology.

Two organizing ideas make this immense subject navigable: shared cellular organization and descent with inherited modification. A neuron and a gliding bacterium differ dramatically in shape and behavior, yet both depend on a cell membrane and metabolism. Their differences become intelligible through specialization and evolution.

Keep observations separate from the mathematical models used to explain them. A measured membrane potential is evidence; an idealized two-state ion-channel gating equation proposes a mechanism. A mathematical model earns trust through predictions and comparisons with measurements. Its assumptions delimit its usefulness, and disagreement can reveal missing biology rather than a faulty measurement.

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 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).

The probability that an ion channel is in a conducting open state under specified conditions. For N equivalent channels, each carrying open-state electric current i, the mean total electric current is $N{P}_{O}i$. An ensemble open fraction and the fraction of time one channel remains open coincide under suitable stationary and ergodic assumptions. A two-state ion-channel gating model is one model of this probability, not its definition.

For two-state ion-channel gating, let ${\alpha }_n(V)$ and ${\beta }_n(V)$ be the rates of entering and leaving the permissive state. Gain minus loss givesAt fixed membrane potential, these coefficients are constant. The deviation $w=n-n_{\infty }$ obeys $\dot{w}=-w/{\tau }_n$, so the initial condition $n(0)=n_0$ yieldsThis approaches the fixed-voltage limit monotonically and stays between $n_0$ and $n_{\infty }$, as a valid probability must.

Convention qualification: the displayed single exponential function describes a voltage-clamp step, not an arbitrary time-dependent action potential. For example, a second imposed voltage step changes both coefficients and requires a new exponential segment. For general $V(t)$ the integrating-factor solution is $n(t)=e^{-A(t)}[n_0+{\int }_0^t{\alpha }_n(V(s))e^{A(s)}ds]$, with $A(t)={\int }_0^t[{\alpha }_n(V(s))+{\beta }_n(V(s))]ds$.

Voltage gating and ion selectivity are different structural functions. A typical voltage-gated potassium channel has four pore-forming subunits; their voltage-sensing regions move charged residues in the membrane electric field and couple that motion to a pore gate. Carbonyl coordination in the selectivity filter compensates potassium dehydration in an ion-specific geometry. The smaller sodium ion is not automatically passed more readily: its hydration and coordination energetics differ. Structural work on a mammalian Kv channel resolves the sensor/pore architecture and an open activation gate.^[ref]

A sodium-channel alpha subunit contains four homologous domains assembled into one pore, with separate voltage sensors. Rapid activation and fast inactivation are distinguishable conformational transitions; an intracellular inactivation apparatus stabilizes nonconducting states after activation. Repolarization favours recovery of availability. A channel can therefore be closed but activatable, or inactivated and temporarily unavailable, despite zero current in both cases.

Several types of evidence test these ideas. Ionic reversal potentials shift when ionic gradients are changed, establishing selectivity and driving force. Single-channel recordings show discrete conducting levels and stochastic dwell times; whole-cell current changes often reflect changed open probability rather than a continuously varying single-channel current. Small gating-charge transients can remain when ionic permeation is blocked, demonstrating sensor movement rather than ion flux. Conditioning and paired-pulse records separate activation, inactivation and recovery. Structural perturbations of voltage sensors and of inactivation machinery alter different kinetic properties, supporting their functional distinction.

These observations constrain models but do not make an individual macroscopic trace a unique structural reconstruction. Different distributions of closed and inactivated states can produce similar averages. In particular, the sustained potassium trace supports little inactivation on the displayed interval; it does not justify assigning every potassium channel a universal absence of an inactivation mechanism.

At fixed voltage let $a$ and $b$ be opening and closing rates. The open probability obeysThis illustrates equal open probability with unequal ion-channel relaxation times: both channels have $p_{\infty }=1/2$, but channel A has ${\tau }_A=1\mathrm{ms}$ and channel B has ${\tau }_B=0.1\mathrm{ms}$. Starting closed, $p_A(t)=\frac{1}{2}(1-e^{-t/(1\mathrm{ms})})$ and $p_B(t)=\frac{1}{2}(1-e^{-t/(0.1\mathrm{ms})})$. If channel number, unitary conductance and driving force agree, both have the same steady macroscopic current; B approaches it ten times faster.

An individual channel remains either open or closed. Its mean closed and open dwell times are $1/a$ and $1/b$, respectively: both are $2\mathrm{ms}$ in A and $0.2\mathrm{ms}$ in B. Thus identical equilibrium open probability does not imply identical kinetics or noise correlation time. Ensemble probability relaxation depends on the sum of rates, while the mean dwell in a given state depends on its exit rate.

A two-state closed–open scheme at fixed voltage relaxes monotonically and cannot by itself produce a sodium-current transient during a maintained step with constant driving force. A minimal sodium scheme therefore includes an unavailable state: $C\rightleftarrows O\rightleftarrows I$, with voltage-dependent activation, inactivation and recovery; closed-state recovery pathways may be added when needed. Opening is initially rapid, occupancy then accumulates in $I$, and recovery on repolarization restores an activatable population.

For the illustrated potassium current, a closed–open model gives sustained opening, but several activation steps can account for its initial delay. A conventional illustrative coarse-grained description iswith each gate following $\dot{q}={\alpha }_q(V)(1-q)-{\beta }_q(V)q$. Fast $m$ activation, subsequent decrease of $h$, and slower increase of $n$ produce the two qualitative records. The powers are phenomenological gate assumptions, not values uniquely inferred from these traces or an exact structural count for every cardiac channel.

A 1-Hz cardiac rhythm has about a second between successive activations, whereas a 100-Hz neuronal rhythm has only $10\mathrm{ms}$; the available resting interval is shorter still after subtracting the action potential. Recovery or deactivation with a timescale comparable to that interval can then leave residual inactivation or potassium activation, giving use-dependent changes and refractoriness. Millisecond or faster relaxation usually completes much more readily during a long diastolic interval. The particular two-state A and B in d would both nearly equilibrate over a full $10\mathrm{ms}$ at their stipulated constant rates; their values alone do not demonstrate high-frequency failure. Real comparison requires voltage-dependent recovery rates and the different cardiac and neuronal waveforms. A frozen-rate model cannot quantitatively predict either firing pattern from frequency alone.

For two-state ion-channel gating, write $C{\rightleftarrows }_{k_c}^{k_o}O$. Gain minus loss of open probability gives the ordinary differential equationThus the ratio of the transition rates determines the final open probability, while their sum determines the response speed. Both given channels have $p_{\infty }=1/2$. For an initially closed population, with $t$ measured in milliseconds,The mean individual open and closed lifetimes are $1/k_c$ and $1/k_o$: both are $2\mathrm{ms}$ for A and $0.2\mathrm{ms}$ for B. With the same channel number, single-channel electrical conductance and driving force, the steady macroscopic electric currents are equal, but B reaches that value ten times faster. If both populations were already at $p_O=1/2$, neither would show a mean relaxation under these fixed rates: a response also requires an initial condition or a change in rates.

The stationary variance of the number open is $N/4$ for independent channels. Its autocovariance is $(N/4)e^{-|t|/\tau }$, so B's fluctuations decorrelate ten times faster even though their instantaneous variance is unchanged. Faster gating allows an ion channel population to follow a brief stimulus and contribute to rapid action potential timing. Slower gating integrates over brief changes and can provide a delayed opposing electric current, useful in setting action potential duration or frequency selectivity. Those examples describe consequences of kinetics; the stipulated voltage-independent rates alone do not define a complete excitable neuron.

The given probability density function is the completion-time density for two independent sequential irreversible steps, with exponential distributions of waiting times at rates $k_a,k_b>0$. Let $T=T_a+T_b$. The convolution isIt is a two-stage hypoexponential distribution. The density vanishes at zero, initially rises as $k_a{k}_{b}t$, peaks, and then decays with a tail governed by the slower rate. Setting its derivative to zero givesNormalization follows either from the convolution of normalized densities or directly from $(k_a{k}_b/(k_a-k_b))(1/k_b-1/k_a)=1$. In the equal-rate limit $k_a=k_b=k$, the apparent singularity is removable andThis is an Erlang distribution with shape two. If one rate is much faster, its short waiting time becomes hard to resolve and the distribution resembles a single exponential distribution after its initial rise.

The observation supports at least two kinetically resolved stages before the recorded event, rather than one memoryless rate-limiting event. Possibilities include substrate binding followed by a conformational or catalytic step. It does not identify which stage is binding, chemistry, product release or rotation. The density is unchanged by exchanging $k_a$ and $k_b$, so temporal order and chemical identity cannot be recovered from this fit alone. Other hidden-state mechanisms can also mimic the same density. Dependence of one fitted rate on ATP concentration, for example, would provide additional evidence about assignment, rather than following from the distribution by itself.

The left panel illustrates b for an initially closed population. The other panels are dimensionless examples of the distributions derived here; none reconstructs the missing source trace.

In the Hodgkin-Huxley model, $V$ is the membrane potential, the voltage inside the axon relative to outside. The space clamp assumption makes it spatially uniform. The gating variables lie between zero and one: $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Thus large $h$ means channels remain available, not that inactivation is strong.

The left side $C_m\dot{V}$ is capacitive electric current per unit membrane area. The imposed $I$ is inward-positive applied electric current. The other terms are outward-positive ionic electric currents: sodium with electrical conductance $g_{Na}{m}^{3}h$, potassium with conductance $g_K{n}^4$, and a passive leak with conductance $g_L$. Each current is its conductance times the difference between $V$ and the corresponding reversal potential $V_{Na}$, $V_K$ or $V_L$. The coefficients $g_{Na},g_K$ are maximal conductances; the gate factors represent independent permissive gates in an ion channel.

For each gate, $k_{\infty }(V)$ is the equilibrium opening/availability fraction at fixed voltage and ${\tau }_k(V)>0$ its exponential relaxation time. The equation follows from two-state ion-channel gating: if $\dot{k}={\alpha }_k(V)(1-k)-{\beta }_k(V)k$, thenVoltage feedback couples these otherwise elementary gating equations to the membrane potential and produces action potentials.

At rest, the cell has relatively high intracellular potassium and high extracellular sodium. The membrane potential is negative inside, largely because selective resting permeability, particularly potassium leak, combines with these electrochemical gradients. The sodium-potassium pump maintains the gradients over time; it is not the immediate regenerative current of an action potential.

Depolarization rapidly activates sodium voltage-gated ion channels, causing inward sodium current and further depolarization. Slower sodium inactivation and delayed potassium activation then reduce inward current and increase outward potassium current, repolarizing the membrane. Persisting potassium conductance can produce an after-hyperpolarization; recovery of sodium availability contributes to the neuronal refractory period. A spike transfers only a small fraction of the bulk ions, so it changes charge separation much more than it changes the overall concentration gradients.

For two-state ion-channel gating, let $X$ be the fraction of permissive gates. Opening transfers probability from $1-X$ at rate ${\alpha }_X(V)$, and closing transfers it back at rate ${\beta }_X(V)$:Independent subunits that must all be permissive multiply their probabilities. The Hodgkin-Huxley model therefore uses $g_{\mathrm{Na}}={\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and $g_{\mathrm{K}}={\bar{g}}_{\mathrm{K}}{n}^4$: $m$ describes sodium activation, $h$ sodium availability, and $n$ potassium activation. Their voltage-dependent time scales produce the sequence of inward and outward current. This channel-level construction is the mechanism in the primary Hodgkin–Huxley membrane-current model.

Two-state ion-channel gating models an individual gate as a random switch between closed and open states. Write $p(t)$ for its open probability; this is the $X$ in the header equation. An individual gate's state is always closed or open, rather than partly open. A population of identical gates can have a fraction open near $p(t)$, and averaging repeated identically prepared trials estimates the same probability. A finite population's observed fraction fluctuates. See Purves and colleagues, Neuroscience, 2nd edition (2001): Ion Channels Underlying Action Potentials for single-channel steps and their population averages.

At fixed membrane potential $V$, let constant rates $\alpha ,\beta >0$ govern closed-to-open and open-to-closed transitions respectively, in ${\mathrm{s}}^{-1}$. This is a two-state continuous-time Markov chain. Over a short interval $\Delta t$, an initially closed gate opens with probability $\alpha \Delta t+o(\Delta t)$, and an initially open gate closes with probability $\beta \Delta t+o(\Delta t)$. Gain minus loss gives the deterministic ordinary differential equation for the probability:Here $p_{\infty }$ is the fixed-voltage limiting open probability, and $\tau$ is a time constant. The ratio of the rates fixes the limit, while their sum fixes the speed. With initial probability $p(0)=p_0\in [0,1]$,For a short derivation, subtract $p_{\infty }$: the deviation obeys $d(p-p_{\infty })/dt=-(p-p_{\infty })/\tau$, yielding the exponential function. After one time constant, $1/e$ of the initial deviation remains. The solution stays between $p_0$ and $p_{\infty }$, so it remains a valid probability. The rate and time constant formulations are described in Gerstner and colleagues, Neuronal Dynamics, §2.2.2: transition rates and channel kinetics.

For a worked toy condition A, choose $\alpha =\beta =50{\mathrm{s}}^{-1}$ and $p_0=0.1$. Then $p_{\infty }=0.5$, $\tau =0.01\mathrm{s}=10\mathrm{ms}$, andIn a separately prepared condition B, choose $\alpha =200{\mathrm{s}}^{-1}$, $\beta =50{\mathrm{s}}^{-1}$ and the same $p_0=0.1$. Now $p_{\infty }=0.8$, $\tau =4\mathrm{ms}$, and $p(10\mathrm{ms})=0.8-0.7e^{-2.5}\approx 0.742540$. Condition B approaches a higher limit more quickly. These are two stipulated constant-rate conditions with $V$ held fixed in each experiment, not an empirical function relating rates to voltage or a sequential switch between conditions. The initial $p_0=0.1$ need not be the limiting probability at the newly imposed voltage.

In the Hodgkin-Huxley model, the rates depend on membrane potential: ${\alpha }_X(V)$ and ${\beta }_X(V)$. Its $X_{\infty }(V)={\alpha }_X(V)/({\alpha }_X(V)+{\beta }_X(V))$ is the limit for voltage held at that $V$. During an action potential, changing $V$ changes the rates and gates feed back onto $V$ through ionic electric currents; substituting the present voltage into the constant-rate solution does not solve that coupled system. Gating describes availability of a conducting pathway. Even an open ion channel carries zero net ionic electric current at its reversal potential.

The useful bridge is from abrupt random molecular events to a smooth population prediction. A channel requiring several independent permissive gates in an ion channel has an open probability equal to their product; this can make channel availability much smaller than each gate's availability. The Hodgkin-Huxley model uses ${\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and ${\bar{g}}_{\mathrm{K}}{n}^4$ for sodium and potassium electrical conductances, respectively.