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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.

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.

The leaky integrate-and-fire model combines a passive equation for membrane potential with a firing rule. Between events, inward-positive applied electric current $I(t)$ obeyswhere $C>0$ is membrane capacitance, $g_L>0$ is the leak conductance, $R$ is the corresponding electrical resistance, and $E_L$ is the leak reversal potential. Here the leak conductance is constant. The time constant $\tau$ describes passive relaxation. When $V$ first reaches a threshold $V_{\theta }$ from below, record an action potential event and reset $V$ to $V_r<V_{\theta }$. In the version used here, hold $V=V_r$ for a fixed neuronal refractory period $t_{\mathrm{ref}}$, then resume integration. See 1.3 Integrate-And-Fire Models.

For constant $I$, with $t=0$ at the start of an integration interval and $V(0)=V_{\mathrm{initial}}$, the exact subthreshold solution isThe equilibrium point of this passive equation is $V_{\infty }=E_L+RI$. Subtracting $V_{\infty }$ leaves $d(V-V_{\infty })/dt=-(V-V_{\infty })/\tau$, explaining the exponential function. This solution applies only until a threshold event or a change in $I$. Starting below threshold, constant input produces repeated events if $V_{\infty }>V_{\theta }$; if $V_{\infty }=V_{\theta }$, the threshold is approached asymptotically and never reached at finite time.

For a worked example, chooseThese imply $g_L=10\mathrm{nS}$ and $C=200\mathrm{pF}$. All these parameters are illustrative model choices, not measurements from a particular neuron. With $I=200\mathrm{pA}$, Ohm's law gives $RI=20\mathrm{mV}$ and hence $V_{\infty }=-45\mathrm{mV}$. ThusAt threshold, $e^{-t/\tau }=1/4$, so the first event occurs after $T=20\ln 4\mathrm{ms}\approx 27.7259\mathrm{ms}$. Each subsequent cycle includes the $2\mathrm{ms}$ neuronal refractory period, givingThis is the ideal sustained firing rate; the first-event latency is $T$, not $T+t_{\mathrm{ref}}$.

With $I=100\mathrm{pA}$ instead, $RI=10\mathrm{mV}$ and $V_{\infty }=-55\mathrm{mV}$. From the same initial state, $V(t)=-55\mathrm{mV}-10\mathrm{mV}{e}^{-t/\tau }$ stays below $-50\mathrm{mV}$: no events occur, however long this current is held. The critical sustained current is $(V_{\theta }-E_L)/R=150\mathrm{pA}$; equality gives asymptotic approach rather than finite-time firing.

The leak gives old inputs progressively less influence, while capacitance prevents an ordinary finite electric current step from changing $V$ instantly. This combination makes the leaky integrate-and-fire model useful for exploring input summation, event timing and network activity at low computational cost. Its fixed parameters and reset omit the evolving voltage-gated ion channels of the Hodgkin-Huxley model, spatial integration in dendrites and mechanisms of neural adaptation. Treating a synapse as a prescribed $I(t)$ also omits its voltage-dependent driving force unless a conductance term is added. See 1.4 Limitations of the Leaky Integrate-and-Fire Model and constant-input firing rate of a leaky integrate-and-fire neuron.

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 firing threshold specifies the condition for initiating an action potential. The leaky integrate-and-fire model uses a prescribed membrane potential $\vartheta$; real initiation involves active voltage-gated ion channels and depends on recent activity and recording site. A threshold drawn on a passive voltage trace is a comparison criterion, not a simulated action potential.

The leaky integrate-and-fire model treats the membrane as a capacitance $C_m$. The leak conductance $g_L$ drives voltage towards its resting reversal $V_L$; the excitatory conductance $g_E$ drives it towards its reversal $V_E$; and $I_e$ is injected current, positive inward. A sufficiently high $V_E$ makes increasing $g_E$ depolarizing over the relevant voltage range, while also increasing total conductance and shortening the integration time.

For constant input and conductances, putThenIf this trajectory reaches $V_{\theta }$, register a spike and reset the voltage to $V_0<V_{\theta }$, optionally enforcing a specified neuronal refractory period. The reset makes repeated threshold crossings possible under a constant suprathreshold drive. If $V_{\infty }\le V_{\theta }$ and the initial voltage is below threshold, no finite-time crossing occurs; equality gives asymptotic approach. For time-varying currents or conductances, integrate the subthreshold differential equation until the next crossing.

Use this model when spike timing, spike counts and network integration matter more than the detailed waveform: it is inexpensive for large recurrent networks or comparisons of current-driven and conductance-driven input. It omits ionic spike initiation, channel-specific adaptation and many dendritic effects, for which a Hodgkin-Huxley model or a spatial model may be needed.

Computational models and biological experiments answer complementary questions. Experiments constrain what a nervous system actually does; a model makes a proposed explanation precise enough to test whether its mechanisms can produce those observations. The useful comparison is between competing, experimentally constrained explanations, rather than between modelling and experimentation as alternatives. A good model exposes assumptions, predicts new observations and identifies measurements that would distinguish it from rivals.

A Hodgkin-Huxley model links action potentials to voltage-dependent conductances and channel kinetics. It is more informative than merely fitting a spike waveform because changes in its conductances predict changes in excitability. Yet a model calibrated for one cell or preparation does not automatically represent another: different channel combinations and spatial geometry can yield similar outputs. Detailed parameters must be constrained by experiments, and apparent agreement does not make them uniquely identifiable. A leaky integrate-and-fire model deliberately sacrifices channel detail to study the effects of connectivity, timing and synapses in a large network. It is useful for such questions even though it cannot explain the microscopic origin of the action-potential waveform.

At a functional level, an LNP model tests which stimulus features predict a neuron's response and how those features map to firing rate. Independent testing, including response correlations rather than only mean rate, reveals what its assumptions miss. It can characterize a receptive field without claiming that the brain explicitly performs its particular fitted calculation. A Hopfield network likewise demonstrates that recurrent interactions can implement content-addressable memory: corrupted cues relax toward stored patterns. Its usefulness is an existence proof and a quantitative account of interference, not evidence that biological memory literally uses symmetric binary connections and the same update schedule.

Developmental models connect observed input statistics with synaptic plasticity. Hebbian learning with normalization or synaptic competition can produce selective weights and segregation of competing inputs. Models of correlated ON and OFF inputs can explain the formation of oriented simple-cell subregions and predict how changing the correlations changes their organization. That makes altered visual experience or patterned stimulation informative tests. Such models still need anatomical, developmental and plasticity constraints; several learning rules may reproduce similar final receptive fields. A primary model of activity-dependent ON/OFF competition illustrates this approach.

Normative models instead ask what representation would perform a specified task efficiently. For example, sparse coding of natural images can produce localized oriented filters resembling simple cell receptive fields. This links environmental statistics to a useful coding objective, but resemblance alone neither proves that the objective is the only explanation nor specifies the biological learning mechanism. Predictions about altered stimulus ensembles, population activity and synaptic changes provide stronger tests. The original sparse-code receptive-field model provides an example.

These distinctions also explain the dangers of models. Excessive flexibility can fit noise; unrealistic assumptions can generate a desired effect by construction; and multiple parameter sets can have nearly indistinguishable outputs. Simulation without analysis may obscure rather than explain a mechanism. Cross-validation, uncertainty analysis, comparisons with simpler models and controlled perturbations are therefore important. Conversely, collecting more measurements without a discriminating hypothesis can leave the same explanatory ambiguity unresolved.

The productive cycle is to propose a constrained model, determine which observations it predicts beyond the fitting data, perform the relevant experiment, and revise or reject the model. Models help select experiments and interpret results; experiments establish their scope and reveal omitted mechanisms. The appropriate level of detail depends on the question: spike-generation mechanisms, circuit computation and developmental organization need not be explained by the same model.

If every interspike interval is at least $t_{\mathrm{abs}}>0$, a sustained spike train has mean firing rate at most $1/t_{\mathrm{abs}}$. Across a window of duration $T$, the event count satisfies $N(T)\le 1+T/t_{\mathrm{abs}}$; dividing by $T$ and taking the long-time limit gives the bound. The extra one accounts for a possible first event. The bound permits much slower firing and says nothing about whether stronger drive can maintain the limiting rate in a real neuron.

For the chosen phenomenological paired-pulse recovery, the hold alone gives $500\mathrm{Hz}$. Add the further assumption that every successful event resets precisely the same recovery curve, with no accumulation from earlier events. Then $0.35\mathrm{nA}$ onset-timed pulses can sustain at most $1000/4.942488\approx 202.327\mathrm{Hz}$ in that model. This repeatable reset is an extra assumption: a two-pulse result alone cannot establish it. A real neuron's sustained rate also depends on neural adaptation, input delivery and evolving ion-channel inactivation. The existing leaky integrate-and-fire model similarly adds integration time to its imposed hold.

A homogeneous Poisson process of rate $\lambda$ has interspike intervals with an exponential distribution, soWith the purely illustrative choices $\lambda =100\mathrm{Hz}$ and $t_{\mathrm{abs}}=0.002\mathrm{s}$, this is $1-e^{-0.2}\approx 0.181269$. Thus about $18.13\%$ of intervals predicted by that model violate this chosen hold. Matching a mean spike train count does not make a stimulus-only Poisson process respect recent firing.

A minimal alternative is a renewal process with independent and identically distributed intervals $S=t_{\mathrm{abs}}+X$, where $X$ has an exponential distribution of rate $\lambda$. Its hazard function is zero during the hold and $\lambda$ afterwards. Its mean interval is $t_{\mathrm{abs}}+1/\lambda$ and long-run firing rate isThe same $100{\mathrm{s}}^{-1}$ rate after a $2\mathrm{ms}$ hold gives $83.333\mathrm{Hz}$, not $100\mathrm{Hz}$. This separate stochastic example has no gradual relative recovery; it illustrates why an available-state rate differs from an observed mean firing rate. Smooth dependence on time since the last event can extend the hazard function, while longer histories require more than this simple renewal process. Time-rescaled spike-train diagnostics can test the resulting history-dependent predictions against recorded spike trains.

During a relative neuronal refractory period, another action potential is possible, but incomplete recovery changes the excitation needed to obtain it. Remaining sodium ion-channel inactivation and elevated potassium electrical conductance both matter. A hyperpolarization after an action potential can increase the voltage excursion needed for firing, while extra electrical conductance can drain an injected pulse faster. Thus a single fixed neuronal firing threshold measured from the instantaneous voltage misses relevant state variables.^[ref]

Recovery can depend on earlier spikes as well as the last one; the Hodgkin-Huxley model and neural adaptation provide richer histories than an imposed reset. A fixed hold in a leaky integrate-and-fire model is a useful simplification with a specified domain, rather than a complete explanation of a relative neuronal refractory period.^[ref]

Temporal summation is overlap of responses to inputs arriving at different times. It can occur at one repeatedly activated synapse or across several synapses; it is not defined by reuse of a particular contact. For prescribed electric current inputs in a passive compartment, residual depolarization from earlier pulses changes the voltage reached by later pulses. The membrane time constant sets how rapidly that residual decays, while the duration of synaptic conductance and spatial filtering by a passive dendritic cable can add other timescales.

For the diagram, write $v=V-E_L$, choose inward-positive imposed electric current $I$, and use the ordinary differential equationA rectangular pulse of amplitude $I_0=300\mathrm{pA}$ and duration $d=5\mathrm{ms}$, starting from rest, ends withAfterwards its contribution decays as $A{e}^{-(t-d)/{\tau }_m}$. The superposition principle therefore gives, at the end of the third pulse with onset spacing $\Delta \ge d$,For $\Delta =5\mathrm{ms}$ this is $15.829\mathrm{mV}$, so the peak membrane potential is $-49.171\mathrm{mV}$. For $\Delta =30\mathrm{ms}$ it is $8.447\mathrm{mV}$, giving $-56.553\mathrm{mV}$. Both protocols inject $3I_{0}d=4.5\mathrm{pC}$ of electric charge; equal charge does not imply equal peak voltage because the leak drains charge while it is delivered.

In the adjacent-pulse case, the illustrative neuronal firing threshold $-50\mathrm{mV}$ is crossed at $10+20\log 2\approx 23.863\mathrm{ms}$. A leaky integrate-and-fire model would emit its first event there and reset, so its later trajectory would differ from the passive diagram. The widely spaced train never reaches that threshold. Thus changing input timing alone can change whether an event occurs. A real neuron's neuronal firing threshold is also state-dependent, and synaptic conductance pulses are not prescribed electric current pulses.^[ref]

Measured responses to repeated presynaptic action potentials may also change through synaptic plasticity. A larger second response can therefore reflect a stronger second input as well as residual postsynaptic voltage. To isolate passive temporal summation, compare known injected electric current waveforms with their single-pulse responses under stable recording conditions; to test actual synapses, measure whether input efficacy changes. These are proposed discriminating comparisons, not additional controls claimed for the slice experiment.^[ref]