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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.

The axon initial segment is the specialized proximal region of an axon near the soma. In many neurons, its voltage-gated ion channels make it the site where an action potential begins. ^[ref]

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 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]

Model a spatially uniform cell membrane as a capacitor of membrane capacitance $C_m$ in parallel with membrane electrical resistance $R_m$ and a fixed leak reversal potential $E_L$. Exclude active ion channels, seal leak current and pipette capacitance. For electric current $I$ injected into the biological cell, electric charge conservation yieldsStarting at $V(0)=E_L$ and applying constant $I$ gives the ordinary differential equation solutionThe steady state rise is $I{R}_m$; the time constant is $R_m{C}_m$. With $I=50\mathrm{pA}$, $R_m=200\mathrm{M\Omega }$, $C_m=50\mathrm{pF}$ and $E_L=-70\mathrm{mV}$, the rise is $10\mathrm{mV}$ and ${\tau }_m=10\mathrm{ms}$. At one time constant the rise is $10(1-e^{-1})\mathrm{mV}\approx 6.321\mathrm{mV}$. The figure's uncompensated pipette voltage adds $I{R}_s=1\mathrm{mV}$ for $R_s=20\mathrm{M\Omega }$. The membrane voltage is continuous at step onset in this model, while the pipette voltage immediately acquires that access drop.

Checking the units is instructive: electrical resistance times capacitance has units $\Omega \mathrm{F}=\mathrm{s}$, and electric current times electrical resistance has units of voltage. Unlike the passive voltage clamp transient under patch-clamp series resistance, this ideal current clamp membrane time constant is set by $R_m{C}_m$, not $R_s{C}_m$. Strong current steps can recruit voltage-gated ion channels; the passive model then fails and cannot generate an action potential. Small pulses of both signs, stable access and repeated baseline measurements help test whether the passive approximation is appropriate.

A synapse is a signaling junction between a neuron and another biological cell. The sending side is presynaptic and the receiving side postsynaptic. At a chemical synapse, released neurotransmitter acts on a receptor in the receiving cell membrane. At an electrical synapse, gap junctions provide a direct path for electric current between cells. See 1.1 Elements of Neuronal Systems.

In a common sequence at a chemical synapse, an arriving action potential opens voltage-gated ion channels selective for calcium. Calcium entry triggers fusion of synaptic vesicles with the presynaptic cell membrane, releasing neurotransmitter into the synaptic cleft. Binding to a receptor changes postsynaptic electrical resistance and conductance, directly through an ion channel or through an intracellular signaling pathway. The resulting change in membrane potential is a synaptic potential. See Ion Channels and the Electrical Properties of Membranes.

A useful conductance approximation writes the inward-positive synaptic electric current aswhere $g_{\mathrm{syn}}\ge 0$ describes the activated electrical resistance and conductance, $V$ is the postsynaptic membrane potential, and $E_{\mathrm{syn}}$ is the reversal potential, the voltage at which this current vanishes. Opening these channels tends to pull $V$ toward $E_{\mathrm{syn}}$; it does not prescribe a current of fixed sign independently of $V$.

For a worked example, let $V=-65\mathrm{mV}$ and $g_{\mathrm{syn}}=1\mathrm{nS}$. A reversal potential of $0\mathrm{mV}$ gives $I_{\mathrm{syn}}=65\mathrm{pA}$, an inward electric current that raises membrane potential. A reversal potential of $-75\mathrm{mV}$ instead gives $-10\mathrm{pA}$, which lowers membrane potential. These are illustrative calculations, not measurements. A reversal potential of $-60\mathrm{mV}$ can raise $V$ from $-65\mathrm{mV}$ yet oppose reaching a threshold of $-50\mathrm{mV}$: a depolarizing input need not be excitatory.

An excitatory synapse favors a subsequent action potential, whereas an inhibitory synapse opposes it. The effect depends on the receptor, the permeating ions, their gradients and the reversal potential; a neurotransmitter does not have one fixed excitatory or inhibitory sign in every context. Even with $E_{\mathrm{syn}}=V$, an added conductance can reduce responses to other inputs by lowering effective electrical resistance, an effect called shunting inhibition. See Purves and colleagues: excitatory and inhibitory postsynaptic potentials. This dependence matters when translating a biological synapse into a signed weight or an imposed electric current in computational neuroscience.

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]