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The common vector field $F$ represents an identical reference cell. The small cell-specific term $ϵG_i$ represents heterogeneity of cellular properties, such as differing ion-current parameters or intrinsic oscillation rates. It is not external random noise: the displayed model is deterministic.

The Kronecker delta factors restrict coupling to the first state component, the membrane potential. Substitution gives a coupling current in the voltage equation proportional to $ϵ{\sum }_j{d}_{ij}(u_j^1-u_i^1)$, while the other state components have no direct intercellular coupling. This is the linear voltage-difference response of gap junctions. Gating and chemical variables can still change indirectly through their own coupled intracellular dynamics. The zero diagonal removes self-coupling, which would contribute zero anyway. Physical reciprocal gap-junction coupling is often symmetric, but zero diagonal alone does not impose reciprocity; that distinction matters in the final part.

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.

A dynamic clamp is a real-time feedback experiment. Measure the membrane potential through an electrode, numerically evolve chosen channel or synapse state variables, calculate the modeled current, and inject that current into the real neuron. A virtual synapse might use $I_{\mathrm{syn}}=g_{\mathrm{syn}}(t)(E_{\mathrm{syn}}-V(t))$; a voltage-dependent virtual channel also updates its gate probabilities from the measured voltage. Repeat at a rate fast relative to the dynamics being modeled. Unlike voltage clamp, this procedure does not hold the voltage at a command value.

Two uses are to add a controlled inhibitory conductance and measure its effect on firing or integration, and to couple two real neurons by artificial reciprocal inhibitory synapses to test the resulting network dynamics. Both artificial conductances and reciprocal inhibitory connections were demonstrated in the primary Sharp–O'Neil–Abbott–Marder dynamic-clamp experiments. Conductance magnitude, time course and reversal voltage can be varied without requiring the cell to express a new channel. Sampling delay, electrode error and the inability of a single injection site to reproduce all dendritic conductance locations are practical limitations.

An action potential is a brief regenerative change in a neuron's membrane potential, followed by repolarization and recovery. In a typical spiking neuron, depolarization recruits voltage-gated ion channels; inward current promotes further depolarization until inactivation and opposing currents terminate the event. The neuronal refractory period makes immediate repetition difficult or impossible. Once initiated, the roughly stereotyped event propagates along the axon. Its amplitude is therefore not generally the primary variable used to represent stimulus strength.

Inputs arriving at synapses on a neuron and its dendrites change membrane currents and produce graded voltage changes. Their spatial and temporal integration can trigger an action potential. At a chemical synapse, an arriving spike promotes release of a neurotransmitter, whose receptors alter the next neuron's conductances. Depending on the synapse and receptor mechanism, the input can promote or suppress the next neuron's firing. Thus information flows through activity patterns interacting with connectivity and synaptic dynamics, rather than by an identical voltage waveform simply passing unchanged across every cell.

In rate coding, a relevant response statistic is the number of spikes in a chosen interval, or the mean firing rate inferred from repeated trials or a population. For example, a sensory variable might be represented by mean counts of two versus five spikes in a $100$ ms window. A downstream neuron integrating many synaptic inputs over that window can distinguish the rates even if individual spike times vary. The window, averaging method and neuron's stimulus tuning must be specified; not every cell increases its rate for every stronger stimulus. Neural adaptation can also make the rate depend on recent stimulus history.

In temporal coding, precise times carry information beyond that count. Two responses containing the same number of spikes can differ in their first-spike latency after stimulus onset. Alternatively, spikes can lock to a particular phase of a periodic input, or relative timing across two neurons can encode coincidence. A decoder that reads only the long-window count discards those distinctions, whereas a temporally sensitive synapse or coincidence detector can use them.

Rate coding uses a chosen frequency or count summary; temporal coding uses the arrangement of spike times relative to a reference. They are complementary possibilities within neural coding, and a spike train can carry information in both. Stochastic variability does not prevent coding, but requires assessing how reliably different response distributions can be distinguished.

Phosphorylation converts ABA perception into regulated ion transport, coordinates calcium-dependent and calcium-independent inputs, and makes stomatal closure reversible. Its role is distributed across a network of kinases, phosphatases and transport targets. ABA does not close the pore by directly supplying a mechanical force.

The opening and closing machinery is a pair of guard cells surrounding a stoma. Accumulation of osmotically active solutes favors water entry and increased turgor pressure; solute loss favors water exit and reduced aperture. During water stress, abscisic acid shifts the balance toward anion and potassium loss. Changes in transport can precede slower changes in gene expression, allowing the existing guard-cell machinery to respond rapidly.

The core ABA switch involves inhibition of an inhibitor. Without the activating ABA signal, selected clade-A protein phosphatase 2C enzymes, including ABI1, restrain ABA-responsive SnRK2 kinases. ABA binding to a PYR-PYL-RCAR ABA receptor promotes inhibition of these phosphatases. This permits activating kinase phosphorylation to persist and supports activation of OST1, also called SnRK2.6. Thus a hormone-dependent change in phosphatase activity changes the phosphorylation state and catalytic activity of downstream signaling proteins. The effect is specific to the relevant receptor, phosphatase and kinase combinations; it is not a claim that all phosphatases inhibit closure or all phosphorylation promotes it.

Biochemical reconstitution supports this causal ordering: defined receptor, phosphatase and kinase components can reconstruct ABA-dependent regulation without requiring an intact plant. This separates a sufficient core perception-and-relay module from the many parallel processes present in a stressed cell. However, a reconstructed molecular or transcriptional readout alone does not prove that the full mechanical response of a stoma has been rebuilt.^[ref]

A crucial transport target is SLAC1. OST1-dependent phosphorylation activates this slow anion channel, allowing anions to leave the guard cell. Anion efflux depolarizes the membrane potential, favoring outward potassium current through GORK. Loss of these solutes reduces the osmotic drive retaining water, so water exits and turgor pressure falls. Activation of an anion channel can therefore initiate a larger electrical and osmotic response without the kinase phosphorylating every ion that moves. Suppression of opening-associated transport further reinforces the shift toward closure.

Channel activation experiments provide an important bridge between the core ABA switch and physiology. Coexpression and electrophysiology showed that OST1 activates SLAC1, whereas phosphatase action antagonizes activation; kinase activity and channel regulatory regions are therefore functionally important. These findings are stronger than simple colocalization. Together with guard-cell genetic and current measurements, they support a route from a phosphorylation-dependent channel change to altered aperture. A heterologous cell nevertheless lacks the complete transport and mechanical environment of a real guard cell.^[ref]

Calcium-dependent signaling converges on this machinery. Calcium-dependent protein kinases can activate SLAC1, and different kinases have different calcium sensitivities. Consequently, phosphorylation can combine hormone-dependent release of the phosphatase brake with a calcium signature. This is a convergence network, not a mandatory linear sequence in which ABA must always cause a measurable global calcium rise before any closure can occur.^[ref] Regulation of plant NADPH oxidases and reactive oxygen species can further couple kinase activity, redox signalling and calcium entry. The physiological contribution of these branches varies with conditions.

Reversal is as important as activation. Phosphatases, calcium clearance and renewed opening-associated transport reset the system when conditions change. Phosphorylation can alter activity, interactions and localization without waiting for protein replacement. The outcome is a tunable balance of kinase and phosphatase activities that connects ABA-dependent sensing to solute flux, water movement and pore mechanics, while integrating additional environmental inputs.

Cells interconvert several forms of free energy. Chemical concentration differences and the chemical potentials of metabolites can drive reactions, including ATP hydrolysis. The usable free-energy difference is that between reactants and products under the actual cellular conditions; breaking a chemical bond alone does not release energy without the rest of the reaction. Oxidation-reduction reactions transfer electrons down an electrochemical potential difference and can drive ion pumping in an electron transport chain. Light absorption can create excited states and charge separation in photosynthesis, eventually supporting chemical synthesis.

An electrochemical gradient stores free energy in both an ion concentration difference and a membrane potential. Chemiosmosis couples downhill ion flow to ATP synthesis by chemiosmosis; ATP-powered pumps can run the conversion the other way. Molecular motors and other mechanochemical enzymes convert chemical free energy into mechanical work or elastic strain, while ion gradients can also power transport and rotary motion. These couplings conserve total energy but dissipate some free energy as heat. Equilibrium thermal fluctuations alone cannot sustain directional work without a maintained nonequilibrium resource.

Use number concentrations and write the electrochemical potential of a sodium ion as $\tilde{\mu }={\mu }^{\circ }+k_{\mathrm{B}}T\ln c+e\psi$. Define the inward sodium-motive force as available work per positive charge for transport from outside to inside. With $V_m={\psi }_i-{\psi }_o$,Its negative is the signed inward free-energy cost per charge; stating this direction removes a common sign ambiguity.

Long-open sodium channels establish electrochemical equilibrium, $k_{\mathrm{B}}T\ln (c_{i0}/c_o)+e{V}_{m0}=0$. This does not generally require equal concentrations: impermeant counterions can support a nonzero equilibrium membrane potential. Let $n_0=c_{i0}V$. For a large fixed-concentration external reservoir, after pumping $n$ ions out, $c_i(n)=(n_0-n)/V$ and $V_m(n)=V_{m0}-ne/C$. The sodium export cost from a capacitive vesicle for exporting another ion isThe initial-equilibrium relation was used in the last step. The first term is the incremental electrical cost and the second is the change in chemical cost from their initial balanced values.

If the intended initial reference is uncharged with $V_{m0}=0$ and $c_{i0}=c_o$, then $n_0=c_{o}V$ and the requested form in the listed variables isThat equal-concentration reference is an additional assumption. For example, $c_{i0}=2c_o$ and $V_{m0}=-(k_{\mathrm{B}}T/e)\ln 2$ also give zero initial sodium flux, but a different ion inventory. Thus the open-channel condition alone cannot determine $n_0$ from $c_{o}V$. If concentrations are molar rather than number densities, use the corresponding Avogadro constant conversion and $RT$ for molar free energies. Treating the capacitor charge discretely gives a charging-energy increment $(n+1/2)e^2/C$ rather than the differential $n{e}^2/C$; the distinction is negligible for the thousands of ions used next.

Use the initial equal-concentration, zero-potential reference, or equivalently quote component changes relative to the initial electrochemical equilibrium. At $n=n_0/2=5000$, with $C=10^{-16}\mathrm{F}$,At room temperature $T=300\mathrm{K}$, the chemical component isThe electrical cost dominates by about $447$. The associated change in membrane potential is $-ne/C=-8.01\mathrm{V}$. Such a large voltage is a consequence of the ideal fixed-capacitance, perfectly impermeable model; an actual lipid membrane would not generally remain intact and nonconducting over this entire extrapolation.

The electrical component is a straight line through zero, $n{e}^2/C$. The chemical component, $k_{\mathrm{B}}T\ln [n_0/(n_0-n)]$, starts at zero, increases convexly, and diverges as $n\to n_0^{-}$ because the remaining internal concentration vanishes. Their sum lies above both: nearly electrical over most of the stated range, but eventually diverging with the chemical term. For a nonzero initial equilibrium potential, the separate absolute electrical and chemical components have opposite constant offsets; the total curve and the quoted changes remain the same. The final few ions require a discrete-particle treatment rather than a literal continuum divergence.

Patch clamp connects an electrode to a small area of cell membrane through a solution-filled patch pipette. A gigaohm seal around the rim makes tiny electric currents through ion channels distinguishable from seal leak current. The beautiful change of scale is from a population's average response to individual opening and closing events in single-channel recording.

Separate two questions. The configuration determines which cell membrane is accessible: an intact patch in cell-attached recording, an excised inside-out patch or outside-out patch, or intracellular access in whole-cell recording. The recording mode determines what is commanded and what is observed: voltage clamp commands an electric potential difference and measures electric current; current clamp commands electric current and measures membrane potential. Whole-cell recording supports both modes. A patch can contain several ion channels; a small recording area does not establish that only one is present.

The 1976 article Single-channel currents recorded from membrane of denervated frog muscle fibres describes an extracellular fire-polished pipette with a $3$–$5\mu \mathrm{m}$ opening and $2$–$5\mathrm{M\Omega }$ electrical resistance. It contained saline and a channel-activating compound. Two separate intracellular electrodes supplied local voltage clamp of the muscle fibre; the patch amplifier measured local electric current. Cooling to $6$–$8{}^{\circ }\mathrm{C}$ and hyperpolarization helped resolve elementary events. Figure 2 shows step-like pulses with a $200\mathrm{Hz}$ filter at $-120\mathrm{mV}$ and $8{}^{\circ }\mathrm{C}$. Discrete amplitudes and overlapping events supported individual ion channel openings rather than a smoothly varying population response. This was neither whole-cell recording through the patch pipette nor current clamp. The specialized conditions improved visibility but limit generalization to warmer preparations; filtering can conceal brief openings.^[ref]

For a new single-channel recording experiment, useful controls include a matched recording without the activating compound, controlled membrane potential, repeated seal tests, and an explicitly stated filter frequency response. These are proposed controls, not claims that every one was performed in that 1976 figure. Changing membrane potential tests the single-channel current-voltage relation; changing the recording filter tests whether a short pulse's apparent amplitude is an instrument artifact. A flat baseline establishes neither an absence of ion channels nor an absence of very brief events.

Choose the following original phenomenological mathematical model solely to calculate a paired-pulse readout. It is not experimental data, a fit to the teaching experiment, or a simulation of the Hodgkin-Huxley model. The conditioning action potential defines time zero. Test pulses have fixed width $0.2\mathrm{ms}$; $I$ denotes their inward electric current amplitude. Impose an absolute neuronal refractory period $t_{\mathrm{abs}}=2\mathrm{ms}$. No finite test amplitude generates an event for $0<\Delta <t_{\mathrm{abs}}$. After that hold, stipulate a required amplitudewith $I_{\infty }=0.2\mathrm{nA}$, dimensionless $A=2$ and recovery time constant ${\tau }_r=3\mathrm{ms}$. Emit a test event exactly when $I\ge I_{\mathrm{req}}(\Delta )$, including equality. For this ideal event readout, assign events to pulse onset and assume the conditioning event is successfully evoked. There is no calculated membrane potential, spike waveform, response latency or stochastic success probability.

The exponential decay represents a combined recovery deficit; it is neither the Hodgkin-Huxley model's $h$ nor a measured potassium electrical conductance. One such curve cannot separately identify sodium recovery and persistent potassium electrical conductance. A constant pulse-width amplitude criterion also does not establish an equivalent rule for other pulse widths or voltage trajectories.

At the hold boundary, $I_{\mathrm{req}}(2\mathrm{ms})=0.2(1+2)=0.6\mathrm{nA}$. At $\Delta =4\mathrm{ms}$,at $\Delta =6\mathrm{ms}$,Thus a $0.35\mathrm{nA}$ test fails at $1\mathrm{ms}$ because of the imposed hold, fails at $4\mathrm{ms}$ because recovery is insufficient for that amplitude, and succeeds at $6\mathrm{ms}$. At $4\mathrm{ms}$, a $0.5\mathrm{nA}$ pulse would instead succeed: failure of the weaker pulse was relative refractoriness.

For $I>I_{\infty }$, solve $I\ge I_{\mathrm{req}}(\Delta )$ using the logarithm. The earliest allowed delay isFor $I=0.35\mathrm{nA}$ this is $2+3\ln (8/3)\approx 4.942488\mathrm{ms}$; for $I=0.5\mathrm{nA}$ it is $2+3\ln (4/3)\approx 2.863046\mathrm{ms}$. If $I\ge 0.6\mathrm{nA}$, the imposed $2\mathrm{ms}$ hold is the only delay restriction. If $I\le I_{\infty }$, no finite delay passes the rule: the required amplitude stays strictly above its asymptotic baseline. A finite experimental “fully recovered” time therefore also needs a tolerance and a readout definition.

Return of a depolarized membrane potential toward a more negative value. During an action potential, reduced inward current and increased outward current commonly contribute. It is distinct from hyperpolarization, which takes the membrane beyond its reference resting value.

A reversal potential $E_{\mathrm{rev}}$ is a membrane potential at which a specified open ion channel, or specified combination of ion channels, carries zero net electric current, with the surrounding conditions fixed; the electric current changes sign across that voltage. Specifying which current matters: an individual ion contribution and a sum of several contributions can reverse at different voltages. Zero net electric current does not mean that all ions have stopped moving. See Purves et al., Neuroscience, 2nd edition (2001): reversal of a mixed-ion channel current.

Throughout these examples, membrane potential is $V=V_{\mathrm{inside}}-V_{\mathrm{outside}}$, and conventional current is inward-positive. Positive electric current corresponds to net positive electric charge entering or net negative electric charge leaving. Negative electric current corresponds to the reverse charge transfer. A convenient Ohmic approximation iswhere $g$ is the available electrical conductance. For fixed $g>0$, $V<E_{\mathrm{rev}}$ gives positive electric current, $V=E_{\mathrm{rev}}$ gives zero, and $V>E_{\mathrm{rev}}$ gives negative electric current; the line has slope $-g$. If $g=0$, the modeled pathway carries zero electric current at every voltage: closing it does not locate its reversal potential. 3.1 Synapses (§3.1: conductance-based synaptic current) uses outward-positive $g(V-E)$; the formula here reverses that sign to match this wiki's inward-positive convention.

For the stipulated single-pathway example from ion channel, $E_{\mathrm{rev}}=-80\mathrm{mV}$ and $g=10\mathrm{nS}$ giveThe open ion channel carries zero net electric current at $-80\mathrm{mV}$ even though $g$ remains positive. Near that point a change of membrane potential changes the electric current, which is why zero current at one voltage does not make an open pathway electrically irrelevant; see shunting inhibition.

For an ion channel passing only one ion species by passive transport, its reversal potential can equal that species' Nernst equilibrium potential, where the chemical and electrical contributions to electrochemical potential balance. At that thermodynamic equilibrium, passage in both directions can continue with no net transport. For an ion channel admitting several ion species, opposing contributions can cancel while each species continues to have net transport. Such a reversal potential is not generally any one species' Nernst equilibrium potential. See the Purves reference above.

To work out that cancellation explicitly, put two Ohmic ion channel populations across the same cell membrane. Stipulate constant electrical conductances and fixed reversal potentials:The potassium and sodium contributions add because both pathways experience the same membrane potential:Thus $I_{\mathrm{total}}=12\mathrm{nS}(E_{\mathrm{eff}}-V)$. This weighted mean follows simply by expanding and regrouping the two current terms; it assumes additive Ohmic contributions and a positive total electrical conductance. At $V=E_{\mathrm{eff}}$, the inward-positive individual electric currents areThe potassium contribution is outward and the sodium contribution inward. Equal and opposite electric currents give $I_{\mathrm{total}}=0$; neither pathway has stopped transporting its ions. The calculation treats two distinct pathways. Applying it to one mixed-species ion channel would additionally require a justified decomposition into independent Ohmic contributions; a general mixed-channel reversal potential cannot be obtained by assuming those hypotheses silently.

This is an instantaneous balance of the specified passive electric currents. It would be the stationary voltage of a passive capacitance model containing only these constant pathways and no applied electric current, but it is not a complete prediction of a living cell's resting membrane potential. Other electric currents, voltage-dependent gating, and pumps involved in active transport can matter. Pumps that transfer net electric charge are electrogenic and contribute their own electric current. See Purves et al.: functional properties of the sodium/potassium pump.

The toy model holds ion concentrations and therefore the stipulated reversal potentials fixed. Continuing inward sodium and outward potassium transport would change finite reservoirs unless active transport or external control maintained them. Transport can be a small fraction of the reservoir's ions over a short interval, making fixed concentrations a useful approximation; it is not a claim that the reservoirs never change. See Gerstner et al., Neuronal Dynamics (2014), §2.1: equilibrium potentials, channel currents and maintained gradients.

A reversal potential makes the effect of opening an ion channel predictable from the present membrane potential. The same open pathway can carry inward or outward electric current as $V$ crosses its reversal potential. The two-pathway example reveals a particularly useful distinction: zero net charge transfer can coexist with continuing exchange of different substances. That distinction connects membrane transport to electric current measurements without mistaking electrical balance for thermodynamic equilibrium.

Shunting inhibition is suppression of another input's effect through increased synaptic conductance: opening additional ion channels lowers effective electrical resistance, reducing the voltage response to a given added electric current. When the added reversal potential equals the resting membrane potential, there can be no voltage change at rest and still be inhibition of a simultaneous input. See 1.4 Limitations of the Leaky Integrate-and-Fire Model.

Use the passive synaptic potential model below with leak $g_L=10\mathrm{nS}$, $E_L=-65\mathrm{mV}$ and capacitance $C=200\mathrm{pF}$. A sustained inward test electric current of $I_{\mathrm{ext}}=100\mathrm{pA}$ alone gives a steady voltage increase of $I_{\mathrm{ext}}/g_L=10\mathrm{mV}$, so $V_{\infty }=-55\mathrm{mV}$. Effective electrical resistance is $100\mathrm{M\Omega }$ and the time constant is $20\mathrm{ms}$.

Now hold a shunting conductance $g_i=20\mathrm{nS}$ open with $E_i=E_L=-65\mathrm{mV}$. With no test electric current, rest stays at $-65\mathrm{mV}$. With the same $100\mathrm{pA}$ test electric current, total conductance becomes $30\mathrm{nS}$, so the steady voltage increase is only $100/30\simeq 3.333\mathrm{mV}$: $V_{\infty }\simeq -61.667\mathrm{mV}$. Effective electrical resistance is $33.333\mathrm{M\Omega }$ and the time constant is $200/30\simeq 6.667\mathrm{ms}$. A faster response can therefore be smaller. At the original rest the shunting electric current is zero; at $V=-55\mathrm{mV}$ it is $20(-65+55)=-200\mathrm{pA}$, opposing the voltage excursion.

The factor of three reduction applies to this constant-electric current, constant-conductance example. A finite pulse, conductance-driven input or spatially extended dendrite need not have the same reduction. The interesting feature is conditional action: the shunt can leave a quiet neuron's voltage unchanged yet reshape its response when another input arrives. A model that represents all inhibition as a fixed outward electric current misses this behavior.

For an ideal open ion channel with constant electrical conductance $g$, membrane potential $V$ and reversal potential $E_{\mathrm{rev}}$, define outward electric current as positive:This is an Ohm's law model of the open state, not a universal channel law or a rule for opening probability. In the synthetic figure, $g=10\mathrm{pS}$ and $E_{\mathrm{rev}}=0\mathrm{mV}$. Because $1\mathrm{pS}\cdot 1\mathrm{mV}=10^{-3}\mathrm{pA}$, the open-state electric currents at $-80$ and $-40\mathrm{mV}$ are $-0.8$ and $-0.4\mathrm{pA}$. Zero electric current at the reversal potential does not imply a closed ion channel. Real ion channels can have nonlinear responses, so the straight line is an explicit idealization.

Start at sodium electrochemical equilibrium with $n_0$ internal ions. Exporting $n$ ions changes the membrane potential by $-ne/C$ and the internal number density to $(n_0-n)/V$. Subtracting the initially balanced chemical and electrical contributions gives the displayed incremental cost. An initial zero potential further implies $n_0=c_{o}V$ for an ideal fixed external reservoir. Nonzero initial equilibrium potential permits unequal concentrations, so that latter equality is an additional assumption.

The space clamp problem is the failure to maintain the commanded membrane potential throughout a spatially extended biological cell. A soma voltage clamp can coexist with substantial voltage changes in distant dendrites or an axon. Reducing patch-clamp series resistance improves access at the recording site, but does not eliminate electrical separation within the neuron.

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

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

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

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]

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

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

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

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

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