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Current clamp prescribes injected electric current while recording membrane potential. A constant command does not force membrane potential to remain constant: a neuron can produce action potentials or synaptic potentials while that command is unchanged. At zero commanded electric current, the amplifier still measures membrane potential; intrinsic ionic electric currents continue. The name concerns current supplied by the instrument, not the cancellation of all currents in the biological cell.

In whole-cell recording, the measured pipette voltage during a current pulse includes the patch-clamp series resistance drop. A modern authored methods article explicitly describes current-pulse protocols and this distortion.^[ref] Check the access contribution with small pulses and distinguish the immediate pipette-voltage jump from the slower membrane capacitance response. A zero-input interval measures a baseline, not proof that the instrument has no voltage offset. The passive example below explains these distinctions without asserting any new physiological measurements.

Membrane capacitance is the capacitance associated with separated electric charge across cell membrane. A local ideal capacitor has charging electric current $I_C=C_{m}dV/dt$. If a known actual membrane voltage change is $\Delta V$, integrating the charging transient after subtracting sustained ionic and leak components gives $C_m=\Delta V^{-1}\int I_{C}dt$. Pipette capacitance, finite patch-clamp series resistance and imperfect transient subtraction can bias the estimate. A capacitive transient is not by itself an ion channel opening.

A membrane potential can coexist with near electroneutrality in the bulk cytosol and external chemical solution. The small imbalance producing the voltage lies near the cell membrane; most positive and negative electric charges balance in the bulk.^[ref]

For a separate original capacitor model, choose a total membrane capacitance $C=5\mathrm{pF}$ and $V_m=-70\mathrm{mV}$. These are stipulated parameters, not measured properties of a particular biological cell, and this voltage is not asserted to equal the preceding $E_{\mathrm{K}}$. Model the separated electric charge relative to an uncharged reference byUsing the elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$, this representsexcess monovalent negative-charge equivalents inside. These can represent a deficit of positive ions, rather than the arrival of that many negative ions.

Independently choose an internal reservoir volume $\mathcal{V}=1\mathrm{pL}=10^{-12}\mathrm{L}$ containing $140\mathrm{mM}=0.140\mathrm{mol}{\mathrm{L}}^{-1}$ potassium, with balancing negative electric charge. With the Avogadro constant $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$, its potassium inventory isThus, if charging were entirely due to outward monovalent potassium transport, about one in $38600$ of these potassium ions would suffice. This is a comparison with the chosen potassium inventory, not a fraction of all charged particles or a measurement of spatial charge density. No shape, surface area or thickness has been inferred from the chosen capacitance and volume. The calculation illustrates why substantial voltage does not require a strongly charged bulk cytosol.

The membrane time constant of a passive uniform compartment is its membrane capacitance $C$ divided by its total fixed electrical conductance $G$. After input removal, voltage deviation from the relevant equilibrium undergoes exponential decay; in one membrane time constant it falls to $e^{-1}$ of its initial value. For leak alone, $G=g_L$; while fixed synaptic conductances are open, $G=g_L+{\sum }_j{g}_j$. Thus shunting inhibition can both reduce voltage responses and shorten the integration window. This is a local passive timescale: a branching dendrite can have several decay modes, and a time-varying synaptic conductance need not admit one constant ${\tau }_m$.

A passive dendritic cable models a uniform dendrite with axial electrical resistance, membrane leak and membrane capacitance, excluding active ion channels. Let $r_a$ be axial electrical resistance per length, $r_m$ membrane electrical resistance times length, and $c_m$ membrane capacitance per length. For voltage deviation $v=V-E_L$ from the leak reversal potential, electric charge conservation gives the partial differential equation$i$ is imposed inward electric current per length. Units are $r_a:\Omega /\mathrm{m}$, $r_m:\Omega \mathrm{m}$ and $c_m:\mathrm{F}/\mathrm{m}$, so ${\tau }_m$ is a time and $\lambda$ a length. On a source-free semi-infinite cable with imposed $v(0)=v_0$ and $v(x)\to 0$ as $x\to \infty$, the steady state isDifferentiate twice to verify the ordinary differential equation ${\lambda }^2{v}^{\prime \prime }-v=0$. At $x=\lambda$, the amplitude is $e^{-1}\approx 0.368$ of the boundary amplitude. This is a stationary attenuation law, not the complete waveform of a brief EPSP. Transient passive spread also broadens and delays responses. Finite ends, branch points and the soma's load alter the solution; distance alone cannot specify every real synaptic potential.^[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.

The spatial panel uses two compartments with membrane potentials $V_s,V_d$, one representing the soma and one a dendrite. Set $v_s=V_s-E_L$, $v_d=V_d-E_L$, with leak reversal potential $E_L=-65\mathrm{mV}$, each membrane capacitance $C=100\mathrm{pF}$, each leak electrical conductance $g_L=5\mathrm{nS}$ and axial electrical conductance $g_c=10\mathrm{nS}$. Electric charge conservation givesThis chosen passive model has two decay time constants, $C/g_L=20\mathrm{ms}$ for the equal-voltage mode and $C/(g_L+2g_c)=4\mathrm{ms}$ for the voltage-difference mode. Starting from rest, a $100\mathrm{pA}$ step into the soma gives, with $u$ the time since onset,while the same step into the dendrite gives$u$ is expressed in $\mathrm{ms}$. Their long-time rises are $12$ and $8\mathrm{mV}$ respectively. Applying both inputs gives $20(1-e^{-u/20})\mathrm{mV}$ exactly: spatial attenuation coexists with linear summation of prescribed electric currents. After the steps end, both modes decay with their respective time constants. This deliberately coarse model is not a reconstructed dendrite or a fit to the experiment above.

The conductance panels use the existing one-compartment synaptic potential equation, with $C=200\mathrm{pF}$, $g_L=10\mathrm{nS}$, $E_L=-65\mathrm{mV}$ and excitatory reversal potential $E_e=0\mathrm{mV}$. For a constant excitatory synaptic conductance $g_e$, the steady state voltage rise and membrane time constant areEach isolated $2\mathrm{nS}$ input has a limiting rise $130/12\approx 10.833\mathrm{mV}$. Adding those isolated rises predicts $21.667\mathrm{mV}$, but opening both together gives $260/14\approx 18.571\mathrm{mV}$. The two conductances share a changing driving force. This is sublinear input summation even though the voltage ordinary differential equation is linear for each prescribed conductance waveform.

Add $g_i=20\mathrm{nS}$ with reversal potential $E_i=E_L$ to the joint $4\mathrm{nS}$ excitation. The limiting rise becomes $260/34\approx 7.647\mathrm{mV}$ and the membrane time constant becomes $200/34\approx 5.882\mathrm{ms}$. The shunt alone leaves rest unchanged. At the joint-excitation-only steady state $V\approx -46.429\mathrm{mV}$, however, its inward electric current would be $20(-65+46.429)\approx -371.429\mathrm{pA}$: it opposes that excursion. The finite $50\mathrm{ms}$ traces in the diagram approach, but need not reach, these limiting voltages. See shunting inhibition for the constant-electric current comparison.

Treat the membrane as a capacitor, and count the additional positive charge on its inner surface. With radius $r=10^{-6}$ m and length $\ell =10^{-4}$ m, including both end caps givesThe membrane capacitance is $C=C_{m}A=6.3460$ pF. Thus $\Delta Q=C\Delta V=5.3941\times 10^{-13}$ C, corresponding to $\Delta N=\Delta Q/e=3.3667\times 10^6$ monovalent sodium ions. Converting charge to amount of substance using the Faraday constant,Omitting the ends changes the answer to $0.01762$ mM, a one-percent correction. A large membrane potential change therefore requires only about $0.12\%$ of the intracellular sodium population to enter. The bulk liquid remains nearly electrically neutral; excess charge resides chiefly near the cell membrane. This explains why an action potential need not exhaust the concentration difference that drives it, and why fixed reversal potentials are a useful approximation during one spike. This membrane capacitive ion budget assumes sodium supplies the minimum net charging requirement. Simultaneous opposing electric currents can require more sodium entry; repeated spikes eventually require the sodium-potassium ATPase to maintain the gradients.

Use signed ion valence $z$ and define $V={\psi }_1-{\psi }_2$. The molar electrochemical potential in side $i$ of an ideal dilute solution isTransfer of an infinitesimal amount $dn$ from side 1 to side 2 changes the total Gibbs free energy by $({\tilde{\mu }}_2-{\tilde{\mu }}_1)dn$. At equilibrium this derivative is zero, soFor a cell with side 1 inside, this is $E_i=(RT/z_{i}F)\ln (c_{i,\mathrm{out}}/c_{i,\mathrm{in}})$. The concentrations are equilibrium free-ion concentrations, not automatically their exact initial values. For nonideal solutions the ratio must be a ratio of thermodynamic activities. The temperature and solvent must be common to both reservoirs, and an ion must actually be able to exchange for this equilibrium condition to be dynamically established.

Equal compartment volumes are unnecessary for deriving the voltage law, but help specify number conservation if concentration changes are appreciable. For equal volume $\mathcal{V}$ and a monovalent cation transfer $n$ from side 1 to 2, $c_1=c_1^0-n/\mathcal{V}$, $c_2=c_2^0+n/\mathcal{V}$. Combined with the membrane capacitance, this gives an implicit finite-reservoir equilibrium rather than treating both concentrations as externally clamped.

Usable biological free energy has several interconvertible forms. At approximately fixed temperature and pressure, chemical reactions are described by chemical potentials, ${\mu }_i={\mu }_i^{\circ }+RT\log a_i$, and ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. ATP hydrolysis, for example, has ${\Delta }_{r}G={\Delta }_r{G}^{\circ \prime }+RT\log (a_{\mathrm{ADP}}{a}_{\mathrm{Pi}}/a_{\mathrm{ATP}})$ under the specified biochemical standard-state convention. The activities and proton/ion convention matter. Breaking a bond alone costs energy; the net free-energy change includes the products, solvent and all bond changes, not a store released simply by snapping one bond.

Redox reactions provide free energy when electrons move between donors and acceptors with a favourable potential difference: $\Delta G=-nF\Delta E$. Electron-transfer chains can couple that drop to ion transport, building an electrochemical gradient. For transfer from an outer to an inner reservoir,Both concentration difference and electrical potential contribute. ATP synthase couples downhill ion flow to nucleotide free energy; driven pumps can perform the reverse conversion and maintain gradients used by transporters and excitability.

Mechanical work appears as $Fdx$ or torque times angle, and can be stored elastically, for example $\kappa x^2/2$. Molecular motors convert chemical free energy into movement or force; mechanically driven reactions can conversely reshape chemical transitions. Electrical energy can also be stored in membrane capacitance, while bulk ionic concentration gradients store additional chemical free energy. Light supplies photon energy $h\nu$, which photosynthetic machinery can convert into redox and chemical potential differences.

Coupling is essential: an overall favourable reaction can drive an unfavourable one only when their processes are linked, rather than merely present in the same cell. At fixed conditions the extracted work is bounded by the consumed free-energy decrease. Dissipation produces heat and entropy, so energy conservation does not mean all energy remains available for work. Thermal fluctuations can drive random motion, but a persistent directional cycle requires a nonequilibrium free-energy source; an equilibrium asymmetric structure alone is not a free motor.

Including both end faces of the perfect cylinder, with $r=1\mu \mathrm{m}$ and $\ell =100\mu \mathrm{m}$,The membrane capacitance is $C=0.01\mathcal{A}=6.346\times 10^{-12}\mathrm{F}$. The required net inward electric charge is $Q=C\Delta V=5.394\times 10^{-13}\mathrm{C}$, or $Q/e\simeq 3.37\times 10^6$ monovalent ions. If sodium supplies that net electric charge, Faraday constant $F=N_{A}e$ givesThe final concentration is about $15.0178\mathrm{mM}$, only $0.119\%$ above the starting value. Neglecting the end faces changes the estimate by $1\%$, giving about $17.6\mu \mathrm{M}$. Over $2\mathrm{ms}$ the capacitive electric charge corresponds to mean net electric current $Q/\Delta t\simeq 0.270\mathrm{nA}$; the duration does not otherwise enter the total electric charge.

This is a net capacitive estimate, not a measurement of total sodium influx. Simultaneous outward potassium electric current makes actual sodium entry larger, and most bulk ions remain near their original concentrations during one spike. Long-term pumps restore gradients after repeated firing; they are not required to replace the entire intracellular sodium pool during the two-millisecond voltage rise.

The equivalent circuit has membrane capacitance in parallel with electrical conductance–battery branches: voltage-dependent sodium and potassium conductances in series with their reversal potentials, a chloride electrical conductance in series with $E_{Cl}$, and other leak conductances. For externally supplied electric current,The model figure in solution includes this circuit and an illustrative passive response. Although chloride electric current is small when $V\simeq E_{Cl}$, a perturbation produces the restoring electric current $\delta I_{Cl}=g_{Cl}\delta V$. A large chloride electrical conductance therefore shunts perturbations, lowers input resistance and shortens the passive time constant, $R_{\mathrm{in}}\simeq 1/\sum g_j$ and ${\tau }_m\simeq C_m/\sum g_j$.

Reducing that electrical conductance increases the voltage response to a given electric current or extracellular-potassium disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if $E_{Cl}$ equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.^[ref]

Later cessation of contraction can be explained by sustained depolarization causing voltage-gated sodium channel ion-channel inactivation and failure to recover: fewer available channels support progressively smaller or less frequent spikes until depolarization block occurs. Accumulated extracellular potassium in restricted spaces and incomplete recovery between spikes can promote that depolarization. This requires the nonlinear voltage-gating dynamics; a passive chloride resistor and capacitor alone cannot generate or terminate an action-potential train. The stated membrane permeability change does not uniquely determine the time to block or prove one particular ionic accumulation history.

The original Panel A is readable: the baseline is roughly $-45\mathrm{mV}$ and the peak about $+40\mathrm{mV}$, giving a voltage excursion near $85\mathrm{mV}$. Using the specific membrane capacitance and surface area,A coarser $100\mathrm{mV}$ spike estimate gives $6.2\times 10^6$ ions, illustrating the precision appropriate to the graph. These are the net monovalent charges needed to charge the capacitor if sodium supplies the inward charge; simultaneous opposing currents can make the actual sodium influx larger.

The cell contains $0.015\mathrm{mol}{\mathrm{L}}^{-1}\times 3\times 10^{-13}\mathrm{L}\times N_A\simeq 2.71\times 10^9$ intracellular sodium ions. The capacitive estimate adds only about $0.20\%$, or $0.029\mathrm{mM}$. Thus a spike does not equilibrate bulk concentrations. The sodium-potassium ATPase maintains the gradients over many spikes, exporting three sodium ions per two potassium ions imported per ATP; the estimated sodium load alone corresponds to at least about $1.8\times 10^6$ pump cycles. Rapid repolarization itself comes mainly from changing channel conductances, while sustained fast firing requires adequate energy supply and gradient restoration. The small stated volume and large area are used as supplied, without assuming a particular cell geometry.

The passive upper limiting membrane potential for a purely sodium-selective pathway is its Nernst equilibrium potential. Equal inside and outside electrochemical potentials, using ideal concentrations, requireThis is the maximum absolute inside-minus-outside voltage under the stated passive assumptions. If the reference resting membrane potential is $-60\mathrm{mV}$, the corresponding increase is about $121\mathrm{mV}$.

Actual action potentials usually peak below $E_{\mathrm{Na}}$ because voltage-gated sodium channels undergo ion-channel inactivation, while outward potassium and leak electric currents oppose inward sodium electric current. Finite membrane capacitance and finite available time also prevent instantaneous equilibration. At $V=E_{\mathrm{Na}}$, the sodium driving force itself is zero.

In the Hodgkin-Huxley model, $V$ is the intracellular-minus-extracellular membrane potential, $C_m$ is the membrane capacitance per unit area, and $I_{\mathrm{app}}$ is the applied inward electric current per unit area. A space clamp makes $V$ spatially uniform, leaving temporal ordinary differential equations. With outward ionic electric current positive, the capacitance balance is $C_m\dot{V}=I_{\mathrm{app}}-I_i$.

The variables $m,n,h$ lie between zero and one. They describe sodium activation, potassium activation and availability against sodium inactivation, respectively. Thus larger $h$ means less inactivation. Their voltage-dependent limiting values are $m_{\infty },n_{\infty },h_{\infty }$, and ${\tau }_m,{\tau }_n,{\tau }_h$ are their positive relaxation time constants.

Each ionic term has the Ohm's law form conductance times driving potential. The sodium term has instantaneous electrical conductance $g_{\mathrm{Na}}{m}^{3}h$, the potassium term has instantaneous electrical conductance $g_{\mathrm{K}}{n}^4$, and the leak term has constant electrical conductance $g_{\mathrm{L}}$. Here the $g$ constants are maximal electrical conductances per area, and $V_{\mathrm{Na}},V_{\mathrm{K}},V_{\mathrm{L}}$ are the corresponding reversal potentials. The factors $m^{3}h$ and $n^4$ arise from independent permissive gates in an ion channel. A conducting channel carries zero net ionic electric current at its reversal potential.

The cylindrical part has length 2 micrometres and radius 0.5 micrometres. Including both hemispherical ends gives area $A=2\pi rL+4\pi r^2=3\pi \times 10^{-12}$ m${}^2$. The membrane capacitance is $C={ϵ}_0{ϵ}_{r}A/d\simeq 4.17\times 10^{-14}$ F. Define the signed inward electrochemical drop per unit positive charge asThere is a factor-of-two ambiguity in the wording of the charge data. If the stated excess means the inner capacitor charge $Q=-40000e$, then $\Delta \psi =Q/C=-0.154$ V and $\Delta p\simeq -0.183$ V at 298 K. If “inside minus outside” is taken literally as $Q_{\mathrm{in}}-Q_{\mathrm{out}}=-40000e$ for opposite capacitor charges, the inner plate has $-20000e$, giving $\Delta \psi =-0.0768$ V and $\Delta p\simeq -0.106$ V. The latter is the literal charge-difference interpretation; the former is the commonly intended excess-charge convention. In either case inward proton motion is favorable, and the positive available driving magnitude is $-\Delta p$.

Patch-clamp series resistance, also called patch-clamp access resistance, is the effective electrical resistance between the pipette's sensed voltage and the cell interior. It includes tip access and any remaining obstruction after break-in. For injected electric current $I$ into the biological cell, an ideal resistive access path givesThus $1\mathrm{nA}$ through $20\mathrm{M\Omega }$ produces a $20\mathrm{mV}$ difference: a substantial possible voltage clamp error. The electric current in this equation is the access-path current, excluding any bypass seal leak current. In current clamp, the same access drop contaminates the pipette voltage during injection. Bridge compensation subtracts an estimated $I{R}_s$; an estimate that changes during recording leaves an error.

An independent passive voltage clamp calculation makes both speed and error visible. Write $v=V-E_L$ and $v_p=V_p-E_L$, with fixed leak reversal potential $E_L$, constant membrane electrical resistance $R_m$ and membrane capacitance $C_m$. Assume an ideal command at the pipette node and omit seal leak current and pipette capacitance. Electric charge conservation givesFor a step $v_p=\Delta V$ from $v(0)=0$, the ordinary differential equation givesWith the figure's parameters and $\Delta V=10\mathrm{mV}$, the time constant is $0.909\mathrm{ms}$, the final membrane rise is $9.091\mathrm{mV}$, and access-path electric current falls from $500$ to $45.45\mathrm{pA}$. Only when $R_m\gg R_s$ does the time constant approach $R_s{C}_m$ and the final voltage approach the command. Actual voltage clamp also depends on amplifier feedback control and the space clamp problem.

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]

Whole-cell recording establishes intracellular electrical access by rupturing the membrane under a sealed patch pipette while retaining the gigaohm seal. Under adequate voltage clamp, the measured electric current combines contributions from accessible ion channels, membrane capacitance and seal leak current. Current clamp instead follows membrane potential while supplying a chosen electric current. Whole-cell recording is useful for linking a biological cell's total electrical response to its ion channels, but neither mode automatically controls distant dendrites or preserves the original cytosol.

In the 1981 methods demonstration, a bovine chromaffin biological cell was tested before and after patch rupture. Small voltage clamp steps produced a much larger capacitive transient after break-in. Integrating it gave $5\mathrm{pF}$ for a $13\mu \mathrm{m}$ cell; its decay indicated patch-clamp series resistance below $20\mathrm{M\Omega }$. Depolarizing commands then revealed ionic electric currents. Separate current clamp recordings showed spontaneous action potentials. The comparison established access to the cell interior, rather than a claim that every pipette command equals membrane potential. Slower transients sometimes indicated incomplete rupture, and these small-cell results do not establish spatial control of a branched neuron.^[ref]