From Vector Calculus to the LIF Neuron

This article walks the chain from vector calculus to the spiking neuron: the operators and theorems needed to read Maxwell’s equations, the four equations themselves in differential and integral form, their collapse to Ohm and capacitance and the RC equation when applied in matter, the same RC equation specialised to a cell membrane plus the threshold-and-reset rule that makes it spike, and five worked examples that exercise the chain end-to-end. The destination is the conductance-based leaky integrate-and-fire (COBA) neuron used throughout the pinglab models and notebooks.

Vector Calculus

A short primer on the vector calculus needed to read Maxwell’s equations in both their differential and integral forms. We work in $\mathbb{R}^3$ with Cartesian coordinates $(x, y, z)$. Scalar fields take a point to a number; vector fields take a point to a vector. The operators below are the alphabet for talking about how those fields change in space, and the two theorems at the end are what turn the local (differential) story into the global (integral) one.

Operators

Inner product

Takes two vectors and returns a scalar.

where $\theta$ is the angle between them. It measures how much $\mathbf{a}$ projects onto $\mathbf{b}$; it is zero exactly when the two are perpendicular. Dot products of the form $\mathbf{F} \cdot d\mathbf{A}$ and $\mathbf{F} \cdot d\boldsymbol{\ell}$ are what pick out the component of a field along a surface normal or path tangent — the building block of flux and circulation.

Intuitively: how much of one arrow lies along the other — a shadow length.

Cross product

Takes two vectors and returns a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span.

The direction follows the right-hand rule. Unlike the dot product, this one is anti-symmetric: $\mathbf{a} \times \mathbf{b} = -\,\mathbf{b} \times \mathbf{a}$. We use it implicitly via the curl operator below.

Intuitively: a vector pointing along the axle of the rotation that would carry $\mathbf{a}$ onto $\mathbf{b}$, with length equal to the area of the parallelogram they span.

Gradient

Takes a scalar field $f(x, y, z)$ to a vector field.

At every point it gives the direction of steepest increase of $f$ and a magnitude equal to the rate of that increase. The symbol $\nabla = \bigl(\partial_x,\, \partial_y,\, \partial_z\bigr)$ is a formal vector of partial-derivative operators; the gradient is what you get when you apply it to a scalar.

Intuitively: standing on a hilly landscape with $f$ as the altitude — the arrow pointing straight up the steepest path.

Jacobian

Takes a vector field $\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^m$ to a matrix of partial derivatives at each point. For $\mathbf{F}(\mathbf{x}) = \bigl(F_1(\mathbf{x}), \dots, F_m(\mathbf{x})\bigr)$,

The $(i, j)$ entry is the rate at which the $i$-th output component changes with respect to the $j$-th input variable. Special cases: when $\mathbf{F}$ is scalar-valued ($m = 1$) the Jacobian is a row vector — the transpose of the gradient. When $\mathbf{F}$ is a coordinate transformation ($m = n$), $|\det J_\mathbf{F}|$ is the local volume-scaling factor and appears as the Jacobian determinant in change-of-variables integrals.

The use that matters most in nb033 is linearisation: near a point $\mathbf{x}_0$, $\mathbf{F}(\mathbf{x}_0 + \delta\mathbf{x}) \approx \mathbf{F}(\mathbf{x}_0) + J_\mathbf{F}(\mathbf{x}_0)\,\delta\mathbf{x}$. For a dynamical system $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$ with a fixed point at $\mathbf{x}^\star$ (where $\mathbf{F}(\mathbf{x}^\star) = 0$), the linearised dynamics are $\dot{\delta\mathbf{x}} = J_\mathbf{F}(\mathbf{x}^\star)\,\delta\mathbf{x}$, and the eigenvalues of $J_\mathbf{F}(\mathbf{x}^\star)$ determine stability and oscillation frequency in the neighbourhood of the fixed point. This is the engine of bifurcation analysis.

Intuitively: the Jacobian is the multivariable derivative — the best linear approximation to a vector-valued function at a point.

Divergence

Takes a vector field $\mathbf{F} = (F_x, F_y, F_z)$ to a scalar field.

It measures the net outflow of $\mathbf{F}$ per unit volume at a point — positive at sources (lines fan outward), negative at sinks (lines converge), zero where the field passes through cleanly.

Intuitively: a tiny soap bubble dropped into the field — divergence is the rate at which it expands (positive) or shrinks (negative) at that point.

Curl

Takes a vector field to a vector field.

At every point it measures the local rotation of $\mathbf{F}$ — its magnitude is the rate of circulation per unit area, its direction is the axis about which the field swirls (right-hand rule).

Intuitively: a paddle wheel placed at this point in a flowing fluid — curl is the axis it spins around and how fast.

Flux

The flux of a vector field $\mathbf{F}$ through a surface $S$ is the integral of its normal component across that surface.

where $\mathbf{n}$ is the unit normal to $S$ at each point and $dA$ is the scalar area element. In 2D the surface degenerates to a curve and $dA \to d\ell$, the arclength element. Each infinitesimal patch contributes $\mathbf{F} \cdot \mathbf{n}\, dA$ — the projection of the field onto the surface’s normal, times the patch area. Positive contributions are field lines crossing out; negative ones are field lines crossing in. The integral sums them with sign.

The choice of which way $\mathbf{n}$ points is part of the orientation conventions below. Flipping $\mathbf{n}$ flips the sign of the integral, so the meaning of “out” has to be fixed up front — outward normal for closed surfaces, user-chosen for open ones.

Intuitively: how many field lines net-cross the surface, counted positively in the chosen direction and negatively the other way.

Orientation conventions

Flux and the integral theorems below all carry signs that depend on how you orient the surface or boundary, and getting these wrong is what makes minus signs in physical laws look mysterious. Two conventions, applied consistently:

• Closed surfaces (a $\partial V$ wrapping a volume $V$) carry the outward normal. The surface element $d\mathbf{A}$ points away from the enclosed volume, so $\mathbf{F} \cdot d\mathbf{A}$ counts flux leaving $V$ as positive and flux entering as negative. This is what makes the divergence theorem a “net outflow” statement.
• Open surfaces with boundary (an $S$ bounded by a curve $\partial S$) carry a normal $d\mathbf{A}$ that the user picks, and the boundary curve $\partial S$ is then oriented by the right-hand rule — curl your right-hand fingers around $\partial S$, and your thumb points along $d\mathbf{A}$. Flip $d\mathbf{A}$ and the direction of integration around $\partial S$ also flips, so both sides of Stokes’ theorem change sign in lockstep.

These conventions are what make signs in the physical laws non-arbitrary. In Maxwell’s equations, the minus in Faraday’s law is the canonical example: once the right-hand rule fixes $\partial S$ relative to $d\mathbf{A}$, Lenz’s law (the induced current opposes the change in flux) forces that sign — get the convention wrong and you’d derive perpetual motion.

Integral theorems

The two theorems below convert a local statement about a field’s divergence or curl into a global statement about flux or circulation across a boundary. They are what let the differential and integral forms of physical conservation laws be two sides of the same coin.

Divergence theorem

For a vector field $\mathbf{F}$ defined on a volume $V$ with closed boundary surface $\partial V$, the integral of the divergence over the volume equals the flux through the boundary.

Intuitively: whatever the field is sourcing inside the volume has to flow out through the boundary. It is a conservation statement about field lines — every line that originates inside $V$ has to cross $\partial V$ on its way out.

Stokes’ theorem

For a vector field $\mathbf{F}$ on an oriented surface $S$ with boundary curve $\partial S$, the flux of the curl through $S$ equals the line integral around $\partial S$.

Intuitively: if the field is swirling across the surface, that swirl has to add up to something circulating around the edge — like measuring a whirlpool by walking its rim rather than counting eddies across its face.

Vector identities

A short reference list of identities that come up repeatedly when manipulating the operators above. Throughout, $f$ and $g$ are scalar fields and $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ are vector fields. The first group are pure-algebra identities about dot/cross products; the rest are differential identities about how $\nabla$ combines with itself or distributes through products.

Triple products.

The scalar triple product is cyclically invariant: it equals the signed volume of the parallelepiped spanned by the three vectors. Swapping any two arguments flips the sign.

The vector triple product, mnemonic “BAC − CAB”. The cross product is non-associative; this identity tells you exactly how it fails.

Second-derivative identities. The Laplacian $\nabla^2 f \equiv \nabla\cdot(\nabla f) = \partial_x^2 f + \partial_y^2 f + \partial_z^2 f$ is the divergence of the gradient of a scalar field.

The curl of any gradient is zero. Equivalently, any field that is the gradient of a scalar potential is automatically curl-free — the basis for the electric scalar potential under electrostatics.

The divergence of any curl is zero. Equivalently, any field that is the curl of another vector is automatically divergence-free — the basis for the magnetic vector potential $\mathbf{B} = \nabla\times\mathbf{A}$ when $\nabla\cdot\mathbf{B} = 0$.

The curl-of-curl identity. This is what turns Maxwell’s two curl equations into the wave equation: taking $\nabla\times$ of Faraday and substituting Ampère–Maxwell on the right invokes this identity on the left, and $\nabla\cdot\mathbf{E} = 0$ in vacuum kills the first term.

Product rules. The differential operators distribute through products in patterns analogous to the ordinary calculus product rule.

The last of these is what proves Poynting’s theorem in electromagnetism: applied to $\mathbf{S} = \mathbf{E} \times \mathbf{B}/\mu_0$ it produces an energy-flux conservation law.

Maxwell’s Equations

The four Maxwell equations are the backbone of classical electromagnetism. They come in two equivalent forms — differential, which states how the fields behave at every point, and integral, which states how they behave across surfaces and along loops. Both forms use the operators and integral theorems collected in Vector Calculus above.

Differential form

Here $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields, $\rho$ is the charge density, $\mathbf{J}$ the current density, and $\epsilon_0$, $\mu_0$ the vacuum permittivity and permeability. Two of the equations involve a divergence and two a curl, and that split is what lets the two integral theorems convert each pair into the integral form below.

Integral form

Here $Q_{\text{enc}}$ is the total charge enclosed by the closed surface $\partial V$, $I_{\text{enc}}$ is the total current passing through the open surface $S$, and the orientation conventions for $d\mathbf{A}$ and $d\boldsymbol{\ell}$ are the ones from the vector-calculus primer.

Linking the two forms

The two forms are linked by the integral theorems from Integral theorems:

• The divergence theorem turns the two $\nabla \cdot$ equations into the two flux statements through closed surfaces. Applied to $\mathbf{F} = \mathbf{E}$, it converts $\nabla \cdot \mathbf{E} = \rho/\epsilon_0$ into the enclosed-charge version of Gauss’s law; the same move applied to $\mathbf{B}$ gives the magnetic Gauss law with zero on the right.
• Stokes’ theorem turns the two $\nabla \times$ equations into the two circulation-vs-flux-derivative statements. Applied to $\mathbf{F} = \mathbf{E}$, it converts Faraday’s $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$ into the integral form above (and the minus sign carries over directly); applied to $\mathbf{F} = \mathbf{B}$, it does the same for the Ampère–Maxwell law.

The minus sign on Faraday’s law is the canonical example of orientation conventions doing real work: once the right-hand rule fixes $\partial S$ relative to $d\mathbf{A}$, Lenz’s law forces that sign, and the differential and integral forms agree on it automatically.

Gauss’s law

Charges are the sources of the electric field. The strength of the source at every point is the local charge density.

Differential form.

Integral form.

The two forms say the same thing at different scales. The differential form is a local identity: at every point in space, the divergence of the electric field equals the charge density there, divided by the vacuum permittivity. The integral form is the global consequence: for any closed surface $\partial V$ you draw, the total outward flux of $\mathbf{E}$ across it is the total charge $Q_{\text{enc}}$ inside, divided by $\epsilon_0$. The link between the two is the divergence theorem applied to $\mathbf{F} = \mathbf{E}$.

Intuitively: electric field lines start on positive charges and end on negative charges. Wrap any closed surface around a chunk of space and count the lines crossing it — positive when leaving, negative when entering — and you’ve measured the net charge inside. No charges inside, no net flux. The shape of the surface doesn’t matter; only what’s enclosed.

Symbols and units (SI):

The constant $\epsilon_0$ is what sets the scale: a single $1\,\mathrm{C}$ point charge produces a total flux of $1/\epsilon_0 \approx 1.13 \times 10^{11}\,\mathrm{V\,m}$ — large because a coulomb is a lot of charge. In everyday terms, $\epsilon_0$ encodes “how stiffly the vacuum opposes electric field lines”: a small $\epsilon_0$ would mean a tiny charge produces an enormous field.

Gauss’s law of magnetism

The magnetic field has no sources or sinks. Every magnetic field line that enters a region eventually leaves it — there is no magnetic equivalent of an isolated charge.

Differential form.

Integral form.

The two forms again say the same thing at different scales, linked by the divergence theorem. The differential form says the divergence of $\mathbf{B}$ is zero everywhere — no points act as sources or sinks. The integral form says that for every closed surface, the net magnetic flux through it is zero — exactly as many field lines leave as enter. The right-hand side of Gauss’s law had a $Q_{\text{enc}}/\epsilon_0$ term; here, there is no equivalent, because no magnetic monopole has ever been observed.

Intuitively: magnetic field lines form closed loops. Wrap any closed surface around a chunk of space and the loops crossing into it must also cross out — net flux is zero.

Symbols and units (SI):

No constant appears on the right-hand side — the law’s content is the equation $= 0$. If a magnetic monopole were ever discovered, it would acquire a $\rho_m/\mu_0$-style term and stop being structurally simpler than its electric cousin. Until then, the magnetic side of Maxwell’s equations is “missing” exactly one of the four source terms electricity carries.

Faraday’s law of induction

A changing magnetic field induces a circulating electric field. The faster the magnetic flux changes through a loop, the larger the induced electric circulation around its boundary.

Differential form.

Integral form.

The two forms are linked by Stokes’ theorem. The differential form is a local identity: the curl of $\mathbf{E}$ at every point equals the negative time-derivative of $\mathbf{B}$ there. The integral form is the global consequence: the total $\mathbf{E}$-circulation around any closed loop $\partial S$ equals the negative rate of change of the $\mathbf{B}$-flux through any surface $S$ spanning that loop. The minus sign is Lenz’s law: the induced electric field drives currents that oppose the change in magnetic flux — a self-stabilising sign forced by the right-hand rule relating $\partial S$ to $d\mathbf{A}$.

Intuitively: shove a magnet through a wire loop and the loop pushes back. The pushing is the induced electric field, the direction it circulates is whichever way opposes whatever you were doing.

Symbols and units (SI):

No constants appear on the right-hand side beyond the minus sign — the relation between $\mathbf{E}$ and $\partial_t \mathbf{B}$ is exact, with no scale factor. The whole content of the law sits in that minus sign and the geometric pairing $\mathbf{E} \leftrightarrow \mathbf{B}$.

Ampère–Maxwell law

The magnetic field circulates around both real currents and changing electric fields. The “displacement-current” term — Maxwell’s addition to Ampère’s original law — is what completes the symmetry with Faraday’s law and makes self-propagating electromagnetic waves possible.

Differential form.

Integral form.

The two forms are linked by Stokes’ theorem, exactly mirroring Faraday’s law but with the electric and magnetic roles swapped (and no minus sign). The differential form says the curl of $\mathbf{B}$ at every point has two sources: a real current density $\mathbf{J}$, and a changing electric field $\partial_t \mathbf{E}$. The integral form says the $\mathbf{B}$-circulation around a loop equals $\mu_0$ times the current piercing any surface spanning it, plus $\mu_0\epsilon_0$ times the rate of change of $\mathbf{E}$-flux through the same surface.

The displacement-current term $\mu_0 \epsilon_0\, \partial_t \mathbf{E}$ was Maxwell’s invention; without it, the original Ampère law $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ is inconsistent with charge conservation (taking $\nabla\cdot$ of both sides would imply $\nabla\cdot \mathbf{J} = 0$ always, which is false whenever charge density changes in time). Adding the term not only patches that inconsistency but also produces the wave equation $\nabla^2 \mathbf{E} = \mu_0 \epsilon_0\, \partial_t^2 \mathbf{E}$ when combined with Faraday’s law in vacuum — the propagation speed comes out as $c = 1/\sqrt{\mu_0 \epsilon_0}$, which is the speed of light.

Intuitively: every current is wrapped by a magnetic field, and so is every region of changing electric field. The second part is what lets light exist: an oscillating $\mathbf{E}$ generates a circulating $\mathbf{B}$ which generates a circulating $\mathbf{E}$, and the pair propagates through empty space at speed $c$.

Symbols and units (SI):

The two constants $\mu_0$ and $\epsilon_0$ encode how the vacuum responds to magnetism and electricity respectively, and they appear here as a product. Their reciprocal-square-root is the speed of light: $c = 1/\sqrt{\mu_0\epsilon_0} \approx 2.998 \times 10^{8}\,\mathrm{m/s}$. That “the speed of light falls out of Maxwell’s equations” is the historical moment electromagnetism and optics merged into one subject.

From Fields to Circuits

Maxwell’s equations describe electromagnetism at every point in space, but for a single wire or capacitor that spatial resolution is overkill. Applied in matter, two of the four equations collapse onto two circuit-level constitutive laws — Ohm and Q=CV — which together give the RC equation, the simplest first-order voltage dynamics in physics. This section walks that collapse end-to-end. The destination is

the equation an RC circuit obeys and the same equation a single point-neuron’s membrane obeys between spikes.

Ohm’s law in matter

In free space Maxwell’s equations relate $\mathbf{E}$, $\mathbf{B}$, $\rho$ and $\mathbf{J}$. In a conductor there is one extra relation, supplied by the material rather than by Maxwell: an electric field $\mathbf{E}$ pushes the free charges around, and the rate at which they move per unit area, per unit time, is the current density $\mathbf{J}$. To linear order,

where $\sigma$ is the conductivity of the material (siemens per metre, $\mathrm{S/m}$). This is a constitutive law — not derivable from Maxwell alone, it tells you how a specific material responds to a field, in the same role the relation $\mathbf{D} = \epsilon\mathbf{E}$ plays for dielectrics.

What conductivity actually is

A field $\mathbf{E}$ exerts a force $q\mathbf{E}$ on every mobile charge carrier, accelerating it. In a true vacuum that carrier would keep accelerating forever; in a real material it scatters off the lattice (or the surrounding fluid) on a short timescale $\tau_c$, the mean free time between collisions. Between collisions it picks up a velocity $\Delta v = (q/m)\mathbf{E}\,\tau_c$, then loses it again. Averaged over many collisions, every carrier acquires a steady drift velocity

where $\mu$ is the mobility of that carrier (units $\mathrm{m^2/(V\,s)}$). With $n$ carriers per unit volume, each carrying charge $q$, the current density is just $n q \mathbf{v}_d$, so

Conductivity is therefore a product of “how many carriers there are” ($n$), “how much charge each one carries” ($q$), and “how easily each one drifts in a field” ($\mu$, or equivalently $\tau_c/m$). A metal has $n \sim 10^{28}\,\mathrm{m^{-3}}$ near-free electrons and $\tau_c \sim 10^{-14}\,\mathrm{s}$, giving $\sigma \sim 10^7\,\mathrm{S/m}$. Salt water has $\sim 10^{26}\,\mathrm{m^{-3}}$ dissolved ions of charge $\pm e$, but the mobility of an ion in water is roughly six orders of magnitude smaller than an electron in a metal, so $\sigma \sim 1\,\mathrm{S/m}$. A neuron’s cytoplasm and the extracellular fluid sit in the same $\sim 1\,\mathrm{S/m}$ ballpark. The lipid bilayer at the membrane is the opposite extreme: essentially no mobile ions at all, $\sigma \sim 10^{-13}\,\mathrm{S/m}$. That is why the membrane behaves as a capacitor in the next section and the fluids on either side behave as wires.

Intuitively: $\sigma$ measures how responsive a material is to a field — how many carriers it offers up, multiplied by how easily each one moves. A field exerts a force per carrier; collisions limit the velocity; the product is the current.

Voltage

To get from the differential statement $\mathbf{J} = \sigma\mathbf{E}$ to the familiar $V = IR$ we first need to nail down what voltage means. Voltage (or potential difference) between two points $A$ and $B$ is the work done per unit charge to carry a test charge from $B$ to $A$ against the electric field:

Units: joules per coulomb $\equiv$ volts ($\mathrm{V}$). The minus sign is bookkeeping: $V_{AB} > 0$ when $A$ is at higher potential than $B$, so a positive charge would want to fall from $A$ to $B$ — like a ball wanting to roll downhill.

Two consequences make voltage a useful circuit-level concept:

• Path-independence. In the electrostatic limit, $\nabla\times\mathbf{E} = 0$ (no time-varying $\mathbf{B}$), so the line integral above depends only on the endpoints, not on the path taken between them. This is what justifies talking about “the voltage at a node” rather than “the voltage along this particular wire.”
• Kirchhoff’s voltage law. Path-independence is equivalent to saying the integral around any closed loop is zero: $\oint \mathbf{E}\cdot d\boldsymbol{\ell} = 0$ → $\sum V = 0$ around the loop. (When $\partial_t \mathbf{B}$ is not zero through the loop, Faraday’s law modifies this — but for the membrane circuits we care about the time-varying magnetic field is negligible.)

For a uniform-cross-section wire of length $L$ with constant $\mathbf{E}$ along it, the integral collapses to $V = E\,L$ — the special case the textbook starts with.

Ohm’s law in integral form

Substituting $V = EL$ and $I = JA$ into $\mathbf{J} = \sigma\mathbf{E}$ for the uniform wire gives

with $R$ the wire’s resistance (ohms, $\Omega = \mathrm{V/A}$). For neurons the two relevant values of $\sigma$ — high for the fluids, $\sim$ zero for the membrane — are what make the rest of this section’s RC picture justified.

Capacitance

A thin insulator sandwiched between two conductors can hold separated charge: positive on one plate, negative on the other. To get the relation between stored charge and voltage, walk Gauss’s law and the voltage definition through a parallel-plate geometry.

Take two plates of area $A$ separated by a gap $d$, holding charge $+Q$ and $-Q$. The integral form of Gauss’s law applied to a pillbox surface straddling one plate gives a uniform field between the plates of magnitude

Integrating that field across the gap gives the voltage between the plates, $V = E\,d = Q d / (\epsilon_0 A)$. Rearranging,

where $C$ is the capacitance (farads, $\mathrm{F} = \mathrm{C/V}$). The relation $Q = CV$ is the load-bearing piece: capacitance is the proportionality between stored charge and the voltage that charge creates.

Differentiating in time gives the current flowing onto the plates,

This is the “current-through-a-capacitor” rule. Charge moving onto the plates is the current; the voltage rises in lockstep, at a rate set by how big the capacitor is.

Intuitively: a capacitor is a tank. Pouring charge in (current) raises its level (voltage) at a rate determined by how big the tank is (capacitance).

The RC equation

Wire a resistor $R$ in parallel with a capacitor $C$, with one end of both held at a reference voltage $V_\infty$ and the other driven by an external current source $I_{\text{ext}}(t)$. Kirchhoff’s current law at the top node says the input current splits between charging the capacitor and leaking through the resistor,

Rearranging gives the RC equation:

With $I_{\text{ext}} = 0$ the voltage relaxes exponentially toward $V_\infty$ on the timescale $\tau$. With a steady $I_{\text{ext}}$ the steady-state shifts to $V_\infty + R\, I_{\text{ext}}$. Whatever the input, the dynamics is always a first-order linear ODE — the same form that appears in the COBA derivation on the models page, where $\tau_{\text{eff}}$ and $V_\infty$ become functions of the open conductances.

The RC equation is also already most of the way to the LIF neuron. The only thing missing for LIF is a threshold + reset rule on top: when $V$ crosses some firing level $\theta$, emit a spike and snap $V$ back to a reset value. The subthreshold dynamics between spikes is exactly the RC equation.

Intuitively: an RC circuit is the simplest dynamical voltage in physics — one state variable, one timescale, one rest level. Everything more elaborate (cable equations, Hodgkin–Huxley, COBA) is decoration on top of this skeleton.

The LIF Neuron

The leaky integrate-and-fire (LIF) neuron is the simplest model of a spiking cell that still preserves the two things that make neurons interesting: a continuous voltage that integrates inputs in time, and a discrete spike output emitted when that voltage crosses a threshold. Subthreshold, an LIF cell is exactly the RC circuit derived in the previous section — Maxwell’s equations applied to a thin insulator between two conducting fluids. The active ingredient is just one extra rule: when the membrane voltage reaches a firing level, emit a spike and reset.

This section walks the construction in four steps. First, the membrane equation — the RC circuit specialised to a cell, with ion channels playing the role of resistors. Second, the reversal potentials at the end of those channels, via the Nernst equation. Third, the threshold-and-reset rule that completes the LIF model. Fourth, the two ways of wiring synaptic inputs in — current-based (CUBA) and conductance-based (COBA) — which together turn one cell into the building block of the pinglab models and notebooks.

The cell membrane as an RC circuit

A neuron’s membrane is a lipid bilayer, $\sim 5$ nm thick, separating an inside compartment (axoplasm, high $K^+$, low $Na^+$) from an outside one (extracellular fluid, high $Na^+$, low $K^+$). Electrically:

• The bilayer is an insulator. No free charge crosses it without help. By Ohm’s law in matter, its conductivity is essentially zero.
• Both the inside and outside fluids are conductors with $\sigma \sim 1\,\mathrm{S/m}$ — many orders of magnitude higher than the bilayer’s.
• That makes the membrane a capacitor in the parallel-plate sense, with capacitance per unit area $C_m \sim 1\,\mu\mathrm{F}/\mathrm{cm}^2$.
• Embedded ion channels open and close to let specific ions through. While open, each channel contributes a conductance $g_{\text{channel}}$ across the membrane.

Stacking all of this for a small patch of membrane and applying Kirchhoff’s current law gives

where the sum is over the channel types $k$ that are currently open, $g_k$ is the total open conductance of that type, and $E_k$ is the channel’s reversal potential — the voltage at which the net current of ion $k$ through its channel is zero. The driving term $(V - E_k)$ makes each open channel pull $V$ toward its own $E_k$ at a rate set by $g_k$.

Collapse all the channels into a single passive “leak” conductance $g_L$ with reversal $E_L$ and the equation reduces to the canonical RC form,

with membrane time constant $\tau_m = C_m / g_L$ and steady state $V_\infty = E_L$. This is the subthreshold dynamics of an LIF neuron — the equation it integrates whenever it is not currently spiking. The only remaining question before we can write down the full model is where the reversal potential $E_L$ comes from — and for that we need to leave electromagnetism for a moment and visit thermodynamics.

Ion gradients and the Nernst equation

Each ion species $k$ sits at a different concentration inside and outside the cell, actively maintained by pumps that burn ATP. Two forces act on an ion that crosses the membrane:

• Diffusion drives the ion from high concentration to low, by random thermal motion — the same statistical mechanics that produced the Maxwell–Boltzmann distribution in Particles in a Box. Fick’s law gives a flux $\mathbf{J}_{\text{diff}} = -D\,\nabla c$.
• Drift acts because the ion has charge $q_k$, so it feels the electric field across the membrane: $\mathbf{J}_{\text{drift}} = \mu\, q_k\, c\, \mathbf{E}$, with $\mu$ the ionic mobility.

For each ion species at equilibrium, these two fluxes cancel. Setting $\mathbf{J}_{\text{diff}} + \mathbf{J}_{\text{drift}} = 0$, using the Einstein relation $D = \mu\, k_B T$ to link diffusion and mobility, and integrating across the membrane gives the Nernst equation:

This is the voltage at which the field-driven drift of ion $k$ exactly cancels its concentration-driven diffusion — its equilibrium, or reversal, potential. For mammalian body temperature ($T \approx 310\,\mathrm{K}$), the typical values are

The leak conductance $g_L$ used in the previous section is a mixture of mostly $K^+$ and $Cl^-$ channels, so the effective $E_L$ for the leak sits in the neighbourhood of $-65\,\mathrm{mV}$ — the cell’s resting potential. Mixed-cation excitatory channels (AMPA receptors) end up with a reversal near $E_e \approx 0\,\mathrm{mV}$; chloride-selective inhibitory channels (GABA-A receptors) sit near $E_i \approx -75\,\mathrm{mV}$ — exactly the values used in the COBA model. They are not free parameters; they fall out of the ion concentrations the cell maintains with its pumps.

Intuitively: every ion feels two restoring forces — thermal diffusion pushing toward equal concentrations, and the electric field across the membrane. The voltage at which they balance is that ion’s reversal potential. The cell spends ATP to keep the concentrations away from equilibrium, and the reversal potentials are the price of that maintenance.

Threshold and reset

The membrane equation derived above is passive: it leaks toward $E_L$, never crosses any threshold, never spikes. A real neuron does fire — when its membrane voltage rises above $\sim -50\,\mathrm{mV}$ a cascade of voltage-gated $Na^+$ and $K^+$ channels opens, drives the membrane up to $\sim +30\,\mathrm{mV}$ and back down in about a millisecond, and emits an action potential. The full mechanism is captured by the Hodgkin–Huxley model, which adds four nonlinear gating variables to the membrane equation.

The LIF idealisation throws all of that out. Instead of modelling the spike-shape dynamics, it just declares: when $V$ reaches a fixed threshold $V_{\text{th}}$, an instantaneous spike is emitted and $V$ is snapped back to a reset voltage $V_{\text{reset}}$. Optionally, the cell enters a brief refractory period $\tau_{\text{ref}}$ during which it is unresponsive — modelling the few-ms recovery time after a biological spike.

This is the only nonlinearity in LIF. The model is otherwise a linear ODE; the spike rule is what makes it neuron-like rather than just a leaky integrator.

Typical biophysical settings:

The choice of where to put the reset $V_{\text{reset}}$ is a modelling decision. Two common conventions:

• Hard reset: $V_{\text{reset}}$ is a fixed value, usually $E_L$. Any overshoot above threshold is discarded.
• Subtract reset: $V \leftarrow V - V_{\text{th}}$. Preserves overshoot, useful when integrating very fast inputs across long timesteps.

These choices show up explicitly in the pinglab CUBA implementation — reset_mode = “zero” vs reset_mode = “subtract”.

The full LIF model

Putting it all together — the subthreshold RC dynamics, the reversal-potential leak, and the threshold-and-reset rule — gives the canonical LIF neuron: