A Working Physics Reference

A mass on a spring feels a restoring force $F=-kx$ (Hooke's law), so $m\ddot x=-kx$, i.e. $$\ddot x+\omega_0^2 x=0,\qquad \omega_0=\sqrt{k/m}.$$ Solution $x(t)=A\cos(\omega_0 t+\phi)$ — pure oscillation at natural frequency $\omega_0$, forever. Energy sloshes back and forth between kinetic and potential, the total staying fixed.

A drag force $-b\dot x$ gives $m\ddot x+b\dot x+kx=0$, usually written $$\ddot x+2\gamma\dot x+\omega_0^2 x=0,\qquad \gamma=\tfrac{b}{2m}.$$ The characteristic equation $r^2+2\gamma r+\omega_0^2=0$ has roots $r=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}$, and those roots decide everything: underdamped ($\gamma\lt\omega_0$, complex roots) rings while decaying; critically damped ($\gamma=\omega_0$) returns fastest with no overshoot; overdamped ($\gamma\gt\omega_0$, two real roots) crawls back without oscillating.

$\ddot x+2\gamma\dot x+\omega_0^2 x=\frac{F_0}{m}\cos(\omega t)$. After transients die, the system oscillates at the drive frequency $\omega$, with an amplitude that peaks near $\omega\approx\omega_0$ — resonance. (Precisely, the displacement amplitude peaks at $\omega_r=\sqrt{\omega_0^2-2\gamma^2}$, a touch below $\omega_0$, and a peak exists at all only for light damping $\gamma<\omega_0/\sqrt2$; in the high-$Q$ regime $\omega_r\to\omega_0$.) The sharpness is set by the quality factor $Q=\omega_0/2\gamma$: high $Q$ = lightly damped = tall narrow resonance peak.

$\ddot\theta+\dfrac{g}{L}\sin\theta=0$. For small swings $\sin\theta\approx\theta$, recovering a harmonic oscillator with $\omega=\sqrt{g/L}$ (period independent of amplitude — Galileo's observation). For large swings the $\sin\theta$ makes it genuinely nonlinear: the period grows with amplitude, the phase portrait develops a separatrix between swinging-back-and-forth and going-over-the-top, and adding drive and damping makes it chaotic.

For an ohmic conductor, current is proportional to voltage: $V=IR$, or in the form neuroscience prefers, $I=gV$ with conductance $g=1/R$ (siemens). Power dissipated: $P=IV=I^2R=V^2/R$.

Charge a capacitor through a resistor and the voltage relaxes exponentially: $$V(t)=V_\infty+\big(V_0-V_\infty\big)e^{-t/\tau},\qquad \tau=RC.$$ After one $\tau$ it has covered $63\%$ ($1-1/e$) of the gap to its target; after $\sim$5$\tau$ it's essentially there. The resistor sets how fast charge can flow onto the capacitor; the capacitor sets how much charge a given voltage needs.

At temperature $T$, the probability of finding a system in a state of energy $E_i$ is $$P_i=\frac{e^{-E_i/k_BT}}{Z},\qquad Z=\sum_j e^{-E_j/k_BT}.$$ $Z$ is the partition function — the normaliser (Volume I, Part 0, step 4) that makes the probabilities sum to 1. Low-energy states are exponentially favoured; raising $T$ flattens the distribution toward uniform.

In thermal equilibrium, each quadratic degree of freedom carries average energy $\tfrac12 k_BT$. So the scale of every spontaneous fluctuation — a molecule's speed, a membrane's voltage jitter — is set by $k_BT\approx4.3\times10^{-21}\ \text{J}$ at body temperature.

A particle taking independent random steps has mean position zero (it doesn't drift), but its mean-squared displacement grows linearly in time: in 1D, $\langle x^2\rangle=2Dt$, where $D$ is the diffusion coefficient. So the typical distance travelled scales as $\sqrt{t}$, not $t$ — diffusion is slow over long distances. Doubling how far you need to go takes four times as long.

First law (flux): particles flow down the concentration gradient, $J=-D\,\dfrac{\partial c}{\partial x}$. Second law (evolution): combine that with conservation and $$\frac{\partial c}{\partial t}=D\,\frac{\partial^2 c}{\partial x^2}.$$ This is the heat equation with concentration in place of temperature: a profile smooths out, peaks flatten, the second derivative (curvature) drives the change.

$D=\mu\,k_BT$, where $\mu$ is mobility (drift velocity per unit force). Spelled out for a sphere of radius $r$ in a fluid of viscosity $\eta$ (Stokes–Einstein), $D=\dfrac{k_BT}{6\pi\eta r}$. The same random collisions that make a particle diffuse are the ones that resist its drift — fluctuation and dissipation are two readings of one process, with $k_BT$ setting the scale.

If a force also acts (say an electric field on a charged ion), the flux is diffusion plus drift: $$J=-D\,\frac{\partial c}{\partial x}\;+\;\mu\,c\,F_{\text{ext}}.$$ For ions in the membrane field this is the Nernst–Planck equation. Setting the net flux to zero — diffusion exactly balancing electrical drift — is what defines the equilibrium (Nernst) potential in Part V.

$k=A\,e^{-E_a/RT}$. Reactions speed up with temperature because the fraction of molecules with enough energy to clear the activation barrier $E_a$ is a Boltzmann factor (Part III). The $e^{-E_a/RT}$ here is the same exponential as $P\propto e^{-E/k_BT}$, scaled to a mole — recognise it and you know temperature dependence will be exponential, not linear.

An ion species feels two pulls: diffusion down its concentration gradient, and the electric field. At the equilibrium (Nernst) potential these cancel — set the total free-energy change of moving one ion across to zero ($zFE + RT\ln\frac{[\text{in}]}{[\text{out}]}=0$) and solve: $$E_{\text{ion}}=\frac{RT}{zF}\ln\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}.$$ For a singly-charged cation at body temperature, $E=61.5\ \text{mV}\times\log_{10}\dfrac{[\text{out}]}{[\text{in}]}$. This is where the membrane's "batteries" — $E_K\approx-80$, $E_{\text{Na}}\approx+60\ \text{mV}$ — come from.

A real membrane is permeable to several ions, so its resting potential is a permeability-weighted blend of their Nernst potentials — the GHK voltage equation: $$V_m=\frac{RT}{F}\ln\frac{P_K[K]_o+P_{\text{Na}}[\text{Na}]_o+P_{Cl}[Cl]_i}{P_K[K]_i+P_{\text{Na}}[\text{Na}]_i+P_{Cl}[Cl]_o}.$$ (The chloride terms are flipped, inside-over-outside, because it's an anion.) Whichever ion is most permeable drags $V_m$ toward its own Nernst potential.

A patch of membrane is a capacitor (the bilayer, $C_m$) in parallel with conductances (the ion channels, each $g_{\text{ion}}$), and each conductance sits in series with its own battery — the ion's Nernst potential $E_{\text{ion}}$. Conservation of current (KCL) at the membrane reads $$C_m\frac{dV}{dt}=-\sum_{\text{ion}} g_{\text{ion}}\,(V-E_{\text{ion}})+I_{\text{ext}}.$$ With fixed (passive) conductances this is the RC relaxation of Part II. Let the conductances depend on voltage and you have Hodgkin–Huxley (Part VII).

Depolarise the membrane past threshold ($\approx-55\ \text{mV}$) and voltage-gated Na⁺ channels open; Na⁺ rushes in, depolarising further, opening more channels — a runaway positive feedback that drives $V$ up toward $E_{\text{Na}}\approx+60\ \text{mV}$ (the upstroke). Two slower processes then end it: Na⁺ channels inactivate and K⁺ channels open, so $V$ falls back toward $E_K$ (the downstroke), often overshooting into a brief hyperpolarisation. The spike is therefore all-or-none (the feedback either ignites or it doesn't) and followed by a refractory period while Na⁺ inactivation recovers.

An action potential reaching the terminal opens voltage-gated Ca²⁺ channels; Ca²⁺ entry triggers vesicles to fuse and dump neurotransmitter into the cleft; it binds receptors on the next cell, opening ligand-gated channels. The result is a postsynaptic conductance change modelled as $I_{\text{syn}}=g_{\text{syn}}(t)\,(V-E_{\text{syn}})$ — Ohm's law again, with the synapse's own reversal potential. If $E_{\text{syn}}$ is above threshold the input is excitatory (an EPSP); if below rest, inhibitory (an IPSP).

Current conservation across the membrane, with the Na⁺ and K⁺ conductances made voltage-dependent through gating variables: $$C_m\frac{dV}{dt}=I_{\text{ext}}-\underbrace{\bar g_{\text{Na}}\,m^3h\,(V-E_{\text{Na}})}_{\text{sodium}}-\underbrace{\bar g_K\,n^4\,(V-E_K)}_{\text{potassium}}-\underbrace{g_L\,(V-E_L)}_{\text{leak}}.$$ Each ionic term is Ohm's law $g(V-E)$ from Part II, with $E$ the Nernst potential from Part V and the conductance scaled by gate variables. The leak is constant; the other two breathe with voltage.

Every gating variable $x\in\{m,h,n\}$ obeys $$\frac{dx}{dt}=\alpha_x(V)(1-x)-\beta_x(V)\,x=\frac{x_\infty(V)-x}{\tau_x(V)}.$$ That right-hand form is exactly the membrane relaxation $\tau\dot y=-(y-\text{target})$ from Part II and Volume I, Part VI: each gate relaxes exponentially toward a voltage-dependent steady state $x_\infty(V)$ with its own voltage-dependent time constant $\tau_x(V)$. Spot the $\tau$, know the shape.

Model a dendrite as a long thin tube: current leaks through the membrane and flows axially down the salty core. Write $r_m$ for the membrane resistance of a unit length (units $\Omega\!\cdot\!\text{cm}$ — note it decreases as the cable lengthens, since more membrane means more leak paths in parallel), $c_m$ for the capacitance per length, and $r_i$ for the axial resistance per length of the core ($\Omega/\text{cm}$). Balancing these gives the cable equation $$\lambda^2\,\frac{\partial^2 V}{\partial x^2}=\tau\,\frac{\partial V}{\partial t}+V,$$ with $V$ measured from rest. It is the diffusion equation of Part IV (a $\partial^2/\partial x^2$ term) plus the leak term $V$ that makes the signal decay as it spreads.

Inject steady current at one point and the steady-state voltage falls off as $$V(x)=V_0\,e^{-x/\lambda}.$$ One space constant $\lambda$ out, the signal is down to $1/e\approx37\%$ — the spatial twin of the $e^{-t/\tau}$ time decay you already know. Typical $\lambda$ is tens to hundreds of microns, which is why distant synaptic inputs arrive at the soma attenuated and smeared, and why far-flung signalling needs the active regeneration of an action potential rather than passive spread.

$\dfrac{dN}{dt}=rN\left(1-\dfrac{N}{K}\right)$: exponential when the population $N$ is small, levelling off as it nears the carrying capacity $K$. The continuous version gives a tidy S-curve; its discrete cousin, the logistic map, period-doubles into chaos (Volume I, Part VII). The same saturating nonlinearity models a firing-rate ceiling.

$\dot x=\alpha x-\beta xy,\quad \dot y=\delta xy-\gamma y$ (prey $x$, predators $y$). The cross terms $xy$ are the law of mass action again — predators and prey "react" when they meet — and they produce sustained cycles: prey boom, predators follow, prey crash, predators starve, repeat. Structurally the same feedback as an autocatalytic chemical oscillator (Part V).

$\dot S=-\beta SI,\ \dot I=\beta SI-\gamma I,\ \dot R=\gamma I$ (susceptible, infected, recovered). The $\beta SI$ infection term is, once more, mass action — infection is a "reaction" between an $S$ and an $I$. The epidemic grows only if the basic reproduction number $R_0=\beta S_0/\gamma\gt1$ — a threshold condition with the same flavour as a neuron's firing threshold.