Continuum thermodynamics applies the first law of thermodynamics and Second law of thermodynamics locally to a deforming material. A region can gain internal energy through mechanical work, heat flux through its boundary and internal heat supply. Those mechanisms must be distinguished from transport caused by the region itself moving.

With specific internal energy $e$, mean mass density $\rho$, symmetric stress tensor $\sigma$, rate-of-strain tensor $D$ and outward heat flux $\mathbf{q}$, a simple continuum satisfiesHere $r$ is heat supplied per unit mass per unit time, and the dot is a material derivative. The Clausius-Duhem inequality additionally requires nonnegative local entropy production. Together the two laws test proposed constitutive equations: an equation can satisfy mechanical balance while incorrectly predicting heat flow from cold to hot without compensating work. Elastic energy storage, viscous dissipation and thermal conduction enter the same accounting framework.

Plasticity in continuum mechanics models permanent deformation after loading passes a material's yield criterion. It supplements the balances of continuum mechanics with an evolution law for plastic strain, and often with internal variables representing work hardening. This separates the recoverable elastic response from changes that remain after unloading.

In one-dimensional small-strain loading, $\epsilon =\sigma /E+{\epsilon }^p$. Unloading without further plastic deformation follows slope $E$, the Young's modulus, and leaves residual strain ${\epsilon }^p$ when $\sigma$ returns to zero. A tensile testing experiment with repeated unloading makes this distinction visible. A yield surface generalizes the threshold to multiaxial stress; a flow rule determines the direction of further plastic strain. The threshold and flow rule are separate ingredients, so a yield criterion alone cannot predict a component's permanent shape.

A composite material contains distinguishable constituent phases whose arrangement contributes to the bulk properties. In a fibre-reinforced example, strong fibres carry much of the axial load while a composite matrix holds them together and transmits shear stress between them. Carbon-fibre-reinforced polymer combines carbon fibres with a polymer composite matrix; glass fibres provide another reinforcement.

The constituent proportions alone do not determine the result. Aligned fibres produce anisotropy, whereas changing their orientations changes which directions are reinforced.^[ref] In an ideal long-fibre composite, the longitudinal rule of mixtures estimates the axial Young's modulus. A comparison of tensile testing along and across the fibres exposes the importance of geometry and fibre-matrix load transfer. The same idea of combining phase properties with geometry appears in effective conductivity, although the relevant constitutive equations differ.

Isotropy is invariance under rotations, so no spatial direction is preferred. For a three-dimensional second-rank conductivity tensor, invariance under every rotation forces $a=a_{s}I$. Isotropy of each constituent does not imply isotropy of an arranged composite microstructure. Statistical isotropy concerns the rotational invariance of its probability law.

Anisotropy means that a physical property depends on direction. A strip cut along the fibres of a long-fibre composite can resist stretching much more strongly than a strip cut across them. The directional difference can come from the crystal structure of a constituent or from the arrangement of otherwise isotropic constituents.

For thermal conduction, a thermal conductivity tensor relates the temperature gradient to the heat flux. Along its principal directions the conductivities can differ, and in a general direction the heat flux need not be parallel to the negative temperature gradient. Anisotropic elasticity similarly requires a tensor-valued constitutive equation rather than a single Young's modulus. Measuring a specimen in several orientations therefore reveals information that one axial measurement cannot; averaging over randomly oriented constituents can reduce macroscopic anisotropy, but the result depends on the actual orientation distribution.

The continuum approximation describes a material using fields such as mass density, velocity field and stress, rather than following its individual molecules. Its averaging volumes must contain enough microscopic constituents to define bulk properties while remaining small relative to the distances over which those properties change appreciably. It supplies the model underlying much of fluid mechanics and elasticity.

A flux records transport of a quantity through a surface per unit time. A flux density $\mathbf{j}$ gives total transport rate ${\int }_S\mathbf{j}\cdot \mathbf{n}dS$, where $\mathbf{n}$ is the unit normal vector. A conservation law relates its divergence to the local change in the quantity's density.

By Ronald B. Goldfarb.

Online text: tsapps.nist.gov/publication/get_pdf.cfm?pub_id=30354

Cited passages: R. B. Goldfarb, Demagnetizing Factors, printed p. 104, §2 opening paragraphs: magnetometric versus fluxmetric sensing, nonuniform cylindrical fields and susceptibility-dependent factors.

An outline font stores glyph shapes as geometric boundaries that can be scaled before drawing. Bézier curves provide editable segments of these boundaries. TrueType outlines use quadratic Bézier curves; Type 2 charstrings used in Compact Font Format outlines use cubic Bézier curves. ^{[ref] [ref]}

Scaling the boundary and choosing output pixels are separate operations. The rasterization step turns the scaled glyph geometry into a displayed image, so the same outline font can serve different output sizes without storing a separate image for every size.

Online text: math.mit.edu/~gs/cse/websections/cse41.pdf

How neural responses represent variables relevant to an organism. A coding claim specifies both which response features carry information and how a downstream system can use them.

Average the stimulus history preceding each observed spike, after removing the stimulus mean. With white Gaussian stimulation and a suitable one-filter encoding model, the result is proportional to the relevant linear filter. An even nonlinearity can make the average vanish despite strong stimulus dependence, so averaging alone is not universally sufficient.

A stimulus passes through a linear temporal or spatial filter, then a nonnegative nonlinear rate function. A conditionally Inhomogeneous Poisson process generates spikes at that rate. The filter and nonlinearity can be fitted from data and tested on held-out trials. Refractoriness and spike-history correlations require extensions beyond this basic cascade.

The symmetric two-neuron rate model has nonnegative activity and saturating recurrent input. Every equilibrium has $x=y$: otherwise monotonicity of $g$ gives a contradiction. With positive parameters it has three distinct equilibria exactly when $am>2\sigma$, namely zero and $z_{\pm }=(m\pm \sqrt{m^2-4{\sigma }^2/a^2})/2$. The lower positive equilibrium is a saddle and the origin and upper equilibrium are stable nodes. The resulting two attracting activity levels model a persistent switch rather than a spiking waveform.

A conductance-based Hodgkin-Huxley model couples membrane potential to voltage-dependent channel-gating variables. It explains generation of action potentials through regenerative and opposing ionic electric currents, while its parameters and channel assumptions determine its biological domain of validity.

In the classical model, $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Each follows two-state ion-channel gating, with its own voltage-dependent rates. The available electrical conductances areThe barred quantities are maximum electrical conductances. The factors have the interpretation of independent permissive gates in an ion channel: three activation factors and a distinct inactivation-availability factor for sodium, and four identical activation factors for potassium. Large $h$ means availability; small $h$ means inactivation. These are model assumptions, not a literal structural description of every ion channel. See Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics (2014), §2.2: model definition and stochastic opening.

With inward-positive electric current, a single-compartment form isHere $C$ is membrane capacitance, $I_{\mathrm{ext}}$ is imposed inward electric current, $g_L$ is leak electrical conductance, and each $E$ is a reversal potential. Use consistent units, and either whole-compartment quantities throughout or quantities per membrane area throughout. Opening changes electrical conductance; the reversal potential and present membrane potential determine the associated electric current. Coupling this equation to the voltage-dependent gates makes a generally nonlinear system of ordinary differential equations; a gate's fixed-voltage limit alone does not specify its evolving state during an action potential.

The original quantitative account is ^[ref]. The contrast with the leaky integrate-and-fire model is useful: here an action potential emerges from evolving ionic electric currents, whereas the simpler model records a threshold event and imposes a reset.

The leaky integrate-and-fire model combines a passive equation for membrane potential with a firing rule. Between events, inward-positive applied electric current $I(t)$ obeyswhere $C>0$ is membrane capacitance, $g_L>0$ is the leak conductance, $R$ is the corresponding electrical resistance, and $E_L$ is the leak reversal potential. Here the leak conductance is constant. The time constant $\tau$ describes passive relaxation. When $V$ first reaches a threshold $V_{\theta }$ from below, record an action potential event and reset $V$ to $V_r<V_{\theta }$. In the version used here, hold $V=V_r$ for a fixed neuronal refractory period $t_{\mathrm{ref}}$, then resume integration. See 1.3 Integrate-And-Fire Models.

For constant $I$, with $t=0$ at the start of an integration interval and $V(0)=V_{\mathrm{initial}}$, the exact subthreshold solution isThe equilibrium point of this passive equation is $V_{\infty }=E_L+RI$. Subtracting $V_{\infty }$ leaves $d(V-V_{\infty })/dt=-(V-V_{\infty })/\tau$, explaining the exponential function. This solution applies only until a threshold event or a change in $I$. Starting below threshold, constant input produces repeated events if $V_{\infty }>V_{\theta }$; if $V_{\infty }=V_{\theta }$, the threshold is approached asymptotically and never reached at finite time.

For a worked example, chooseThese imply $g_L=10\mathrm{nS}$ and $C=200\mathrm{pF}$. All these parameters are illustrative model choices, not measurements from a particular neuron. With $I=200\mathrm{pA}$, Ohm's law gives $RI=20\mathrm{mV}$ and hence $V_{\infty }=-45\mathrm{mV}$. ThusAt threshold, $e^{-t/\tau }=1/4$, so the first event occurs after $T=20\ln 4\mathrm{ms}\approx 27.7259\mathrm{ms}$. Each subsequent cycle includes the $2\mathrm{ms}$ neuronal refractory period, givingThis is the ideal sustained firing rate; the first-event latency is $T$, not $T+t_{\mathrm{ref}}$.

With $I=100\mathrm{pA}$ instead, $RI=10\mathrm{mV}$ and $V_{\infty }=-55\mathrm{mV}$. From the same initial state, $V(t)=-55\mathrm{mV}-10\mathrm{mV}{e}^{-t/\tau }$ stays below $-50\mathrm{mV}$: no events occur, however long this current is held. The critical sustained current is $(V_{\theta }-E_L)/R=150\mathrm{pA}$; equality gives asymptotic approach rather than finite-time firing.

The leak gives old inputs progressively less influence, while capacitance prevents an ordinary finite electric current step from changing $V$ instantly. This combination makes the leaky integrate-and-fire model useful for exploring input summation, event timing and network activity at low computational cost. Its fixed parameters and reset omit the evolving voltage-gated ion channels of the Hodgkin-Huxley model, spatial integration in dendrites and mechanisms of neural adaptation. Treating a synapse as a prescribed $I(t)$ also omits its voltage-dependent driving force unless a conductance term is added. See 1.4 Limitations of the Leaky Integrate-and-Fire Model and constant-input firing rate of a leaky integrate-and-fire neuron.

For a directed three-neuron rate ring $\tau {\dot{x}}_i=-x_i+W_{i,i-1}\tanh (\beta x_{i-1})$, the linearization has characteristic equation $(1+\tau \lambda {)}^3=\Gamma$, where $\Gamma ={\beta }^3{W}_{13}{W}_{32}{W}_{21}$. Positive loop gain has strict linear asymptotic stability for $\Gamma <1$. Negative loop gain has it for $-8<\Gamma <0$; at $\Gamma =-8$ a complex pair crosses the imaginary axis at frequency $\sqrt{3}/\tau$. The saturating cubic gives a supercritical Hopf bifurcation under an ordinary loop-gain variation. This directed system does not inherit the energy-descent property of a symmetric Hopfield network.

A dynamic clamp measures a real neuron's voltage, updates a conductance model in real time and injects the resulting current, for example $I=g(t)(E-V)$. It implements a virtual conductance without holding the voltage fixed. It can add artificial synapses, couple real neurons into a hybrid network or test how a voltage-dependent conductance affects excitability. Sampling delay and the injection site limit fidelity to spatially distributed biological channels.