Physics

The equations, their parameters and the source of every number are in MATH_REFERENCE.md. This page states what the model is, so you can judge whether it fits your question.

All internal arithmetic is in SI units: volts, amperes, seconds, farads, siemens, mol/m³, and m²/s for membrane diffusion constants. Millivolts appear only at the API boundary and in plots.

The ion-level model

Each cell carries the intracellular concentration of Na⁺, K⁺, Cl⁻, Ca²⁺ and two impermeant species, plus two gate variables per expressed channel. The membrane potential follows from net charge,

\[ V_{mem} = \frac{F\,\mathrm{vol}}{C_m A} \sum_k z_k c_k, \]

which is about 16 V per mol/m³, so voltage is a small difference of large numbers. That is why the solvers use a pure absolute tolerance on concentrations.

Fluxes across the membrane use the Goldman-Hodgkin-Katz electrodiffusion expression. The Na/K-ATPase and the Ca-ATPase are thermodynamically constrained pumps. Gap junctions carry ions between neighbours and are gated by the transjunctional voltage, so a large voltage difference closes them and a domain isolates itself.

These equations are ported from BETSE (BSD-2-Clause); the file and function each one came from is named in MATH_REFERENCE.md, and the notice is in THIRD_PARTY_NOTICES.md.

Channels

bioelectrics.ionic.LIBRARY holds Kv1.5, Kir2.1, HCN2 and Cav1.2 ported from BETSE with their original experimental references, leak channels, and two phenomenological CAM-type channels used to produce bistability.

Steady states and their derivatives

A bioelectric pattern memory is a stable steady state. find_steady_state solves \(f(x) = 0\) by pseudo-transient continuation, using an exact sparse Jacobian built with forward-mode autograd and a distance-2 colouring of the cell graph. On a grid that costs \(5n\) Jacobian-vector products regardless of tissue size.

steady_state_implicit differentiates the result with the implicit function theorem, so a gradient costs one sparse linear solve rather than backpropagation through the whole relaxation.

Solvers

Method	Use
dopri5	the default, adaptive, safe with strong channel expression
rosenbrock	linearly implicit ROS2, for stiff problems
rk4, euler	fixed step; they raise an error rather than return NaN

Model zoo

bioelectrics.zoo.cam2016 implements Cervera, Alcaraz & Mafe (2016) with reproduction tests that assert the published fixed points and the behaviour of its figures 4 and 5. See Validation for the numbers, and for the two models that cannot be reproduced from the public record.