Integration

A neuron model is a system of differential equations: the membrane potential, gating variables, and ion concentrations all evolve in time according to a right-hand side \(f(t, y)\). Integration is the act of advancing that system forward in time. braincell keeps this concern in its own module, braincell.quad, shared by both cell types.

This page is the conceptual overview. For hands-on use — comparing solvers, writing your own — see the full Numerical Integration guide.

The protocol: describing the equations

Two small abstractions describe what to integrate:

• braincell.DiffEqState — a state variable that carries its own derivative (and, for cable diffusion, a diffusion term).

• braincell.DiffEqModule — a module that exposes a right-hand side over such states.

Cells implement this protocol, so any solver knows how to step them. You rarely touch it unless you are building a new dynamical component.

The registry: choosing a solver

braincell.quad holds a registry of solvers, each selectable by name. You pick one when constructing a cell:

import braincell
hh = braincell.SingleCompartment(size, solver='ind_exp_euler')

The available integrators span the usual families:

Family	Members (by name)
Explicit Runge–Kutta	euler, midpoint, rk2, rk3, rk4, heun2, heun3, ssprk3, ralston2/3/4
Exponential Euler	exp_euler, ind_exp_euler, exp_exp_euler
Implicit	backward_euler, implicit_euler, implicit_exp_euler, implicit_rk4
Composite / cable	cn_exp_euler, cn_rk4 (Crank–Nicolson), splitting, staggered, dhs_voltage

You can list everything available at runtime:

import braincell.quad as quad
sorted(quad.all_integrators)

Choosing wisely: stiffness

Biophysical neuron models are stiff — they mix very fast (sodium spike) and slow (calcium, adaptation) timescales. Explicit solvers like rk4 must take tiny time steps to stay stable on stiff systems, which is slow.

Tip

For single-compartment HH-style models, exponential-Euler variants (exp_euler, ind_exp_euler) handle the stiff gating kinetics stably at practical step sizes and are a good default. For multi-compartment cable equations, the staggered / Crank–Nicolson solvers are designed for the spatial coupling.

Differentiability

Because every solver is written in JAX, the entire time loop is differentiable. Gradients of an output (a spike count, a voltage trace, a loss) with respect to parameters (conductances, time constants) flow straight back through the integration — this is what makes gradient-based model fitting possible.

Writing your own integrator

The registry is open: decorate a stepping function with @register_integrator and it becomes selectable by name like any built-in. The Advanced Integration guide walks through this.

See also

• Numerical Integration — the full integration guide (overview, protocol, solvers, advanced).

• braincell.quad module — every integrator and its signature.

• Discretization — what the cable solvers operate on.