braincell.quad.exp_euler_step

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braincell.quad.exp_euler_step#

braincell.quad.exp_euler_step(target, *args)[source]#

Advance one step with the (coupled) exponential Euler method.

The exponential Euler method targets semi-linear ODEs of the form

\[\frac{dy}{dt} = A(y_n)\, y + g(t, y),\]

where the local Jacobian \(A(y_n) = \partial f / \partial y\) is treated implicitly via the matrix exponential while the residual \(g\) is held frozen at \(y_n\). The update reads

\[y_{n+1} = y_n + \Delta t \, \varphi_1(\Delta t \, A)\, f(t_n, y_n), \qquad \varphi_1(z) = \frac{e^{z} - 1}{z}.\]

Equivalently, with \(A\) the local linearization,

\[y_{n+1} = A^{-1}\!\left(e^{\Delta t A} - I\right) f(t_n, y_n) + y_n.\]

Because the linear part is integrated exactly, the scheme is A-stable for the local linearization and remains accurate where forward Euler would blow up. It is the workhorse explicit-style integrator for Hodgkin-Huxley type membranes.

Unlike ind_exp_euler_step(), this routine treats the entire state vector as a single coupled block: it builds the dense local Jacobian over all DiffEqState leaves, computes its matrix exponential, and applies it in one shot. That captures cross-state coupling exactly to first order in \(\Delta t\) but costs \(O(M^3)\) per step in the state dimension \(M\).

Parameters:

  • target (DiffEqModule) – The neuron model to advance. Must be an HHTypedNeuron subclass — currently SingleCompartment (states are stacked along [n_neuron, n_state]) or braincell.Cell (states are concatenated along [n_neuron, n_compartment * n_state]).

  • *args – Extra positional arguments forwarded to target’s pre_integral(), compute_derivative(), and post_integral() hooks (typically the input current for this step).

Returns:

target’s differential states are updated in place.

Return type:

None

Raises:

  • AssertionError – If target is not an HHTypedNeuron.

  • ValueError – If target is an HHTypedNeuron of an unsupported subtype.

See also

ind_exp_euler_step

State-by-state variant that linearizes each DiffEqState independently.

backward_euler_step

Linearized backward Euler counterpart.

Notes

The current time and step size are read from the active brainstate.environ context.

Examples

>>> import brainstate
>>> import brainunit as u
>>> from braincell.quad import exp_euler_step
>>> with brainstate.environ.context(t=0. * u.ms, dt=0.025 * u.ms):
...     exp_euler_step(my_neuron, input_current)
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