Variable timestep integration (CVODE)

As opposed to fixed timestep integration, variable timestep integration (CVODE in NEURON parlance) uses the SUNDIALS package to solve a DERIVATIVE or KINETIC block using a variable timestep. This allows for faster computation times if the function in question does not vary too wildly.

Implementation in NMODL

The code generation for CVODE is activated only if exactly one of the following is satisfied:

there is one KINETIC block in the mod file
there is one DERIVATIVE block in the mod file
a PROCEDURE block is solved with the after_cvode, cvode_t, or cvode_t_v methods

In NMODL, all KINETIC blocks are internally first converted to DERIVATIVE blocks. The DERIVATIVE block is then converted to a CVODE block, which contains two parts; the first part contains the update step for non-stiff systems (functional iteration), while the second part contains the update step for stiff systems (additional step using the Jacobian). For more information, see CVODES documentation, eqs. (4.8) and (4.9). Given a DERIVATIVE block of the form:

DERIVATIVE state {
    x_i' = f(x_1, ..., x_n)
}

the structure of the CVODE block is then roughly:

CVODE state[n] {
    Dx_i = f_i(x_1, ..., x_n)
}{
    Dx_i = Dx_i / (1 - dt * J_ii(f))
}

where N is the total number of ODEs to solve, and J_ii(f) is the diagonal part of the Jacobian, i.e.

$J_{ii}(f) = \frac{ \partial f_i(x_1, \ldots, x_n) }{\partial x_i}$

As an example, consider the following DERIVATIVE block:

DERIVATIVE state {
    X' = - X
}

Where X is a STATE variable with some initial value, specified in the INITIAL block. The corresponding CVODE block is then:

CVODE state[1] {
    DX = - X
}{
    DX = DX / (1 - dt * (-1))
}

NOTE: in case there are CONSERVE statements in KINETIC blocks, as they are merely hints to NMODL, and have no impact on the results, they are removed from CVODE blocks before the codegen stage.