Bifurcation-Preserving Discrete Reduction of Type-I Neuron Dynamics: Threshold, Timing and Step-Size Admissibility

Map-based neuron models replace the integration of a stiff
conductance-based system by a cheap state update and are therefore
attractive for large-scale network simulation. Most are
phenomenological: their parameters are fitted coefficients rather than
conductances, and no statement is available about which features of a
parent model they preserve. Nullcline-guided maps retain biophysical
parameters, which raises the natural question of which quantitative
properties of the parent model can be preserved analytically. We ask when a one-dimensional map derived from a
Type-I (saddle-node on invariant circle) conductance-based neuron
preserves the quantitative threshold and timing structure of its
parent, and we answer with explicit, checkable conditions.
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We show that the subthreshold dynamics of the
$I_{\mathrm{Na,p}}+I_{K}$ model admit a scalar reduction
$C\sigma(V)\dot V=I-I_{ss}(V)$, where $I_{ss}$ is the steady-state
current--voltage relation and $\sigma$ carries the first-order
correction from the recovery variable. Two explicit local diagnostics
characterise the reduction near the resting fold: a gating threshold
$\varepsilon_{*}$, with
$\operatorname{tr}J(V_c)=(\varepsilon_{*}-\varepsilon)/\tau$ at the
fold, so that $\varepsilon>\varepsilon_{*}$ is exactly the local
spectral condition for Type-I onset; and a parameter
$\chi=\tau I_{ss}'/(C\varepsilon)$, vanishing at the fold, which
controls the error in the linear relaxation rate at rest.
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Discretising the reduction with an explicit step, and treating the
action potential as an event of fixed physical duration, yields a
hybrid map whose fixed points coincide, on a fixed subthreshold
voltage interval, with the parent equilibria there for every applied
current, and whose interspike interval reproduces the Type-I
bottleneck law including its prefactor, with a discretisation error
that is $O(\Delta t)$ uniformly as rheobase is approached. Stability
of a resting fixed point is preserved exactly when
$0<\Delta t<2C\sigma(V_r)/I_{ss}'(V_r)$; if this holds uniformly along
the prescribed resting branch up to rheobase, the map's operational
rheobase equals the relevant local maximum of $I_{ss}$, while a step
violating it can produce a spurious flip instability below the parent
rheobase.
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The construction therefore delivers a reduced model that advances by a
single scalar update per step, carries the parent model's conductances
and reversal potentials explicitly, and comes with proved statements
about what it preserves and where it ceases to apply. We also measure
what the reduction costs, and find the advantage to be
regime-dependent: the map's work is fixed at $O(T/\Delta t)$ updates
while an adaptive solver's responds to the local time scale, so near
onset the solver is the cheaper of the two, and the map retains a
reduction error that no step size removes.