When a differential equation blows up — when its solution reaches infinity in finite time — the standard numerical response is to refine the timestep. As the singularity approaches, take smaller and smaller steps, tracking the solution as it climbs toward infinity. The refinement is reactive: the computation speeds up because the dynamics speed up.
The alternative is to slow down time itself.
Time dilation replaces the physical time variable with a dilated clock that runs proportionally slower where activity is high. The singularity that was unreachable in finite physical time becomes reachable in infinite dilated time — and since numerical methods can march forward indefinitely, the blow-up transforms from a computational catastrophe into a tractable steady state.
The paper applies this to neural firing models, where the blow-up is the spike: a neuron's voltage diverges to infinity (in the mathematical idealization), then resets. Standard adaptive timestepping chases each spike with increasingly fine resolution, spending computational effort on the steepest part of the trajectory. Time dilation instead stretches the spike into a gentle plateau, solvable with fixed-size steps. The computational cost shifts from resolving the singularity to maintaining the dilated timescale — and the dilated version is cheaper.
The structural insight: many computational intractabilities are artifacts of the coordinate system, not of the physics. The blow-up is a property of the time variable, not of the solution. In dilated coordinates, the same physical process unfolds smoothly. The singularity does not disappear — it is redistributed across infinite dilated time, where “infinite” is computationally tractable because the increment is fixed.
This is distinct from standard coordinate transformations in physics (which remove coordinate singularities like the event horizon in Schwarzschild coordinates). Here, the physical singularity is real — the voltage genuinely diverges — but the numerical difficulty is coordinate-dependent. The right response is not to compute faster in the face of divergence but to change the clock so that divergence takes forever. The singularity is not resolved. It is deferred, infinitely.