In large networks of excitatory neurons, synchronous firing events happen almost instantaneously. In mean-field models, this manifests as a finite-time blow-up — the firing rate goes to infinity in finite time, a mathematical singularity that represents every neuron firing at once.
Standard numerical methods crash at the singularity. This paper resolves it with a physical trick: dilate time. Rescale the time variable in proportion to the firing activity, so that as the firing rate diverges, the time axis stretches to accommodate it. What was an instantaneous catastrophe in real time becomes a smooth process in dilated time.
The implementation is a hybrid scheme. Away from the blow-up, the simulation runs in normal time. A mesh-independent switching criterion detects when synchrony approaches, and the simulation transitions to dilated time. After the synchronization event, a semi-analytical “moving Gaussian” method tracks the post-collapse distribution of neurons, then the simulation reverts to normal time.
The framework captures both steady-state behavior and periodic firing patterns — sequences of blow-ups followed by recovery, followed by the next blow-up. The results match particle-based simulations (which track individual neurons) without requiring the extreme time-step restrictions that particle methods demand.
The deeper principle: a singularity in one coordinate system is often regular in another. The blow-up is an artifact of the original time parameterization, not a fundamental obstacle. Choosing a time variable that respects the dynamics — running the clock at the same speed as the system — dissolves the singularity without changing the physics.