On preserving metric properties of integrate-and-fire sampling

The leaky integrate-and-fire model (LIF), which consists of a leaky integrator followed by a threshold-based comparator, is analyzed from a mathematical metric analysis point of view. The question is addressed whether metric properties are preserved under this non-linear operator that maps input signals to spike trains, or, synonymously, event sequences. By measuring the distance between input signals by means of Hermann Weyl's discrepancy norm and applying its discrete counterpart to measure the distance between event sequences, it is proven that LIF approximately preserves the metric. It turns out that in this setting, for arbitrarily small thresholds, LIF is an asymptotic isometry.