Title: Perfect spike detection via time reversal,
Abstract: Spiking neuronal networks are usually simulated with three main simulation
schemes: the classical time-driven and event-driven schemes, and the more
recent hybrid scheme. All three schemes evolve the state of a neuron through a
series of checkpoints: equally spaced in the first scheme and determined
neuron-wise by spike events in the latter two. The time-driven and the hybrid
scheme determine whether the membrane potential of a neuron crosses a threshold
at the end of of the time interval between consecutive checkpoints. Threshold
crossing can, however, occur within the interval even if this test is negative.
Spikes can therefore be missed. The present work derives, implements, and
benchmarks a method for perfect retrospective spike detection. This method can
be applied to neuron models with affine or linear subthreshold dynamics. The
idea behind the method is to propagate the threshold with a time-inverted
dynamics, testing whether the threshold crosses the neuron state to be evolved,
rather than vice versa. Algebraically this translates into a set of
inequalities necessary and sufficient for threshold crossing. This test is
slower than the imperfect one, but faster than an alternative perfect tests
based on bisection or root-finding methods. Comparison confirms earlier results
that the imperfect test rarely misses spikes (less than a fraction $1/10^8$ of
missed spikes) in biologically relevant settings. This study offers an
alternative geometric point of view on neuronal dynamics.