Perfect Detection of Spikes in the Linear Sub-threshold Dynamics of Point Neurons

Spiking neuronal networks are usually simulated with one of three main 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 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 offers an alternative geometric point of view on neuronal dynamics, and 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 can be optimized in several ways. Comparison confirms earlier results that the imperfect tests rarely miss spikes (less than a fraction 1/108 of missed spikes) in biologically relevant settings.