This title appears in the Scientific Report : 2015 

Detection of spiking events in continuous-time spiking neuron models
Krishnan, Jeyashree (Corresponding author)
Mana, PierGianLuca / Helias, Moritz / Kunkel, Susanne / Napoli, Di / Diesmann, Markus
Computational and Systems Neuroscience; INM-6
3rd HBP Winter School, Manchester, UK, Manchester (UK), 2016-01-11 - 2016-01-15
Theory, modelling and simulation
The leaky integrate-and-fire neuron model is one of the commonly used spiking neuron models that can mimic the dynamics of neurons to high accuracy. This model consists of a system of first order linear differential equations with which the sub-threshold dynamics can be exactly integrated. Any excursion of the membrane potential above threshold leads to a spike, immediately after which the membrane potential is clamped to zero and input is sent to other neurons.Theoretical descriptions of neuron models require that the numerical implementations be in agreement with the exact solutions of the mathematical model. In time-driven simulators like NEST, the state of the neurons is updated in discrete time steps, and the spikes are detected and emitted only at the end of each step. Such a discrete-time neuronal network simulation leads to the following problems:- Spikes get constrained to the time grid, therefore they do not carry a precise time stamp, leading to artifical synchronization.- Depending on the computational step size, it may happen that the neuron voltage is below threshold at the beginning and end of the timestep with the excursion happening within the step, thereby causing a spike miss. - In addition, grid-constrained spiking causes an integration error that decreases only linearly with the computational step size.The purpose of this work is to formulate and implement efficient techniques that can be included in the neuron models to handle events at every point on the time grid by computing the precise spike times. We have constructed a precise numerical implementation of a particular variant of the leaky integrate-and-fire neuron model that does not miss any spikes. This implementation relies on the computation of the exact time to the maximum of potential in closed form based on the Lambert-W function. This model can catch otherwise missed spikes for large computational step sizes but is in principle computationally expensive because the cost comes from both the frequency of the test and the time for its calculation.To decrease the computational cost of the spike test, we have constructed an additional test that can predict the occurrence of threshold crossing in continuous-time allowing us to calculate the time to maximum from the previous test only when necessary. This test is based on the analysis of the trajectories in state-space governed by the system of equations describing this model. This has helped us construct a series of tests that can predict before propagation whether or not an event is expected in the future, thereby helping us achieve both high accuracy and less computational time.