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@INPROCEEDINGS{Krishnan:281077,
      author       = {Krishnan, Jeyashree and Mana, PierGianLuca and Helias,
                      Moritz and Kunkel, Susanne and Di Napoli, Edoardo and
                      Diesmann, Markus},
      title        = {{D}etection of spiking events in continuous-time spiking
                      neuron models},
      school       = {RWTH Aachen},
      reportid     = {FZJ-2016-00782},
      year         = {2015},
      abstract     = {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.},
      month         = {Jan},
      date          = {2016-01-11},
      organization  = {3rd HBP Winter School, Manchester, UK,
                       Manchester (UK), 11 Jan 2016 - 15 Jan
                       2016},
      subtyp        = {Other},
      cin          = {INM-6 / JSC},
      cid          = {I:(DE-Juel1)INM-6-20090406 / I:(DE-Juel1)JSC-20090406},
      pnm          = {574 - Theory, modelling and simulation (POF3-574) /
                      Simulation and Data Laboratory Quantum Materials (SDLQM)
                      (SDLQM)},
      pid          = {G:(DE-HGF)POF3-574 / G:(DE-Juel1)SDLQM},
      typ          = {PUB:(DE-HGF)24},
      url          = {https://juser.fz-juelich.de/record/281077},
}