Comparison of methods to calculate exact spike times in integrate-and-fire neurons with exponential currents
Alexander Hanuschkin
1
Susanne Kunkel
1
Moritz Helias
1
Abigail Morrison
0
Markus Diesmann
0
1
0
Computational Neuroscience Group, RIKEN Brain Science Institute
,
Wako City, Saitama 351-0198
,
Japan
1
Bernstein Center for Computational Neuroscience, Albert-Ludwigs-University
,
79104 Freiburg
,
Germany
Acknowledgements Partially funded by DIP F1.2, BMBF Grant 01GQ0420 to the Bernstein Center for Computational Neuroscience Freiburg, and EU Grant 15879 (FACETS). References 1. Hansel D, Mato G, Meunier C, Neltner L: On numerical simulations of integrate-and-fire neural networks. Neural Comput 1998, 10(2):467-483. 2. Morrison A, Straube S, Plesser HE, Diesmann M: Exact subthreshold integration with continuous spike times in discrete-time neural network simulations. Neural Comput 2007, 19(1):47-79. 3. Gewaltig M-O, Diesmann M: NEST. Scholarpedia 2007, 2(4):1430. 4. Rotter S, Diesmann M: Exact digital simulation of time-invariant linear systems with applications to neuronal modeling. Biol Cybern 1999, 81(5-6):381-402. 5. Brette R: Exact simulation of integrate-and-fire models with exponential currents. Neural Comput 2007, 19(10):2604-2609. 6. Brunel N: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 2000, 8(3):183-208.
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Discrete-time neuronal network simulation strategies
typically constrain spike times to a grid determined by the
computational step size. This approach can have the effect
of introducing artificial synchrony [1]. However,
timecontinuous approaches can be computationally
demanding, both with respect to calculating future spike times and
to event management, particularly for large network sizes.
To address this problem, Morrison et al. [2] presented a
general method of handling off-grid spiking in
combination with exact subthreshold integration in discrete time
driven simulations [3,4]. Within each time step an
eventdriven environment is emulated to process incoming
spikes, whereas the timing of outgoing spikes is based on
interpolation. Therefore, the computation step size is a
decisive factor for both integration error and simulation
time.
An alternative approach for calculating the exact spike
times of integrate-and-fire neurons with exponential
currents was recently published by Brette [5]. The problem of
accurate detection of the first threshold crossing of the
membrane potential is converted into finding the largest
root of a polynomial. Common numerical means like
Descartes' rule and Sturm's theorem are applicable.
Although this approach was developed in the context of
event-driven simulations, we take advantage of its ability
to predict future threshold crossings in the time-driven
environment of NEST [3]. We compare the accuracy of the
two approaches in single-neuron simulations and the
efficiency in a balanced random network of 10,000 neurons
[6]. We show that the network simulation time when
using the polynomial method depends only weakly on
the computational step size, and the single neuron
integration error is independent of it. Although the
polynomial method attains the maximum precision expected
from double numerics for all input rates and computation
step sizes, the interpolation method is more efficient for
input rates above a critical value. For applications where a
lesser degree of precision is acceptable, the interpolation
method is more efficient for all input rates.
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