Equilibrium and response properties of the integrate-and-fi re neuron in discrete time

The integrate-and-fire neuron with exponential postsynaptic potentials is a frequently employed model to study neural networks. Simulations in discrete time still have highest performance at moderate numerical errors, which makes them first choice for long-term simulations of plastic networks. Here we extend the population density approach to investigate how the equilibrium and response properties of the leaky integrate-and-fire neuron are affected by time discretization. We present a novel analytical treatment of the boundary condition at threshold, taking both discretization of time and finite synaptic weights into account. We uncover an increased membrane potential density just below threshold as the decisive property that explains the deviations found between simulations and the classical diffusion approximation. Temporal discretization and finite synaptic weights both contribute to this effect. Our treatment improves the standard formula to calculate the neuron's equilibrium firing rate. Direct solution of the Markov process describing the evolution of the membrane potential density confirms our analysis and yields a method to calculate the firing rate exactly. Knowing the shape of the membrane potential distribution near threshold enables us to devise the transient response properties of the neuron model to synaptic input. We find a pronounced non-linear fast response component that has not been described by the prevailing continuous time theory for Gaussian white noise input.

[1]  Tim Gollisch,et al.  Modeling Single-Neuron Dynamics and Computations: A Balance of Detail and Abstraction , 2006, Science.

[2]  Stefan Rotter,et al.  Higher-Order Statistics of Input Ensembles and the Response of Simple Model Neurons , 2003, Neural Computation.

[3]  Guido van Rossum,et al.  Python Programming Language , 2007, USENIX Annual Technical Conference.

[4]  Markus Diesmann,et al.  The spread of rate and correlation in stationary cortical networks , 2003, Neurocomputing.

[5]  Markus Diesmann,et al.  Maintaining Causality in Discrete Time Neuronal Network Simulations , 2007 .

[6]  Lawrence Sirovich,et al.  Dynamics of Neuronal Populations: The Equilibrium Solution , 2000, SIAM J. Appl. Math..

[7]  Wulfram Gerstner,et al.  A benchmark test for a quantitative assessment of simple neuron models , 2008, Journal of Neuroscience Methods.

[8]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[9]  L Schimansky-Geier,et al.  Transmission of noise coded versus additive signals through a neuronal ensemble. , 2001, Physical review letters.

[10]  Moritz Helias,et al.  Structural Plasticity Controlled by Calcium Based Correlation Detection , 2008, Frontiers Comput. Neurosci..

[11]  Nicolas Brunel,et al.  Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons , 2000, Journal of Computational Neuroscience.

[12]  A. Burkitt,et al.  Shot noise in the leaky integrate-and-fire neuron. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Lawrence Sirovich,et al.  Dynamics of neuronal populations: eigenfunction theory; some solvable cases , 2003, Network.

[14]  W. Senn,et al.  Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. , 2003, Journal of neurophysiology.

[15]  D. Hansel,et al.  How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs , 2003, The Journal of Neuroscience.

[16]  S. Rotter,et al.  Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units , 2009, PLoS Comput. Biol..

[17]  Charles R. MacCluer,et al.  The Many Proofs and Applications of Perron's Theorem , 2000, SIAM Rev..

[18]  A. Siegert On the First Passage Time Probability Problem , 1951 .

[19]  H. Sompolinsky,et al.  Chaos in Neuronal Networks with Balanced Excitatory and Inhibitory Activity , 1996, Science.

[20]  Germán Mato,et al.  On Numerical Simulations of Integrate-and-Fire Neural Networks , 1998, Neural Computation.

[21]  M. de Kamps,et al.  A simple and stable numerical solution for the population density equation. , 2003, Neural computation.

[22]  Virginia Giorno,et al.  AN OUTLINE OF THEORETICAL AND ALGORITHMIC APPROACHES TO FIRST PASSAGE TIME PROBLEMS WITH APPLICATIONS TO BIOLOGICAL MODELING , 1999 .

[23]  Stefan Rotter,et al.  State space analysis of synchronous spiking in cortical neural networks , 2001, Neurocomputing.

[24]  R. Stein A THEORETICAL ANALYSIS OF NEURONAL VARIABILITY. , 1965, Biophysical journal.

[25]  Ad Aertsen,et al.  Stable propagation of synchronous spiking in cortical neural networks , 1999, Nature.

[26]  Markus Diesmann,et al.  The mechanism of synchronization in feed-forward neuronal networks , 2008 .

[27]  Wulfram Gerstner,et al.  Phenomenological models of synaptic plasticity based on spike timing , 2008, Biological Cybernetics.

[28]  Eugene M. Izhikevich,et al.  Which model to use for cortical spiking neurons? , 2004, IEEE Transactions on Neural Networks.

[29]  Carson C. Chow,et al.  Aperiodic stochastic resonance. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  M. J. Richardson,et al.  Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Magnus J. E. Richardson,et al.  Spike-train spectra and network response functions for non-linear integrate-and-fire neurons , 2008, Biological Cybernetics.

[32]  Fredric M. Wolf,et al.  Action Potential Onset Dynamics and the Response Speed of Neuronal Populations , 2004, Journal of Computational Neuroscience.

[33]  Bartlett W. Mel,et al.  Impact of Active Dendrites and Structural Plasticity on the Memory Capacity of Neural Tissue , 2001, Neuron.

[34]  Walter Senn,et al.  Minimal Models of Adapted Neuronal Response to In VivoLike Input Currents , 2004, Neural Computation.

[35]  L. Ricciardi,et al.  The Ornstein-Uhlenbeck process as a model for neuronal activity , 1979, Biological Cybernetics.

[36]  R. Kempter,et al.  Hebbian learning and spiking neurons , 1999 .

[37]  Frances S. Chance,et al.  Effects of synaptic noise and filtering on the frequency response of spiking neurons. , 2001, Physical review letters.

[38]  Jaime de la Rocha,et al.  Supplementary Information for the article ‘ Correlation between neural spike trains increases with firing rate ’ , 2007 .

[39]  Nicolas Brunel,et al.  Fast Global Oscillations in Networks of Integrate-and-Fire Neurons with Low Firing Rates , 1999, Neural Computation.

[40]  Markus Diesmann,et al.  Exact Subthreshold Integration with Continuous Spike Times in Discrete-Time Neural Network Simulations , 2007, Neural Computation.