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Journal of Computational Neuroscience

, Volume 39, Issue 1, pp 29–51 | Cite as

Statistical structure of neural spiking under non-Poissonian or other non-white stimulation

  • Tilo Schwalger
  • Felix Droste
  • Benjamin Lindner
Article

Abstract

Nerve cells in the brain generate sequences of action potentials with a complex statistics. Theoretical attempts to understand this statistics were largely limited to the case of a temporally uncorrelated input (Poissonian shot noise) from the neurons in the surrounding network. However, the stimulation from thousands of other neurons has various sorts of temporal structure. Firstly, input spike trains are temporally correlated because their firing rates can carry complex signals and because of cell-intrinsic properties like neural refractoriness, bursting, or adaptation. Secondly, at the connections between neurons, the synapses, usage-dependent changes in the synaptic weight (short-term plasticity) further shape the correlation structure of the effective input to the cell. From the theoretical side, it is poorly understood how these correlated stimuli, so-called colored noise, affect the spike train statistics. In particular, no standard method exists to solve the associated first-passage-time problem for the interspike-interval statistics with an arbitrarily colored noise. Assuming that input fluctuations are weaker than the mean neuronal drive, we derive simple formulas for the essential interspike-interval statistics for a canonical model of a tonically firing neuron subjected to arbitrarily correlated input from the network. We verify our theory by numerical simulations for three paradigmatic situations that lead to input correlations: (i) rate-coded naturalistic stimuli in presynaptic spike trains; (ii) presynaptic refractoriness or bursting; (iii) synaptic short-term plasticity. In all cases, we find severe effects on interval statistics. Our results provide a framework for the interpretation of firing statistics measured in vivo in the brain.

Keywords

Interspike-interval statistics Stochastic integrate-and-fire neuron Non-renewal process Temporal correlations Spontaneous activity 

Notes

Acknowledgments

Research was supported by the European Research Council (Grant Agreement no. 268689, MultiRules), the BMBF (FKZ: 01GQ1001A), and the research training group GRK1589/1.

Conflict of interests

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Tilo Schwalger
    • 1
    • 2
  • Felix Droste
    • 2
    • 3
  • Benjamin Lindner
    • 2
    • 3
  1. 1.Brain Mind InstituteÉcole Polytechnique Féderale de Lausanne (EPFL) Station 15LausanneSwitzerland
  2. 2.Bernstein Center for Computational NeuroscienceBerlinGermany
  3. 3.Department of PhysicsHumboldt Universität zu BerlinBerlinGermany

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