Biological Cybernetics

, Volume 112, Issue 6, pp 539–545 | Cite as

Non-monotonic accumulation of spike time variance during membrane potential oscillations

  • Eric S. Kuebler
  • Matias Calderini
  • André Longtin
  • Nicolas Bent
  • Philippe Vincent-Lamarre
  • Jean-Philippe ThiviergeEmail author
Original Article


A spike-phase neural code has been proposed as a mechanism to encode stimuli based on the precise timing of spikes relative to the phase of membrane potential oscillations. This form of coding has been reported in both in vivo and in vitro experiments across several regions of the brain, yet there are concerns that such precise timing may be compromised by an effect referred to as variance accumulation, wherein spike timing variance increases over the phase of an oscillation. Here, we provide a straightforward explanation of this effect based on the theoretical spike time variance. The proposed theory is consistent with recordings of mitral neurons. It shows that spike time variance can increase in a nonlinear fashion with spike number, in a way that is dependent upon the frequency and amplitude of the oscillation. Further, non-monotonic accumulation of variance can arise from different combinations of oscillation parameters. Nonlinear accumulation sometimes leads to lower variance than that of a mean rate-matched homogeneous Poisson process, particularly for spikes that occur in later phases of oscillation. However, such an advantage is limited to a narrow range of oscillation amplitudes and frequencies. These results suggest fundamental constraints on spike-phase coding, and reveal how certain spikes in a sequence may exhibit increased firing time precision relative to their neighbors.


Neural oscillations Poisson model Spike variance 



This work was supported by grants to J.P.T. from the Natural Sciences and Engineering Council of Canada (NSERC Grant No. 210977 and No. 210989), the Canadian Institutes of Health Research (CIHR Grant No. 6105509), and the University of Ottawa Brain and Mind Institute (uOBMI), as well as a graduate scholarship to E.S.K. from NSERC. AL also acknowledges support from NSERC. The authors are thankful to Alfonso Renart, Andreas Schaefer, and Maurice Chacron for useful discussions.

Compliance with Ethical Standards

Conflict of interest

The authors declare no competing financial interests.

Supplementary material

422_2018_782_MOESM1_ESM.m (3 kb)
Supplementary material 1 (m 3 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of PsychologyUniversity of OttawaOttawaCanada
  2. 2.Department of PhysicsUniversity of OttawaOttawaCanada
  3. 3.Center for Neural DynamicsUniversity of OttawaOttawaCanada

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