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@ARTICLE{Stapmanns:872733,
      author       = {Stapmanns, Jonas and Kühn, Tobias and Dahmen, David and
                      Luu, Tom and Honerkamp, Carsten and Helias, Moritz},
      title        = {{S}elf-consistent formulations for stochastic nonlinear
                      neuronal dynamics},
      journal      = {Physical review / E covering statistical, nonlinear,
                      biological, and soft matter physics},
      volume       = {101},
      number       = {4},
      issn         = {1063-651X},
      address      = {Woodbury, NY},
      publisher    = {Inst.},
      reportid     = {FZJ-2020-00211},
      pages        = {042124},
      year         = {2020},
      abstract     = {Neural dynamics is often investigated with tools from
                      bifurcation theory. However, many neuron models are
                      stochastic, mimicking fluctuations in the input from unknown
                      parts of the brain or the spiking nature of signals. Noise
                      changes the dynamics with respect to the deterministic
                      model; in particular classical bifurcation theory cannot be
                      applied. We formulate the stochastic neuron dynamics in the
                      Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism
                      and present the fluctuation expansion of the effective
                      action and the functional renormalization group (fRG) as two
                      systematic ways to incorporate corrections to the mean
                      dynamics and time-dependent statistics due to fluctuations
                      in the presence of nonlinear neuronal gain. To formulate
                      self-consistency equations, we derive a fundamental link
                      between the effective action in the Onsager-Machlup (OM)
                      formalism, which allows the study of phase transitions, and
                      the MSRDJ effective action, which is computationally
                      advantageous. These results in particular allow the
                      derivation of an OM effective action for systems with
                      non-Gaussian noise. This approach naturally leads to
                      effective deterministic equations for the first moment of
                      the stochastic system; they explain how nonlinearities and
                      noise cooperate to produce memory effects. Moreover, the
                      MSRDJ formulation yields an effective linear system that has
                      identical power spectra and linear response. Starting from
                      the better known loopwise approximation, we then discuss the
                      use of the fRG as a method to obtain self-consistency beyond
                      the mean. We present a new efficient truncation scheme for
                      the hierarchy of flow equations for the vertex functions by
                      adapting the Blaizot, Méndez, and Wschebor approximation
                      from the derivative expansion to the vertex expansion. The
                      methods are presented by means of the simplest possible
                      example of a stochastic differential equation that has
                      generic features of neuronal dynamics.},
      cin          = {INM-6 / IAS-6 / INM-10 / IAS-4 / IKP-3},
      ddc          = {530},
      cid          = {I:(DE-Juel1)INM-6-20090406 / I:(DE-Juel1)IAS-6-20130828 /
                      I:(DE-Juel1)INM-10-20170113 / I:(DE-Juel1)IAS-4-20090406 /
                      I:(DE-Juel1)IKP-3-20111104},
      pnm          = {574 - Theory, modelling and simulation (POF3-574) / 571 -
                      Connectivity and Activity (POF3-571) / MSNN - Theory of
                      multi-scale neuronal networks (HGF-SMHB-2014-2018) /
                      RenormalizedFlows - Transparent Deep Learning with
                      Renormalized Flows (BMBF-01IS19077A) / HBP SGA2 - Human
                      Brain Project Specific Grant Agreement 2 (785907) / HBP SGA3
                      - Human Brain Project Specific Grant Agreement 3 (945539)},
      pid          = {G:(DE-HGF)POF3-574 / G:(DE-HGF)POF3-571 /
                      G:(DE-Juel1)HGF-SMHB-2014-2018 /
                      G:(DE-Juel-1)BMBF-01IS19077A / G:(EU-Grant)785907 /
                      G:(EU-Grant)945539},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {pmid:32422832},
      UT           = {WOS:000527130200002},
      doi          = {10.1103/PhysRevE.101.042124},
      url          = {https://juser.fz-juelich.de/record/872733},
}