Journal Article

FZJ-2020-00211

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Self-consistent formulations for stochastic nonlinear neuronal dynamics

Stapmanns, J.FZJ* ; Kühn, T. (Corresponding author)FZJ* ; Dahmen, D.FZJ* ; Luu, T.FZJ* ; Honerkamp, C. ; Helias, M.FZJ*

2020
Inst. Woodbury, NY

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Please use a persistent id in citations: http://hdl.handle.net/2128/26002  doi:10.1103/PhysRevE.101.042124

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.

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Contributing Institute(s):

1. Computational and Systems Neuroscience (INM-6)
2. Theoretical Neuroscience (IAS-6)
3. Jara-Institut Brain structure-function relationships (INM-10)
4. Theorie der Starken Wechselwirkung (IAS-4)
5. Theorie der starken Wechselwirkung (IKP-3)

Research Program(s):

1. 574 - Theory, modelling and simulation (POF3-574) (POF3-574)
2. 571 - Connectivity and Activity (POF3-571) (POF3-571)
3. MSNN - Theory of multi-scale neuronal networks (HGF-SMHB-2014-2018) (HGF-SMHB-2014-2018)
4. RenormalizedFlows - Transparent Deep Learning with Renormalized Flows (BMBF-01IS19077A) (BMBF-01IS19077A)
5. HBP SGA2 - Human Brain Project Specific Grant Agreement 2 (785907) (785907)
6. HBP SGA3 - Human Brain Project Specific Grant Agreement 3 (945539) (945539)

Appears in the scientific report 2020

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Database coverage:
; ; ; Clarivate Analytics Master Journal List ; Current Contents - Physical, Chemical and Earth Sciences ; Ebsco Academic Search ; IF < 5 ; JCR ; SCOPUS ; Science Citation Index ; Science Citation Index Expanded ; Web of Science Core Collection

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