Description: Self-consistent formulations for stochastic nonlinear neuronal dynamics

This title appears in the Scientific Report : 2019

Self-consistent formulations for stochastic nonlinear neuronal dynamics

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 input signals. Such noise in the input, however, changes the dynamics with respect to the...

Personal Name(s):	Stapmanns, Jonas (Corresponding author)
	Kühn, Tobias (Corresponding author) / Dahmen, David / Luu, Tom / Honerkamp, Carsten (Last author) / Helias, Moritz (Last author)
Contributing Institute:	Computational and Systems Neuroscience; INM-6 Theorie der Starken Wechselwirkung; IAS-4 Theorie der starken Wechselwirkung; IKP-3 Jara-Institut Brain structure-function relationships; INM-10 Theoretical Neuroscience; IAS-6
Imprint:	2018
Document Type:	Preprint
Research Program:	Human Brain Project Specific Grant Agreement 1 Supercomputing and Modelling for the Human Brain Theory of multi-scale neuronal networks Connectivity and Activity Theory, modelling and simulation Human Brain Project Specific Grant Agreement 2
Link:	OpenAccess OpenAccess
	Publikationsportal JuSER

Please use the identifier: http://hdl.handle.net/2128/21891 in citations.

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 input signals. Such noise in the input, however, changes the dynamics with respect to the deterministic model. We formulate the stochastic neuron dynamics in the Martin-Siggia-Rose De Dominicis-Janssen (MSDRJ) 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 lends itself to direct physical interpretation, and the MSRDJ effective action, which is computationally advantageous. These results in particular allow the extension of the OM formalism to 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 (BMW) 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.