Self-consistent formulations for stochastic nonlinear neuronal dynamics

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 bifurcation theory cannot be applied. We formulate stochastic neuronal 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.

Equivalent to published version, including two minor typo fixes in Eq (6) and Fig 6. All conclusions unchanged. 7 figures

References in corpus (13)

• Exact evolution equation for the effective potential
• The large deviation approach to statistical mechanics
• Landau-Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization
• Two types of criticality in the brain
• Path Integral Approach to Random Neural Networks
• Dynamics of interacting particle systems: stochastic process and field theory
• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• Correlations, fluctuations and stability of a finite-size network of coupled oscillators
• Echoes in correlated neural systems
• Frequency regulators for the nonperturbative renormalization group: A general study and the model A as a benchmark
• Field theories and exact stochastic equations for interacting particle systems
• Truncated-Unity Parquet Equations: Application to the Repulsive Hubbard Model
• Finite size effects for spiking neural networks with spatially dependent coupling

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