  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, M\'endez 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.
