Towards a field theory for neuronal networks

Mean-field descriptions of neuronal networks yield stabilityconstraints that guide efficient model development (Schuecker etal. 2017). Linear response theory allows the study of spontaneousfluctuations and responses to weak stimula (Bos et al. 2016).But existing approaches cannot be systematically extended beyond weakdepartures from stationarity, ignore non-linear apsects of neuronalinteraction, and describe only population-averagedactivities. Massively parallel recordings of neuronal activity,however, expose a large cell-to-cell variability outside the realm ofa population-level description.We here present recent developments towards a systematic theory offluctuating and correlated activity in neuronal networks.Leaving the population description, we present a cumulant expansionthat includes higher order statistics and non-linearities toaccurately predict mean activities and fluctuations on the level ofindividual neurons (Dahmen et al. 2016): It exposes the intrinsicallygenerated wide distribution of pairwise correlations and explains therich repertoire of network responses to external stimulations.The framework readily extends to the non-stationary setting, explainingthe temporal modulation of correlations in oscillatory network states(Kuehn et al. 2017).Moreover, the approach reduces the non-linear neuronal dynamics to aneffective stochastic differential equation, which enables theinvestigation by field theoretical methods (Schuecker et al. 2016):Treatment of disordered connectivity (Dahmen et al. 2016b, Dahmen etal. 2017 CNS), analysis of phase transitions (Goedeke et al. 2016,Schuecker et al. 2017b CNS), and systematic corrections to mean-fieldresults.Partially supported by the Helmholtz Association: Young investigator'sgroup VH-NG-1028, portfolio theme SMHB, the Jülich Aachen ResearchAlliance (JARA), and 604102 (HBP)