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Advanced theoretical approaches to collective network phenomena (Helias, Farkhooi)

Abstract

Collective phenomena  are the heart of many dynamical properties  of neuronal networks: neuronal avalanches at critical states, global oscillations, activity waves in the local field potential, correlated activity between pairs of cells in massively parallel recordings. Yet, they arise only from the concerted Interaction of many elements and are hence inherently challenging to describe and understand mathematically. Systems of similar complexity as neuronal networks have with big success been investigated in other fields of science, such as many particle physics, field theory, theory of disordered  systems. Also mathematical  statistics has developed powerful tools to describe stochastic processes that often form the core of neuronal network models.

We here compose a workshop of speakers who have applied methods from mathematics and statistical physics to foster our understanding  of such collective states, for example using large deviation techniques, theory of disordered systems, martingales, field theory, the renormalization group and methods for many particle systems and critical phenomena. The workshop aims to reflect on and discuss the current state of these endeavors. in particular we want to point out strengths, weaknesses, and domains of application of the respective approaches, highlight recent trends and pinpoint the experimental phenomena  that challenge existing theoretical  models.

Recent years witnessed increased understanding of weakly correlated, stationary and oscillatory states by help of conventional perturbative methods and mean-field theory. But only rather recently, network activity has been formulated in the language of statistical field theory and path Integrals [reviewed in e.g. Chow & Buice 2015, Hertz, Sollich, Roudl 2016], opening up new routes to apply powerful methods from statistical physics, field theory, and mathematics to the dynamics of neuronal networks. in particular, these tools provide means to describe neural activity beyond the population Ievel and allow Interpretations of the heterogeneity across neurons in massively parallel spike data that can nowadays be obtained by multi-electrode recordings. We now need a phase of active knowledge transfer to make maximal use of these analogies to foster progress in computational neuroscience on the challenging problem of the Interaction between large numbers of neurons.

Schedule