|
Abstract |
Neuronal networks are many particle systems with interesting physical
properties: They operate far from thermodynamic equilibrium and show
correlated states of collective activity that result from the
interaction of large numbers of relatively simple units [1]. We here
present recent progress towards a quantitative understanding of such
systems by application of non-equilibrium statistical mechanics.
Mean-field theory and linear response theory capture many qualitative
properties of the "ground state" of recurrent networks [2]. A
fundamental quantity required is the single neuron transfer function.
Formally, it constitutes an escape problem driven by colored noise.
We recently applied boundary layer theory to obtain a reduction to the
technically much simpler white noise problem [3]. It allows us, for
example, to formulate a theory of finite-size fluctuations in layered
neuronal networks [4]. Verification of such theoretical predictions
is fundamentally hindered by sub-sampling: We only see a tiny fraction
of all neurons within the living brain at a time. Employing tools
from disordered systems (spin glasses) combined with an auxiliary
field formulation, we overcome this issue by deriving a mean-field
theory that is valid beyond the commonly-made self-averaging
assumption. It predicts that the heterogeneity of the network
connectivity enables a novel sort of critical dynamics which unfolds
in a low-dimensional subspace [5]. The functional consequences are
analyzed by importing tools from field theory of stochastic
differential equations. We obtain closed-form expressions for the
transition to chaos and for the sequential memory capacity of the
network by help of replica calculations [6]. We find that cortical
networks operate in a hitherto unreported regime that combines
instability on short time scales with asymptotically non-chaotic
dynamics; a regime which has optimal memory capacity.
As an outlook we present two directions in which field-theoretical
methods enable insights into network dynamics: First, a novel
diagrammatic expansion of the effective action around non-Gaussian
solvable theories [7]; we exemplify this method by finally providing
the long-searched for diagrammatic formulation of the
Thouless-Anderson-Palmer mean-field theory of the Ising model.
Second, the application of the functional renormaliztion group to
neuronal dynamics [8]. It enables the systematic study of second order
phase transitions in such networks. |
