268 research outputs found
Transverse Meissner Physics of Planar Superconductors with Columnar Pins
The statistical mechanics of thermally excited vortex lines with columnar
defects can be mapped onto the physics of interacting quantum particles with
quenched random disorder in one less dimension. The destruction of the Bose
glass phase in Type II superconductors, when the external magnetic field is
tilted sufficiently far from the column direction, is described by a poorly
understood non-Hermitian quantum phase transition. We present here exact
results for this transition in (1+1)-dimensions, obtained by mapping the
problem in the hard core limit onto one-dimensional fermions described by a
non-Hermitian tight binding model. Both site randomness and the relatively
unexplored case of bond randomness are considered. Analysis near the mobility
edge and near the band center in the latter case is facilitated by a real space
renormalization group procedure used previously for Hermitian quantum problems
with quenched randomness in one dimension.Comment: 23 pages, 22 figure
Beyond-mean-field theory for the statistics of neural coordination
Understanding the coordination structure of neurons in neuronal networks is
essential for unraveling the distributed information processing mechanisms in
brain networks. Recent advancements in measurement techniques have resulted in
an increasing amount of data on neural activities recorded in parallel,
revealing largely heterogeneous correlation patterns across neurons. Yet, the
mechanistic origin of this heterogeneity is largely unknown because existing
theoretical approaches linking structure and dynamics in neural circuits are
mostly restricted to average connection patterns. Here we present a systematic
inclusion of variability in network connectivity via tools from statistical
physics of disordered systems. We study networks of spiking leaky
integrate-and-fire neurons and employ mean-field and linear-response methods to
map the spiking networks to linear rate models with an equivalent
neuron-resolved correlation structure. The latter models can be formulated in a
field-theoretic language that allows using disorder-average and replica
techniques to systematically derive quantitatively matching beyond-mean-field
predictions for the mean and variance of cross-covariances as functions of the
average and variability of connection patterns. We show that heterogeneity in
covariances is not a result of variability in single-neuron firing statistics
but stems from the sparse realization and variable strength of connections, as
ubiquitously observed in brain networks. Average correlations between neurons
are found to be insensitive to the level of heterogeneity, which in contrast
modulates the variability of covariances across many orders of magnitude,
giving rise to an efficient tuning of the complexity of coordination patterns
in neuronal circuits
Hidden connectivity structures control collective network dynamics
A common approach to model local neural circuits is to assume random
connectivity. But how is our choice of randomness informed by known network
properties? And how does it affect the network's behavior? Previous approaches
have focused on prescribing increasingly sophisticated statistics of synaptic
strengths and motifs. However, at the same time experimental data on parallel
dynamics of neurons is readily accessible. We therefore propose a complementary
approach, specifying connectivity in the space that directly controls the
dynamics - the space of eigenmodes. We develop a theory for a novel ensemble of
large random matrices, whose eigenvalue distribution can be chosen arbitrarily.
We show analytically how varying such distribution induces a diverse range of
collective network behaviors, including power laws that characterize the
dimensionality, principal components spectrum, autocorrelation, and
autoresponse of neuronal activity. The power-law exponents are controlled by
the density of nearly critical eigenvalues, and provide a minimal and robust
measure to directly link observable dynamics and connectivity. The density of
nearly critical modes also characterizes a transition from high to low
dimensional dynamics, while their maximum oscillation frequency determines a
transition from an exponential to power-law decay in time of the correlation
and response functions. We prove that the wide range of dynamical behaviors
resulting from the proposed connectivity ensemble is caused by structures that
are invisible to a motif analysis. Their presence is captured by motifs
appearing with vanishingly small probability in the number of neurons. Only
reciprocal motifs occur with finite probability. In other words, a motif
analysis can be blind to synaptic structures controlling the dynamics, which
instead become apparent in the space of eigenmode statistics
Integration of continuous-time dynamics in a spiking neural network simulator
Contemporary modeling approaches to the dynamics of neural networks consider
two main classes of models: biologically grounded spiking neurons and
functionally inspired rate-based units. The unified simulation framework
presented here supports the combination of the two for multi-scale modeling
approaches, the quantitative validation of mean-field approaches by spiking
network simulations, and an increase in reliability by usage of the same
simulation code and the same network model specifications for both model
classes. While most efficient spiking simulations rely on the communication of
discrete events, rate models require time-continuous interactions between
neurons. Exploiting the conceptual similarity to the inclusion of gap junctions
in spiking network simulations, we arrive at a reference implementation of
instantaneous and delayed interactions between rate-based models in a spiking
network simulator. The separation of rate dynamics from the general connection
and communication infrastructure ensures flexibility of the framework. We
further demonstrate the broad applicability of the framework by considering
various examples from the literature ranging from random networks to neural
field models. The study provides the prerequisite for interactions between
rate-based and spiking models in a joint simulation
Learning Interacting Theories from Data
One challenge of physics is to explain how collective properties arise from
microscopic interactions. Indeed, interactions form the building blocks of
almost all physical theories and are described by polynomial terms in the
action. The traditional approach is to derive these terms from elementary
processes and then use the resulting model to make predictions for the entire
system. But what if the underlying processes are unknown? Can we reverse the
approach and learn the microscopic action by observing the entire system? We
use invertible neural networks (INNs) to first learn the observed data
distribution. By the choice of a suitable nonlinearity for the neuronal
activation function, we are then able to compute the action from the weights of
the trained model; a diagrammatic language expresses the change of the action
from layer to layer. This process uncovers how the network hierarchically
constructs interactions via nonlinear transformations of pairwise relations. We
test this approach on simulated data sets of interacting theories. The network
consistently reproduces a broad class of unimodal distributions; outside this
class, it finds effective theories that approximate the data statistics up to
the third cumulant. We explicitly show how network depth and data quantity
jointly improve the agreement between the learned and the true model. This work
shows how to leverage the power of machine learning to transparently extract
microscopic models from data
Asymptotic Dynamics in Quantum Field Theory
A crucial element of scattering theory and the LSZ reduction formula is the
assumption that the coupling vanishes at large times. This is known not to hold
for the theories of the Standard Model and in general such asymptotic dynamics
is not well understood. We give a description of asymptotic dynamics in field
theories which incorporates the important features of weak convergence and
physical boundary conditions. Applications to theories with three and four
point interactions are presented and the results are shown to be completely
consistent with the results of perturbation theory.Comment: 18 pages, 3 figure
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