19 research outputs found
Well-Posedness and Symmetries of Strongly Coupled Network Equations
We consider a diffusion process on the edges of a finite network and allow
for feedback effects between different, possibly non-adjacent edges. This
generalizes the setting that is common in the literature, where the only
considered interactions take place at the boundary, i. e., in the nodes of the
network. We discuss well-posedness of the associated initial value problem as
well as contractivity and positivity properties of its solutions. Finally, we
discuss qualitative properties that can be formulated in terms of invariance of
linear subspaces of the state space, i. e., of symmetries of the associated
physical system. Applications to a neurobiological model as well as to a system
of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change
Non-equilibrium dynamics of stochastic point processes with refractoriness
Stochastic point processes with refractoriness appear frequently in the
quantitative analysis of physical and biological systems, such as the
generation of action potentials by nerve cells, the release and reuptake of
vesicles at a synapse, and the counting of particles by detector devices. Here
we present an extension of renewal theory to describe ensembles of point
processes with time varying input. This is made possible by a representation in
terms of occupation numbers of two states: Active and refractory. The dynamics
of these occupation numbers follows a distributed delay differential equation.
In particular, our theory enables us to uncover the effect of refractoriness on
the time-dependent rate of an ensemble of encoding point processes in response
to modulation of the input. We present exact solutions that demonstrate generic
features, such as stochastic transients and oscillations in the step response
as well as resonances, phase jumps and frequency doubling in the transfer of
periodic signals. We show that a large class of renewal processes can indeed be
regarded as special cases of the model we analyze. Hence our approach
represents a widely applicable framework to define and analyze non-stationary
renewal processes.Comment: 8 pages, 4 figure
Laplacians with point interactions -- expected and unexpected spectral properties
We study the one-dimensional Laplace operator with point interactions on the
real line identified with two copies of the half-line . All
possible boundary conditions that define generators of -semigroups on
are characterized.
Here, the Cayley transform of the boundary conditions plays an important role
and using an explicit representation of the Green's functions, it allows us to
study invariance properties of semigroups
Discovering universal statistical laws of complex networks
Different network models have been suggested for the topology underlying
complex interactions in natural systems. These models are aimed at replicating
specific statistical features encountered in real-world networks. However, it
is rarely considered to which degree the results obtained for one particular
network class can be extrapolated to real-world networks. We address this issue
by comparing different classical and more recently developed network models
with respect to their generalisation power, which we identify with large
structural variability and absence of constraints imposed by the construction
scheme. After having identified the most variable networks, we address the
issue of which constraints are common to all network classes and are thus
suitable candidates for being generic statistical laws of complex networks. In
fact, we find that generic, not model-related dependencies between different
network characteristics do exist. This allows, for instance, to infer global
features from local ones using regression models trained on networks with high
generalisation power. Our results confirm and extend previous findings
regarding the synchronisation properties of neural networks. Our method seems
especially relevant for large networks, which are difficult to map completely,
like the neural networks in the brain. The structure of such large networks
cannot be fully sampled with the present technology. Our approach provides a
method to estimate global properties of under-sampled networks with good
approximation. Finally, we demonstrate on three different data sets (C.
elegans' neuronal network, R. prowazekii's metabolic network, and a network of
synonyms extracted from Roget's Thesaurus) that real-world networks have
statistical relations compatible with those obtained using regression models
How Structure Determines Correlations in Neuronal Networks
Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks