358 research outputs found
Finite-size and correlation-induced effects in Mean-field Dynamics
The brain's activity is characterized by the interaction of a very large
number of neurons that are strongly affected by noise. However, signals often
arise at macroscopic scales integrating the effect of many neurons into a
reliable pattern of activity. In order to study such large neuronal assemblies,
one is often led to derive mean-field limits summarizing the effect of the
interaction of a large number of neurons into an effective signal. Classical
mean-field approaches consider the evolution of a deterministic variable, the
mean activity, thus neglecting the stochastic nature of neural behavior. In
this article, we build upon two recent approaches that include correlations and
higher order moments in mean-field equations, and study how these stochastic
effects influence the solutions of the mean-field equations, both in the limit
of an infinite number of neurons and for large yet finite networks. We
introduce a new model, the infinite model, which arises from both equations by
a rescaling of the variables and, which is invertible for finite-size networks,
and hence, provides equivalent equations to those previously derived models.
The study of this model allows us to understand qualitative behavior of such
large-scale networks. We show that, though the solutions of the deterministic
mean-field equation constitute uncorrelated solutions of the new mean-field
equations, the stability properties of limit cycles are modified by the
presence of correlations, and additional non-trivial behaviors including
periodic orbits appear when there were none in the mean field. The origin of
all these behaviors is then explored in finite-size networks where interesting
mesoscopic scale effects appear. This study leads us to show that the
infinite-size system appears as a singular limit of the network equations, and
for any finite network, the system will differ from the infinite system
A scalable computational framework for establishing long-term behavior of stochastic reaction networks
Reaction networks are systems in which the populations of a finite number of
species evolve through predefined interactions. Such networks are found as
modeling tools in many biological disciplines such as biochemistry, ecology,
epidemiology, immunology, systems biology and synthetic biology. It is now
well-established that, for small population sizes, stochastic models for
biochemical reaction networks are necessary to capture randomness in the
interactions. The tools for analyzing such models, however, still lag far
behind their deterministic counterparts. In this paper, we bridge this gap by
developing a constructive framework for examining the long-term behavior and
stability properties of the reaction dynamics in a stochastic setting. In
particular, we address the problems of determining ergodicity of the reaction
dynamics, which is analogous to having a globally attracting fixed point for
deterministic dynamics. We also examine when the statistical moments of the
underlying process remain bounded with time and when they converge to their
steady state values. The framework we develop relies on a blend of ideas from
probability theory, linear algebra and optimization theory. We demonstrate that
the stability properties of a wide class of biological networks can be assessed
from our sufficient theoretical conditions that can be recast as efficient and
scalable linear programs, well-known for their tractability. It is notably
shown that the computational complexity is often linear in the number of
species. We illustrate the validity, the efficiency and the wide applicability
of our results on several reaction networks arising in biochemistry, systems
biology, epidemiology and ecology. The biological implications of the results
as well as an example of a non-ergodic biological network are also discussed.Comment: 31 pages, 9 figure
Size distributions of shocks and static avalanches from the Functional Renormalization Group
Interfaces pinned by quenched disorder are often used to model jerky
self-organized critical motion. We study static avalanches, or shocks, defined
here as jumps between distinct global minima upon changing an external field.
We show how the full statistics of these jumps is encoded in the
functional-renormalization-group fixed-point functions. This allows us to
obtain the size distribution P(S) of static avalanches in an expansion in the
internal dimension d of the interface. Near and above d=4 this yields the
mean-field distribution P(S) ~ S^(-3/2) exp(-S/[4 S_m]) where S_m is a
large-scale cutoff, in some cases calculable. Resumming all 1-loop
contributions, we find P(S) ~ S^(-tau) exp(C (S/S_m)^(1/2) -B/4 (S/S_m)^delta)
where B, C, delta, tau are obtained to first order in epsilon=4-d. Our result
is consistent to O(epsilon) with the relation tau = 2-2/(d+zeta), where zeta is
the static roughness exponent, often conjectured to hold at depinning. Our
calculation applies to all static universality classes, including random-bond,
random-field and random-periodic disorder. Extended to long-range elastic
systems, it yields a different size distribution for the case of contact-line
elasticity, with an exponent compatible with tau=2-1/(d+zeta) to
O(epsilon=2-d). We discuss consequences for avalanches at depinning and for
sandpile models, relations to Burgers turbulence and the possibility that the
above relations for tau be violated to higher loop order. Finally, we show that
the avalanche-size distribution on a hyper-plane of co-dimension one is in
mean-field (valid close to and above d=4) given by P(S) ~ K_{1/3}(S)/S, where K
is the Bessel-K function, thus tau=4/3 for the hyper plane.Comment: 34 pages, 30 figure
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