38,353 research outputs found
Gaussian estimates for a heat equation on a network
We consider a diffusion problem on a network on whose nodes we impose
Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove
well-posedness of the associated initial value problem, and we exploit the
theory of sub-Markovian and ultracontractive semigroups in order to obtain
upper Gaussian estimates for the integral kernel. We conclude that the same
diffusion problem is governed by an analytic semigroup acting on all -type
spaces as well as on suitable spaces of continuous functions. Stability and
spectral issues are also discussed. As an application we discuss a system of
semilinear equations on a network related to potential transmission problems
arising in neurobiology.Comment: In comparison with the already published version of this paper (Netw.
Het. Media 2 (2007), 55-79), a small gap in the proof of Proposition 3.2 has
been fille
Signatures of criticality arise in simple neural population models with correlations
Large-scale recordings of neuronal activity make it possible to gain insights
into the collective activity of neural ensembles. It has been hypothesized that
neural populations might be optimized to operate at a 'thermodynamic critical
point', and that this property has implications for information processing.
Support for this notion has come from a series of studies which identified
statistical signatures of criticality in the ensemble activity of retinal
ganglion cells. What are the underlying mechanisms that give rise to these
observations? Here we show that signatures of criticality arise even in simple
feed-forward models of retinal population activity. In particular, they occur
whenever neural population data exhibits correlations, and is randomly
sub-sampled during data analysis. These results show that signatures of
criticality are not necessarily indicative of an optimized coding strategy, and
challenge the utility of analysis approaches based on equilibrium
thermodynamics for understanding partially observed biological systems.Comment: 36 pages, LaTeX; added journal reference on page 1, added link to
code repositor
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