Logarithmic diffusion and porous media equations: a unified description

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a lorentzian form, consequently this equation characterizes a super diffusion like a L\'evy kind. In addition is obtained an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.Comment: 5 pages. To appear in Phys. Rev.

Existence of radial stationary solutions for a system in combustion theory

In this paper, we construct radially symmetric solutions of a nonlinear noncooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory

Streamer Propagation as a Pattern Formation Problem: Planar Fronts

Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a non-ionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations.Comment: 4 pages, revtex, 3 ps file

Finite size effects near the onset of the oscillatory instability

A system of two complex Ginzburg - Landau equations is considered that applies at the onset of the oscillatory instability in spatial domains whose size is large (but finite) in one direction; the dependent variables are the slowly modulated complex amplitudes of two counterpropagating wavetrains. In order to obtain a well posed problem, four boundary conditions must be imposed at the boundaries. Two of them were already known, and the other two are first derived in this paper. In the generic case when the group velocity is of order unity, the resulting problem has terms that are not of the same order of magnitude. This fact allows us to consider two distinguished limits and to derive two associated (simpler) sub-models, that are briefly discussed. Our results predict quite a rich variety of complex dynamics that is due to both the modulational instability and finite size effects

Mapping the Spatio-Temporal Pattern of the Mammalian Target of Rapamycin (mTOR) Activation in Temporal Lobe Epilepsy

Growing evidence from rodent models of temporal lobe epilepsy (TLE) indicates that dysregulation of the mammalian target of rapamycin (mTOR) pathway is involved in seizures and epileptogenesis. However, the role of the mTOR pathway in the epileptogenic process remains poorly understood. Here, we used an animal model of TLE and sclerotic hippocampus from patients with refractory TLE to determine whether cell-type specific activation of mTOR signaling occurs during each stage of epileptogenesis. In the TLE mouse model, we found that hyperactivation of the mTOR pathway is present in distinct hippocampal subfields at three different stages after kainate-induced seizures, and occurs in neurons of the granular and pyramidal cell layers, in reactive astrocytes, and in dispersed granule cells, respectively. In agreement with the findings in TLE mice, upregulated mTOR was observed in the sclerotic hippocampus of TLE patients. All sclerotic hippocampus (n = 13) exhibited widespread reactive astrocytes with overactivated mTOR, some of which invaded the dispersed granular layer. Moreover, two sclerotic hippocampus exhibited mTOR activation in some of the granule cells, which was accompanied by cell body hypertrophy. Taken together, our results indicate that mTOR activation is most prominent in reactive astrocytes in both an animal model of TLE and the sclerotic hippocampus from patients with drug resistant TLE

Anticipating and Managing Future Trade-offs and Complementarities between Ecosystem Services

This paper shows how, with the aid of computer models developed in close collaboration with decision makers and other stakeholders, it is possible to quantify and map how policy decisions are likely to affect multiple ecosystem services in future. In this way, potential trade-offs and complementarities between different ecosystem services can be identified, so that policies can be designed to avoid the worst trade-offs, and where possible, enhance multiple services. The paper brings together evidence from across the Rural Economy and Land Use Programme’s Sustainable Uplands project for the first time, with previously unpublished model outputs relating to runoff, agricultural suitability, biomass, heather cover, age, and utility for Red Grouse (Lagopus scotica), grass cover, and accompanying scenario narratives and video. Two contrasting scenarios, based on policies to extensify or intensify land management up to 2030, were developed through a combination of interviews and discussions during site visits with stakeholders, literature review, conceptual modeling, and process-based computer models, using the Dark Peak of the Peak District National Park in the UK as a case study. Where extensification leads to a significant reduction in managed burning and grazing or land abandonment, changes in vegetation type and structure could compromise a range of species that are important for conservation, while compromising provisioning services, amenity value, and increasing wildfire risk. However, where extensification leads to the restoration of peatlands damaged by former intensive management, there would be an increase in carbon sequestration and storage, with a number of cobenefits, which could counter the loss of habitats and species elsewhere in the landscape. In the second scenario, land use and management was significantly intensified to boost UK self-sufficiency in food. This would benefit certain provisioning services but would have negative consequences for carbon storage and water quality and would lead to a reduction in the abundance of certain species of conservation concern. The paper emphasizes the need for spatially explicit models that can track how ecosystem services might change over time, in response to policy or environmental drivers, and in response to the changing demands and preferences of society, which are far harder to anticipate. By developing such models in close collaboration with decision makers and other stakeholders, it is possible to depict scenarios of real concern to those who need to use the research findings. By engaging these collaborators with the research findings through film, it was possible to discuss adaptive options to minimize trade-offs and enhance the provision of multiple ecosystem services under the very different future conditions depicted by each scenario. By preparing for as wide a range of futures as possible in this way, it may be possible for decision makers to act rapidly and effectively to protect and enhance the provision of ecosystem services in the face of unpredictable future change.Additional co-authors: Nanlin Jin, Brian J Irvine, Mike J Kirkby, William E Kunin, Christina Prell, Claire H Quinn, Bill Slee, Sigrid Stagl, Mette Termansen, Simon Thorp, and Fred Worral

Single-neuron dynamics in human focal epilepsy

Epileptic seizures are traditionally characterized as the ultimate expression of monolithic, hypersynchronous neuronal activity arising from unbalanced runaway excitation. Here we report the first examination of spike train patterns in large ensembles of single neurons during seizures in persons with epilepsy. Contrary to the traditional view, neuronal spiking activity during seizure initiation and spread was highly heterogeneous, not hypersynchronous, suggesting complex interactions among different neuronal groups even at the spatial scale of small cortical patches. In contrast to earlier stages, seizure termination is a nearly homogenous phenomenon followed by an almost complete cessation of spiking across recorded neuronal ensembles. Notably, even neurons outside the region of seizure onset showed significant changes in activity minutes before the seizure. These findings suggest a revision of current thinking about seizure mechanisms and point to the possibility of seizure prevention based on spiking activity in neocortical neurons

The Interaction between Early Life Epilepsy and Autistic-Like Behavioral Consequences: A Role for the Mammalian Target of Rapamycin (mTOR) Pathway

Early life seizures can result in chronic epilepsy, cognitive deficits and behavioral changes such as autism, and conversely epilepsy is common in autistic children. We hypothesized that during early brain development, seizures could alter regulators of synaptic development and underlie the interaction between epilepsy and autism. The mammalian Target of Rapamycin (mTOR) modulates protein translation and is dysregulated in Tuberous Sclerosis Complex, a disorder characterized by epilepsy and autism. We used a rodent model of acute hypoxia-induced neonatal seizures that results in long term increases in neuronal excitability, seizure susceptibility, and spontaneous seizures, to determine how seizures alter mTOR Complex 1 (mTORC1) signaling. We hypothesized that seizures occurring at a developmental stage coinciding with a critical period of synaptogenesis will activate mTORC1, contributing to epileptic networks and autistic-like behavior in later life. Here we show that in the rat, baseline mTORC1 activation peaks during the first three postnatal weeks, and induction of seizures at postnatal day 10 results in further transient activation of its downstream targets phospho-4E-BP1 (Thr37/46), phospho-p70S6K (Thr389) and phospho-S6 (Ser235/236), as well as rapid induction of activity-dependent upstream signaling molecules, including BDNF, phospho-Akt (Thr308) and phospho-ERK (Thr202/Tyr204). Furthermore, treatment with the mTORC1 inhibitor rapamycin immediately before and after seizures reversed early increases in glutamatergic neurotransmission and seizure susceptibility and attenuated later life epilepsy and autistic-like behavior. Together, these findings suggest that in the developing brain the mTORC1 signaling pathway is involved in epileptogenesis and altered social behavior, and that it may be a target for development of novel therapies that eliminate the progressive effects of neonatal seizures