On volterra integral equations on time scales

The purpose of the present paper is to study some properties of solutions of Volterra integral equations on time scales. We generalize to a time scale some known properties concerning continuity and convergence of solutions from the continuous case.

Reference evapotranspiration models using different time scales in the Jaboticabal region of Sao Paulo, Brazil

The aim of this paper is to compare 18 reference evapotranspiration models to the standard Penman-Monteith model in the Jaboticabal, Sao Paulo, region for the following time scales: daily, 5-day, 15-day and seasonal. A total of 5 years of daily meteorological data was used for the following analyses: accuracy (mean absolute percentage error, Mape), precision (R-2) and tendency (bias) (systematic error, SE). The results were also compared at the 95% probability level with Tukey's test. The Priestley-Taylor (1972) method was the most accurate for all time scales, the Tanner-Pelton (1960) method was the most accurate in the winter, and the Thornthwaite (1948) method was the most accurate of the methods that only used temperature data in the equations.

Ordinary fractional differential equations are in fact usual entire ordinary differential equations on time scales

Despite the huge number of works considering fractional derivatives or derivatives on time scales some basic facts remain to be evaluated. Here we will be showing that the fractional derivative of monomials is in fact an entire derivative considered on an appropriate time scale.

Generalizing the dynamic field theory of spatial cognition across real and developmental time scales

Within cognitive neuroscience, computational models are designed to provide insights into the organization of behavior while adhering to neural principles. These models should provide sufficient specificity to generate novel predictions while maintaining the generality needed to capture behavior across tasks and/or time scales. This paper presents one such model, the Dynamic Field Theory (DFT) of spatial cognition, showing new simulations that provide a demonstration proof that the theory generalizes across developmental changes in performance in four tasks—the Piagetian A-not-B task, a sandbox version of the A-not-B task, a canonical spatial recall task, and a position discrimination task. Model simulations demonstrate that the DFT can accomplish both specificity—generating novel, testable predictions—and generality—spanning multiple tasks across development with a relatively simple developmental hypothesis. Critically, the DFT achieves generality across tasks and time scales with no modification to its basic structure and with a strong commitment to neural principles. The only change necessary to capture development in the model was an increase in the precision of the tuning of receptive fields as well as an increase in the precision of local excitatory interactions among neurons in the model. These small quantitative changes were sufficient to move the model through a set of quantitative and qualitative behavioral changes that span the age range from 8 months to 6 years and into adulthood. We conclude by considering how the DFT is positioned in the literature, the challenges on the horizon for our framework, and how a dynamic field approach can yield new insights into development from a computational cognitive neuroscience perspective.

Biogenic calcium phosphate transformation in soils over millennial time scales

Changes in bioavailability of phosphorus (P) during pedogenesis and ecosystem development have been shown for geogenic calcium phosphate (Ca-P). However, very little is known about long-term changes of biogenic Ca-P in soil. Long-term transformation characteristics of biogenic Ca-P were examined using anthropogenic soils along a chronosequence from centennial to millennial time scales. Phosphorus fractionation of Anthrosols resulted in overall consistency with the Walker and Syers model of geogenic Ca-P transformation during pedogenesis. The biogenic Ca-P (e.g., animal and fish bones) disappeared to 3% of total P within the first ca. 2,000 years of soil development. This change concurred with increases in P adsorbed on metal-oxides surfaces, organic P, and occluded P at different pedogenic time. Phosphorus K-edge X-ray absorption near-edge structure (XANES) spectroscopy revealed that the crystalline and therefore thermodynamically most stable biogenic Ca-P was transformed into more soluble forms of Ca-P over time. While crystalline hydroxyapatite (34% of total P) dominated Ca-P species after about 600-1,000 years, beta-tricalcium phosphate increased to 16% of total P after 900-1,100 years, after which both Ca-P species disappeared. Iron-associated P was observable concurrently with Ca-P disappearance. Soluble P and organic P determined by XANES maintained relatively constant (58-65%) across the time scale studied. Disappearance of crystalline biogenic Ca-P on a time scale of a few thousand years appears to be ten times faster than that of geogenic Ca-P.

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.

Polymer surfaces on small length- and short time-scales

The behaviour of a polymer depends strongly on the length- and time scale as well as on the temperature rnat which it is probed. In this work, I describe investigations of polymer surfaces using scanning probe rnmicroscopy with heatable probes. With these probes, surfaces can be heated within seconds down to rnmicroseconds. I introduce experiments for the local and fast determination of glass transition and melting rntemperatures. I developed a method which allows the determination of glass transition and melting rntemperatures on films with thicknesses below 100 nm: A background measurement on the substrate was rnperformed. The resulting curve was subtracted from the measurement on the polymer film. The rndifferential measurement on polystyrene films with thicknesses between 35 nm and 160 nm showed rncharacteristic signals at 95 ± 1 °C, in accordance with the glass transition of polystyrene. Pressing heated rnprobes into polymer films causes plastic deformation. Nanometer sized deformations are currently rninvestigated in novel concepts for high density data storage. A suitable medium for such a storage system rnhas to be easily indentable on one hand, but on the other hand it also has to be very stable towards rnsurface induced wear. For developing such a medium I investigated a new approach: A comparably soft rnmaterial, namely polystyrene, was protected with a thin but very hard layer made of plasma polymerized rnnorbornene. The resulting bilayered media were tested for surface stability and deformability. I showed rnthat the bilayered material combines the deformability of polystyrene with the surface stability of the rnplasma polymer, and that the material therefore is a very good storage medium. In addition we rninvestigated the glass transition temperature of polystyrene at timescales of 10 µs and found it to be rnapprox. 220 °C. The increase of this characteristic temperature of the polymer results from the short time rnat which the polymer was probed and reflects the well-known time-temperature superposition principle. rnHeatable probes were also used for the characterization of silverazide filled nanocapsules. The use of rnheatable probes allowed determining the decomposition temperature of the capsules from few rnnanograms of material. The measured decomposition temperatures ranged from 180 °C to 225 °C, in rnaccordance with literature values. The investigation of small amounts of sample was necessary due to the rnlimited availability of the material. Furthermore, investigating larger amounts of the capsules using rnconventional thermal gravimetric analysis could lead to contamination or even damage of the instrument. rnBesides the analysis of material parameters I used the heatable probes for the local thermal rndecomposition of pentacene precursor material in order to form nanoscale conductive structures. Here, rnthe thickness of the precursor layer was important for complete thermal decomposition. rnAnother aspect of my work was the investigation of redox active polymers - Poly-10-(4-vinylbenzyl)-10H-rnphenothiazine (PVBPT)- for data storage. Data is stored by changing the local conductivity of the material rnby applying a voltage between tip and surface. The generated structures were stable for more than 16 h. It rnwas shown that the presence of water is essential for succesfull patterning.

Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons

Neural dynamic processes correlated over several time scales are found in vivo, in stimulus-evoked as well as spontaneous activity, and are thought to affect the way sensory stimulation is processed. Despite their potential computational consequences, a systematic description of the presence of multiple time scales in single cortical neurons is lacking. In this study, we injected fast spiking and pyramidal (PYR) neurons in vitro with long-lasting episodes of step-like and noisy, in-vivo-like current. Several processes shaped the time course of the instantaneous spike frequency, which could be reduced to a small number (1-4) of phenomenological mechanisms, either reducing (adapting) or increasing (facilitating) the neuron's firing rate over time. The different adaptation/facilitation processes cover a wide range of time scales, ranging from initial adaptation (<10 ms, PYR neurons only), to fast adaptation (<300 ms), early facilitation (0.5-1 s, PYR only), and slow (or late) adaptation (order of seconds). These processes are characterized by broad distributions of their magnitudes and time constants across cells, showing that multiple time scales are at play in cortical neurons, even in response to stationary stimuli and in the presence of input fluctuations. These processes might be part of a cascade of processes responsible for the power-law behavior of adaptation observed in several preparations, and may have far-reaching computational consequences that have been recently described.

Brain connectivity at different time-scales measured with EEG

We present an overview of different methods for decomposing a multichannel spontaneous electroencephalogram (EEG) into sets of temporal patterns and topographic distributions. All of the methods presented here consider the scalp electric field as the basic analysis entity in space. In time, the resolution of the methods is between milliseconds (time-domain analysis), subseconds (time- and frequency-domain analysis) and seconds (frequency-domain analysis). For any of these methods, we show that large parts of the data can be explained by a small number of topographic distributions. Physically, this implies that the brain regions that generated one of those topographies must have been active with a common phase. If several brain regions are producing EEG signals at the same time and frequency, they have a strong tendency to do this in a synchronized mode. This view is illustrated by several examples (including combined EEG and functional magnetic resonance imaging (fMRI)) and a selective review of the literature. The findings are discussed in terms of short-lasting binding between different brain regions through synchronized oscillations, which could constitute a mechanism to form transient, functional neurocognitive networks.

Linking brain connectivity across different time scales with electroencephalogram, functional magnetic resonance imaging, and diffusion tensor imaging

Structural and functional connectivity are intrinsic properties of the human brain and represent the amount of cognitive capacities of individual subjects. These connections are modulated due to development, learning, and disease. Momentary adaptations in functional connectivity alter the structural connections, which in turn affect the functional connectivity. Thus, structural and functional connectivity interact on a broad timescale. In this study, we aimed to explore distinct measures of connectivity assessed by functional magnetic resonance imaging and diffusion tensor imaging and their association to the dominant electroencephalogram oscillatory property at rest: the individual alpha frequency (IAF). We found that in 21 healthy young subjects, small intraindividual temporal IAF fluctuations were correlated to increased blood oxygenation level-dependent signal in brain areas associated to working memory functions and to the modulation of attention. These areas colocalized with functionally connected networks supporting the respective functions. Furthermore, subjects with higher IAF show increased fractional anisotropy values in fascicles connecting the above-mentioned areas and networks. Hence, due to a multimodal approach a consistent functionally and structurally connected network related to IAF was observed.

Resolving diffusion in clay minerals at different time scales: Combination of experimental and modeling approaches

Since no single experimental or modeling technique provides data that allow a description of transport processes in clays and clay minerals at all relevant scales, several complementary approaches have to be combined to understand and explain the interplay between transport relevant phenomena. In this paper molecular dynamics simulations (MD) were used to investigate the mobility of water in the interlayer of montmorillonite (Mt), and to estimate the influence of mineral surfaces and interlayer ions on the water diffusion. Random Walk (RW) simulations based on a simplified representation of pore space in Mt were used to estimate and understand the effect of the arrangement of Mt particles on the meso- to macroscopic diffusivity of water. These theoretical calculations were complemented with quasielastic neutron scattering (QENS) measurements of aqueous diffusion in Mt with two pseudo-layers of water performed at four significantly different energy resolutions (i.e. observation times). The size of the interlayer and the size of Mt particles are two characteristic dimensions which determine the time dependent behavior of water diffusion in Mt. MD simulations show that at very short time scales water dynamics has the characteristic features of an oscillatory motion in the cage formed by neighbors in the first coordination shell. At longer time scales, the interaction of water with the surface determines the water dynamics, and the effect of confinement on the overall water mobility within the interlayer becomes evident. At time scales corresponding to an average water displacement equivalent to the average size of Mt particles, the effects of tortuosity are observed in the meso- to macroscopic pore scale simulations. Consistent with the picture obtained in the simulations, the QENS data can be described using a (local) 3D diffusion at short observation times, whereas at sufficiently long observation times a 2D diffusive motion is clearly observed. The effects of tortuosity measured in macroscopic tracer diffusion experiments are in qualitative agreement with RW simulations. By using experimental data to calibrate molecular and mesoscopic theoretical models, a consistent description of water mobility in clay minerals from the molecular to the macroscopic scale can be achieved. In turn, simulations help in choosing optimal conditions for the experimental measurements and the data interpretation. (C) 2014 Elsevier B.V. All rights reserved.

Time scales in cognitive neuroscience

Cognitive neuroscience boils down to describing the ways in which cognitive function results from brain activity. In turn, brain activity shows complex fluctuations, with structure at many spatio-temporal scales. Exactly how cognitive function inherits the physical dimensions of neural activity, though, is highly non-trivial, and so are generally the corresponding dimensions of cognitive phenomena. As for any physical phenomenon, when studying cognitive function, the first conceptual step should be that of establishing its dimensions. Here, we provide a systematic presentation of the temporal aspects of task-related brain activity, from the smallest scale of the brain imaging technique's resolution, to the observation time of a given experiment, through the characteristic time scales of the process under study. We first review some standard assumptions on the temporal scales of cognitive function. In spite of their general use, these assumptions hold true to a high degree of approximation for many cognitive (viz. fast perceptual) processes, but have their limitations for other ones (e.g., thinking or reasoning). We define in a rigorous way the temporal quantifiers of cognition at all scales, and illustrate how they qualitatively vary as a function of the properties of the cognitive process under study. We propose that each phenomenon should be approached with its own set of theoretical, methodological and analytical tools. In particular, we show that when treating cognitive processes such as thinking or reasoning, complex properties of ongoing brain activity, which can be drastically simplified when considering fast (e.g., perceptual) processes, start playing a major role, and not only characterize the temporal properties of task-related brain activity, but also determine the conditions for proper observation of the phenomena. Finally, some implications on the design of experiments, data analyses, and the choice of recording parameters are discussed.