Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations.

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

Noise induced aperiodic rotations of particles trapped by a non-conservative force.

We describe a mechanism whereby random noise can play a constructive role in the manifestation of a pattern, aperiodic rotations, that would otherwise be damped by internal dynamics. The mechanism is described physically in a theoretical model of overdamped particle motion in two dimensions with symmetric damping and a non-conservative force field driven by noise. Cyclic motion only occurs as a result of stochastic noise in this system. However, the persistence of the cyclic motion is quantified by parameters associated with the non-conservative forcing. Unlike stochastic resonance or coherence resonance, where noise can play a constructive role in amplifying a signal that is otherwise below the threshold for detection, in the mechanism considered here, the signal that is detected does not exist without the noise. Moreover, the system described here is a linear system.

Predicting first traversal times for virions and nanoparticles in mucus with slowed diffusion.

Particle-tracking experiments focusing on virions or nanoparticles in mucus have measured mean-square displacements and reported diffusion coefficients that are orders of magnitude smaller than the diffusion coefficients of such particles in water. Accurate description of this subdiffusion is important to properly estimate the likelihood of virions traversing the mucus boundary layer and infecting cells in the epithelium. However, there are several candidate models for diffusion that can fit experimental measurements of mean-square displacements. We show that these models yield very different estimates for the time taken for subdiffusive virions to traverse through a mucus layer. We explain why fits of subdiffusive mean-square displacements to standard diffusion models may be misleading. Relevant to human immunodeficiency virus infection, using computational methods for fractional subdiffusion, we show that subdiffusion in normal acidic mucus provides a more effective barrier against infection than previously thought. By contrast, the neutralization of the mucus by alkaline semen, after sexual intercourse, allows virions to cross the mucus layer and reach the epithelium in a short timeframe. The computed barrier protection from fractional subdiffusion is some orders of magnitude greater than that derived by fitting standard models of diffusion to subdiffusive data.

The electrotonic structure of pyramidal neurons contributing to prefrontal cortical circuits in macaque monkeys is significantly altered in aging.

Whereas neuronal numbers are largely preserved in normal aging, subtle morphological changes occur in dendrites and spines, whose electrotonic consequences remain unexplored. We examined age-related morphological alterations in 2 types of pyramidal neurons contributing to working memory circuits in the macaque prefrontal cortex (PFC): neurons in the superior temporal cortex forming "long" projections to the PFC and "local" projection neurons within the PFC. Global dendritic mass homeostasis, measured by 3-dimensional scaling analysis, was conserved with aging in both neuron types. Spine densities, dendrite diameters, lengths, and branching complexity were all significantly reduced in apical dendrites of long projection neurons with aging, but only spine parameters were altered in local projection neurons. Despite these differences, voltage attenuation due to passive electrotonic structure, assuming equivalent cable parameters, was significantly reduced with aging in the apical dendrites of both neuron classes. Confirming the electrotonic analysis, simulated passive backpropagating action potential efficacy was significantly higher in apical but not basal dendrites of old neurons. Unless compensated by changes in passive cable parameters, active membrane properties, or altered synaptic properties, these effects will increase the excitability of pyramidal neurons, compromising the precisely tuned activity required for working memory, ultimately resulting in age-related PFC dysfunction.

Dynamics of chromatic visual system processing differ in complexity between children and adults.

Measures of chromatic contrast sensitivity in children are lower than those of adults. This may be related to immaturities in signal processing at or near threshold. We have found that children's VEPs in response to low contrast supra-threshold chromatic stimuli are more intra-individually variable than those recorded from adults. Here, we report on linear and nonlinear analyses of chromatic VEPs recorded from children and adults. Two measures of signal-to-noise ratio are similar between the adults and children, suggesting that relatively high noise is unlikely to account for the poor clarity of negative and positive peak components in the children's VEPs. Nonlinear analysis indicates higher complexity of adults' than children's chromatic VEPs, at levels of chromatic contrast around and well above threshold.

The correlation dimension: a useful objective measure of the transient visual evoked potential?

Visual evoked potentials (VEPs) may be analyzed by examination of the morphology of their components, such as negative (N) and positive (P) peaks. However, methods that rely on component identification may be unreliable when dealing with responses of complex and variable morphology; therefore, objective methods are also useful. One potentially useful measure of the VEP is the correlation dimension. Its relevance to the visual system was investigated by examining its behavior when applied to the transient VEP in response to a range of chromatic contrasts (42%, two times psychophysical threshold, at psychophysical threshold) and to the visually unevoked response (zero contrast). Tests of nonlinearity (e.g., surrogate testing) were conducted. The correlation dimension was found to be negatively correlated with a stimulus property (chromatic contrast) and a known linear measure (the Fourier-derived VEP amplitude). It was also found to be related to visibility and perception of the stimulus such that the dimension reached a maximum for most of the participants at psychophysical threshold. The latter suggests that the correlation dimension may be useful as a diagnostic parameter to estimate psychophysical threshold and may find application in the objective screening and monitoring of congenital and acquired color vision deficiencies, with or without associated disease processes.

A biofilm and organomineralisation model for the growth and limiting size of ooids.

Ooids are typically spherical sediment grains characterised by concentric layers encapsulating a core. There is no universally accepted explanation for ooid genesis, though factors such as agitation, abiotic and/or microbial mineralisation and size limitation have been variously invoked. Here we examine the possible influence of microbial organomineralisation on the formation of some naturally occurring ooids. We develop a mathematical model for ooid growth, inspired by work on avascular brain tumours, that assumes mineralisation in a biofilm to form a central core which then nucleates the progressive growth of concentric laminations. The model predicts a limiting size with the sequential width variation of growth rings comparing favourably with those observed in experimentally grown ooids generated from biomicrospheres. In reality, this model pattern may be complicated during growth by syngenetic aggrading neomorphism of the unstable mineral phase, followed by diagenetic recrystallisation that further complicates the structure. Our model provides a potential key to understanding the genetic archive preserved in the internal structures of some ooids.

Nonconservative dynamics of optically trapped high-aspect-ratio nanowires.

We investigate the dynamics of high-aspect-ratio nanowires trapped axially in a single gradient force optical tweezers. A power spectrum analysis of the dynamics reveals a broad spectral resonance of the order of kHz with peak properties that are strongly dependent on the input trapping power. A dynamical model incorporating linear restoring optical forces, a nonconservative asymmetric coupling between translational and rotational degrees of freedom, viscous drag, and white noise provides an excellent fit to experimental observations. A persistent low-frequency cyclical motion around the equilibrium trapping position, with a frequency distinct from the spectral resonance, is observed from the time series data.