Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions.

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.

Fractional cable models for spiny neuronal dendrites.

Cable equations with fractional order temporal operators are introduced to model electrotonic properties of spiny neuronal dendrites. These equations are derived from Nernst-Planck equations with fractional order operators to model the anomalous subdiffusion that arises from trapping properties of dendritic spines. The fractional cable models predict that postsynaptic potentials propagating along dendrites with larger spine densities can arrive at the soma faster and be sustained at higher levels over longer times. Calibration and validation of the models should provide new insight into the functional implications of altered neuronal spine densities, a hallmark of normal aging and many neurodegenerative disorders.

Anomalous subdiffusion with multispecies linear reaction dynamics.

We have introduced a set of coupled fractional reaction-diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics. The model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics. The effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms. One consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species. The general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail. We have carried out Monte Carlo simulations corresponding to these special cases and we find excellent agreement with theory.

Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations.

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the CTRW model to include more general reaction dynamics.

Functionally relevant measures of spatial complexity in neuronal dendritic arbors.

We introduce a set of scaling exponents for characterizing global 3D morphologic properties of mass distribution, branching and taper in neuronal dendritic arbors, capable of distinguishing functionally relevant changes in dendritic complexity that standard Sholl analysis and fractal analysis cannot. We demonstrate that the scaling exponent for mass distribution, d(M), comprises a sum of independent scaling exponents for branching, d(N), and taper, d(T). The accuracy of experimental measurements of the scaling exponents was verified using computer generated self-similar binary trees of known fractal dimension, and with prescribed amounts of branching and taper. The theory was applied to measuring 3D spatial complexity in the apical and basal dendritic trees of two functionally distinct types of macaque monkey neocortical pyramidal neurons: long corticocortical projection neurons from superior temporal cortex to area 46 of the prefrontal cortex (PFC), and local projection neurons within area 46 of the PFC. Two distinct scaling subregions (proximal and medial) were identified in both apical and basal trees of the two neuron types, and scaling exponents were fitted. A small but significant difference in mass scaling in the proximal region distinguished long from local projection neurons. Interestingly, both classes of neuron exhibited a homeostatic pattern of mass distribution across the two regions: despite large differences between proximal and medial regions in branching and tapering exponents, these effects were compensatory, resulting in a uniform, slow reduction of mass with distance from the soma, over both scaling regions of the apical and basal trees. Given a uniformly excitable membrane, the electrotonic properties of dendritic arbors depend entirely upon mass distribution, and its relative contributions from dendritic branching and taper. By capturing each of these complex morphologic properties in a single, globally descriptive parameter, the new 3D scaling exponents introduced in this study permit efficient morphometric characterization of complex dendritic arbors in the fewest possible parameters, that can be directly related to their electrotonic properties, and hence to neuronal function.

New techniques for imaging, digitization and analysis of three-dimensional neural morphology on multiple scales.

Cognitive impairment in normal aging and neurodegenerative diseases is accompanied by altered morphologies on multiple scales. Understanding of the role of these structural changes in producing functional deficits in brain aging and neuropsychiatric disorders requires accurate three-dimensional representations of neuronal morphology, and realistic biophysical modeling that can directly relate structural changes to altered neuronal firing patterns. To date however, tools capable of resolving, digitizing and analyzing neuronal morphology on both local and global scales, and with sufficient throughput and automation, have been lacking. The precision of existing image analysis-based morphometric tools is restricted at the finest scales, where resolution of fine dendritic features and spine geometry is limited by the skeletonization methods used, and by quantization errors arising from insufficient imaging resolution. We are developing techniques for imaging, reconstruction and analysis of neuronal morphology that capture both local and global structural variation. To minimize quantization error and evaluate more precisely the fine geometry of dendrites and spines, we introduce a new shape analysis technique, the Rayburst sampling algorithm that uses the original grayscale data rather than the segmented images for precise, continuous radius estimation, and multidirectional radius sampling to represent non-circular branch cross-sections and anisotropic structures such as dendritic spine heads, with greater accuracy. We apply the Rayburst technique to 3D neuronal shape analysis at different scales. We reconstruct and digitize entire neurons from stacks of laser-scanning microscopy images, as well as globally complex structures such as multineuron networks and microvascular networks. We also introduce imaging techniques necessary to recover detailed information on three-dimensional mass distribution and surface roughness of amyloid beta plaques from human Alzheimer's disease patients and from the Tg2576 mouse that expresses the "Swedish" mutation of the amyloid precursor protein. By providing true three-dimensional morphometry of complex histologic structures on multiple scales, the tools described in this report will enable multiscale biophysical modeling studies capable of testing potential mechanisms by which altered dendritic structure, spine geometry and network branching patterns that occur in normal aging and in many brain disorders, determine deficits of functions such as working memory and cognition.

Turing pattern formation in fractional activator-inhibitor systems.

Activator-inhibitor systems of reaction-diffusion equations have been used to describe pattern formation in numerous applications in biology, chemistry, and physics. The rate of diffusion in these applications is manifest in the single parameter of the diffusion constant, and stationary Turing patterns occur above a critical value of d representing the ratio of the diffusion constants of the inhibitor to the activator. Here we consider activator-inhibitor systems in which the diffusion is anomalous subdiffusion; the diffusion rates are manifest in both a diffusion constant and a diffusion exponent. A consideration of this problem in terms of continuous-time random walks with sources and sinks leads to a reaction-diffusion system with fractional order temporal derivatives operating on the spatial Laplacian. We have carried out an algebraic stability analysis of the homogeneous steady-state solution in fractional activator-inhibitor systems, with Gierer-Meinhardt reaction kinetics and with Brusselator reaction kinetics. For each class of reaction kinetics we identify a Turing instability bifurcation curve in the two-dimensional diffusion parameter space. The critical value of d , for Turing instabilities, decreases monotonically with the anomalous diffusion exponent between unity (standard diffusion) and zero (extreme subdiffusion). We have also carried out numerical simulations of the governing fractional activator-inhibitor equations and we show that the Turing instability precipitates the formation of complex spatiotemporal patterns. If the diffusion of the activator and inhibitor have the same anomalous scaling properties, then the surface profiles of these patterns for values of d slightly above the critical value varies from smooth stationary patterns to increasingly rough and nonstationary patterns as the anomalous diffusion exponent varies from unity towards zero. If the diffusion of the activator is anomalous subdiffusion but the diffusion of the inhibitor is standard diffusion, we find stable stationary Turing patterns for values of d well below the threshold values for pattern formation in standard activator-inhibitor systems.

Pilot study on the factors that influence learning by general practice registrars in central Australia.

INTRODUCTION: To counter a medical workforce shortage in rural and remote areas of Australia an increasing number of general practitioners are being trained in rural and remote areas. General practice (GP) registrars train in general practice as working apprentices alongside GP supervisors. GP registrars are allocated a training advisor to oversee their progress throughout their training. Central Australian GP registrars expressed concern to their training advisor regarding certain work partnerships with their GP supervisors. The study was carried out in response to these concerns, which were raised during a shortage of GPs in the area. The aim of the study was to explore factors in the interaction between GP registrars and GP supervisors in the context of their practices that impact on the quality of GP registrar learning in Central Australia. METHOD: A qualitative research method was used to explore the subtleties and issues in relationships between GP registrars, their GP supervisors and their practices. The interview schedule comprised pairs of polarised, provocative statements to generate discussion. Topics for the interview schedule were derived from the data from training advisor visits and the literature. GP registrars in Central Australia who had completed at least one six-month term in general practice were eligible for the study. Five female GP registrars participated in the study. Interviews were recorded, transcribed and checked by the participants before the interview material became the research record. RESULTS: The interview schedule generated considerable discussion as planned. The structures that determine GP income were seen as a barrier to GP registrar learning in Central Australia. The registrars reported that the fee-for-service model prevented them capitalising on learning opportunities both inside and outside their designated general practice. The GP registrars considered their training was compromised by the need to provide clinical service during a time of workforce shortage. Adaptation to a practice was seen as an important skill for GP registrars to learn, providing this did not compromise a registrar's own ethical and professional values. Learning was optimised by agreement between GP registrars and GP supervisors on the teaching subjects, and a mix of opportunistic and planned teaching sessions. Geographical isolation was perceived to have had a significant negative impact on GP registrar learning but one GP registrar discussed how this could be turned into a positive factor. CONCLUSION: GP registrars reported learning best by providing a clinical service with ready access to a supportive GP supervisor. Workforce pressures in Central Australia at the time of this pilot study reduced the GP supervisors' ability to support GP registrars, especially in a fee-for-service model of health care. GP registrars should be placed in practices where they will receive experience, training and education rather than be allocated to areas of workforce shortage. Changes to the remuneration system for GP registrars and GP supervisors could be considered to enable GP registrars to capitalise on the learning opportunities in remote clinical practice.

Quantitative analysis of the dendritic morphology of corticocortical projection neurons in the macaque monkey association cortex.

The polymodal association areas of the primate cerebral cortex are heavily interconnected and play a crucial role in cognition. Area 46 of the prefrontal cortex in non-human primates receives direct inputs from several association areas, among them the cortical regions lining the superior temporal sulcus. We examined whether projection neurons providing such a corticocortical projection differ in their dendritic morphology from pyramidal neurons projecting locally within area 46. Specific sets of corticocortical projection neurons were identified by in vivo retrograde transport in young macaque monkeys. Full dendritic arbors of retrogradely labeled neurons were visualized in brain slices by targeted intracellular injection of Lucifer Yellow, and reconstructed three-dimensionally using computer-assisted morphometry. Total dendritic length, numbers of segments, numbers of spines, and spine density were analyzed in layer III pyramidal neurons forming long projections (from the superior temporal cortex to prefrontal area 46), as well as local projections (within area 46). Sholl analysis was also used to compare the complexity of these two groups of neurons. Our results demonstrate that long corticocortical projection neurons connecting the temporal and prefrontal cortex have longer, more complex dendritic arbors and more spines than pyramidal neurons projecting locally within area 46. The more complex dendritic arborization of such neurons is likely linked to their participation in cortical networks that require extensive convergence of multiple afferents at the cellular level.

Ultrastructure of vestibular commissural neurons related to velocity storage in the monkey.

The angular vestibulo-ocular reflex maintains gaze during head movements. It is thought to be mediated by two components: direct and velocity storage pathways. The direct angular vestibulo-ocular reflex is conveyed by a three neuron chain from the labyrinth to the ocular motoneurons. The indirect pathway involves a more complex neural network that utilizes a portion of the vestibular commissure. The purpose of the present study was to identify the ultrastructural characteristics of commissural neurons in the medial vestibular nucleus that are related to the velocity storage component of the angular vestibulo-ocular reflex. Ultrastructural studies of degenerating medial vestibular nucleus neurons were conducted in monkeys following midline section of rostral medullary commissural fibers with subsequent behavioral testing. After this lesion, oculomotor and vestibular functions attributable to velocity storage were abolished, whereas the direct angular vestibulo-ocular reflex pathway remained intact. Since this damage was functionally discrete, degenerating neurons were interpreted as potential participants in the velocity storage network. Ultrastructural observations indicate that commissural neurons related to velocity storage are small and medium sized cells having large nuclei with deep indentations and relatively little cytoplasm, which are located in the lateral crescents of rostral medial vestibular nucleus. The morphology of degenerating dendritic profiles varied. Some contained numerous round or tubular mitochondria in a pale cytoplasmic matrix with few other organelles, while others had few mitochondria but many cisterns and vacuoles in dense granular cytoplasm. The commissural nature of these cells was further suggested by the presence of two different types of degenerating axon terminals in the rostral medial vestibular nucleus: those with a moderate density of large spherical synaptic vesicles, and those with pleomorphic, primarily ellipsoid synaptic vesicles. The recognition of two types of degenerating terminals further supports our interpretation that at least two morphological types of commissural neurons participate in the velocity storage network. The degenerating boutons formed contacts with a variety of postsynaptic partners. In particular, synapses were observed between degenerating boutons and non-degenerating dendrites, and between intact terminals and degenerating dendrites. However, degenerating pre- and postsynaptic elements were rarely observed in direct contact, suggesting that additional neurons are interposed in the indirect pathway commissural system. On the basis of these ultrastructural observations, it is concluded that vestibular commissural neurons involved in the mediation of velocity storage have distinguishing ultrastructural features and synaptology, that are different from those of direct pathway neurons.

Spatial orientation of the angular vestibulo-ocular reflex.

During vestibular nystagmus, optokinetic nystagmus (OKN), and optokinetic afternystagmus (OKAN), the axis of eye rotation tends to align with the vector sum of linear accelerations acting on the head. This includes gravitational acceleration and the linear accelerations generated by translation and centrifugation. We define the summed vector of gravitational and linear accelerations as gravito-inertial acceleration (GIA) and designate the phenomenon of alignment as spatial orientation of the angular vestibuloocular reflex (aVOR). On the basis of studies in the monkey, we postulated that the spatial orientation of the aVOR is dependent on the slow (velocity storage) component of the aVOR, not on the short latency, compensatory aVOR component, which is in head-fixed coordinates. Experiments in which velocity storage was abolished by midline medullary section support this postulate. The velocity storage component of the aVOR is likely to be generated in the vestibular nuclei, and its spatial orientation was shown to be controlled through the nodulus and uvula of the vestibulocerebellum. Separate regions of the nodulus/uvula appear to affect the horizontal and vertical/torsional components of the response differently. Velocity storage is weaker in humans than in monkeys, but responds in a similar fashion in both species. We postulate that spatial orientation of the aVOR plays an important role in aligning gaze with the GIA and in maintaining balance during angular locomotion.

Effects of tilt of the gravito-inertial acceleration vector on the angular vestibuloocular reflex during centrifugation.

Effects of tilt of the gravito-inertial acceleration vector on the angular vestibuloocular reflex during centrifugation. Interaction of the horizontal linear and angular vestibuloocular reflexes (lVOR and aVOR) was studied in rhesus and cynomolgus monkeys during centered rotation and off-center rotation at a constant velocity (centrifugation). During centered rotation, the eye velocity vector was aligned with the axis of rotation, which was coincident with the direction of gravity. Facing and back to motion centrifugation tilted the resultant of gravity and linear acceleration, gravito-inertial acceleration (GIA), inducing cross-coupled vertical components of eye velocity. These components were upward when facing motion and downward when back to motion and caused the axis of eye velocity to reorient from alignment with the body yaw axis toward the tilted GIA. A major finding was that horizontal time constants were asymmetric in each monkey, generally being longer when associated with downward than upward cross coupling. Because of these asymmetries, accurate estimates of the contribution of the horizontal lVOR could not be obtained by simply subtracting horizontal eye velocity profiles during facing and back to motion centrifugation. Instead, it was necessary to consider the effects of GIA tilts on velocity storage before attempting to estimate the horizontal lVOR. In each monkey, the horizontal time constant of optokinetic after-nystagmus (OKAN) was reduced as a function of increasing head tilt with respect to gravity. When variations in horizontal time constant as a function of GIA tilt were included in the aVOR model, the rising and falling phases of horizontal eye velocity during facing and back to motion centrifugation were closely predicted, and the estimated contribution of the compensatory lVOR was negligible. Beating fields of horizontal eye position were unaffected by the presence or magnitude of linear acceleration during centrifugation. These conclusions were evaluated in animals in which the low-frequency aVOR was abolished by canal plugging, isolating the contribution of the lVOR. Postoperatively, the animals had normal ocular counterrolling and horizontal eye velocity modulation during off-vertical axis rotation (OVAR), suggesting that the otoliths were intact. No measurable horizontal eye velocity was elicited by centrifugation with angular accelerations

Clues for the development of food-based dietary guidelines: how are dietary targets being achieved by UK consumers?

Expert scientific advice to the UK Government has been translated into eight general dietary guidelines, which form the core of population-based dietary advice in the UK and are supplemented by a food selection guide showing the types and proportions of foods needed for a balanced and healthy diet. Data from the Dietary and Nutritional Survey of British Adults were used to identify statistically significant differences between subgroups of the study population that met, or failed to meet, population nutritional goals for intakes of total fat, saturated fat and dietary fibre. Several eating habits--including greater consumption of starchy foods (particularly wholemeal varieties), greater consumption of fruit and the substitution of reduced-fat milk for whole-fat milk--were shared by the subgroups that met each of the nutritional goals. This analysis provides clues for any future refinement of food-based dietary guidelines.

Providing evidence-based answers to clinical questions. A pilot information service for general practitioners.

OBJECTIVE: To pilot a clinical information service for general practitioners. METHODS: A representative sample of 31 GPs was invited to submit clinical questions to a local academic department of general practice. Their views on the service and the usefulness of the information were obtained by telephone interview. RESULTS: Over one month, nine GPs (29% of the sample, 45% of those stating an interest), submitted 20 enquiries comprising 45 discrete clinical questions. The median time to search for evidence, appraise it and write answers to each enquiry was 2.5 hours (range, 1.0-7.4 hours). The median interval between receipt of questions and dispatch of answers was 3 days (range, 1-12 days). CONCLUSIONS: The GPs found the answers useful in clinical decision making; in four out of 20 cases patient management was altered.

Time trends in human dietary exposure to PCDDs, PCDFs and PCBs in the UK.

Total Diet Study samples collected in 1982 and 1992 were analysed. Estimated dietary intakes of PCDDs, PCDFs and PCBs by UK consumers were found to have fallen substantially during this period and are now considerably below the Tolerable Daily Intake (TDI). Composite samples of human milk were also analysed for PCDDs, PCDFs and PCBs. Estimated combined dietary intakes of PCDDs, PCDFs and PCBs by breast fed infants in 1993-94 via breast milk fell from 170 pg TEQ/kg bodyweight/day at 2 months to 39 pg TEQ/kg bodyweight/day at 10 months.

Control of spatial orientation of the angular vestibuloocular reflex by the nodulus and uvula.

Spatial orientation of the angular vestibuloocular reflex (aVOR) was studied in rhesus monkeys after complete and partial ablation of the nodulus and ventral uvula. Horizontal, vertical, and torsional components of slow phases of nystagmus were analyzed to determine the axes of eye rotation, the time constants (Tcs) of velocity storage, and its orientation vectors. The gravito-inertial acceleration vector (GIA) was tilted relative to the head during optokinetic afternystagmus (OKAN), centrifugation, and reorientation of the head during postrotatory nystagmus. When the GIA was tilted relative to the head in normal animals, horizontal Tcs decreased, vertical and/or roll time constants (Tc(vert/roll)) lengthened according to the orientation of the GIA, and vertical and/or roll eye velocity components appeared (cross-coupling). This shifted the axis of eye rotation toward alignment with the tilted GIA. Horizontal and vertical/roll Tcs varied inversely, with T(chor) being longest and T(cvert/roll) shortest when monkeys were upright, and the reverse when stimuli were around the vertical or roll axes. Vertical or roll Tcs were longest when the axes of eye rotation were aligned with the spatial vertical, respectively. After complete nodulo-uvulectomy, T(chor) became longer, and periodic alternating nystagmus (PAN) developed in darkness. T(chor) could not be shortened in any of paradigms tested. In addition, yaw-to-vertical/roll cross-coupling was lost, and the axes of eye rotation remained fixed during nystagmus, regardless of the tilt of the GIA with respect to the head. After central portions of the nodulus and uvula were ablated, leaving lateral portions of the nodulus intact, yaw-to-vertical/roll cross-coupling and control of Tc(vert/roll) was lost or greatly reduced. However, control of Tchor was maintained, and T(chor) continued to vary as a function of the tilted GIA. Despite this, the eye velocity vector remained aligned with the head during yaw axis stimulation after partial nodulo-uvulectomy, regardless of GIA orientation to the head. The data were related to a three-dimensional model of the aVOR, which simulated the experimental results. The model provides a basis for understanding how the nodulus and uvula control processing within the vestibular nuclei responsible for spatial orientation of the aVOR. We conclude that the three-dimensional dynamics of the velocity storage system are determined in the nodulus and ventral uvula. We propose that the horizontal and vertical/roll Tcs are separately controlled in the nodulus and uvula with the dynamic characteristics of vertical/roll components modulated in central portions and the horizontal components laterally, presumably in a semicircular canal-based coordinate frame.

Contribution of vestibular commissural pathways to spatial orientation of the angular vestibuloocular reflex.

During nystagmus induced by the angular vestibuloocular reflex (aVOR), the axis of eye velocity tends to align with the direction of gravito-inertial acceleration (GIA), a process we term "spatial orientation of the aVOR." We studied spatial orientation of the aVOR in rhesus and cynomolgus monkeys before and after midline section of the rostral medulla abolished all oculomotor functions related to velocity storage, leaving the direct optokinetic and vestibular pathways intact. Optokinetic afternystagmus and the bias component of off-vertical-axis rotation were lost, and the aVOR time constant was reduced to a value commensurate with the time constants of primary semicircular canal afferents. Spatial orientation of the aVOR, induced either during optokinetic or vestibular stimulation, was also lost. Vertical and roll aVOR time constants could no longer be lengthened in side-down or supine/prone positions, and static and dynamic tilts of the GIA no longer produced cross-coupling from the yaw to pitch and yaw to roll axes. Consequently, the induced nystagmus remained entirely in head coordinates after the lesion, regardless of the direction of the resultant GIA vector. Gains of the aVOR and of optokinetic nystagmus to steps of velocity were unaffected or slightly increased. These results are consistent with a model in which the direct aVOR pathways are organized in semicircular canal coordinates and spatial orientation is restricted to the indirect (velocity storage) pathways.

Modeling the organization of the linear and angular vestibulo-ocular reflexes.

A one-dimensional mathematical model of the compensatory linear vestibuloocular reflex (lVOR) was developed. The model was based on the concept that to effect oculomotor compensation, linear head acceleration sensed by the otoliths must be integrated twice to form the angular position-related signal required by the motoneurons. This contradicts the postulate that linear acceleration is differentiated to generate "jerk," which is then used to drive the compensatory lVOR. The transfer characteristics of different otolith afferent classes were modeled by a transfer function with a common modal structure and different degrees of compensation. Both the time and frequency domain behavior of regular and irregular otolith afferents were simulated. The outputs of the various afferent classes were superposed by a linear filter to generate the velocity command which drives the oculomotor velocity-position integrator. The model was used to simulate the dominant gain and phase characteristics of the compensatory lVOR in monkey and the dynamic characteristics of the compensatory human lVOR response for brief periods of linear acceleration on a sled. The model was then combined with the velocity storage-based model of the angular vestibulo-ocular reflex (aVOR) to simulate the eye velocity response to centrifugation in monkey and man. The model suggests that the orientation response that modifies the time constants of the velocity storage integrator is the dominant aspect of the response to linear acceleration in monkey. Human responses, on the other hand, are dominated by an effect of the beating field, which modifies the eye velocity command to the oculomotor system.