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.

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.

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.

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.

A fast, portable desaccading program.

This short communication reports a program for detecting and removing saccades, quick phases and blink signals from eye movement records to allow determination of slow phase eye velocity (SPV) during vestibular stimulation. The program is written in C and is simple, fast, and independent of computer type.

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.

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.