47 research outputs found
Autocatalytic Loop, Amplification and Diffusion: A Mathematical and Computational Model of Cell Polarization in Neural Chemotaxis
The chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response
A developmental approach to predicting neuronal connectivity from small biological datasets: a gradient-based neuron growth model.
PMCID: PMC3931784 Open Access article: BB/G006652/1 and BB/G006369/1.Relating structure and function of neuronal circuits is a challenging problem. It requires demonstrating how dynamical patterns of spiking activity lead to functions like cognitive behaviour and identifying the neurons and connections that lead to appropriate activity of a circuit. We apply a "developmental approach" to define the connectome of a simple nervous system, where connections between neurons are not prescribed but appear as a result of neuron growth. A gradient based mathematical model of two-dimensional axon growth from rows of undifferentiated neurons is derived for the different types of neurons in the brainstem and spinal cord of young tadpoles of the frog Xenopus. Model parameters define a two-dimensional CNS growth environment with three gradient cues and the specific responsiveness of the axons of each neuron type to these cues. The model is described by a nonlinear system of three difference equations; it includes a random variable, and takes specific neuron characteristics into account. Anatomical measurements are first used to position cell bodies in rows and define axon origins. Then a generalization procedure allows information on the axons of individual neurons from small anatomical datasets to be used to generate larger artificial datasets. To specify parameters in the axon growth model we use a stochastic optimization procedure, derive a cost function and find the optimal parameters for each type of neuron. Our biologically realistic model of axon growth starts from axon outgrowth from the cell body and generates multiple axons for each different neuron type with statistical properties matching those of real axons. We illustrate how the axon growth model works for neurons with axons which grow to the same and the opposite side of the CNS. We then show how, by adding a simple specification for dendrite morphology, our model "developmental approach" allows us to generate biologically-realistic connectomes
Geometry and field theory in multi-fractional spacetime
We construct a theory of fields living on continuous geometries with
fractional Hausdorff and spectral dimensions, focussing on a flat background
analogous to Minkowski spacetime. After reviewing the properties of fractional
spaces with fixed dimension, presented in a companion paper, we generalize to a
multi-fractional scenario inspired by multi-fractal geometry, where the
dimension changes with the scale. This is related to the renormalization group
properties of fractional field theories, illustrated by the example of a scalar
field. Depending on the symmetries of the Lagrangian, one can define two
models. In one of them, the effective dimension flows from 2 in the ultraviolet
(UV) and geometry constrains the infrared limit to be four-dimensional. At the
UV critical value, the model is rendered power-counting renormalizable.
However, this is not the most fundamental regime. Compelling arguments of
fractal geometry require an extension of the fractional action measure to
complex order. In doing so, we obtain a hierarchy of scales characterizing
different geometric regimes. At very small scales, discrete symmetries emerge
and the notion of a continuous spacetime begins to blur, until one reaches a
fundamental scale and an ultra-microscopic fractal structure. This fine
hierarchy of geometries has implications for non-commutative theories and
discrete quantum gravity. In the latter case, the present model can be viewed
as a top-down realization of a quantum-discrete to classical-continuum
transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and
improved (especially section 4.5), typos corrected, references added; v4:
further typos correcte
MEASURING THE FRACTAL STRUCTURE OF INTERSTELLAR CLOUDS
To study the structure of interstellar clouds we used the so-called perimeter-area relation to estimate fractal dimensions. We studied the reliability of the method by applying it to artificial fractals and discuss some of the problems and pitfalls. Results for two different cloud types (high-velocity clouds (HVCs) and infrared cirrus) are summarized. We find dimensions 1.2 <D <1.55, somewhat higher than found in previous, similar studies.</p