The finite volume spectral element method to solve Turing models in the biological pattern formation

It is well known that reaction-diffusion systems describing Turing models can display very rich pattern formation behavior. Turing systems have been proposed for pattern formation in various biological systems, e.g. patterns in fish, butterflies, lady bugs and etc. A Turing model expresses temporal behavior of the concentrations of two reacting and diffusing chemicals which is represented by coupled reaction-diffusion equations. Since the base of these reaction-diffusion equations arises from the conservation laws, we develop a hybrid finite volume spectral element method for the numerical solution of them and apply the proposed method to Turing system generated by the Schnakenberg model. Also, as numerical simulations, we study the variety of spatio-temporal patterns for various values of diffusion rates in the problem.     

Turing patterns with Turing machines: emergence and low-level structure formation

Despite having advanced a reaction–diffusion model of ordinary differential equations in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Turing has never been considered to have approached a definition of cellular automata. However, his treatment of morphogenesis, and in particular a difficulty he identified relating to the uneven distribution of certain forms as a result of symmetry breaking, are key to connecting his theory of universal computation with his theory of biological pattern formation. Making such a connection would not overcome the particular difficulty that Turing was concerned about, which has in any case been resolved in biology. But instead the approach developed here captures Turing’s initial concern and provides a low-level solution to a more general question by way of the concept of algorithmic probability, thus bridging two of his most important contributions to science: Turing pattern formation and universal computation. I will provide experimental results of one-dimensional patterns using this approach, with no loss of generality to a n-dimensional pattern generalisation.     

LBGK Simulations of Turing Patterns in CIMA Model

A lattice Bhatnagar–Gross–Krook (LBGK) model for reaction-diffusion systems is presented. This model provides a mesoscopic approach to the dynamics of spatially-distributed reacting systems. Pure diffusion phenomena are computed and the results are agreement with the theoretical predictions. This method is also applied to formation of Turing patterns in chloride-iodide-malonic acid (CIMA) reactive model. We get hexagonal structures and stripes which are agreement with other numerical results and experimental results.     

Morphogenesis, morphology and men: pattern formation from embryo to mind

In 1952, Alan Turing published his last work on the concept of embryonic morphogenesis, propounding a computational framework for pattern formation within the developing embryo. This concept of morphogenesis and the concept of embryo pattern formation based on chemical diffusion patterns were corroborated with the discovery of the Homeobox or Hox genes. In the following decades, Hox gene research has expanded and is now shown to underlie the variety of morphological novelties that we experience in nature, the patterning of structural aspects of different organs including the brain and also mutant animals that may in the future give rise to novel speciation. Turing had the foresight and vision and with his work created the field of computational biology and mathematical modeling in biological systems. In this paper, we will discuss the concept of Hox genes, their role in patterning the embryo, how it relates to Turing’s concept of morphogenesis and what further insights they may provide.     

Parallel Turing machines on four-dimensional input tapes

The parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. On the other hand, due to the advances in many application areas such as motion picture processing, computer animation, virtual reality systems, and so forth, it has become increasingly apparent that the study of four-dimensional patterns is of crucial importance. Therefore, we think that the study of four-dimensional automata as a computational model of four-dimensional pattern processing is also meaningful. In this article, we propose a four-dimensional parallel Turing machine (4-PTM), and investigate some of its properties based on hardware complexity.     

Nonlinear Galerkin methods for a system of PDEs with Turing instabilities

We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.     

Interactive segmentation of the cerebral lobes with fuzzy inference in 3T MR images.

Measurement of volume and surface area of the frontal, parietal, temporal and occipital lobes from magnetic resonance (MR) images shows promise as a method for use in diagnosis of dementia. This article presents a novel computer-aided system for automatically segmenting the cerebral lobes from 3T human brain MR images. Until now, the anatomical definition of cerebral lobes on the cerebral cortex is somewhat vague for use in automatic delineation of boundary lines, and there is no definition of cerebral lobes in the interior of the cerebrum. Therefore, we have developed a new method for defining cerebral lobes on the cerebral cortex and in the interior of the cerebrum. The proposed method determines the boundaries between the lobes by deforming initial surfaces. The initial surfaces are automatically determined based on user-given landmarks. They are smoothed and deformed so that the deforming boundaries run along the hourglass portion of the three-dimensional shape of the cerebrum with fuzzy rule-based active contour and surface models. The cerebrum is divided into the cerebral lobes according to the boundaries determined using this method. The reproducibility of our system with a given subject was assessed by examining the variability of volume and surface area in three healthy subjects, with measurements performed by three beginners and one expert user. The experimental results show that our system segments the cerebral lobes with high reproducibility.     

大脑中神经回路的电位发放比较

通过对大脑神经中存在的四个回路：嗅球网络、小脑皮层、海马结构和丘脑皮层,建立基于解剖的神经元回路模型,对四种神经元回路中神经元的动作电位进行数值模拟,通过外加刺激来研究四种回路的网络特性,得到丰富的发放模式,分析比较这四种回路中信息传递特性,模拟大脑中神经元网络信号传递特性.     

Traveling bumps and their collisions in a two-dimensional neural field

A neural field is a continuous version of a neural network model accounting for dynamical pattern forming from populational firing activities in neural tissues. These patterns include standing bumps, moving bumps, traveling waves, target waves, breathers, and spiral waves, many of them observed in various brain areas. They can be categorized into two types: a wave-like activity spreading over the field and a particle-like localized activity. We show through numerical experiments that localized traveling excitation patterns (traveling bumps), which behave like particles, exist in a two-dimensional neural field with excitation and inhibition mechanisms. The traveling bumps do not require any geometric restriction (boundary) to prevent them from propagating away, a fact that might shed light on how neurons in the brain are functionally organized. Collisions of traveling bumps exhibit rich phenomena; they might reveal the manner of information processing in the cortex and be useful in various applications. The trajectories of traveling bumps can be controlled by external inputs.     

Null steering of linear antenna arrays with use of modified touring ant colony optimization algorithm

A new and flexible method is presented that is based on the ant colony optimization algorithm for the pattern synthesis of linear antenna arrays with the prescribed nulls. Nulling of the pattern is achieved by controlling only the amplitude of each array element. To show the versatility of the present method, some design specifications such as the side lobe level, null depth, and dynamic range ratio are considered by introducing a set of weighting factors in the cost function constructed for the ant colony optimization algorithm. Several illustrative examples of Chebyshev patterns with the imposed single, multiple, and broad nulls are given. © 2002 Wiley Periodicals, Inc. Int J RF and Microwave CAE 12: 375–383, 2002. Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mmce10034     

Turing structures and turbulence in a chemical reaction-diffusion system

Turing structures and turbulence are analyzed within the framework of a model for the chlorine dioxide—iodine—malonic acid reaction. This reaction has recently provided the first experimental results on the formation of stationary chemical patterns under laboratory conditions. The model is distinguished from previously studied simple reaction-diffusion models by showing a strongly subcritical transition to stripes. This makes it possible for a great variety of localized structures to emerge and be stable. The dynamics of phase singularities and the formation and breakdown of chemical spirals are investigated.     

时滞控制神经网络的稳定性和Turing斑图结构

本文讨论了一类时滞反应扩散神经网络模型.利用时滞来控制系统的稳定性、分岔和Turing斑图.研究结果表明,在一定条件下,时滞不仅能影响系统的稳定性和周期震荡性,还能影响系统的Turing不稳定性.数值模拟验证了理论分析的正确性,同时还说明了时滞能改变斑图的结构.     

Computer Vision and Pattern Recognition Techniques for 2-D and 3-D MR Cerebral Cortical Segmentation (Part I): A State-of-the-Art Review

Extensive growth in functional brain imaging, perfusion-weighted imaging, diffusion-weighted imaging, brain mapping and brain scanning techniques has led tremendously to the importance of the cerebral cortical segmentation, both in 2-D and 3-D, from volumetric brain magnetic resonance imaging data sets. Besides that, recent growth in deformable brain segmentation techniques in 2-D and 3-D has brought the engineering community, such as the areas of computer vision, image processing, pattern recognition and graphics, closer to the medical community, such as to neuro-surgeons, psychiatrists, oncologists, neuro-radiologists and internists. This paper is an attempt to review the state-of-the-art 2-D and 3-D cerebral cortical segmentation techniques from brain magnetic resonance imaging based on three main classes: region-based, boundary/surface-based and fusion of boundary/surface-based with region-based techniques. In the first class, region-based techniques, we demonstrated more than 18 different techniques for segmenting the cerebral cortex from brain slices acquired in orthogonal directions. In the second class, boundary/surface-based, we showed more than ten different techniques to segment the cerebral cortex from magnetic resonance brain volumes. Particular emphasis will be placed by presenting four state-of-the-art systems in the third class, based on the fusion of boundary/surface-based with region-based techniques outlined in Part II of the paper, also called regional-geometric deformation models, which take the paradigm of partial differential equations in the level set framework. We also discuss the pros and cons of various techniques, besides giving the mathematical foundations for each sub-class in the cortical taxonomy. Received: 25 August 2000, Received in revised form: 28 March 2001, Accepted: 28 March 2001     

Cortical hierarchies, sleep, and the extraction of knowledge from memory

Odin the Allfather had in his service two great ravens. These ravens' names were Hugin (Thought) and Munin (Memory) and every morning at dawn they would fly off over Midgard (the world) in search of news and information to learn more about humans and their activities. At sundown, they would return to Odin where they would perch one on each of Odin's shoulders, and whisper into his ears all that they had seen and heard.Experience, stored in the brain as memory, is the raw material for intelligence and thought. It has been suggested that at sundown (i.e., during sleep) the brain adjusts its own synaptic matrix to enable adaptive responses to future events by a process of gradient descent optimization, involving repeated reactivations of recent and older memories and gradual adjustment of the synaptic weights. Memory retrieval, thought, and the generation of adaptive behavioral responses involve globally coordinated trajectories through the neuronal state-space, mediated by appropriate synaptic linkages. Artificial neural networks designed to implement even the most rudimentary forms of memory and knowledge extraction and adaptive behavior incorporate massively and symmetrically interconnected nodes; yet, in the cerebral cortex, the probability of a synaptic connection between any two arbitrarily chosen cells is on the order of 10⁻⁶, i.e., so close to zero that a naive modeler might neglect this parameter altogether. The probability of a symmetric connection is even smaller (10⁻¹²). How then, are thought and memory even possible? The solution appears to have been in the evolution of a modular, hierarchical cortical architecture, in which the modules are internally highly connected but only weakly interconnected with other modules. Appropriate inter-modular linkages are mediated indirectly via common linkages with higher level modules collectively known as association cortex. The hippocampal formation in the temporal lobe is the highest level of association cortex. It generates sequentially coupled patterns unique to the location and content of experience, but which do not contain the actual stored data. Rather, the patterns serve as pointers or ‘links’ to the data. Spontaneous reactivation of these linking patterns during sleep may enable the retrieval of recent sequences of experience stored in the lower levels of the cortex and the gradual extraction of knowledge from them. In this essay I explore these ideas, their implications, and the neuroscientific evidence for them.     

Fusion of Region and Boundary/Surface-Based Computer Vision and Pattern Recognition Techniques for 2-D and 3-D MR Cerebral Cortical Segmentation (Part-II): A State-of-the-Art Review

Extensive growth in functional brain imaging, perfusion-weighted imaging, diffusion-weighted imaging, brain mapping and brain scanning techniques has led tremendously to the importance of cerebral cortical segmentation both in 2-D and 3-D from volumetric brain magnetic resonance imaging data sets. Besides that, recent growth in deformable brain segmentation techniques in 2-D and 3-D has brought the engineering community, such as the areas of computer vision, image processing, pattern recognition and graphics, closer to the medical community, such as to neuro-surgeons, psychiatrists, oncologists, neuro-radiologists and internists. In Part I of this research (see Suri et al [1]), an attempt was made to review the state-of-the-art in 2-D and 3-D cerebral cortical segmentation techniques from brain magnetic resonance imaging based on two main classes: region- and boundary/surface-based. More than 18 different techniques for segmenting the cerebral cortex from brain slices acquired in orthogonal directions were shown using region-based techniques. We also showed more than ten different techniques to segment the cerebral cortex from magnetic resonance brain volumes using boundary/surface-based techniques. This paper (Part II) focuses on presenting state-of-the-art systems based on the fusion of boundary/surface-based with region-based techniques, also called regional-geometric deformation models, which takes the paradigm of partial differential equations in the level set framework. We also discuss the pros and cons of these various techniques, besides giving the mathematical foundations for each sub-class in the cortical taxonomy. Special emphasis is placed on discussing the advantages, validation, challenges and neuro-science/clinical applications of cortical segmentation. Received: 25 August 2000, Received in revised form: 28 March 2001, Accepted: 28 March 2001     

On the complexity of ω-type Turing acceptors

Turing machines are considered as recognizers of sets of infinite (ω-type) sequences, so called ω-languages. The basic results on such ω-type Turing acceptors were presented in a preceding paper. This paper focuses on the theory of deterministic ω-type Turing acceptors (ω-DTA's) which turns out to be crucially different from the ‘classical’ theory of Turing machines. It is shown that there exists no ω-DTA which is universal for all ω-DTA's. Two infinite complexity hierarchies for ω-DTA's are established, the ‘states hierarchy’, corresponding to the number of states in the machine, and the ‘designated sets hierarchy’, corresponding to the number of designated sets of states used in the recognition. Concrete examples of ω-languages characterizing each of the complexity classes are exhibited. Two additional examples of interesting ω-languages are presented:

1. (i) An ω-language which is ‘inherently non-deterministic’, i.e. can be recognized by a non-deterministic Turing acceptor but by no deterministic acceptor.

2. (ii) An ω-language which cannot be recognized even by a non-deterministic Turing acceptor.

The above examples are constructed without using diagonalization. Oscillating ω-DTA's, i.e. ω-DTA's which are allowed to oscillate on ω-inputs, are also considered and are shown to be strictly more powerful than non-oscillating ω-DTA's, yet strictly less powerful than non-deterministic ω-Turing acceptors.     

基于小波相干性算法的孤独症儿童脑电评估

孤独症是一种先天的大脑发育障碍性疾病,孤独症儿童的早期评估诊断尤为重要.脑电图(Eleetroencephalography,EEG)是大脑神经细胞电生理活动在大脑皮层或头皮表面的总体反映.EEG信号中包含了大量的生理与疾病信息,可为某些脑疾病提供诊断依据.本文按照国际10-20系统标准电极分布将全脑划分为5个脑区,采...     

Determination of ECoG information flow activity based on Granger causality and Hilbert transformation

Analysis of directional information flow patterns among different regions of the brain is important for investigating the relation between ECoG (electrocorticographic) and mental activity. The objective is to study and evaluate the information flow activity at different frequencies in the primary motor cortex. We employed Granger causality for capturing the future state of the propagation path and direction between recording electrode sites on the cerebral cortex. A grid covered the right motor cortex completely due to its size (approx. 8 cm × 8 cm) but grid area extends to the surrounding cortex areas. During the experiment, a subject was asked to imagine performing two activities: movement of the left small finger and/or movement of the tongue. The time series of the electrical brain activity was recorded during these trials using an 8 × 8 (0.016–300 Hz band with) ECoG platinum electrode grid, which was placed on the contralateral (right) motor cortex. For detection of information flow activity and communication frequencies among the electrodes, we have proposed a method based on following steps: (i) calculation of analytical time series such as amplitude and phase difference acquired from Hilbert transformation, (ii) selection of frequency having highest interdependence for the electrode pairs for the concerned time series over a sliding window in which we assumed time series were stationary, (iii) calculation of Granger causality values for each pair with selected frequency. The information flow (causal influence) activity and communication frequencies between the electrodes in grid were determined and shown successfully. It is supposed that information flow activity and communication frequencies between the electrodes in the grid are approximately the same for the same pattern. The successful employment of Granger causality and Hilbert transformation for the detection of the propagation path and direction of each component of ECoG among different sub-cortex areas were capable of determining the information flow (causal influence) activity and communication frequencies between the populations of neurons successfully.     

Gradual spatial pattern formation of homogeneous robot group

This paper proposes a gradual formation of a spatial pattern for a homogeneous robot group. The autonomous formation of spatial pattern is one of key technologies for the advancement of cooperative robotic systems because a pattern formation can be regarded as function differentiation of a multi-agent system. When multiple autonomous robots without a given local task cooperatively work for a global objective, the function differentiation is the first and indispensable step. For example, each member of cooperative insects or animals can autonomously recognize own local tasks through mutual communication with local members. There were a lot of papers that reported a spatial pattern formation of multiple robots, but the global information was supposed to be available in their approaches. It is however almost impractical assumption for a small robot to be equipped with an advanced sensing system for global localization due to robot’s scale and sensor size. The local information-based algorithm for the pattern formation is desired even if each robot is not equipped with a global localization sensor.We therefore propose a gradual pattern formation algorithm, i.e., a group of robots improves complexity of their pattern from to a simple pattern to a goal pattern like a polygon. In the algorithm, the Turing diffusion-driven instability theory is used so that it could differentiate roles of each robot in a group based only on local information. In experiment, we demonstrate that robots can make a few polygon patterns from a circle pattern by periodically differentiating robot’s roles into a vertex or a side. We show utilities of the proposed gradual pattern formation algorithm for multiple autonomous robots based on local information through some experiments.     

基于大脑神经元放电的脑-机接口技术

基于大脑运动皮层神经元放电的脑-机接口通过记录大脑运动皮层神经元的放电信号控制瘫痪肢体或假肢运动,其软硬件核心为神经元群体解码和神经元放电活动的检测。解码方法分为推理算法和分类器方法,检测方法通过在大脑运动皮层区植入长效电极记录单个或群体神经元的放电活动。分析表明,脑-机接口技术应在更多脑区域上植入长效电极达到更好控制设备的目的,各类解码算法应通过联合并加入反馈信号提高对神经元信号的解码效果。