Neuronal differentiation and synapse formation occur in space and time with fractal dimension.
TL;DR: The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension and defines a simple in vitro biological model for biophysical and biochemical studies on natural neural networks.
Abstract: The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension. The interacting cells create first a dynamic system with its own attractor, (i.e., a fragment of time and space where the dynamic processes occur and where no further evolution of the system is possible at all owing to the action of the intrasystemic forces unless some extrasystemic forces act upon it). This attractor is then modified in the active manner by the differentiating cells until the system attains a degenerated stationary state and differentiation ends. The fractal structure of the system is also lost in the course of tumor progression. Our data indicate that the cellular system can attain the degenerated stationary state, leaving the attractor with a fractal dimension directly or undergoing diversification into many attractors and going through the areas of deterministic chaos. Since evolution of the cellular system is driven by the cooperative dynamic processes, as reflected by the changes of the mean fractal dimension between the intervals of the Gompertzian curve, it is likely that cells differentiate into neurons and create synapses with a conjugated probability and non-Gaussian distribution rather than with the classical probability and the Gaussian distribution. These findings can help to optimize features of artificial neural networks. They also define a simple in vitro biological model for biophysical and biochemical studies on natural neural networks.
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Journal Article•
TL;DR: The emerging discipline known as chaos theory is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for ''chaos'' as it applies to general dynamical systems.
Abstract: Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.
85 citations
TL;DR: The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found.
Abstract: The origin of the Gompertz function G(t)=G(0)e(b/a(1-e(-at))) widely applied to fit the biological and medical data, particularly growth of organisms, organs, and tumors is analyzed. It is shown that this function is a solution of a time-dependent counterpart of the Schrodinger equation for the Morse oscillator with anharmonicity constant equal to 1. The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found. These are eigenstates of the annihilation operator identified with the operator of growth, whereas eigenstates of the creation operator represent the Gompertzian states of regression. The coherent formation of the specific growth patterns in the Gompertzian systems appears as a result of the nonlocal long-range cooperation between the microlevel (the individual cell) and the macrolevel (the system as a whole).
49 citations
TL;DR: A topological inevitability of some developmental events through the use of classical topological concepts is discussed and a topological imperative as a certain set of topological rules that constrains and directs embryogenesis is revealed.
Abstract: The review presents a topological interpretation of some morphogenetic events through the use of well-known mathematical concepts and theorems Spatial organization of the biological fields is analyzable in topological terms Topological singularities inevitably emerging in biological morphogenesis are retained and transformed during pattern formation It is the topological language that can provide strict and adequate description of various phenomena in developmental and evolutionary transformations The relationship between local and global orders in metazoan development, ie, between local morphogenetic processes and integral developmental patterns, is established A topological inevitability of some developmental events through the use of classical topological concepts is discussed This methodology reveals a topological imperative as a certain set of topological rules that constrains and directs embryogenesis A breaking of spatial symmetry of preexisting pattern plays a critical role in biological morphogenesis in development and evolution
34 citations
Cites background from "Neuronal differentiation and synaps..."
...The quasifractal structure of neurons appears to correlate with chaotic processes in the nervous system (Goldberger et al., 1990; Goldberger, 1997, 2006; Waliszewski and Konarski, 2002)....
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01 Jan 2005
TL;DR: The Gompertz function as discussed by the authors describes global dynamics of many natural processes including growth of normal and malignant tissues, and the fractal structure of time-space is a prerequisite condition for the coupling between time and space.
Abstract: The Gompertz function describes global dynamics of many natural processes including growth of normal and malignant tissues. On one hand, the Gompertz function defines a fractal. The fractal structure of time-space is a prerequisite condition for the coupling and Gompertzian growth. On the other hand, the Gompertz function is a probability function. Its derivative is a probability density function. Gompertzian dynamics emerges as a result of the co-existence of at least two antagonistic processes with the complex coupling of their probabilities. This dynamics implicates a coupling between time and space through a linear function of their logarithms. The spatial fractal dimension is a function of both scalar time and the temporal fractal dimension. The Gompertz function reflects the equilibrium between regular states with predictable dynamics and chaotic states with unpredictable dynamics; a fact important for cancer chemoprevention. We conclude that the fractal-stochastic dualism is a universal natural law of biological complexity.
28 citations
Cites background from "Neuronal differentiation and synaps..."
...It was suggested that at least two different dynamic processes determine dynamics of tumor growth [ 3 , 4, 8, 9]. If so, a combination of two such processes should generate the sigmoid Gompertz curve....
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...Dynamics of a variety of natural phenomena, such as magnetic hysteresis [1], kinetics of enzymatic reactions, (e.g., PCR), oxygenation of hemoglobin, intensity of photosynthesis as a function of CO2 concentration (reviewed in [2]), drug dose-response curve [ 3 ], dynamics of growth, (e.g., bacteria, normal eukaryotic organisms, and cancer) [3-8] is described by the universal sigmoidal function of time known as the Gompertz function (see ......
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...... intensity of photosynthesis as a function of CO2 concentration (reviewed in [2]), drug dose-response curve [3], dynamics of growth, (e.g., bacteria, normal eukaryotic organisms, and cancer) [3-8] is described by the universal sigmoidal function of time known as the Gompertz function (see equation 1). A similar function describes a rate of differentiation of the aggregated cancer cells as a function of retinoid concentration [ 3 ]....
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...From equation 1 and equation 15, and from the fact that the Gompertz function is a fractal [ 3 , 8], we get equation 16:...
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...Dynamics of a variety of natural phenomena, such as magnetic hysteresis [1], kinetics of enzymatic reactions, (e.g., PCR), oxygenation of hemoglobin, intensity of photosynthesis as a function of CO2 concentration (reviewed in [2]), drug dose-response curve [3], dynamics of growth, (e.g., bacteria, normal eukaryotic organisms, and cancer) [ 3-8 ] is described by the universal sigmoidal function of time known as the Gompertz function (see ......
[...]
TL;DR: The Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential, which indicates that distribution of intrasystemic forces is both non-linear and asymmetric.
Abstract: The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.
27 citations
References
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Book•
01 Jan 1992
TL;DR: Jacek M. Zurada is a Professor with the Electrical and Computer Engineering Department at the University of Louisville, Kentucky and has published over 350 journal and conference papers in the areas of neural networks, computational intelligence, data mining, image processing and VLSI circuits.
Abstract: Jacek M. Zurada received his MS and Ph.D. degrees (with distinction) in electrical engineering from the Technical University of Gdansk, Poland. Since 1989 he has been a Professor with the Electrical and Computer Engineering Department at the University of Louisville, Kentucky. He was Department Chair from 2004 to 2006. He has published over 350 journal and conference papers in the areas of neural networks, computational intelligence, data mining, image processing and VLSI circuits. INTRODUCTION TO ARTIFICIAL NEURAL SYSTEMS
2,883 citations
01 Dec 1992
TL;DR: The emerging discipline known as chaos theory is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for "chaos" as it applies to general dynamical systems.
Abstract: : The emerging discipline known as "chaos theory" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for "chaos" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.
1,220 citations
794 citations
TL;DR: No evidence for retinoic acid toxicity is found, suggesting that the effect of the drug was to induce the development of neurons and glia rather than to select against cells differentiating along other developmental pathways.
Abstract: Murine embryonal carcinoma cells can differentiate into a varied spectrum of cell types. We observed the abundant and precocious development of neuronlike cells when embryonal carcinoma cells of various pluripotent lines were aggregated and cultured in the presence of nontoxic concentrations of retinoic acid. Neuronlike cells were also formed in retinoic acid-treated cultures of the embryonal carcinoma line, P19, which does not differentiate into neurons in the absence of the drug. The neuronal nature of these cells was confirmed by their staining with antiserum directed against neurofilament protein in indirect immunofluorescence experiments. Retinoic acid-treated cultures also contained elevated acetylcholinesterase activity. Glial cells, identified by immunofluorescence analysis of their intermediate filaments, and a population of fibroblastlike cells were also present in retinoic acid-treated cultures of P19 cells. We did not observe embryonal carcinoma, muscle, or epithelial cells in these cultures. Neurons and glial cells appeared in cultures exposed to retinoic acid for as little as 48 h. We found no evidence for retinoic acid toxicity, suggesting that the effect of the drug was to induce the development of neurons and glia rather than to select against cells differentiating along other developmental pathways.
785 citations