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Clark KB. Neural Field Continuum Limits and the Structure–Function Partitioning of Cognitive–Emotional Brain Networks. BIOLOGY 2023; 12:biology12030352. [PMID: 36979044 PMCID: PMC10045557 DOI: 10.3390/biology12030352] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Revised: 01/07/2023] [Accepted: 02/13/2023] [Indexed: 02/25/2023]
Abstract
In The cognitive-emotional brain, Pessoa overlooks continuum effects on nonlinear brain network connectivity by eschewing neural field theories and physiologically derived constructs representative of neuronal plasticity. The absence of this content, which is so very important for understanding the dynamic structure-function embedding and partitioning of brains, diminishes the rich competitive and cooperative nature of neural networks and trivializes Pessoa’s arguments, and similar arguments by other authors, on the phylogenetic and operational significance of an optimally integrated brain filled with variable-strength neural connections. Riemannian neuromanifolds, containing limit-imposing metaplastic Hebbian- and antiHebbian-type control variables, simulate scalable network behavior that is difficult to capture from the simpler graph-theoretic analysis preferred by Pessoa and other neuroscientists. Field theories suggest the partitioning and performance benefits of embedded cognitive-emotional networks that optimally evolve between exotic classical and quantum computational phases, where matrix singularities and condensations produce degenerate structure-function homogeneities unrealistic of healthy brains. Some network partitioning, as opposed to unconstrained embeddedness, is thus required for effective execution of cognitive-emotional network functions and, in our new era of neuroscience, should be considered a critical aspect of proper brain organization and operation.
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Affiliation(s)
- Kevin B. Clark
- Cures Within Reach, Chicago, IL 60602, USA;
- Felidae Conservation Fund, Mill Valley, CA 94941, USA
- Campus and Domain Champions Program, Multi-Tier Assistance, Training, and Computational Help (MATCH) Track, National Science Foundation’s Advanced Cyberinfrastructure Coordination Ecosystem: Services and Support (ACCESS), https://access-ci.org/
- Expert Network, Penn Center for Innovation, University of Pennsylvania, Philadelphia, PA 19104, USA
- Network for Life Detection (NfoLD), NASA Astrobiology Program, NASA Ames Research Center, Mountain View, CA 94035, USA
- Multi-Omics and Systems Biology & Artificial Intelligence and Machine Learning Analysis Working Groups, NASA GeneLab, NASA Ames Research Center, Mountain View, CA 94035, USA
- Frontier Development Lab, NASA Ames Research Center, Mountain View, CA 94035, USA & SETI Institute, Mountain View, CA 94043, USA
- Peace Innovation Institute, The Hague 2511, Netherlands & Stanford University, Palo Alto, CA 94305, USA
- Shared Interest Group for Natural and Artificial Intelligence (sigNAI), Max Planck Alumni Association, 14057 Berlin, Germany
- Biometrics and Nanotechnology Councils, Institute for Electrical and Electronics Engineers (IEEE), New York, NY 10016, USA
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Firmbach M, Bäcker A, Ketzmerick R. Partial barriers to chaotic transport in 4D symplectic maps. CHAOS (WOODBURY, N.Y.) 2023; 33:013125. [PMID: 36725645 DOI: 10.1063/5.0130682] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2022] [Accepted: 12/12/2022] [Indexed: 06/18/2023]
Abstract
Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase space. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map, we establish a partial barrier based on what we call a cantorus-NHIM-a normally hyperbolic invariant manifold with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM, we utilize periodic NHIMs to quantify the corresponding flux.
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Affiliation(s)
- Markus Firmbach
- Institut für Theoretische Physik and Center for Dynamics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Arnd Bäcker
- Institut für Theoretische Physik and Center for Dynamics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Roland Ketzmerick
- Institut für Theoretische Physik and Center for Dynamics, Technische Universität Dresden, 01062 Dresden, Germany
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Ertl J, Marquardt M, Schumacher M, Rommel P, Main J, Bayer M. Signatures of Exciton Orbits in Quantum Mechanical Recurrence Spectra of Cu_{2}O. PHYSICAL REVIEW LETTERS 2022; 129:067401. [PMID: 36018667 DOI: 10.1103/physrevlett.129.067401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2021] [Revised: 02/10/2022] [Accepted: 07/14/2022] [Indexed: 06/15/2023]
Abstract
The seminal work by Kazimierczuk et al. [Nature 514, 343 (2014)10.1038/nature13832] has shown the existence of highly excited exciton states in a regime, where the correspondence principle is applicable and quantum mechanics turns into classical mechanics; however, any interpretation of exciton spectra based on a classical approach to excitons is still missing. Here, we close this gap by computing and comparing quantum mechanical and semiclassical recurrence spectra of cuprous oxide. We show that the quantum mechanical recurrence spectra exhibit peaks, which, by application of semiclassical theories and a scaling transformation, can be directly related to classical periodic exciton orbits. The application of semiclassical theories to exciton physics requires the detailed analysis of the classical exciton dynamics, including three-dimensional orbits, which strongly deviate from hydrogenlike Keplerian orbits. Our findings illuminate important aspects of excitons in semiconductors by directly relating the quantum mechanical band structure splittings of excitons to the corresponding classical exciton dynamics.
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Affiliation(s)
- Jan Ertl
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
- Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany
| | - Michael Marquardt
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Moritz Schumacher
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Patric Rommel
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Jörg Main
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Manfred Bayer
- Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany
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Onken F, Lange S, Ketzmerick R, Bäcker A. Bifurcations of families of 1D-tori in 4D symplectic maps. CHAOS (WOODBURY, N.Y.) 2016; 26:063124. [PMID: 27368789 DOI: 10.1063/1.4954024] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The regular structures of a generic 4d symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1d-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations, no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example, we study two coupled standard maps by visualizing the elliptic and hyperbolic 1d-tori in a 3d phase-space slice, local 2d projections, and frequency space. The observed bifurcations are consistent with the analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.
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Affiliation(s)
- Franziska Onken
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Steffen Lange
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
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Lange S, Richter M, Onken F, Bäcker A, Ketzmerick R. Global structure of regular tori in a generic 4D symplectic map. CHAOS (WOODBURY, N.Y.) 2014; 24:024409. [PMID: 24985463 DOI: 10.1063/1.4882163] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
For the case of generic 4d symplectic maps with a mixed phase space, we investigate the global organization of regular tori. For this, we compute elliptic 1-tori of two coupled standard maps and display them in a 3d phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4d phase space. The 1-tori occur in two types of one-parameter families: (α) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (β) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincaré-Birkhoff theorem for 2d maps, or (ii) the family may form large bends. In combination, these results allow for describing the hierarchical structure of regular tori in the 4d phase space analogously to the islands-around-islands hierarchy in 2d maps.
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Affiliation(s)
- S Lange
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - M Richter
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - F Onken
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - A Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - R Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
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Richter M, Lange S, Bäcker A, Ketzmerick R. Visualization and comparison of classical structures and quantum states of four-dimensional maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022902. [PMID: 25353540 DOI: 10.1103/physreve.89.022902] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2013] [Indexed: 06/04/2023]
Abstract
For generic 4D symplectic maps we propose the use of 3D phase-space slices, which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example, we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods, such as 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the comparison of Husimi functions of eigenstates of 4D maps with classical phase-space structures. This confirms the semiclassical eigenfunction hypothesis for 4D maps.
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Affiliation(s)
- Martin Richter
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Steffen Lange
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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Koh YW, Takatsuka K. Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion. Phys Rev E 2008; 76:066205. [PMID: 18233902 DOI: 10.1103/physreve.76.066205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2007] [Revised: 08/22/2007] [Indexed: 11/07/2022]
Abstract
Methods to search for periodic orbits are usually implemented with the Newton-Raphson type algorithms that extract the orbits as fixed points. When used to find periodic orbits in flows, however, many such approaches have focused on using mappings defined on the Poincaré surfaces of section, neglecting components perpendicular to the surface of section. We propose a Newton-Raphson based method for Hamiltonian flows that incorporates these perpendicular components by using the full monodromy matrix. We investigated and found that inclusion of these components is crucial to yield an efficient process for converging upon periodic orbits in high dimensional flows. Numerical examples with as many as nine degrees of freedom are provided to demonstrate the effectiveness of our method.
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Affiliation(s)
- Yang Wei Koh
- Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902, Tokyo, Japan.
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Shojiguchi A, Li CB, Komatsuzaki T, Toda M. Fractional behavior in multidimensional Hamiltonian systems describing reactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:056205. [PMID: 18233737 DOI: 10.1103/physreve.76.056205] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2007] [Indexed: 05/25/2023]
Abstract
The fractional behavior is presented for a minimal Hamiltonian system of three degrees of freedom which describes reaction processes. The model has a double-well potential where the Arnold web within the well is nonuniform. The survival probability within the well exhibits power law decay in addition to exponential decay. Moreover, the trajectories of the power law decay exhibit 1/f spectra and subdiffusion in the action space, while the trajectories of the exponential decay show Lorentzian spectra and normal diffusion. Transient features of these statistical properties reveal the dynamical connection, i.e., how trajectories approach to (depart from) the Arnold web from (to) the region around the potential saddle. In particular, a wavelet analysis enables us to extract transient features of the resonances. Based on these results, we suggest that resonance junctions including higher-order resonances are important for understanding the dynamical origins of the fractional behavior in reaction processes.
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Affiliation(s)
- Akira Shojiguchi
- Physics Department, Nara Women's University, Kita-Uoya-Nishimachi, Nara 630-8506, Japan.
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Efstathiou K, Sadovskií D, Zhilinskií B. Classification of perturbations of the hydrogen atom by small static electric and magnetic fields. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.1843] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000
Physica
142
, 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone.
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Affiliation(s)
- K Efstathiou
- Instituut voor Wiskunde en Informatica, Rijksuniversiteit GroningenGroningen 9747 AC, The Netherlands
| | - D.A Sadovskií
- UMR du CNRS 8101, Université du Littoral59140 Dunkerque, France
| | - B.I Zhilinskií
- UMR du CNRS 8101, Université du Littoral59140 Dunkerque, France
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