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Index

3-body problem, 646, 649

Abelian topological ground state, 547accelerometer, 617achieved, 25, 378, 412across, 133action initiation, 395action of a Lie group, 594action potential, 45activation derivative, 99, 237activation dynamics, 100, 237, 400activator, 231Adaline, 80Adams–Bashforth–Moulton integrator,

612adaptive control, 47adaptive filter, 100, 237adaptive functional measure, 25adaptive Kalman filtering, 47adaptive NLS, 20adaptive quantum computation, 183adaptive quantum systems, 19adaptive resonance theory, 104adaptive sensory-motor control, 105adiabatic evolution, 330adiabatic exchange, 513adiabatic invariant, 513adiabatic process, 513adiabatic transport, 185adjoint, 594adjoint functors, 575adjoint representation, 598admissibility condition, 625admissibility constant, 625

admissible vector, 196

afferent nerves, 57

affine transformation, 660

Airy equation, 314

algorithmic approach, 401

Ambrose–Singer theorem, 265

amplitude, 171, 622

analogous ART2 system, 2

anatomical MRI, 71

Andronov, 654

Andronov–Hopf bifurcation, 654

angular frequency, 622

ANN dynamorphism, 80

ANN evolution, 79

Anosov diffeomorphism, 655

Anosov flow, 655

Anosov map, 655

anticommutation relations, 177

antikink, 309

anyonic excitations, 185

anyonic statistics, 16

anyons, 14, 514, 515

area-preserving map, 673

Arnold cat map, 655

array of N Josephson junctions, 484

arrow of time, 361

artificial neural network, 1

associative composition, 567

associative memory, 48

associativity of morphisms, 568

asymptotic stability, 322

atlas, 591

atmospheric convection, 665

V.G. Ivancevic, T.T. Ivancevic, Quantum Neural Computation,Intelligent Systems, Control and Automation: Science and Engineering 40,DOI 10.1007/978-90-481-3350-5, c© Springer Science+Business Media B.V. 2010

914 Index

attack function, 105attractor, 664auditory cortex, 44auto- and cross-correlations, 123auto-overlap, 240autocatalator, 693autonomic nervous system, 54average energy, 257averaging decision mechanics, 402Axiom-A systems, 656axon, 51

Backlund transform, 290backpropagation, 86, 400Baker–Campbell–Haussdorf formula,

593bandlimited functions, 624basic biomechanical unit, 134basin of attraction, 664, 673basket cell, 147battle of the sexes, 716behavioral composition, 415Bell state, 5Bell’s inequality, 438Bell’s theorem, 438Belouzov–Zhabotinski reaction, 701Bernoulli map, 655Bernoulli shift dynamics, 259, 261Bernoulli systems, 653Berry phase, 520best response, 716Betti numbers, 570bi-partite quantum system, 443bi-stable perception, 37bidirectional associative memory, 244bifurcation diagram, 671bifurcation point, 676billiard, 653binary (dyadic) dilation, 633binary (dyadic) position, 633binary signals, 99binding by assembly, 36binding by convergence, 36binding by synchrony, 36binding problem, 36binding unit, 36binocular rivalry model, 39bio-diversity, 670biomechanics, 617

biomorph, 695biomorphic systems, 695biophotons, 39bipolar neurons, 52bipolar signals, 99Birkhoff, 651, 652Birkhoff curve shortening flow, 652Bloch, 70Bloch sphere, 5block entropy, 679Bohr–Sommerfeld quantization, 513Bohr’s correspondence principle, 268Boltzmann, 647Boltzmann’s constant, 22Bose–Einstein condensate, 18, 36, 313Bose–Einstein condensation, 512Bose–Hubbard model, 548bosons, 512bound states, 319boundary, 600boundary operator, 601bounded-energy solutions, 339bounded-energy strong solution, 339Boussinesq equation, 301BQP, 181braid group, 16, 185, 514, 589braid relation, 17braid theory, 14braided monoidal category, 580braids, 14brain plasticity, 46, 49Bratteli diagram, 216breather solution, 326, 341, 343Brownian dynamics, 392Burgers dynamical system, 264butterfly effect, 666

cache, 4canonical ensemble, 22canonical micro-circuits, 32Cantor set, 658capacity dimension, 694, 704carrying capacity, 669Cartwright, 654Cash–Karp modification, 774, 783Cauchy problem for KdV, 307cavity resonators, 39central biodynamic adjunction, 49central nervous system, 53

Index 915

central processing unit, 3cerebellar cortex, 43cerebellum, 69cerebral cortex, 31, 43cerebral hemispheres, 68chaos, 649chaos control, 702chaos theory, 258, 651chaotic behavior, 258Chapman–Kolmogorov equation, 394Chapman–Kolmogorov integro-

differential equation, 395,429

characteristic Lyapunov exponents, 703charging energy, 498chemical kinetics, 693Chern–Simons theory, 518Christoffel symbols, 592Chua–Matsumoto circuit, 701Chua’s circuit, 691classical brain, 20classical perceptron, 21classical solutions, 315classical strategies, 451climbing fibre, 146closed, 600closed form, 600closure property, 601CNOT (Controlled NOT), 440CNOT-gate, 17co-active neuro-fuzzy inference system,

94co-existence of alternatives, 157coarse system, 654coarse-graining, 22, 605coboundary, 600cocycle, 600code, 361code subspace, 185cognition, 414cognitive binding problem, 35cognitive information processing, 104Cohen–Grossberg activation equations,

2, 105Cohen–Grossberg theorem, 3, 105coherent quantum oscillations, 481cohomological, 201cohomology class, 203, 603cohomology group, 603

Coiflet wavelet, 121, 122collapse of the tunneling state, 35combining noisy sensor outputs, 611commutator, 172compact attractor, 337compact support, 620complete synchronization, 106complex vector space, 157complex-valued ANNs, 97complex-valued Gaussian, 111complex-valued order parameter, 112complexity class BQP, 6complexity measure, 26complexity of an algorithm, 440computational basis states, 7computational summand, 186condensed, 360conditional Lyapunov exponents, 113configuration integral, 23, 605conformal power, 333conformal transformations, 214conformal z-map, 694conjugate gradient method, 89conjugate transpose, 594connection, 174connectionism, 1, 30connectionist, 81connectionist brain theory, 30consciousness, 39consequences, 402conservation law, 668conservative Hamiltonian chaos, 423conserved energy, 331, 333conserved momentum, 332continual-time systems, 611continuity equation, 176, 204continuous Fourier transform, 624continuous Hopfield circuit, 2continuous Hopfield network, 239continuous limit, 308continuous model for neural activity in

cortical structures, 228continuous wavelet transform, 195, 196,

623controlled phase shifter gates, 7convective Benard fluid flow, 665convex optimization problem, 95convolution, 628convolution theorem, 622

916 Index

Cooley–Tukey algorithm, 626Cooper pair box, 481Cooper pairs, 479cooperative game theory, 713coordination game, 716corrector, 612cost function, 612counter-propagation network, 3covariant derivative, 207covariant Dirac equation, 178covariant force law, 420Crick–Koch binding hypothesis, 49critical, 327critical current, 482, 483critical dynamics, 19critical regularity, 329critical slowing down, 244critique of pure reason, 36cross-correlation, 622cross-overlap, 240crowd cognitive controller, 422cubic NLS, 327cumulative distribution function, 391current state, 398, 411, 412current–phase relation, 485curvature, 174curved arrows, 567cycle, 600, 681cyto-sceletal networks, 350

damped pendulum, 641, 685data compression, 628Daubechies 4 tap wavelet, 631Daubechies wavelet, 121de-noising filter, 124decision problem, 182decoherence, 159, 524deep nuclei, 147definition of a Nash equilibrium, 719defocussing NLS, 327degree of anisotropy, 76delta-rule, 84dendrites, 50dendron, 33depolarization, 59desired, 25, 378, 412desired state, 398, 411deterministic chaos, 665deterministic chaotic system, 647, 648

deterministic diffusion, 283Deutsch algorithm, 441diencephalon, 69diffeomorphism, 658difference equation, 670different, 462differential Hebbian law, 105diffusion anisotropy, 73diffusion coefficient, 71diffusion equation, 223diffusion flux, 223diffusion magnetic resonance imaging,

71diffusion MRI, 71diffusion tensor, 71diffusion-weighted images, 74digital signal processing, 624, 634dilations, 633Dirac, 177Dirac bra-ket notation, 7Dirac equation, 177Dirac interaction picture, 163Dirac-delta distribution, 624Dirac’s perturbation method, 165directional derivative, 424discrete, 120discrete Amari equation, 228discrete breathers, 500discrete Fourier transform, 170, 620,

626discrete Laplacian, 496discrete sine-Gordon equation, 494discrete wavelet transform, 627discrete-time models, 670discrete-time steps, 651discrete-time systems, 611dispersion, 316dispersion relation, 317dispersive wave equations, 313dissipation, 366dissipative chaos, 423dissipative solitons, 219distorted plane waves, 319distribution function, 391divide and conquer algorithm, 626DNA computers, 8dominant strategy, 448dominant strategy equilibrium, 715Donaldson–Witten theory, 201

Index 917

down-sampling operator, 628driven pendulum, 686dual series, 330dual system, 612Duffing, 654Duffing oscillator, 689Duffing–Van der Pol equation, 689dyadic translations, 633dynamical homeostasis, 19dynamical invariant, 653dynamical planar graph, 421dynamical system, 647, 675dynamics, 639dynamics and topology of brain

networks, 26

early selection, 401echo attenuation, 75echoplanar imaging, 75economics, 720eddy, 261edge of chaos, 19, 219eigenfunction, 320Einstein–Hilbert functional, 251elastic pendulum, 692electric current density, 175electro-weak interaction, 597embedding methods, 89encoding, 139energy, 339energy space, 339energy surfaces, 316ensemble code, 126entangled attractors, 423entangled quantum states, 438entangled state, 365entanglement, 5, 9, 438entangling power, 440enteric nervous system, 55entropic motor control, 386entropy, 257, 513, 653EPR, 438equilibrium, 513, 715equilibrium in dominant strategies, 448equipartition of energy, 292equivalence relation, 472ergodic hypothesis, 647, 652ergodic hypothesis of Boltzmann, 647error-correcting codes, 12

error-correction procedures, 14escape rate, 684Euclidean chart, 590Euclidean image, 590Euclidean path integral, 24Euler, 646Euler characteristic, 649Euler–Lagrange equation, 274, 331Euler–Poincare characteristic, 603every game with a finite number of

players and finite number ofstrategies has at least one mixedstrategy Nash equilibrium, 717

evolution PDE, 329evolutionary biology, 726evolutionary computation, 95exact, 600exact form, 600excitation contraction coupling, 377excitatory, 62existence & uniqueness theorems for

ODEs, 675exocytosis, 33exponentially faster, 522extended Kalman filter, 89, 615extended observable system, 612extensive form, 714

Fabry–Perot interferometer, 539fan-beam surveillance radar, 607Faraday, 175Faraday tensor, 174fast Fourier transform, 623, 627fast Hadamard transform, 7fast wavelet transform, 627, 633father wavelet, 631fault-tolerant computation, 12fault-tolerant quantum computation,

511feedforward neural network, 80Feigenbaum cascade, 667Feigenbaum constant, 667Feigenbaum number, 671fermions, 512Feschbach resonance, 551Feynman diagrams, 168Feynman path integral, 168Feynman–Vernon formalism, 363Fibonacci anyons, 17

918 Index

Fibonacci(2n), 186

field, 173, 639

field operator, 173

figure-ground vase, 37

filter bank, 629

filtering, 614

finite Fourier transform, 626

first quantization, 160

first-order-search methods, 88

Fisher’s equation, 225

FitzHough–Nagumo model, 114

FitzHugh–Nagumo neuron, 233

fixed-point, 657, 662, 664

fixed-point attractors, 228

flame front, 264

Floquet exponents, 507

flow, 262, 659, 675

flow pattern, 642

flow-line, 423

fluctuation, 258

flux-flow regime, 498

focal equilibrium, 457

focal point effect, 456

focusing NLS, 327

Fokker–Planck equation, 224, 244, 394

folding, 657

for games without dominant strategies,716

forced nonlinear oscillators, 642

forced Van der Pol oscillator, 687

forgetful memories, 241

formalism of jet bundles, 642

four reasonable requirements, 719

Fourier analysis, 271

Fourier basis functions, 620

Fourier transform, 194, 319, 325, 619,622, 625

Fourier windowing, 630

FPU-paradox, 292

FPU-recurrence, 291

fractal, 631, 694

fractal attractor, 666

fractal dimension, 626, 664

fractal pattern, 671

fractal set, 657

fractional dimension, 694

fractional quantum Hall effect, 16, 18,184, 512

fractional quantum Hall ground states,185

fractionally charged quasi-particles, 16frame functions, 621free wave equation, 313frequency domain, 622frequency spectrum, 620frequency-to-voltage converter, 482Frolich waves, 37frontal lobe, 43frustrated XY models, 489fully recurrent networks, 96functional analysis, 621functional equation, 624functional manifold, 264functional MRI, 71fundamental group, 589fundamental representation, 597fusion, 516fusion channels, 516fusion rule, 516

Galilean invariance, 334, 338game matrix, 714game situations, 468game theory, 713game tree, 714games, 713gamma oscillations, 115gates, 440gauge field, 173, 174gauge theory, 173gauge transformation, 173gauge-invariant phase difference, 501Gauss–Bonnet formula, 246Gaussian distribution, 283Gaussian functional integral, 204Gaussian noise, 613Gaussian random vectors, 283Gaussian soliton, 255Gaussian wave-packet, 778Gaussian white noise, 118, 634Gell-Mann matrices, 598general sense, 700generalized feedforward network, 92generalized Gaussian, 100generalized Hebbian rule, 110generalized Henon map, 673

Index 919

generalized Korteveg–de Vries (gKdV)equation, 314

generalized solution, 700generalized synchronization, 106generic system evolution, 40genetic algorithm, 87, 95genetic control, 95geometrization conjecture, 246George Birkhoff, 647Giaever tunneling junction, 481Gibbs ensemble, 166Gibbs measure, 23Gibbs statistical density function, 166Ginzburg–Landau equation, 263Glauber dynamics, 241glia, 53globally-coupled Kuramoto model, 499globular cluster, 672Goldstone theorem, 361Golgi cells, 147Golgi tendon organs, 144Gottesman–Knill theorem, 441governing equilibrium dynamics, 713gradient descent method, 87gradient information, 82gradient of the performance surface, 82grand conjecture, 341graph, 591Green functions, 194grey matter, 43ground state, 329group action, 586group homomorphism, 596group identity element, 593group inversion, 593group multiplication, 593group velocity, 317groups, 469growth rate, 669growth-rate functions, 726gyroscope, 617

H-cobordism theorem, 656Haar wavelet, 620, 633Haar wavelet transform, 627Hadamard, 653Hadamard gate, 440Hadamard transform, 7Hadamard transform gates, 7

halo orbit, 653Hamilton, 646Hamiltonian, 315, 338Hamiltonian path problem, 8Hamiltonian system, 673Hamming distance, 18Hamming hypercube, 239harmonic, 202harmonic oscillator, 778Harr wavelet, 121Hartman–Grobman theorem, 243Hayashi, 654heat equation, 773heat flow, 513Hebb rules, 47Hebbian innovation, 242Hebbian learning, 374Hebbian theory, 63Hecht–Nielsen counterpropagation

network, 102Heisenberg model, 548Heisenberg picture, 163Heisenberg representation, 165Heisenberg uncertainty principle, 387,

621, 628Heisenberg uncertainty relation, 162Henon map, 652, 672, 712Henon strange attractor, 672her, 456Hermitian, 595Hermitian inner (scalar) product, 158Hermitian matrix, 595Hermitian transpose, 594heuristic approach, 401high-pass filter, 628high-temperature superconductivity,

512Hilbert basis, 633Hilbert space, 95, 157, 339, 621, 647Hodge star, 201Hodgkin–Huxley, 60Hodgkin–Huxley equation, 2Hodgkin–Huxley model, 114Hodgkin–Huxley neural model, 33Holder regularity index, 338holographic hypothesis, 360holonomy, 184homeostasis, 67homoclinic point, 650, 659, 663

920 Index

homoclinic tangle, 649, 662homological algebra, 570homology class, 603homology group, 570homomorphism, 589homotopy methods, 89Hooke’s law, 291Hopf, 654Hopf bifurcation, 232Hopf-like bifurcation, 691Hopfield model, 47Hopfield synaptic matrix, 240Hubbard model, 548, 550human biodynamics engine, 381, 382human brain, 43human heart beat and respiration, 106human memory, 138human vestibular system, 617hurricane, 260hybrid dynamical system of variable

structure, 699hybrid systems, 700hyperbolic fixed-point, 664hyperbolic system, 676hyperbolic tangent threshold activation

functions, 81hyperpolarization, 59

ideal, 125, 461image, 600imitative, 726impulse response, 628in the sense of Filippov, 700independent component analysis

networks, 94individual, 469inertial sensors, 617inferior cerebellar peduncle, 147information, 702information theory, 624, 654inhibitor, 231inhibitory, 62inner product, 633inner-product space, 7innovation, 25instability, 642integrable function, 621integrate-and-fire model, 114integration, 675

intention, 397intention formation, 395inter-spike-interval, 115interior, 724interpolation, 624intuition, 648invariant set, 657, 660inverse, 119inverse discrete Fourier transform, 626inverse DWT, 120inverse dynamic analysis of human

movement, 617inverse Fourier transform, 622inverse scattering, 323inverse scattering problem, 326inverted driven pendulum, 692inviscid Burgers’ equation, 305ion trap, 6irreducible representations, 513Ising anyons, 517Ising Hamiltonian, 238, 240Ising model, 548Ising system, 800Ising-spin Hopfield network, 102isometry, 595iterated map, 651iterative maps, 676Ito stochastic integral, 393

Jacobi identity, 180, 210Jones polynomial, 17, 186, 534Jordan and Elman networks, 93Josephson constant, 482Josephson current, 479Josephson current–phase relation, 483Josephson effect, 479Josephson interferometer, 479Josephson junction, 478, 479Josephson junction ladder, 489Josephson tunneling, 479, 482Josephson voltage–phase relation, 483Josephson-junction quantum computer,

480

Kadanoff spin-blocking procedure, 195Kalman filter, 607, 611Kalman filtering problem, 614Kalman linear-quadratic regulator, 611Kalman regulator, 611

Index 921

Kalman–Bucy filter, 612Kalman-quaternion filter, 618Kaplan–Yorke dimension, 704Karhunen–Loeve covariance matrix,

101, 240KdV–Burgers’ equation, 307Kelvin inversion, 333Kepler, 649kernel, 596, 600kink, 309kink-phason resonance, 497Kitaev model, 548Klein–Gordon equation, 293knot theory, 14knowledge of the manifold, 36Kohonen continuous self organizing

map, 103Kohonen self-organizing map, 93Kolmogorov, 653, 655Kolmogorov–Arnold–Moser theorem,

268Kolmogorov–Sinai, 704Kolmogorov–Sinai entropy, 114, 271,

679, 702, 705Korteveg–de Vries (KdV) equation, 290,

311, 314Kraus operators, 450Kronecker delta, 633Krylov, 654Kuramoto model, 28, 109Kuramoto order parameter, 507Kuramoto–Sivashinsky equation, 307

ladder, 500lag error, 609lag synchronization, 106Lagrange, 646Lagrange’s points, 652Lagrangian, 176laminar flow, 259Landau gauge, 496Landau-level, 526Laplace–Beltrami operator, 221Laplace’s equation, 224Laplacian operator, 19largest eigen-diffusivity, 76largest Lyapunov exponent, 702Laughlin state, 530Lauterbur, 70

Lax pair equation, 324Lax pairs, 324learning dynamics, 100, 101, 237, 238learning rate, 25, 83, 378, 400, 412learning rate scheduling, 83least means square algorithm, 83Lebesgue integrability, 621Lebesgue integrable function, 622Lebesgue integral, 621Left, 78Levenberg–Marquardt algorithm, 89Levi-Civita connection, 428, 592Levi-Civita covariant derivative, 427,

592Levin–Wen model, 548Lie derivative, 698Lie group, 593Lie structure equations, 265Lie–Poisson bracket, 309life space foam, 396lifting scheme, 628light, 152limit cycle, 642, 664, 686linear, 611linear homotopy, 90linear homotopy ODE, 700linear homotopy segment, 700linear optimal filtering problem, 611linear Schrodinger equation, 313linear state feedback control law, 611linear superposition, 152links, 527Liouville equation, 394Liouville theorem, 355Lissajous curves, 693Littlewood, 654local Bernstein adaptation process, 411locally exact, 600locally-coupled Kuramoto model, 499,

508locally-optimal solution, 369logarithmically divergent solitons, 326logic gates, 14logical qutrit, 184logistic equation, 669logistic growth, 669logistic map, 670, 672, 706long-range connections, 508long-range correlation, 360

922 Index

long-range order, 480long-term consequences, 402long-term memory, 100, 138, 237Lorenz attractor, 699Lorenz equations, 693Lorenz mask, 666Lorenz system, 666, 672, 676Lotka–Volterra ensemble dynamics, 244low-pass filter, 628low-regularity well-posedness theory,

315Lyapunov, 654Lyapunov dimension, 704Lyapunov exponent, 271, 664, 678Lyapunov function, 102Lyapunov stability, 675

Mach–Zehnder interferometer, 156machine learning, 1magnetic resonance imaging, 70magnetometer, 618main direction of diffusivities, 76maintaining the action, 395Majorana fermion, 542Malthus model, 669Malthusian parameter, 669, 671Mandelbrot and Julia sets, 694manifold structure, 591Mansfield, 70map, 670map from strategy profiles to pay-offs,

714map sink, 664Markov assumption, 395, 429Markov chain, 392, 680Markov partitions, 655Markov process, 680Markov property, 23Markov stochastic process, 392, 429mass-action interpretation, 724master equation, 394match-based learning, 104matching pennies, 717matrices, 177matrix cost function, 88maximum entropy, 22, 99, 237maximum principle, 248, 255Maxwell, 175, 647Maxwell–Haken laser equations, 693

McCulloch–Pitts neurons, 80McCumber parameter, 504mean diffusivity, 75mean square error, 81mean-field theory, 18measurable functions, 623measurement equation, 613measurement noise, 613measurement sensitivity matrix, 613medulla oblongata, 69Meissner-states, 492method of lines, 345, 774, 783, 788methods of regularization, 194microtubules, 37, 349midbrain, 69middle cerebellar peduncle, 148millimetric image resolution, 71minimizing the error, 81Mittag–Leffler, 650mixed strategy, 454, 717modified Korteveg–de Vries (mKdV)

equation, 314modular feedforward networks, 93modular functors, 181momentum learning, 86momentum operator, 173monetary, 716Monte Carlo method, 672Moore–Penrose pseudoinverse, 240Moore–Read Pfaffian wave-function,

533Moore–Read state, 534Moore’s law, 181Morse, 653Morse theory, 653, 656mossy fibers, 148mother wavelet, 116, 119, 623, 625motion action principle, 398motor cortex, 44motor dynamorphism, 134motor nerve fibers, 54motor servo, 144motor-learning, 134Mott insulators, 549moves, 714multi-soliton, 341multi-soliton solutions, 344multi-spiral strange attractor, 691multilayer perceptron, 80, 90

Index 923

multipolar neurons, 52multisoliton states, 323mutual overlap, 240

Nash equilibrium, 448, 716, 724Nash’s program, 721nature, 714Navier–Stokes equations, 258, 263, 665neocortex, 31neocortical biophysics, 34neocortical dynamics, 33network of networks, 347neural action potential, 377neural attractor dynamics, 228neural communication, 45neural network, 47neural path integral, 377neural state-space, 244neural-networks complexity, 26neuro-computing, 1neurons, 50NeuroQuantology, 30new value, 25, 378, 398Newland transform, 628Newton, 645Newton–Raphson method, 89Newtonian mechanics, 673Newtonian method, 87Newtonian-like action, 398Newton’s Principia, 268NLS-flow, 337no-cloning theorem, 441Noetherian ring, 588Noether’s theorem, 316noise, 25, 378, 412non-Abelian anyons, 18, 511non-Abelian braiding statistics, 516non-Abelian quantum Hall states, 511non-Abelian statistics, 511non-autonomous 2D continuous

systems, 642non-autonomous system, 640non-compact Lie group, 344non-cooperative game theory, 713non-equilibrium phase transitions, 244non-wandering set, 658non-zero sum game, 448nonlinear, 127nonlinear Klein–Gordon equation, 335

nonlinear oscillators, 647nonlinear resonances, 342nonlinear Schrodinger equation (NLS),

20, 294, 311, 313, 327nonlinear wave equation, 331nonlinearity, 288normal, 714normal mode, 291normalization condition, 159normalized Ricci flow, 250normalized state, 158nuclear magnetic resonance, 6Nyquist range, 626

observable system, 612observables, 199observer design, 612occipital lobe, 43ODEs, 676Oja–Hebb learning rule, 100olfactory cortex, 44operator product expansion, 215optical cavity, 6optical neural networks, 8optimal answer, 454optimal estimator problem, 612optimal policy, 412orbit, 344, 659, 675, 681orbit Hilbert space, 169orbital stability, 322order parameter, 329order parameter equation, 243order parameters, 288orthogonal matrix, 595orthogonality, 158orthonormal wavelet, 633oscillator neural networks, 109overdamped junction, 500overdamped ladder, 509, 510overdamped limit, 483overlap, 240

parallel distributed processing, 1parameter space, 18parasympathetic nervous system, 55Pareto optimal, 449Pareto optimality, 719parietal lobe, 43partition function, 22, 205, 257, 655

924 Index

path that requires minimal memory,401

path-integral, 513pattern matching process, 104pattern vector, 21pattern-recognition, 83Pauli gate, 441Pauli matrices, 598Pauli principle, 512Pauli sigma matrices, 178Pavlov, 63Pavlov’s conditional-reflex learning, 63pay-off, 714pdf, 375peduncles, 147penny flipover, 471perception energy, 39perceptron, 83perceptron learning algorithm, 21perfect & imperfect Nash equilibria, 721performance surface, 82period-doubling bifurcation, 667, 671,

707periodic orbit, 270, 650, 662periodic orbit theory, 264, 683periodic phase synchronization, 106periodic solutions, 642peripheral nerve fibers, 54peripheral nervous system, 53permutation group, 513perturbation problem, 329perturbation theory, 267, 329petite conjecture, 341Pfaffian, 533Pfaffian state, 534Pfaffian wave-function, 534phase coherence, 480phase difference, 107, 483, 496phase invariance, 338phase point, 641phase portrait, 642phase space, 268, 271phase synchronization, 106phase transitions, 604phase velocity, 317phase-flow, 641, 675phase-space flow, 651phases, 395photoelectric effect, 151

photons, 152physically-controllable systems of

nonlinear oscillators, 499Pickover’s biomorphs, 696Planck’s constant, 154Planck’s quantum hypothesis, 151plane wave, 317plat closure, 187Poincare, 647Poincare conjecture, 220, 656Poincare map, 673Poincare section, 271, 649, 651, 672,

682, 689Poincare–Bendixson theorem, 642, 648Poincare–Hopf index theorem, 648point at infinity, 153Poisson equation, 166Poisson evolution equation, 309Poisson Lie groups, 308Poisson manifold, 308, 309poly-time algorithm, 186polynomial-time algorithm, 182pons, 69Pontryagin, 654population code, 126population models, 668population-statistic, 722position and velocity innovations, 609positive leading Lyapunov exponent,

703postsynaptic potential, 238Prandtl number, 666predictability time, 703prediction, 614predictor, 612principal component analysis, 116principal component analysis networks,

94principle of cognitive optical illusions,

38Prisoner’s dilemma, 442, 714pristine, 508probability amplitude, 12, 159, 165, 166probability density, 195probability density function, 23, 373probability distribution function, 503process equation, 613process noise, 613processing speed, 398, 401

Index 925

product topology, 262protozoan morphology, 695pruning, 681pseudoconformal symmetry, 334pseudolocality theorem, 256psycho-physical crowd dynamics, 413punctures, 185Purcell, 70pure strategy, 454

QNN of QNNs, 347quadratic cost function, 611quadrature mirror filter, 628quantum action functional, 272quantum algorithm, 441quantum bits, 170quantum brain, 20, 374quantum chaos, 267, 268quantum chaos systems, 281quantum circuit, 7, 17, 171, 181quantum computation, 4, 522quantum computer, 5, 171, 479quantum computer algorithms, 7quantum computing, 4, 170quantum cryptography, 437, 439, 460quantum dimer model, 547quantum dot, 6quantum error correction, 512quantum field theory, 168quantum Fourier transform, 7, 170quantum gates, 7, 12quantum groups, 518quantum Hamilton’s equations, 162quantum information, 437quantum logic gate, 17quantum loop gases, 18quantum mechanics, 621quantum media, 313quantum Monty Hall, 471quantum neural computation, 30quantum neural network, 11, 19quantum observable, 171quantum optics, 437quantum parallelism, 441quantum phase space, 268quantum state ket-vector, 153quantum statistics, 512quantum strategy, 447quantum superposition, 6

quantum superposition equation, 19quantum teleportation, 438quantum trigger, 33quantum truel, 471quantum tunneling effect, 33quantum Turing machine, 17, 181quantum-mechanical wave function, 479quantum-probability, 417quasi-particle tunneling, 33quasi-particles, 519quaternion attitude estimation filter,

618qubit, 3, 5, 12, 170, 182, 479quotient space, 602

Rabi frequency, 284Rabi oscillation, 284radial basis function network, 94radiation, 342radiation state, 342random, 717random matrix theory, 283random partial recursive function

halting, 23random variable, 391random walk, 392rate, 637rate code, 126rate of error growth, 702rate of relaxation, 228rationalistic, 722Rayleigh–Benard convection, 677, 701re-polarization, 59reaction–diffusion, 219reactive neurodynamics, 105recalled, 725reciprocal activation, 145reciprocal inhibition, 144recurrence time, 293recurrent neural networks, 18recurrent QNN, 374recursive homotopy dynamics, 399recursive solution, 611recursively hierarchical structure, 43reduced curvature 1-form, 266refinement equation, 633reflectance pattern, 103reflex, 65, 135regional energy minimization, 78

926 Index

reinforcement learning, 26, 412relative degree, 698relative phase gate, 182relative semions, 547relaxation oscillator, 687reliable predictor, 669removable singularity, 624repeller, 683replicator dynamics, 726represent quantum field theory, 24resistive loading, 483resistively & capacitively-shunted

junction, 501resistively-shunted Josephson junctions,

495resistively-shunted junction, 500resonance, 481return map, 683Reynolds number, 259Ricci flow, 428Ricci flow equation, 220Ricci solitons, 255Ricci tensor, 428, 593Richardson extrapolation, 737Riemann curvature, 428, 592Riemann sphere, 153Riemannian metric tensor, 398Riesz basis, 621ripple, 621risk dominance, 723Romberg integration, 737Rosenblatt, 83Rossler, 690Rossler system, 690rotating-wave approximation, 284route to chaos, 655route to turbulence, 261Rubin face, 37Rudolphine tables, 649Ruelle, 655rules of the game, 443Runge–Kutta–Fehlberg integrator, 774,

783

saltatory conduction, 60sampling theorem, 632scalar curvature, 247scaling function, 624scattering data, 325

scattering space, 344scattering state, 340Schrodinger equation, 19, 39, 152, 154,

224, 294, 318, 778Schrodinger operator, 320Schrodinger’s picture, 152, 163, 165Schwann cells, 53Schwartz function, 321Schwarz-type, 199search, 81selective attention, 401selectivity, 133self-adjoint, 595self-limiting process, 669self-organized, 25, 378, 412self-similar phenomena, 194semidirect product, 597sensitivity, 132sensor fusion in hybrid systems, 616sensory, 377sensory memory, 138sensory-motor integration, 619sequential (threshold) dynamics, 238set of instructions, 714Shannon, 654Shannon entropy, 26Shor’s algorithm, 6, 170Shor’s factorization algorithm, 512short exact sequence, 596short-term consequences, 402short-term memory, 100, 138, 237short-time Fourier transform, 116, 623signal, 25, 378, 412signal coding, 628signal velocity, 100, 237signal-to-noise ratio, 115signal-to-noise ratio, 123simulated annealing, 87Sinai, 655sinc function, 624sine-Gordon (SG) equation, 261, 293,

308Sinfire neural network, 117sinus cardinalis, 624skew-Hermitian, 596slope parameter, 81Smale, 652Smale horseshoe map, 656small world, 501

Index 927

small-world geometry, 28

small-world networks, 26, 499

smoothing, 614

Sobolev inequality, 332

social convergence, 415

social hierarchy, 27

social network, 27

social situations, 713

solid state physics, 437

solitary wave, 312

soliton, 288, 312, 332, 334

soliton resolution conjecture, 323, 341

soliton solution, 329

solution, 700

solution in maxi-min-strategies, 449

somatic nervous system, 54

somatosensory cortex, 44

space of all weighted paths, 400

spatiotemporal networks, 105

special unitary group, 596

spectral theorem, 595

spin-wave modes, 495

spinal nerves, 59

spindle receptors, 144

spine synapse, 33

split-stepping, 306

squeezing, 657

stability, 642

stable and unstable manifold, 659

stable manifold, 673

Standard model, 597

state, 639

state estimator, 612

state vector, 613

static backpropagation, 80

stationary soliton solutions, 341

stationary solutions, 332

statistically, 636

steepest descent method, 82

step size, 25, 83, 378, 400, 412

stimulus, 378

stochastic filtering problem, 373

stochastic integral, 393

stochastic system, 640

stochastic-gradient order parameterequations, 243

stock option, 551

storage, 139

strange attractor, 648, 655, 664, 665,666, 689

strategic, 714strategic form, 714strategic interaction, 713strategy, 443, 714strategy profile, 714stream of photons, 155strengths, 398stretch-and-fold, 689stretching, 657stroboscopic section, 662strong coupling expansion, 330structural stability, 675structurally stable, 655structure constants, 215sub-critical, 327Subband coding algorithm, 630subsumption architectures, 411sum over gauge orbits, 213sum over histories, 400super-critical, 327supercell thunderstorms, 260superconducting-normal transition, 489superconductivity, 479superior cerebellar peduncle, 148superposition, 5supervised gradient descent learning,

25, 378, 412supervised network, 80support vector machine, 94survival probability, 683symbolic dynamics, 655, 659, 660, 681symmetry breaking, 604symmetry breaking instability, 244symmetry group, 344sympathetic nervous system, 55symplectic form, 333symplectic structure, 331synaptic junctions, 46synaptic potential, 377synchronization in chaotic oscillators,

106synchronization of coupled nonlinear

oscillators, 498syncytium, 37synergetics, 243system dynamorphism, 41system with uncertain dynamics, 611

928 Index

Takens, 655

target equations of motion, 609

targeting, 701

temporal associative memories, 241

temporal code, 126

temporal dynamical systems, 101, 237

temporal lobe, 43

temporary phase-locking, 106

tensor product, 182

tensor-field, 675

termination, 395

The Last Problem, 717

the outcome that maximizes theproduct of the players’ utilities,719

theorem on local existence anduniqueness, 339

theoretical ecology, 670

theory of turbulence, 655

there exists a unique solution, 719

thermo-dynamic-like partition function,400

tight-binding model, 549

time-dependent Schrodinger equation,157

time-frequency, 332

time-frequency analysis, 620, 626

time-independent Schrodinger equation,319

time-lagged recurrent networks, 96

time-ordering operator, 488

time-phase plot, 642

Toda lattice, 293

topological, 199

topological defects, 184

topological insulators, 18

topological phase of matter, 519

topological properties, 14

topological quantum computer, 14

topological quantum field theories, 512

topological qubit, 16

topology, 647

toric code, 547

tornado, 260

trace formula, 270

trace operator, 597

tracking-filter prediction equations, 609

trajectory, 641, 675, 681

transcendental unity of apperception,36

transient chaos, 261transient signal components, 620transient spike inputs, 116transition amplitude, 165transition entropy, 400transition functions, 590transition matrix, 613transition propagator, 396, 397translation group, 194trapped ion quantum computer, 9trapped particles, 14trapped quantum particles, 14traveling wave solution, 290turbulence, 258, 676turbulent flow, 259Turing bifurcation, 232twiddle factors, 626twisted, 201two interpretations, 722two player game, 443two-person zero-sum strategic game,

444two-player quantum game, 447

un-damped pendulum, 684uncertainty principle, 116undecimated wavelet transform, 628underdamped junction, 501underdamped ladder array, 509unipolar neurons, 52unitary, 595unitary evolution, 152unitary matrix, 7, 594unitary quantum evolution, 154unitary shift operator, 344unitary topological modular functor,

186universal approximation theorem, 90universal Turing machine, 3unpredictability, 649unstable manifold, 673unstable periodic orbits, 684unsupervised, 25, 378, 412Upanishads, 25, 30, 188utility, 716utility functionals, 447utility functions, 443

Index 929

vacuum state, 365Van der Pol, 654Van der Pol oscillator, 657, 698variation formulas, 249vector momentum, 334vector-field, 423, 675vertebrate brain, 68virial inequality, 330visual cortex, 44voltage-to-frequency converter, 482volume, 249von Neumann, 647von Neumann architecture, 3von Neumann bottleneck, 4von Neumann computer, 3von Neumann’s quantum density

function, 166vortex, 259vorticity dynamics, 260

Walsh functions, 7, 630wave d’Lambertian operator, 331wave psi-function, 157wave-function collapse, 39wave-particle duality of matter, 151wavelet, 621, 631

wavelet analysis, 620wavelet coefficients, 633wavelet transform, 620wavelet transformation, 122wavelet-based QFT, 194weakly-connected neural network, 49,

242Weierstrass, 650weights, 412well-defined input–output function, 440when players move, 714whirling modes, 495whirling regime, 497white matter, 43Wiener, 654Wiener process, 394Wigner distribution, 269, 274Wigner function, 362Winfree-type phase models, 498Witten–Chern–Simons theory, 185Witten-type, 201working memory, 138world lines, 14

zero-sum game, 448Zhu–Botina–Rabitz functional, 282