In-phase and anti-phase synchronization in noisy Hodgkin-Huxley neurons

We numerically investigate the influence of intrinsic channel noise on the dynamical response of delay-coupling in neuronal systems. The stochastic dynamics of the spiking is modeled within a stochastic modification of the standard Hodgkin-Huxley model wherein the delay-coupling accounts for the finite propagation time of an action potential along the neuronal axon. We quantify this delay-coupling of the Pyragas-type in terms of the difference between corresponding presynaptic and postsynaptic membrane potentials. For an elementary neuronal network consisting of two coupled neurons we detect characteristic stochastic synchronization patterns which exhibit multiple phase-flip bifurcations: The phase-flip bifurcations occur in form of alternate transitions from an in-phase spiking activity towards an anti-phase spiking activity. Interestingly, these phase-flips remain robust in strong channel noise and in turn cause a striking stabilization of the spiking frequency

Amplitude death phenomena in delay--coupled Hamiltonian systems

Hamiltonian systems, when coupled {\it via} time--delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular for sufficiently strong coupling there can be amplitude death (AD), namely the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase--flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmoniccoupled oscillators, in particular the H\'enon-Heiles system.Comment: To be appeared in Phys. Rev. E (2013

Three Aspects of Photoionization in Ultrashort Pulses

This document is Sajad Azizi's doctoral thesis titled 'Three Aspects of Photoionization in Ultrashort Pulses.' The research was conducted under the supervision of Prof. Dr. Jan Michael Rost at the Max Planck Institute for the Physics of Complex Systems.:Contents 1 Introduction 1 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Fundamental Concepts 5 2.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Introduction to strong field ionization . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 From the photoelectric effect to strong-field ionization . . . . . . . . . 6 2.3 Non-relativistic time-dependent Hamiltonian . . . . . . . . . . . . . . . . . . 10 2.3.1 Dipole approximation and choice of gauges . . . . . . . . . . . . . . . 11 2.3.2 Interaction of an electron with a classical field . . . . . . . . . . . . . 12 2.4 Ultrashort laser pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 Fourier-limited pulse: Gaussian envelope . . . . . . . . . . . . . . . . . 17 2.4.2 Modulated pulse: sinusoidal phase modulation . . . . . . . . . . . . . 18 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Controlling Non-adiabatic Photoionization with Ultrashort Pulses 20 3.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Introduction to non-adiabatic ionization . . . . . . . . . . . . . . . . . . . . . 21 3.2.1 Intuitive picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.2 Mathematical picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Non-adiabatic ionization with tailored laser pulses . . . . . . . . . . . . . . . 25 3.3.1 Ionization by single Gaussian pulses . . . . . . . . . . . . . . . . . . . 26 3.3.2 Sensitivity of non-adiabatic photoionization to the modulation phase . 28 3.3.3 The role of the catalyzing state . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 Second-order perturbation theory . . . . . . . . . . . . . . . . . . . . . 32 3.3.5 Pulse optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Time-dependent Perturbation Theory for Ultrashort Pulses 37 4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Introduction to time-dependent perturbation theory . . . . . . . . . . . . . . 39 4.2.1 Higher order time-dependent perturbation theory . . . . . . . . . . . . 42 4.2.2 Perturbation theory in shaped short laser pulse . . . . . . . . . . . . . 45 4.3 Application I: non-adiabatic ionization . . . . . . . . . . . . . . . . . . . . . . 47 4.3.1 Slowly varying envelope approximation . . . . . . . . . . . . . . . . . 50 4.3.2 Zero-photon transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.3 Zero-photon ionization probability . . . . . . . . . . . . . . . . . . . . 55 4.3.4 Oscillation in zero-photon transition . . . . . . . . . . . . . . . . . . . 58 4.4 Application II: interference stabilization . . . . . . . . . . . . . . . . . . . . . 59 4.4.1 Third-order time-dependent perturbation theory . . . . . . . . . . . . 61 4.4.2 Ionization probability and stabilization . . . . . . . . . . . . . . . . . . 62 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5 Molecular Photoionization Time Delay 65 5.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 Introduction to time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.1 Time delay in scattering scenario . . . . . . . . . . . . . . . . . . . . 69 5.2.2 Asymptotic behavior of ⟨r⟩ . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3 Photoionization time delay from a scattering theory perspective . . . . . . . 74 5.3.1 Asymptotic solutions and scattering matrix . . . . . . . . . . . . . . . 75 5.3.2 Energy normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.3 Boundary condition and final molecular wavefunction . . . . . . . . . 79 5.3.4 Matrix element and photoionization time delay . . . . . . . . . . . . . 81 5.3.5 Two-center system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Photoionization time delay from a wavepacket perspective . . . . . . . . . . 87 5.4.1 Partial time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.2 Photoelectron wavepacket and photoionization time delay . . . . . . . 92 5.4.3 Anisotropic potential and half-collision checking . . . . . . . . . . . . 94 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Conclusions and Outlook 97 A Renormalized Numerov Method 101 A.1 Introduction to Numerov method . . . . . . . . . . . . . . . . . . . . . . . . 103 A.1.1 Eigenvalue calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A.2 Johnson renormalized Numerov method . . . . . . . . . . . . . . . . . . . . . 106 A.2.1 Proper initialization for extreme values of potential . . . . . . . . . . . 108 A.2.2 Matching point and bound states solutions . . . . . . . . . . . . . . . 110 A.2.3 Discretized continuum states solutions . . . . . . . . . . . . . . . . . . 111 A.2.4 Continuum states solutions . . . . . . . . . . . . . . . . . . . . . . . . 112 B Derivation of the Asymptotic Behavior of ⟨r⟩ 114 C Classical Time Delay 117 D Temporal Airy Pulse 119 E Numerical Details of Perturbation Theory 122 F Atomic Unit

Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Positive trigonometric polynomials for strong stability of difference equations

We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5

Predictor-Feedback Stabilization of Multi-Input Nonlinear Systems

We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by utilizing arguments based on Lyapunov functionals or estimates on solutions. We specialize our methodology to linear systems for which the predictor-feedback control laws are available explicitly and for which global exponential stability is achievable. A detailed example is provided dealing with the stabilization of the nonholonomic unicycle, subject to two different input delays affecting the speed and turning rate, for the illustration of our methodology.Comment: Submitted to IEEE Transactions on Automatic Control on May 19 201