Analysis and Control of Corn plex Nonlinear Processes pptx

The topics of our research encompass - spatio-temporal pattern formation in semiconductors, geophysical, astro- physical, chemical and electrochemical, biological and ecological system,

Analysis and Control of

in Physics, CHemistry and Biology

WORLD SCIENTIFIC LECTURE NOTES IN COMPLEX SYSTEMS

Editor-in-Chief: A S Mikhailov, Fritz Haber Institute, Berlin, Germany

H Cerdeira, ICTP, Triest, Italy

B Huberman, Hewlett-Packard, Palo Alto, USA

K Kaneko, University of Tokyo, Japan

Ph Maini, Oxford University, UK

AIMS AND SCOPE

The aim of this new interdisciplinary series is to promote the exchange of information between scientists working in different fields, who are involved in the study of complex systems, and to foster education and training of young scientists entering this rapidly developing research area

The scope of the series is broad and will include: Statistical physics of large nonequilibrium systems; problems of nonlinear pattern formation in chemistry; complex organization of intracellular processes and biochemical networks of a living cell; various aspects of cell-to-cell communication; behaviour of bacterial colonies; neural networks;

functioning and organization of animal populations and large ecological systems; modeling complex social phenomena; applications of statistical mechanics to studies

of economics and financial markets; multi-agent robotics and collective intelligence; the emergence and evolution of large-scale communication networks; general mathemati-

cal studies of complex cooperative behaviour in large systems

Nonlinear Dynamics: From Lasers to Butterflies

Emergence of Dynamical Order: Synchronization Phenomena in

Complex Systems

Networks of Interacting Machines

Lecture Notes on Turbulence and Coherent Structures in Fluids,

Plasmas and Nonlinear Media

World Scientific Lecture Notes

in Complex Systems - Vol 5

Free University Berlin, Germany

University of Potsdam, Germany

Technical University Berlin, Germany

in Physics, Chemistry and Biology

Published by

World Scientific Publishing Co Re Ltd

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World Scientific Lecture Notes in Complex Systems - Vol 5

ANALYSIS AND CONTROL OF COMPLEX NONLINEAR PROCESSES

IN PHYSICS, CHEMISTRY AND BIOLOGY

Copyright 0 2007 by World Scientific Publishing Co Re Ltd

All rights reserved This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher

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Dedicated t o W e r n e r Ebeling a n d Gerhard Ertl

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Preface

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science Nonlinear dynamics as manifested by determin- istic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc This research has induced a new terminology in science connected with new questions, problems, solutions and methods New scenarios have emerged for spatio- temporal structures in dynamical systems far from equilibrium Their anal- ysis and possible control are intriguing and challenging aspects of the cur- rent research

The duality of fundamental and applied research is a focal point of its main attractivity and fascination Basic topics and foundations are always linked to concrete and precise examples Models and measurements of complex nonlinear processes evoke and provoke new fundamental questions and diversify and broaden the mathematical concepts and tools In return, new mathematical approaches to modeling and analysis enlarge the scope and efficiency of applied research

Fundamental research on complex physical systems, as well as applied in- terdisciplinary investigations of nonlinear dynamics, are valued highly in research programs and education of universities and research institutes in Berlin and Potsdam (Germany) Initiated by Werner Ebeling (Humboldt- University at Berlin) and Gerhard Ertl (Fritz Haber Institute Berlin) the

vii

viii Preface

collaborative research has led to the foundation of the collaborative research center 555 of the German research society (DFG), 9 years ago This center has stimulated cooperations between the different research groups in the region

Recent progress on nonlinear complex processes in Berlin and Potsdam is presented in this volume The topics of our research encompass

- spatio-temporal pattern formation in semiconductors, geophysical, astro- physical, chemical and electrochemical, biological and ecological system,

- the fundamental understanding of elementary structures and their inter- actions in deterministic and stochastic dynamics in two and three spatial dimensions,

- the role of global and nonlocal feedback and external forces in control

of spatio-temporal pattern formation, with applications, e.g., to nonlinear chemical systems and semiconductor nanostructures,

- stochastic and coherence resonance and noise induced behaviour as man- ifestation of the constructive role of noise in supporting the existence of spatial and temporal structures with examples from semiconductors, chem- ical and neuronal systems and climatic dynamics,

-the microscopic stochastic modeling of dynamic cellular and intracellu- lar processes as the propagation of Ca-waves above membranes and the action by neurons,

- the understanding of chaos synchronization and of stochastic synchro- nizat ion,

- the manifold role and the many applications of delayed feedback in chaos control,

- the evidence of synchronization phenomena in population-, climatic and oceanic dynamics as well as their importance in optoelectronic devices All chapters were written in the spirit to convince students and outsiders

Preface ix

of the attractivity of studying complex nonlinear processes as well as t o provide researchers of the field with new details and results Thus all con- tributions contain longer introductions of the topic followed by new findings

of our research

The editors are much indebted to David A Strehober for his helpful tech- nical assistance in the preparation of the book Further on we acknowledge the proof reading of several parts, as well as collecting of the subject in- dex by Simon Fugmann, Felix Miiller, Michael Rading and Tilo Schwalger The German research foundation (DFG-Sfb 555 “Complex Nonlinear Pro- cesses”) has generously supported our research during the last 9 years, and the edition of this book Special thanks are going to Udo Erdmann, the secretary of Sfb 555 We are also grateful to the editor of the World Sci- entific Lecture Notes in Complex Systems series, Alexander S Mikhailov, and to Senior Editor Lakshmi Narayan (Ms) for their help and congenial processing of the edition

Berlin and Potsdam, December 7, 2006

L Schimansky-Geier, B Fiedler, J Kurths and E Scholl

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Contents

1 Noise-induced effects in excitable systems with local and

X R Sailer, V Beato, L Schirrmnsky-Geier and H Engel

2 Synchronization in periodically driven discrete systems 43

T Prager and L Schimansky-Geier

3 Spiral wave dynamics: Reaction and diffusion versus kinematics 69

B Fiedler, M Georgi and N Jangle

4 Cellular calcium oscillations: From bifurcation analysis

t o experiment

A Z Politi, L D Gaspers, A Briimmer, A P Thomas

and T Hofer

5 Pattern formation in semiconductors under the influence

of time-delayed feedback control and noise

E Scholl, J Hizanidis, P Hovel and G Stegemann

6 Dynamics of coupled semiconductor lasers

L Recke, M Wolfrum and S Yanchuk

115

135

185

xi

xii Contents

7 Trapping of phase fronts and twisted spirals in periodi-

cally forced oscillatory media

0 Rudzick and A S Makhailov

8 Visualizing pitting corrosion on stainless steel

M Domhege, C Punckt and H H Rotermund

213

225

9 Unified approach to feedback-mediated control of spiral

V S Zykov and H Engel

10 Radiative driven instabilities

M Hegmann and E Sedlmayr

2 73

11 Building oscillations bottom up: Elemental time scales

R Thul and M Falcke

12 Continuous wavelet spectral analysis of climate dynamics 325

D Maraun, J Kurths and M , Holschneider

13 Synchronization of complex systems: Analysis and control 347

M Rosenblum and A Pikovsky

14 Critical states of seismicity - Implications from a phys-

ical model for the seismic cycle

G Zoller, M Holschneider and J Kurths

15 Predator-prey oscillations, synchronization and pattern

formation in ecological systems

B Blasius and R Tonjes

371

397

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Chapter 1

Noise-induced effects in excitable systems with local and

global coupling

Xaver R Sailer' Valentina Beato2 Lutz Schimansky-Geierl)* and

Harald Engel'>+

Institut fur Physik Humboldt Universitat zu Berlin

Newtonstrasse 15 0-12489 Berlin Germany

Institut fur Theoretische Physik Technische Universitiit Berlin Hardenbergstrajle 36 D-10623 Berlin Germany

* alsgaphysik hu- berlin de h engel@physik tu- berlin de

Contents

1.1 Introduction

1.2 Excitability: What is it and how can we model it?

1.2.1 General concept

1.2.2 A simple model - the FitzHugh-Nagumo system

1.2.3 The Oregonator model for the light-sensitive Belousov-Zhabotinsky re- action

1.3 Stochastic methods

1.3.2 Stochastic processes: White and colored noises

1.3.3 The Fokker-Planck equation h

1.4 Stochastic excitable elements

1.4.1 The Langevin approach: Phase portraits under fluctuations

1.4.2 The Fokker-Planck approach: Numerical solutions

1.4.3 The phenomenon of coherence resonance

1.4.4 Coherence resonance with respect to the correlation time

Excitable elements with coupling

1.5.1 Local coupling: Noise induced nucleations

1.5.2 Propagation of trigger waves in the presence of noise

1.5.3 Pattern formation in dichotomously driven, locally coupled FitzHugh- Nagumo systems

References

1.3.1 Langevin equation

1.3.4 Moment dynamics

1.5 1.5.4 Global coupling

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9

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15

15

16

18

21

26

26

29

31

35

38

1

2 2 2 4

2 X R Sailer V Beato et al

1.1 Introduction

Excitability is a well established phenomenon with applications in various fields of science Systems that, looked at superficially, have little in com- mon, like neurons [as], chemical reactions [4], wildfires [3], lasers [52], and

many others share this property The variety and diversity of the many examples demonstrates the importance of the concept of excitability Fluctuations are ubiquitous in nature In excitable systems they are considered to be of uttermost importance since they are able to "activate" these systems Especially if the magnitude of the fluctuations becomes comparable to activation barrier of the system their influence will be deter- minant This is true for example in neurons where fluctuating input and fluctuating membrane conductances are commonly agreed upon to func- tionally impact on their dynamics

We investigate the influence of fluctuations on excitable dynamics exper- imentally and theoretically This study is organized as follows In section

1.2 we give the general concept of excitability The most important model

for our work, the FitzHugh-Nagumo model, is introduced and its key prop- erties are given In section 1.3 we treat the analytical techniques that we used Section 1.4 centers around single individual elements An experiment

on a chemical excitable medium, the light-sensitive Belousov-Zhabotinsky reaction is introduced We show the setup and explain the experimental implementation The results from an experiment with fluctuating excitabil- ity parameter are given Results from investigations of coupled systems are given in section 1.5 Globally as well as locally coupled systems are dis- cussed

1.2 Excitability: What is it and how can we model it?

1.2.1 General concept

Consider a dynamical system with a stable fixed point Since the fixed point

is stable a small perturbation (or stimulus) from the fixed point will decay

If a perturbation exceeding a certain threshold is applied, an excitable system responds not with a decay of the perturbation but with a large excursion in phase space until finally re-approaching the stable fixed point The form and duration of the excitation is scarcely affected by the exact form of the perturbation Until the system is close to the fixed point again,

a new perturbation does hardly affect the system The corresponding time

Noise-induced effects an excitable systems 3

is called the refractory time

is given in Fig

A graphical representation of the scheme

In a neuron, for example, the stable fixed point 1.1

External perturbation Response of the system

Fig 1.1 Response (right) of an excitable system to different stimuli (left) The system

has a stable fixed point Panel a): A sub-threshold perturbation generates a small system

response Panel b): A super-threshold perturbation initiates an excitation loop Panel

c): The system can not be excited by a super-threshold perturbation applied during

the refractory state Panel d): Two successive super-threshold perturbations generate

excitations only if both are applied to the system in the rest state

could represent a steady potential drop over the cell membrane A super-

threshold perturbation (for example via input from other neurons) leads to

a large electric response of the neuron, a so-called action potential or spike

After some time of insensitivity to new signals the potential drop returns

to its steady value

The precise time from which on a new perturbation can excite the sys-

tem depends on the stimulus One therefore usually talks about the relative

refractory time

An excitable system is sometimes modeled to posses three distinct

states: The state in which the system is in the vicinity of the fixed point

is called the rest state, the state just after excitation is called the firing

4 X R Sailer, V Beato et al

state, and the state just before the system is close to the fixed point again

is called the refractory state The firing and the refractory state differ by

a high (firing) and a low (refractory) value of the activation variable which has a major influence on pattern formation in coupled excitable systems (For further details see project A4.)

1.2.2 A simple model - the FitzHugh-Nagumo s y s t e m

In this part we introduce the FitzHugh-Nagumo (FHN) system Orig- inally derived from the Hodgkin-Huxley model for the giant nerve fiber

of a squid, it has extended its application beyond neuron dynamics to all kinds of excitable systems and has by now become an archetype model for systems exhibiting excitability [l, 14, 361 First, we neglect coupling and fluctuations and use the following form:

the position of the so-called nullclines, the two functions y(x) that are de- termined by setting time derivatives d x l d t = 0 and d y / d t = 0 Depending

on the parameters the FHN system has different dynamical regimes Fig

1 2 shows phase space portraits together with the nullclines and timeseries for three qualitatively different cases

In the upper row we see the excitable regime The solid lines represent the nullclines of the system, the dashed line a typical trajectory Each dash represents a fixed time interval, i.e where the system moves faster through phase space the dashes become longer The system possesses one fixed point (intersection of the nullclines) which is stable Small perturbations decay A super-threshold perturbation leads to a large response (spike) after which the system returns to the fixed point After that a new perturbation is possible if the system from outside is brought again over the threshold

In the middle row the oscillatory regime is illustrated The systems exhibits continuous oscillations Perturbations have a t this stage little in- fluence on the dynamics

In the lower row we find the bistable parameter regime The system possesses two stable fixed points A first overcritical external perturbation

Noise-induced effects in excitable s y s t e m s 5

row), oscillatory (middle row) or bistable (lower row) kinetics t = 0.05

Depending on the parameters the FHN can either exhibit excitable (upper

from one of them leads the system into the basin of attraction of the other one The basin of attraction of a fixed point is that part of phase space from which systems without further perturbation evolve towards this point

6 X R Sailer, V Beato et al.

Once the system is in the vicinity of this second fixed point a new externalperturbation leads to a new excursion which ends at the initial point.The transition from the oscillatory to the excitable parameter regime

can, for example, be achieved via an increase of the parameter b The

system then undergoes a Hopf bifurcation and the sole fixed point loosesstability The location of the Hopf bifurcation in parameter space alsodepends on the parameter e which governs the separation of the timescales

An illustration for different e-values is shown in Fig 1.3 With increasing f

Fig 1.3, Trajectories of the FitzHugh-Nagurno model for sub- and super-threshold perturbations (red and blue curves respectively) Each point plotted at constant time

intervals t, = i;_i + At with At = 0.005, Kinetic parameters b = 1.4, a = 1/3 and

7 = 2 Left: high time scale separation for e = 0.01 Right: low time scale separation for t = 0.1.

the separation of the timescales of the fast activator and the slow inhibitorweakens In Fig, 1.3 WE show typical trajectories for sub- and super-threshold perturbations (respectively red and blue curves) for two different

values of the parameter e In the left panel (high timescale separation) n small perturbation (6x, Sx) to the stable point (XQ, j/o) triggers an excitation

loop It consists in a quick increase of the activator until the right branch

of the x-nullcline is reached, followed by a slow inhibitor production: the

excited state After this, the activator x decreases quickly to the left branch

of the ar-nullcliiie, and again moves slowly along it towards the fixed point

(xo,yo)'- The system is in the refractory state On the right panel we show a

case with low time scale separation between the two system variables The

same perturbation (Sx, 5y) now fails to excite the system The excitation

loop in this case does not follow the a>nullcline As a consequence, theexcited and refractory states are not clearly defined anymore and the systemresponds only to strong perturbations

Noise-induced effects an excitable systems 7

1.2.3 The Oregonator model for the light-sensitive

Belousov-Zhabotinsky reaction

The model that we take into account has been firstly proposed by H J Krug and coworkers in 1990 [30], to properly account for the photochemically- induced production of inhibitor bromide in the Belousov-Zhabotinsky reac-

tion (BZ) catalyzed with the ruthenium complex Ru(bpy)g+ [31, 321 The

Oregonator model was proposed in 1974 [16] on the basis Tyson-Fife re- duction of the more complicated Field-Koros-Noyes mechanism [15] for the

BZ reaction, the following modified model has been derived:

assumed to be linearly dependent on it, d[Br-]/dtoc4 [32] E , E’ and q are scaling parameters, and f is a stoichiometric constant [47] This model can

be reduced to the two-component one by adiabatic elimination of the fast variable w (in the limit 6’ << E ) [47] In this case one gets the following two-component version of the Oregonator kinetics

8 X R Sailer, V Beato et al

1.3 Stochastic methods

Excitable systems as considered here are many particle systems far from equilibrium Hence variables as voltage drop (neurons), light intensity (lasers) or densities (chemical reactions) are always subject to noise and fluctations Their sources might be of quite different origin, first the thermal motion of the molecules, the discreteness of chemical events and the quantum uncertainness create some unavoidable internal fluctuations But in excitable systems, more importantly, the crucial role is played by external sources of fluctuations which act always in nonequilibrium and are not counterbalanced by dissipative forces Hence their intensity and corre- lation times and lengths can be considered as independent variables and, subsequently, as new control parameters of the nonlinear dynamics Normally they can be controlled from outside as, via a random light illumination (chemical reactions) [26] or the pump light (lasers) [49] The inclusion of fluctuations in the description of nonlinear systems is done by two approaches [50] On the one hand side one adds fluctuat- ing sources in the nonlinear dynamics, transforming thus the differential equations into stochastic differential equations The second way is the consideration of probability densities for the considered variables and the formulation of their evolution laws Both concepts are introduced shortly

in the next two subsections

We underline that the usage of stochastic methods in many particle physics was initiated by Albert Einstein in 1905 working on heavy parti- cles immersed in liquids and which are thus permanently agitated by the molecules of the surrounding liquid Whereas Einstein formulated an evo- lution law for the probability P ( T , t ) to find the particle in a certain position

T at time t Paul Langevin formulated a stochastic equation of motion, i.e

a stochastic differential equation for the time dependent position r ( t ) itself

1 3 1 Langevin equation

subject to external noise A corresponding differential equation for the time evolution of ~ ( t ) is called a Langevin-equation and includes random parts We specify to the situation where randomness is added linearly to modify the time derivative of ~ ( t ) , i.e

Noise-induced effects in excitable s y s t e m s 9

Therein the function f(x) is called the deterministic part and stands for the

dynamics as introduced previously when considering the FitzHugh Nagumo system or the Oregonator The new term describes the influence of the randomness or noise and [ ( t ) is a stochastic process which requires further definition It enters linearly and is weighted by the function g ( 2 ) Hence, the increment dx during d t at time t gets a second, stochastic contribution

being independent of the deterministic one In case g(x) is a constant the

noise acts additively, otherwise it is called multiplicative (parametric) noise and the influence of the noise depends on the actual state x ( t ) of the system

The solution of eq 1.4 depends on the sample of [ ( t ) Formally one can interprete the latter as a time dependent parameter and the variable x ( t )

is found by integration over time We note that integrals over [ ( t ) need a stochastic definition and are defined via the existence of the moments [50] For this purpose the moments of [ ( t ) have to be given

In practical applications one uses Gaussian sources [ ( t ) or the so called Markovian random telegraph process For both the forhulation of the mean and the correlation function is sufficient to define the stochastic process Later on we will define the value support of [ ( t ) (Gaussian or dichotomic) and will give the mean and the correlation function, i.e

and

Here we reduced to stationary noise sources Without loss of generality the mean is set to zero For the later on considered types of noise this formula- tion is sufficient to obtain general answers for ensembles and their averages

of the stochastic excitable system Thus we can formulate evolution laws for the probability densities and the other moments We note that the generalization to cases with more than one noise sources is straightforward and crosscorrelations between the noise source have to be defined

1.3.2 Stochastic processes: W h i t e and colored noises

Next we consider several noise sources They are Gaussian sources if their support of values is due to a Gaussian distribution Contrary the dichotomic telegraph process assumes two values , i.e = A and Ez = A' We will always assume A' = -A

The second classification of the noise classifies its temporal correlations

In case of white noise the noise is uncorrelated in time which corresponds

10 X R Sailer, V Beato et al

to K ( T ) = 2 0 6 ( 7 ) being Dirac’s &function Here D scales the intensity

of the noise The power-spectrum of the noise is the Fourier transform

of the correlation function In case of the &function it is a constant and independent of the frequency what was the reason t o call it white noise All other noise sources with frequency-dependent power spectrum are thus colored

which stands for the trajectory of a Brownian particle The integral during

In case of a Brownian particle D is the spatial diffusion coefficient

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 221

We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle We suppose the particle having a mass m equal to unity, and we assume the force due t o the hits with thermal activated molecules of the fluid to be a stochastic variable Moreover, due t o the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered All this yields the following equation

= -yv(t) + F ( t ) ,

d t

where y is the friction constant The random force F ( t ) is supposed to

be independent of the velocity v ( t ) of the Brownian particle and to have zero mean Moreover, the random force is supposed t o be extremely rapidly varying compared to v ( t ) Hence we assume that F ( t ) = and E is Gaussian white noise

Integrating with respect to the time we get:

v(t) = v(0)e-Yt + e-Yt s,” e Y S F ( s ) d s (1.10)

Noise-induced effects in excitable s y s t e m s 11

We suppose the random force to be Gaussian, and a linear operator does not change this property Thus the velocity v ( t ) is Gaussian as well if the initial condition v(0) is a random Gaussian variable independent of the random force The mean value of the velocity reads

( v ( t ) ) = (v(0))e-Yt + e-Yt l e Y s ( F ( s ) ) d s = ( v ( 0 ) ) e - Y t (1.11) For the calculation of the correlation function of the velocity we exploit the assumption that

( F ( t ) F ( s ) ) = 2D6(t - s ) (1.12) The correlation function of the velocity, considering At > 0, is given by

Another stochastic process which yields a non-vanishing correlation time

is the dichotomous random telegraph process [20, 221 It can be described

by a phase according to

where the phase increases at each random time ti by an angle 7 r Thus the process has zero mean and variance A2 Introducing the Heaviside step function B we can write the phase as

Hence the stationary correlation function, obtained for the limit t is

12 X R Sailer, V Beato et al

With this assumption the process expressed in Eq 1.16 has the correlation function

(1.19) where i- is the correlation time of the process v* [20, 221 Thus the random

telegraph process, as well as the Ornstein-Uhlenbeck process shown before, presents an exponentially decaying correlation function

1.3.3 The Fokker-Planck equation

In case of Gaussian white noise the probability density obeys a diffusion equation with a drift In particular, the probability density is the condi- tioned average [50]

P ( Z , tlZO, t o ) = @(Z - 4 t ) ) ) (1.20) that the sampled trajectories are started in zo at time t o Then starting from the Chapman-Kolmogorov equation for Markovian processes one finds

Both the drift term K l ( z ) and the diffusion coefficient K ~ ( z ) are nonlinear functions of the state variable Z They are defined as the moments of the conditioned increments per unit time, i.e

with n = 1 , 2 These moments can be calculated from the Langevin equa- tion (1.4) For additive noise g(z) = l it results in

Kl(Z) = f(.) 1 K2(Z) = D (1.23)

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too ir- regular for Riemann integrals to be applied Application of Stieltjes inte- gration yields a dependence of the moments on how the limit to white noise

is taken If [ ( t ) is the limit of the Ornstein-Uhlenbeck -process with T + 0 (Stratonovich sense) the coefficients read [50]

K ( ) = S(.) + D g ' ( Z ) g ( Z ) , Kz(Z) = 0 g " x ) (1.24)

Noise-induced effects in excitable s y s t e m s 13

Excitable systems have more than one dynamic variables Hence the probability density will depend on these variables, i.e P ( x , y, ,t) In case of Gaussian white noise the evolution operator remains a diffusion equation with drift, but in a higher dimensional phase space We will deal with additive noise in the FitzHugh-Nagumo model in chapter 1.4.2 For colored noise sources the derivation of evolution equations for the probability densities is more difficult In a Markovian embedding i.e if the Ornstein-Uhlenbeck process is defined via white noise (cf chapter 1.3.2) and v ( t ) is part of the phase space one again gets a Fokker-Planck equation for the density P ( z , y, u, t ) Similarly, one finds in case of the telegraph process balance equations for P ( x , y, A, t ) and P ( x , y, , A', t ) which are the densities of the two possible values of the noise A and A' They yield

a drift term from the deterministic part and B jump part which describes the hopping between the two noise values

& = f(G, (4) + d x : ? ) & ( t ) i = 1 N (1.25) Here, (x) = C,=l zZ is the mean value of the system Special focus will be

on systems where the function f has the form

f(x.213) = f(G) + K ( ( x ) - x 3 ) (1.26) with coupling constant K tZ ( t ) represents Gaussian white noise determined

N

by

( & ( t ) ) = 0 ( W l ) < , ( t 2 ) ) = 2 m , , q t z - tl) (1.27)

We interpret Eq 1.25 in the Stratonovich sense

We are especially interested in the mean of the ensemble (x) We there- fore average over Eq 1.25 and make a Taylor's expansion of the right hand side around (x) We obtain:

14 X R Sailer, V Beato et al

the superscript in f ( " ) denotes the n-th derivative of f with respect to

it's argument In equation 1.28 we have made use of the central moments

of the ensemble we need their dynamics, too It is given by:

(1.29)

Eq 1.25 is not the most general form we can treat with the method of the moment dynamics Especially models with more than one dynamical variable, like the FHN system, are important to us In this case (Let us call the second variable y) we have to introduce the mixed central moments

pn,m = ((x - ( ~ ) ) ~ ( y - (y))") (and equivalently for more variables)

If we look closely a t Eqs 1.28 and 1.29 we notice that in general they incorporate infinite sums It is only for polynomials f and g that the sums break off at some final value Even if we deal with polynomials and the sums break off we notice that the dynamics of the n-th central moment generally depends on other, higher moments The system of equations 1.29 forms an infinite set of coupled ordinary differential equations It is only for linear functions f and g that the system decouples

For more complicated functions we need to apply an approximation to the system of equations 1.28 and 1.29 There are two main ways to do this One is to neglect the central moments from a certain order on The other one is to neglect cumulants from a certain order on, instead The infinite set

of equations then reduces to a finite one Most of the approximation meth- ods are incompatible with a normalizable, nonnegative probability distri- bution It is only the trivial method to neglect all moments or equivalently all cumulants above zeroth order and therefore neglect all fluctuations and

go with the deterministic description and the Gaussian approximation that

avoid this problem For a complete description of any other probability distributions infinitely many cumulants or moments have t o be taken into account The Gaussian approximation consists of neglecting all cumulants above second order:

The dynamical description then reduces to the dynamics for the mean and the variance (for more dynamical variables to the means, the variances and the covariances)

Noise-induced effects in excitable systems 15

1.4 Stochastic excitable elements

1.4.1 The Langevin approach: Phase portraits under

fluc-tuations

Fig 1.4 Stochastic trajectories for the FitaHugh-Nagumo model at low noise intensity

<7 2 = 0.05 [left column), and high noise intensity cr 2 = 0.45 (right column) Panels A):

f = 0.01, 60 = 2 Panels B): c - 0.01, bo = 1.4 Panels C): c = 0.1, S>0 = 1-4 The

parameter -v is kepi f i x t ' i l equal to 2, the parameter a = 1/3.

Low noise intensity

Excitability case A High noise intensity

Excitability case B

Excitability case C

16 X R Sailer, V Beato et al

When the parameter t h a t controls the excitation threshold of an ex- citable element fluctuates, then we end up with a system of coupled equa- tions of Langevin type In the case of the FitzHugh-Nagumo system this situation is modeled by the following Eqs.:

show different realizations for the FitzHugh-Nagumo Eqs 1.31, that permit

us t o describe its essential properties

0 Small fluctuations result in sub-threshold perturbations, conse- quently the system explores only a small portion of the phase space near the rest state (xo, y0)bo This is shown both in panel A l ) and

C l ) of Fig 1.4 In panel A l ) the system simply relaxes to the instantaneous fixed points ( 2 0 , y ~ ) b < ( ~ ) In panel C l ) , due t o a high excitability, small stable limit cycles are induced by noise a If the intensity of the fluctuations is increased the system can occasion- ally escape the vicinity of the fixed point and performs excitation loops, compare panel A l ) with A.2) or C l ) with C.2)

0 For fixed noise intensity, a process, which a t low excitability fails

to bring the Eqs 1.31 out of the fixed point vicinity, suffices to induce excitation loops a t higher excitability, compare panels A l ) , B.1) and C.l)

0 The number of states visited by the system during excitations in- creases with the noise strength, panels B l ) and B.2) for example Moreover fluctuations of high intensity significantly affect the tra- ject,ory of the excitation loops, see panel (3.2)

1.4.2 The Fokker-Planck approach: Numerical solutions

In the previous section the stochastic FitzHugh-Nagumo system has been treated using the Langevin eqs 1.31 Alternatively it can be described by

the Fokker-Planck equation ( F P E ) (cf subsec 1.3.3) In the case of the

"This is because the value bo is near the Hopf bifurcation value

Noise-induced effects in excitable s y s t e m s 17

FHN system an analytic solution of this equation cannot be given Here

we therefore use a numeric approach The equation under study reads:

We find qualitative changes in the combination of the extrema and saddle points of the stationary probability density For low noise we see a single maximum centered near the fixed point of the deterministic system (cf the intersection of the nullclines in fig 1.4) For increasing noise the probability density is mainly located close to two elongated maxima These maxima represent the outer branches of the cubic nullcline The probability distribution looks crater-like Systems with the corresponding parameters spend most of their time along the phase space trajectory of a deterministic excited system Once they enter the vicinity of the fixed point they are quickly reexcited The corresponding timeseries are characterized by a large coherence For further increasing noise the minimum of the probability density along with one saddle point vanishes The corresponding system is not so closely bound to the deterministic trajectory any more

We want to mention that the probability density further off the maxima becomes extremely small for small noise intensities so that numerical errors will eventually dominate the obtained results In particular we cannot ex- clude a second maximum for the low noise case in Fig 1.5 However we have also performed simulations with varying E (separation of the timescales) For high E (small separation) we find states with clearly one maximum only

We thus find, depending on the noise intensity and the separation of the timescale three qualitatively different regimes In these regimes differ-

18 X R Sailer, V Beato et al.

Fig, 1.5 Stationary probability distribution of the FitzHugh-Nagumo System hibitor noise intensity is varied (given above the panels) Other parameters: e = 0.1, -y = 2., 6 =1.4 [29]

In-ent combinations of maxima, minima, and saddle points in the stationaryprobability distribution can be observed

We have also performed simulations with noise in the activator variable.The obtained results differ only quantitatively from the ones presented forinhibitor noise

1.4.3, The phenomenon of coherence resonance

First encountered in oscillatory systems as "stochastic resonance without

periodic forcing' [17], and later as "internal stochastic resonance" [23], it

Noise-induced effects in excitable s y s t e m s 19

is with the work of A Pikovsky and J Kurths on the FitzHugh-Nagumo model [40] that this phenomenon got its present name coherence resonance

(CR) and that it was associated with excitable systems In this work the authors showed that the regularity at which an excitable element fires under white noise driving, has a non-monotonous resonant dependence on the noise intensity, and that there exists an optimal noise intensity at which

a sequence of noise-induced excitations is most regular This phenomenon has been studied since more than one decade and has been observed in a huge variety of systems of quite different nature as, for example, anti-CR

in excitable systems with feedback [33], CR in coupled chaotic oscillators [54], internal CR in variable size patches of a cell membrane [44, 451, system size CR in globally coupled FitzHugh-Nagumo elements [46], array- enhanced CR in a model for Ca2+ release [12], CR at the onset of a saddle- node-bifurcation of limit cycles [34] and of period-doubling bifurcations period-doubling bifurcation [38] , and spatial CR in a spatially extended system near a pattern-forming instability [ll] (for a comprehensive review see [35])

To characterize the level of coherence of noise-induced excitations we analyze the time evolution of the activator concentration II: in the FitzHugh- Nagumo model, see Fig 1.6 In this representation the excitation loops shown previously in Fig 1.4 become spikes spaced out by intervals during which the system performs noisy relaxation oscillations around its stable state The phenomenon of coherence resonance manifests itself in the three realizations of z ( t ) for different noise intensities given in Fig 1.6 For very low noise intensity (upper panel) an excitation is a rare event which happens

at random times In the panel at the bottom, for high noise intensity, the systems fires more easily but still rather randomly In the panel in the center instead, at an optimal noise intensity, the system fires almost periodically The typical oscillation period for the system is given by the mean in- terspike time interval (ISI) ( t p ) between two successive noise-induced ex- citations over many realizations, see enlargements in Fig 1.6 To it we associate as error the standard deviation If the system fires regularly, say for simplicity periodically, then the error associated to t , is zero and conse- quently the ratio of the standard deviation srd(t,) to its mean value (t,),

i.e the normalized fluctuations

(1.34)

is equal to zero On the other hand, if the firing is incoherent and takes

20 X R Sailer, V Beato et oi.

Fig 1.6 Noise-induced excitations in the stochastic FitzHugh-Nagumo model under

inhibitory white noise driving Parameters for all plots: 6g — 1-05, f = 0.01, -y = 1.4,

a = 1/3 Left top: <r2 = 0.001 Left center: <r 2 = 0.009 Left bottom: er 2 = 0.064.

In the right panels are shown enlargements of the trajectories plotted on the left side.

Time scales t p , t e and t a are discussed in the next two pages.

noise intensity We plot this dependence for the FitzHugh-Nagumo model

in Fig 1.7

The phenomenon of CR is due to the presence of two different teristic time scales in the system which are affected by noise in differentmariner One is the time during which the system just fluctuates around

charac-the stationary state, which is needed to activate an excitation We call

is the typical duration of an excitation loop, compare Fig 1,6 Noise of

in-Noise-induced effects in excitable systems 2]

Fig 1.7 Normalized fluctuations of the inter-spike interval versus the noise intensity for the FitzHugh-Nagumo model Black curve reproduces the result shown in [40] Pa- rameters 6 0 = 1.05, e = 0.01, -v = 1.4, a - 1/3 in Eqs 1.31 The fixed point is a stable

focus Same results with bo = 1-2 shown by the red curve, where the fixed point is a stable node.

regime the excitation loops, that at low noise intensities possess well definedtrajectories, loose regularity with increasing noise intensity, compare panels

at the bottom of Fig 1.6 This phenomenon is also shown in panels B.Iand B.2 of Fig 1.4 There the trajectories of the noise-induced excitationsspread out in the phase space for increasing noise intensity The transition

minimum Then the coherence of the system is highest

1.4.4 Coherence, resonance with respect to the correlation

time

Most of the research on CR was focused on the case of white noise Whitenoise is a good approximation as long as the intrinsic time scales of thedeterministic system are much larger than the correlation time of the ex-ternal fluctuations This is the case for example in neuvonal dynamicswhere a neuron can be externally forced by another randomly burstingneuron In general, when deterministic and stochastic; time scales arc notwell separated from each other, not only the amplitude but also the tempo-ral correlation is expected to influence noise-induced phenomena ;us CR In

X R Sailer, V Beato et al

this subsection we show that also the correlation time of an external noisesignal is a control parameter of the coherence of the system We show thisfirst experimentally employing the light-sensitive Belousov-Zhabotinsky re-action and then we confirm this result numerically performing calculationswith the two-component Oregonator model

In Fig 1.8 we show schematically the set-up in use in our experiment,for details please see [7] The set-up adopted for our experiments has as

central element an open gel-reactor, which allows to maintain constant

non-equilibrium conditions during the measurements, see Fig 1.8 With it,and by means of computer-based spectrophotometry [37], we analyze waveactivity in the BZ medium -with, sufficiently high spatial and temporal res-olution

Fig 1,8 Top: Sketch of the experimental setup Below: Photograph of the reactor and, behind it, the polarizer, the blue filter and the video projector.

N o i s e - i n d u c e d effects in excitable s y s t e m s 23

Through a noise signal, precisely a random telegraph signal, we induce nucleations of target patterns in the BZ medium Under constant illumina- tion 10 no wave nucleation occurs, but the medium can support excitation waves At light intensity 10 - A 1 the medium is oscillatory and phase waves are induced, which become trigger waves a,s they propaga.te t,owards the surrounding medium maintained a t high light intensity The system is excitable a t light intensity 10 + A 1 and supports traveling patters In case of dichotomous fluctuating light, nucleations occur randomly The regularity

of these phenomena is measured recording the activity at a given point of the gel In this way we get a series of noise-induced nucleations at random time intervals, see data reported in Fig 1.9

Already a t a rough glance on the noise-induced spikes reported in Fig 1.9 it is possible to recognize that the coherence is dependent on T The spikes plotted in the middle panel show a better regularity than the other two above and below it

To get a deeper understanding of this phenomenon we perform exper- iments for correlation times T in the range between 2 and 60 seconds and

we estimate the time t, between two successive nucleations, and calculate its mean and the standard deviation In Fig 1.10 the coherence of the nucleation events is quantified through the normalized fluctuations of t,,

(1.35)

Rp presents a minimum a t 7 Y 20 s, fingerprint of highest coherence Hence our experimental data give a clear evidence of the existence in the light- sensitive BZ reaction of an optimal correlation time of the fluctuating light driving at which the highest coherence is induced [7]

To validate numerically the above reported results, we perform calcula- tions with the two-component Oregonator model introduced in subsection 1.2.3, see Eqs 1.3 Here also we are interested in how fluctuations with non-vanishing correlation time in the excitability parameter affect the co- herence of the system response Thus we assume the parameter d , pro- portional t o the light-intensity, to be an exponentially correlated stochastic variable expressed as

being the stochastic process q* ( t ) a random telegraph signal We choose

$0 in the excitable regime and analyze how the coherence of the Orego- nator system changes (Eqs 1.3) as the correlation time of the process q*

24 X R Sailer, V Beato et al

in gray levels (point chosen in the constant background) From top to bottom: T = 5 s ,

T = 15 s , T = 40 s [7]

varies We choose the parameters 40 and A such that $- = $ o ( l - A)

to the oscillatory regime Then for $ ( t ) = q5+ = + o ( l + A) the system

is excitable, therefore from any initial condition it reaches the stationary state and remains there forever

The results for RP(7) obtained for different values of A, see Fig 1.11, demonstrate that under a random telegraph signal the coherence of noise- induced excitation is enhanced by an optimal choice of the correlation time Here, the optimal correlation time TOPt decreases as the noise amplitude

A increases Further simulations not shown here, confirm that this phe- nomenon holds for a wide range of the bifurcation parameter 40, covering almost the whole excitable regime We emphasize that for not well sep- arated time scales, noise-induced excitations are possible even if both 4-

Noise-induced effects in excitable systems •2~>

Fig 1.10 Coherence resonance with respect to the correlation time in the light-sensitive

BZ reaction The normalized fluctuations of the inter-spike times ( r , arc reported versus the correlation time of the random telegraph signal, [7]

Fig 1,11 Normalized fluctuations of the inter-spike time versus the correlation time of the noise for the Oegonator model, Eqs 1.3 with <po = 0.0075, Black curve A = 0.1, green A = 0.8, gray A = 0.6, blue A = 0,4, red A = 0,3 Each point is an average over S-10 3 inter-spike intervals [7]