##### Document Text Contents

Page 2

Springer Complexity

Springer Complexity is a publication program, cutting across all traditional disciplines

of sciences as well as engineering, economics, medicine, psychology and computer

sciences, which is aimed at researchers, students and practitioners working in the field

of complex systems.ComplexSystemsare systems that comprisemany interactingparts

with the ability to generate a new quality of macroscopic collective behavior through

self-organization, e.g., the spontaneous formation of temporal, spatial or functional

structures. This recognition, that the collective behavior of the whole system cannot be

simply inferred from the understanding of the behavior of the individual components,

has led to various new concepts and sophisticated tools of complexity. The main

concepts and tools – with sometimes overlapping contents and methodologies – are

the theories of self-organization, complex systems, synergetics, dynamical systems,

turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural

networks, cellular automata, adaptive systems, and genetic algorithms.

The topics treated within Springer Complexity are as diverse as lasers or fluids in

physics, machine cutting phenomena of workpieces or electric circuits with feedback

in engineering, growth of crystals or pattern formation in chemistry, morphogenesis in

biology, brain function in neurology, behavior of stock exchange rates in economics, or

the formation of public opinion in sociology. All these seemingly quite different kinds

of structure formation have a number of important features and underlying structures

in common. These deep structural similarities can be exploited to transfer analytical

methods and understanding from one field to another. The Springer Complexity pro-

gram therefore seeks to foster cross-fertilization between the disciplines and a dialogue

between theoreticians and experimentalists for a deeper understanding of the general

structure and behavior of complex systems.

The program consists of individual books, books series such as “Springer Series

in Synergetics”, “Institute of Nonlinear Science”, “Physics of Neural Networks”, and

“Understanding Complex Systems”, as well as various journals.

Page 124

6.6 The Phase-Locked State of N Neurons. Two Delay Times 117

We introduce the new variable x by means of

φ̇ = c+ x(t); c = C/γ (6.90)

so that (6.85) is transformed into

ẋ(t) + γx(t) = A1f(φ(t− τ1)) +A2f(φ(t− τ2)) . (6.91)

In the following we first assume that the r.h.s. is a given function of time t.

Because of the δ-function character of f , we distinguish between the following

four cases, where we incidentally write down the corresponding solutions of

(6.91)

I : tn + τ

′

1 < t < tn + τ

′

2 : x(t) = e

−γ(t−tn−τ1)x(tn + τ

′

1 + ) , (6.92)

II : tn + τ

′

2 ∓ : x(tn + τ

′

2 + ) = x(tn + τ

′

2 − ) +A2 , (6.93)

III: tn + τ

′

2 < t < tn+1 + τ

′

1 : x(t) = e

−γ(t−tn−τ

′

2)x(tn + τ

′

2 + ) , (6.94)

IV: tn+1 + τ

′

1 ± : x(tn+1 + τ

′

1 + ) = x(tn+1 + τ

′

1 − ) +A1 . (6.95)

Combining the results (6.92)–(6.95), we find the following recursion relation

x(tn+1 + τ

′

1 + ) = e

−γ∆x(tn + τ

′

1 + ) + e

−γ(∆+τ ′1−τ

′

2)A2 +A1 . (6.96)

Under the assumption of a steady state, we may immediately solve (6.96)

and obtain

x(tn+1 + τ

′

1 + ) =

(

1− e−γ∆

)−1 (

A1 + e

−γ(∆+τ ′1−τ

′

2)A2

)

. (6.97)

The only unknown quantity is ∆. To this end, we require, as usual,

tn+1∫

tn

φ̇dt = 2π , (6.98)

i.e. that φ increases by 2π. In order to evaluate (6.98) by means of (6.91), we

start from (6.91), which we integrate on both sides over time t

tn+1∫

tn

(ẋ(t) + γx(t))dt =

tn+1∫

tn

(A1f(φ(t− τ1)) +A2f(φ(t− τ2))) dt (6.99)

Because of the steady-state assumption, we have

tn+1∫

tn

ẋ(t)dt = x(tn+1)− x(tn) = 0 (6.100)

Page 125

118 6. The Lighthouse Model. Many Coupled Neurons

so that (6.99) reduces to

γ

tn+1∫

tn

x(t)dt = A1 +A2 . (6.101)

Using this result as well as (6.90) in (6.98), we obtain

c∆+ (A1 +A2)/γ = 2π (6.102)

which can be solved for the time interval ∆ to yield

∆ =

1

c

(2π − (A1 +A2)/γ) . (6.103)

We can also determine the values of x(t) in the whole interval by using the

relations (6.92)–(6.95). Because the time interval ∆ must be positive, we may

suspect that ∆ = 0 or, according to (6.103),

(2π − (A1 +A2)/γ) = 0 (6.104)

represents the stability limit of the stationary phase-locked state. We will

study this relationship in Sect. 6.10.

6.7 Stability of the Phase-Locked State.

Two Delay Times*

In order to study this problem, we assume as initial conditions

φ̇j(0) = φj(0) = 0 (6.105)

and integrate (6.4) over time, thus obtaining

φ̇j(t) + γφj(t) =

∑

k,

Ajk,

J(φk(t− τ

)) + Cjt+Bj(t) , (6.106)

where J has been defined in (6.19). We include the fluctuating forces, put

Bj(t) =

t∫

0

F̂j(σ)dσ , (6.107)

and assume that Cj in (6.7) is time-independent. The phase-locked state

obeys

φ̇(t) + γφ(t) =

∑

k

Ajk,

J(φ(t− τ

)) + Ct . (6.108)

Page 247

The Physics of Structure Formation

Theory and Simulation

Editors: W. Guttinger, G. Dangelmayr

Computational Systems – Natural and

Artificial Editor: H. Haken

From Chemical to Biological

Organization Editors: M. Markus,

S. C. Müller, G. Nicolis

Information and Self-Organization

A Macroscopic Approach to Complex

Systems 2nd Edition By H. Haken

Propagation in Systems Far from

Equilibrium Editors: J. E. Wesfreid,

H. R. Brand, P. Manneville, G. Albinet,

N. Boccara

Neural and Synergetic Computers

Editor: H. Haken

Cooperative Dynamics in Complex

Physical Systems Editor: H. Takayama

Optimal Structures in Heterogeneous

Reaction Systems Editor: P. J. Plath

Synergetics of Cognition

Editors: H. Haken, M. Stadler

Theories of Immune Networks

Editors: H. Atlan, I. R. Cohen

Relative Information Theories

and Applications By G. Jumarie

Dissipative Structures in Transport

Processes and Combustion

Editor: D. Meinköhn

Neuronal Cooperativity

Editor: J. Krüger

Synergetic Computers and Cognition

A Top-Down Approach to Neural Nets

By H. Haken

Foundations of Synergetics I

Distributed Active Systems 2nd Edition

By A. S. Mikhailov

Foundations of Synergetics II

Complex Patterns 2nd Edition

By A. S. Mikhailov, A. Yu. Loskutov

Synergetic Economics By W.-B. Zhang

Quantum Signatures of Chaos

2nd Edition By F. Haake

Rhythms in Physiological Systems

Editors: H. Haken, H. P. Koepchen

Quantum Noise 2nd Edition

By C. W. Gardiner, P. Zoller

Nonlinear Nonequilibrium

Thermodynamics I Linear and Nonlinear

Fluctuation-Dissipation Theorems

By R. Stratonovich

Self-organization and Clinical

Psychology Empirical Approaches

to Synergetics in Psychology

Editors: W. Tschacher, G. Schiepek,

E. J. Brunner

Nonlinear Nonequilibrium

Thermodynamics II Advanced Theory

By R. Stratonovich

Limits of Predictability

Editor: Yu. A. Kravtsov

On Self-Organization

An Interdisciplinary Search

for a Unifying Principle

Editors: R. K. Mishra, D. Maaß, E. Zwierlein

Interdisciplinary Approaches

to Nonlinear Complex Systems

Editors: H. Haken, A. Mikhailov

Inside Versus Outside

Endo- and Exo-Concepts of Observation

and Knowledge in Physics, Philosophy

and Cognitive Science

Editors: H. Atmanspacher, G. J. Dalenoort

Ambiguity in Mind and Nature

Multistable Cognitive Phenomena

Editors: P. Kruse, M. Stadler

Modelling the Dynamics

of Biological Systems

Editors: E. Mosekilde, O. G. Mouritsen

Self-Organization in Optical Systems

and Applications in Information

Technology 2nd Edition

Editors: M.A. Vorontsov, W. B. Miller

Principles of Brain Functioning

A Synergetic Approach to Brain Activity,

Behavior and Cognition

By H. Haken

Synergetics of Measurement, Prediction

and Control By I. Grabec, W. Sachse

Predictability of Complex Dynamical Systems

By Yu. A. Kravtsov, J. B. Kadtke

Interfacial Wave Theory of Pattern Formation

Selection of Dentritic Growth and Viscous

Fingerings in Hele–Shaw Flow By Jian-Jun Xu

Asymptotic Approaches in Nonlinear Dynamics

New Trends and Applications

By J. Awrejcewicz, I. V. Andrianov,

L. I. Manevitch

Page 248

Brain Function and Oscillations

Volume I: Brain Oscillations.

Principles and Approaches

Volume II: Integrative Brain Function.

Neurophysiology and Cognitive Processes

By E. Başar

Asymptotic Methods for the Fokker–Planck

Equation and the Exit Problem in Applications

By J. Grasman, O. A. van Herwaarden

Analysis of Neurophysiological Brain

Functioning Editor: Ch. Uhl

Phase Resetting in Medicine and Biology

Stochastic Modelling and Data Analysis

By P. A. Tass

Self-Organization and the City By J. Portugali

Critical Phenomena in Natural Sciences

Chaos, Fractals, Selforganization and Disorder:

Concepts and Tools By D. Sornette

Spatial Hysteresis and Optical Patterns

By N. N. Rosanov

Nonlinear Dynamics

of Chaotic and Stochastic Systems

Tutorial and Modern Developments

By V. S. Anishchenko, V. V. Astakhov,

A. B. Neiman, T. E. Vadivasova,

L. Schimansky-Geier

Synergetic Phenomena in Active Lattices

Patterns, Waves, Solitons, Chaos

By V. I. Nekorkin, M. G. Velarde

Brain Dynamics

Synchronization and Activity Patterns in

Pulse-Coupled Neural Nets with Delays and

Noise By H. Haken

From Cells to Societies

Models of Complex Coherent Action

By A. S. Mikhailov, V. Calenbuhr

Brownian Agents and Active Particles

Collective Dynamics in the Natural and Social

Sciences By F. Schweitzer

Nonlinear Dynamics of the Lithosphere

and Earthquake Prediction

By V. I. Keilis-Borok, A. A. Soloviev (Eds.)

Nonlinear Fokker-Planck Equations

Fundamentals and Applications

By T. D. Frank

Patters and Interfaces in Dissipative Dynamics

By L. M. Pismen

Springer Complexity

Springer Complexity is a publication program, cutting across all traditional disciplines

of sciences as well as engineering, economics, medicine, psychology and computer

sciences, which is aimed at researchers, students and practitioners working in the field

of complex systems.ComplexSystemsare systems that comprisemany interactingparts

with the ability to generate a new quality of macroscopic collective behavior through

self-organization, e.g., the spontaneous formation of temporal, spatial or functional

structures. This recognition, that the collective behavior of the whole system cannot be

simply inferred from the understanding of the behavior of the individual components,

has led to various new concepts and sophisticated tools of complexity. The main

concepts and tools – with sometimes overlapping contents and methodologies – are

the theories of self-organization, complex systems, synergetics, dynamical systems,

turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural

networks, cellular automata, adaptive systems, and genetic algorithms.

The topics treated within Springer Complexity are as diverse as lasers or fluids in

physics, machine cutting phenomena of workpieces or electric circuits with feedback

in engineering, growth of crystals or pattern formation in chemistry, morphogenesis in

biology, brain function in neurology, behavior of stock exchange rates in economics, or

the formation of public opinion in sociology. All these seemingly quite different kinds

of structure formation have a number of important features and underlying structures

in common. These deep structural similarities can be exploited to transfer analytical

methods and understanding from one field to another. The Springer Complexity pro-

gram therefore seeks to foster cross-fertilization between the disciplines and a dialogue

between theoreticians and experimentalists for a deeper understanding of the general

structure and behavior of complex systems.

The program consists of individual books, books series such as “Springer Series

in Synergetics”, “Institute of Nonlinear Science”, “Physics of Neural Networks”, and

“Understanding Complex Systems”, as well as various journals.

Page 124

6.6 The Phase-Locked State of N Neurons. Two Delay Times 117

We introduce the new variable x by means of

φ̇ = c+ x(t); c = C/γ (6.90)

so that (6.85) is transformed into

ẋ(t) + γx(t) = A1f(φ(t− τ1)) +A2f(φ(t− τ2)) . (6.91)

In the following we first assume that the r.h.s. is a given function of time t.

Because of the δ-function character of f , we distinguish between the following

four cases, where we incidentally write down the corresponding solutions of

(6.91)

I : tn + τ

′

1 < t < tn + τ

′

2 : x(t) = e

−γ(t−tn−τ1)x(tn + τ

′

1 + ) , (6.92)

II : tn + τ

′

2 ∓ : x(tn + τ

′

2 + ) = x(tn + τ

′

2 − ) +A2 , (6.93)

III: tn + τ

′

2 < t < tn+1 + τ

′

1 : x(t) = e

−γ(t−tn−τ

′

2)x(tn + τ

′

2 + ) , (6.94)

IV: tn+1 + τ

′

1 ± : x(tn+1 + τ

′

1 + ) = x(tn+1 + τ

′

1 − ) +A1 . (6.95)

Combining the results (6.92)–(6.95), we find the following recursion relation

x(tn+1 + τ

′

1 + ) = e

−γ∆x(tn + τ

′

1 + ) + e

−γ(∆+τ ′1−τ

′

2)A2 +A1 . (6.96)

Under the assumption of a steady state, we may immediately solve (6.96)

and obtain

x(tn+1 + τ

′

1 + ) =

(

1− e−γ∆

)−1 (

A1 + e

−γ(∆+τ ′1−τ

′

2)A2

)

. (6.97)

The only unknown quantity is ∆. To this end, we require, as usual,

tn+1∫

tn

φ̇dt = 2π , (6.98)

i.e. that φ increases by 2π. In order to evaluate (6.98) by means of (6.91), we

start from (6.91), which we integrate on both sides over time t

tn+1∫

tn

(ẋ(t) + γx(t))dt =

tn+1∫

tn

(A1f(φ(t− τ1)) +A2f(φ(t− τ2))) dt (6.99)

Because of the steady-state assumption, we have

tn+1∫

tn

ẋ(t)dt = x(tn+1)− x(tn) = 0 (6.100)

Page 125

118 6. The Lighthouse Model. Many Coupled Neurons

so that (6.99) reduces to

γ

tn+1∫

tn

x(t)dt = A1 +A2 . (6.101)

Using this result as well as (6.90) in (6.98), we obtain

c∆+ (A1 +A2)/γ = 2π (6.102)

which can be solved for the time interval ∆ to yield

∆ =

1

c

(2π − (A1 +A2)/γ) . (6.103)

We can also determine the values of x(t) in the whole interval by using the

relations (6.92)–(6.95). Because the time interval ∆ must be positive, we may

suspect that ∆ = 0 or, according to (6.103),

(2π − (A1 +A2)/γ) = 0 (6.104)

represents the stability limit of the stationary phase-locked state. We will

study this relationship in Sect. 6.10.

6.7 Stability of the Phase-Locked State.

Two Delay Times*

In order to study this problem, we assume as initial conditions

φ̇j(0) = φj(0) = 0 (6.105)

and integrate (6.4) over time, thus obtaining

φ̇j(t) + γφj(t) =

∑

k,

Ajk,

J(φk(t− τ

)) + Cjt+Bj(t) , (6.106)

where J has been defined in (6.19). We include the fluctuating forces, put

Bj(t) =

t∫

0

F̂j(σ)dσ , (6.107)

and assume that Cj in (6.7) is time-independent. The phase-locked state

obeys

φ̇(t) + γφ(t) =

∑

k

Ajk,

J(φ(t− τ

)) + Ct . (6.108)

Page 247

The Physics of Structure Formation

Theory and Simulation

Editors: W. Guttinger, G. Dangelmayr

Computational Systems – Natural and

Artificial Editor: H. Haken

From Chemical to Biological

Organization Editors: M. Markus,

S. C. Müller, G. Nicolis

Information and Self-Organization

A Macroscopic Approach to Complex

Systems 2nd Edition By H. Haken

Propagation in Systems Far from

Equilibrium Editors: J. E. Wesfreid,

H. R. Brand, P. Manneville, G. Albinet,

N. Boccara

Neural and Synergetic Computers

Editor: H. Haken

Cooperative Dynamics in Complex

Physical Systems Editor: H. Takayama

Optimal Structures in Heterogeneous

Reaction Systems Editor: P. J. Plath

Synergetics of Cognition

Editors: H. Haken, M. Stadler

Theories of Immune Networks

Editors: H. Atlan, I. R. Cohen

Relative Information Theories

and Applications By G. Jumarie

Dissipative Structures in Transport

Processes and Combustion

Editor: D. Meinköhn

Neuronal Cooperativity

Editor: J. Krüger

Synergetic Computers and Cognition

A Top-Down Approach to Neural Nets

By H. Haken

Foundations of Synergetics I

Distributed Active Systems 2nd Edition

By A. S. Mikhailov

Foundations of Synergetics II

Complex Patterns 2nd Edition

By A. S. Mikhailov, A. Yu. Loskutov

Synergetic Economics By W.-B. Zhang

Quantum Signatures of Chaos

2nd Edition By F. Haake

Rhythms in Physiological Systems

Editors: H. Haken, H. P. Koepchen

Quantum Noise 2nd Edition

By C. W. Gardiner, P. Zoller

Nonlinear Nonequilibrium

Thermodynamics I Linear and Nonlinear

Fluctuation-Dissipation Theorems

By R. Stratonovich

Self-organization and Clinical

Psychology Empirical Approaches

to Synergetics in Psychology

Editors: W. Tschacher, G. Schiepek,

E. J. Brunner

Nonlinear Nonequilibrium

Thermodynamics II Advanced Theory

By R. Stratonovich

Limits of Predictability

Editor: Yu. A. Kravtsov

On Self-Organization

An Interdisciplinary Search

for a Unifying Principle

Editors: R. K. Mishra, D. Maaß, E. Zwierlein

Interdisciplinary Approaches

to Nonlinear Complex Systems

Editors: H. Haken, A. Mikhailov

Inside Versus Outside

Endo- and Exo-Concepts of Observation

and Knowledge in Physics, Philosophy

and Cognitive Science

Editors: H. Atmanspacher, G. J. Dalenoort

Ambiguity in Mind and Nature

Multistable Cognitive Phenomena

Editors: P. Kruse, M. Stadler

Modelling the Dynamics

of Biological Systems

Editors: E. Mosekilde, O. G. Mouritsen

Self-Organization in Optical Systems

and Applications in Information

Technology 2nd Edition

Editors: M.A. Vorontsov, W. B. Miller

Principles of Brain Functioning

A Synergetic Approach to Brain Activity,

Behavior and Cognition

By H. Haken

Synergetics of Measurement, Prediction

and Control By I. Grabec, W. Sachse

Predictability of Complex Dynamical Systems

By Yu. A. Kravtsov, J. B. Kadtke

Interfacial Wave Theory of Pattern Formation

Selection of Dentritic Growth and Viscous

Fingerings in Hele–Shaw Flow By Jian-Jun Xu

Asymptotic Approaches in Nonlinear Dynamics

New Trends and Applications

By J. Awrejcewicz, I. V. Andrianov,

L. I. Manevitch

Page 248

Brain Function and Oscillations

Volume I: Brain Oscillations.

Principles and Approaches

Volume II: Integrative Brain Function.

Neurophysiology and Cognitive Processes

By E. Başar

Asymptotic Methods for the Fokker–Planck

Equation and the Exit Problem in Applications

By J. Grasman, O. A. van Herwaarden

Analysis of Neurophysiological Brain

Functioning Editor: Ch. Uhl

Phase Resetting in Medicine and Biology

Stochastic Modelling and Data Analysis

By P. A. Tass

Self-Organization and the City By J. Portugali

Critical Phenomena in Natural Sciences

Chaos, Fractals, Selforganization and Disorder:

Concepts and Tools By D. Sornette

Spatial Hysteresis and Optical Patterns

By N. N. Rosanov

Nonlinear Dynamics

of Chaotic and Stochastic Systems

Tutorial and Modern Developments

By V. S. Anishchenko, V. V. Astakhov,

A. B. Neiman, T. E. Vadivasova,

L. Schimansky-Geier

Synergetic Phenomena in Active Lattices

Patterns, Waves, Solitons, Chaos

By V. I. Nekorkin, M. G. Velarde

Brain Dynamics

Synchronization and Activity Patterns in

Pulse-Coupled Neural Nets with Delays and

Noise By H. Haken

From Cells to Societies

Models of Complex Coherent Action

By A. S. Mikhailov, V. Calenbuhr

Brownian Agents and Active Particles

Collective Dynamics in the Natural and Social

Sciences By F. Schweitzer

Nonlinear Dynamics of the Lithosphere

and Earthquake Prediction

By V. I. Keilis-Borok, A. A. Soloviev (Eds.)

Nonlinear Fokker-Planck Equations

Fundamentals and Applications

By T. D. Frank

Patters and Interfaces in Dissipative Dynamics

By L. M. Pismen