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TitleBrain Dinamics (Synergistics) - H. Haken (Springer, 2007) WW
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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

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