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Tamás Vicsek

Eötvös Loránd University Budapest, Biological Physics, Faculty Member

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Papers by Tamás Vicsek

Cooperative Formation of Chiral Patterns during Growth of Bacterial Colonies

Bacterial colonies can develop chiral morphology in which the colony consists of twisted branches... more Bacterial colonies can develop chiral morphology in which the colony consists of twisted branches, all with the same handedness. Microscopic observations of the chiral growth are presented. We propose that the observed (macroscopic) chirality results from the microscopic chirality of the flagella (via handedness in tumbling) together with orientation interaction between the bacteria. The above assumptions are tested using a generalized version of the communicating walkers model.

Experiments on Laplacian growth

Cluster-Cluster Aggregation

Fractal Measures

chapter in Fractal Growth Phenomena

Novel type of phase transition in a system of self-propelled particles

A simple model with a novel type of dynamics is introduced in order to investigate the emergence ... more A simple model with a novel type of dynamics is introduced in order to investigate the emergence
of self-ordered motion in systems of particles with biologically motivated interaction. In our model
particles are driven with a constant absolute velocity and at each time step assume the average direction
of motion of the particles in their neighborhood with some random perturbation (g) added. We present
numerical evidence that this model results in a kinetic phase transition from no transport (zero average
velocity, ~v, ( = 0) to finite net transport through spontaneous symmetry breaking of the rotational
symmetry. The transition is continuous, since ~v, ~ is found to scale as (71, —g)t with p = 0.45.

Collective Motion

by Tamás Vicsek and Anna Lazar

Wereview the observations and the basic laws describing the essential aspects of collective motio... more Wereview the observations and the basic laws describing the essential aspects of collective
motion — being one of the most common and spectacular manifestation of coordinated
behavior. Our aim is to provide a balanced discussion of the various facets of this highly
multidisciplinary field, including experiments, mathematical methods and models for
simulations, so that readers with a variety of background could get both the basics and
a broader, more detailed picture of the field. The observations we report on include
systems consisting of units ranging from macromolecules through metallic rods and
robots to groups of animals and people. Some emphasis is put on models that are simple
and realistic enough to reproduce the numerous related observations and are useful for
developing concepts for a better understanding of the complexity of systems consisting
of many simultaneously moving entities. As such, these models allow the establishing of
a few fundamental principles of flocking. In particular, it is demonstrated, that in spite of
considerable differences, a number of deep analogies exist between equilibrium statistical
physics systems and those made of self-propelled (in most cases living) units. In both cases
only a few well defined macroscopic/collective states occur and the transitions between

Fluctuations and scaling in biology (book, Oxford)

Contents Preface ... more Contents
Preface
5
Introduction
7
1 Basic concepts (T. Vicsek)
11
1.1 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.1 Noise versus ﬂuctuations . . . . . . . . . . . . . . . . . . . . . 13
1.1.2 Molecular motors driven by noise and ﬂuctuations . . . . . . . 14
1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 Critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 Scaling of event sizes: Avalanches . . . . . . . . . . . . . . . . 17
1.2.3 Scaling of patterns and sequences: Fractals . . . . . . . . . . . 19
1.2.4 Scaling in group motion: Flocks . . . . . . . . . . . . . . . . . 22
2 Introduction to complex patterns, ﬂuctuations and scaling 25
2.1 Fractal geometry (T. Vicsek) . . . . . . . . . . . . . . . . . . . . . .
. 26
2.1.1 Fractals as mathematical and biological objects . . . . . . . . 27
2.1.2 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Useful rules . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 31
2.1.4 Self-similar and self-aﬃne fractals . . . . . . . . . . . . . . . . 33
2.1.5 Multifractals . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.6 Methods for determining fractal dimensions . . . . . . . . . . 36
2.2 Stochastic processes (I. Der´´enyi) . . . . . . . . . . . . . . . . . . . . 39
2.2.1 The physics of microscopic objects . . . . . . . . . . . . . . . 39
2.2.2 Kramers formula and Arrhenius law . . . . . . . . . . . . . . 41
2.3 Continuous phase transitions (Z. Csah´´ok) . . . . . . . . . . . . . . . . 43
2.3.1 The Potts model . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.2 Mean-ﬁeld approximation . . . . . . . . . . . . . . . . . . . . 46
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1

2
CONTENTS
3 Self-organised criticality (SOC) (Z. Csah´ok) 53
3.1 SOC model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 54
3.2 Applications in biology . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 SOC model of evolution . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 SOC in lung inﬂation . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Patterns and correlations
67
4.1 Bacterial colonies (A. Czir´ok) . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.2 Bacteria in colonies . . . . . . . . . . . . . . . . . . . . . . . .
68
4.1.3 Compact morphology . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.4 Branching morphology . . . . . . . . . . . . . . . . . . . . . . 86
4.1.5 Chiral and rotating colonies . . . . . . . . . . . . . . . . . . . 104
4.2 Statistical analysis of DNA sequences (T. Vicsek) . . . . . . . . . . . 112
4.2.1 DNA walk . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 113
4.2.2 Word frequency analysis . . . . . . . . . . . . . . . . . . . . . 117
4.2.3 Vector space techniques . . . . . . . . . . . . . . . . . . . . . 117
4.3 Analysis of brain electrical activity: the dimensional complexity of the electroencephalogram
(M. Moln´ar) . . . . . . . . . . . . . . . . . . . 120
4.3.1 Neurophysiological basis of the electroencephalogram and event- related potentials . . .
. . . . . . . . . . . . . . . . . . . . . . 120
4.3.2 Linear and non-linear methods for the analysis of the EEG . . 122
4.3.3 Examples for the application of PD2 to EEG and ERP analysis 125
4.3.4 Dimensional analysis of ERPs . . . . . . . . . . . . . . . . . . 128
4.3.5 Clinical applications . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 135
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5 Microscopic mechanisms of biological motion (I. Der´enyi, T. Vicsek) 147
5.1 Characterisation of motor proteins . . . . . . . . . . . . . . . . . . . 147
5.1.1 Cytoskeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.2 Muscle contraction . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.3 Rotary motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1.4 Motility assay . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Fluctuation driven transport . . . . . . . . . . . . . . . . . . . . . . . 160
5.2.1 Basic ratchet models . . . . . . . . . . . . . . . . . . . . . . . 162
5.2.2 Brief overview of the models . . . . . . . . . . . . . . . . . . . 163
5.2.3 Illustration of the second law of thermodynamics . . . . . . . 164

CONTENTS
3
5.3 Realistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.1 Kinesin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.2 Myosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.3.3 ATP synthase . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.4 Collective eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4.1 Finite sized particles in a “rocking ratchet” . . . . . . . . . . . 192
5.4.2 Finite sized particles in a “ﬂashing ratchet” . . . . . . . . . . 199
5.4.3 Collective behaviour of rigidly attached particles . . . . . . . . 205
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6 Collective motion
217
6.1 Flocking: collective motion of self-propelled particles (A. Czir´ok, T. Vicsek) . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.1.1 Models and simulations . . . . . . . . . . . . . . . . . . . . . . 218
6.1.2 Scaling properties . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1.3 Further variants of SPP models . . . . . . . . . . . . . . . . . 228
6.1.4 Mean-ﬁeld theory for lattice models . . . . . . . . . . . . . . . 234
6.1.5 Continuum equations for the 1d system . . . . . . . . . . . . . 240
6.1.6 Hydrodynamic formulation for 2D . . . . . . . . . . . . . . . . 247
6.1.7 The existence of long-range order . . . . . . . . . . . . . . . . 251
6.2 Correlated Motion of Pedestrians (D. Helbing, P. Molna´r) . . . . . . 254
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6.2.2 Pedestrian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 255
6.2.3 Trail Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Segregation mechanisms of tissue cells: from experimental data to models

Complex Adaptive Systems Modeling, 2013

Considerable advance has been made in recent years in the research field of pattern formation by ... more Considerable advance has been made in recent years in the research field of pattern formation by segregation of tissue cells. Research has become more quantitative partly due to more in-depth analysis of experimental data and the emergence modeling approaches. In this review we present experimental observations, including some of our new results, on various aspects of two and three dimensional segregation events and then summarize the computational modeling approaches.

Community structure and ethnic preferences in school friendship networks

Recently developed,concepts,and,techniques,of analyzing,complex,systems,provide new,insight into ... more Recently developed,concepts,and,techniques,of analyzing,complex,systems,provide new,insight into the structure of social networks. Uncovering,recurrent preferences,and,organizational,principles in such networks,is a key issue to characterize them. We investigate school friendship networks from the Add Health database. Applying threshold analysis, we find that the friendship networks,do not form a single connected,component,through,mutual,strong nominations,within a school, while under weaker conditions such interconnectedness is present. We extract the

Modeling the Emergence of Modular Leadership Hierarchy During the Collective Motion of Herds Made of Harems

Journal of Statistical Physics, 2014

ABSTRACT Gregarious animals need to make collective decisions in order to keep their cohesiveness... more ABSTRACT Gregarious animals need to make collective decisions in order to keep their cohesiveness. Several species of them live in multilevel societies, and form herds composed of smaller communities. We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected sub-groups (e.g. harems) by combining self organization and social dynamics. It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision. Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortob\&#39;agy, Hungary). We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal.

Universal Patterns of Collective Motion from Minimal Models of Flocking

2008 Second IEEE International Conference on Self-Adaptive and Self-Organizing Systems, 2008

This paper is concerned with the basic laws describing the essential aspects of collective motion... more This paper is concerned with the basic laws describing the essential aspects of collective motion being one of the most common and spectacular manifestation of coordinated actions. Our purpose is to discuss models that are both simple and realistic enough to reproduce the observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many organisms as well as such non-living objects as interacting robots. Understanding the interrelation of these systems has the potential of improving the interpretation of collective behavioral patterns in both living and nonliving systems by learning about similar phenomena in the two domains of nature.

DYNAMICS OF GROWING SELF-AFFINE SURFACES

Topics in Modern Statistical Physics, 1992

The Fractal Nature of Common Patterns

NATO ASI Series, 1993

Chemomodulation of Cellular Movement and Collective Formation of Vortices by Swarming Bacteria

Bacterial colonies have developed sophisticated modes of cooperative behavior which enable them t... more Bacterial colonies have developed sophisticated modes of cooperative behavior which enable them to respond to adverse growth conditions. It has been shown that such behavior can be manifested in formation of complex colonial patterns. Certain Bacillus species exhibit collective migration, "turbulent like" flow and emergence of whirlpools during colonial development. Here we present experimental observations of collective behavior and a generic model to explain such behavior. The model incorporates self-propelled and interacting "particles" (swarmers). We show that velocity interaction between the particles can lead to a synchronized movement. To explain vortices formation, we propose a plausible mechanism involving a special chemotactic response (rotational chemotaxis) which is based on speed modulations according to the concentration of a chemoattractant. This mechanism differs from that exhibited by swimming bacteria. We show that the chemomodulation of swarmers' speed together with the velocity interactions impose a torque on the collective motion and can lead to formation of vortices. The inclusion of both attractive and repulsive rotational chemotaxis in the model captures the salient features of the observed growth patterns~

Collective Dynamics of Two-Mode Stochastic Oscillators

Physical Review Letters, 2001

We study a system of two-mode stochastic oscillators coupled through their collective output. As ... more We study a system of two-mode stochastic oscillators coupled through their collective output. As a function of a relevant parameter four qualitatively distinct regimes of collective behavior are observed. In an extended region of the parameter space the periodicity of the collective output is enhanced by the considered coupling. This system can be used as a new model to describe synchronizationlike phenomena in systems of units with two or more oscillation modes. The model can also explain how periodic dynamics can be generated by coupling largely stochastic units. Similar systems could be responsible for the emergence of rhythmic behavior in complex biological or sociological systems.

Anomalous segregation dynamics of self-propelled particles

New Journal of Physics, 2015

A number of novel experimental and theoretical results have recently been obtained on active soft... more A number of novel experimental and theoretical results have recently been obtained on active soft matter, demonstrating the various interesting universal and anomalous features of this kind of driven systems. Here we consider the adhesion difference-driven segregation of actively moving units, a fundamental but still poorly explored aspect of collective motility. In particular, we propose a model in which particles have a tendency to adhere through a mechanism which makes them both stay in touch and synchronize their direction of motion - but the interaction is limited to particles of the same kind. The calculations corresponding to the related differential equations can be made in parallel, thus a powerful GPU card allows large scale simulations. We find that in a very large system of particles, interacting without explicit alignment rule, three basic segregation regimes seem to exist as a function of time: i) at the beginning the time dependence of the correlation length is analogous to that predicted by the Cahn-Hillard theory, ii) next rapid segregation occurs characterized with a separation of the different kinds of units being faster than any previously suggested speed, finally, iii) the growth of the characteristic sizes in the system slows down due to a new regime in which self-confined, rotating, splitting and re-joining clusters appear. Our results can explain recent observations of segregating tissue cells in vitro.

METHODS FOR DETERMINING FRACTAL DIMENSIONS

Fractal Growth Phenomena, 1989

Social Group Dynamics in Networks

Understanding Complex Systems, 2009

The rich set of interactions between individuals in the society results in complex community stru... more The rich set of interactions between individuals in the society results in complex community structure, capturing highly connected circles of friends, families, or professional cliques in a social network. Due to the frequent changes in the activity and communication patterns of individuals, the associated social and communication network is subject to constant evolution. The cohesive groups of people in such networks can grow by recruiting new members, or contract by loosing members; two (or more) groups may merge into a single community, while a large enough social group can split into several smaller ones; new communities are born and old ones may disappear. We discuss a new algorithm based on a clique percolation technique, that allows to investigate in detail the time dependence of communities on a large scale and as such, to uncover basic relationships of the statistical features of community evolution. According to the results, the behaviour of smaller collaborative or friendship circles and larger communities, e.g., institutions show significant differences. Social groups containing only a few members persist longer on average when the fluctuations of the members is small. In contrast, we find that the condition for stability for large communities is continuous changes in their membership, allowing for the possibility that after some time practically all members are exchanged.

Evolution of the social network of scientific collaborations

Physica A: Statistical Mechanics and its Applications, 2002

The co-authorship network of scientists represents a prototype of complex evolving networks. In a... more The co-authorship network of scientists represents a prototype of complex evolving networks. In addition, it o ers one of the most extensive database to date on social networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an 8-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. Three complementary approaches allow us to obtain a detailed characterization. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, a ecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution.

Networks in life: scaling properties and eigenvalue spectra

Physica A: Statistical Mechanics and its Applications, 2002

We analyze growing networks ranging from collaboration graphs of scientists to the network of sim... more We analyze growing networks ranging from collaboration graphs of scientists to the network of similarities deÿned among the various transcriptional proÿles of living cells. For the explicit demonstration of the scale-free nature and hierarchical organization of these graphs, a deterministic construction is also used. We demonstrate the use of determining the eigenvalue spectra of sparse random graph models for the categorization of small measured networks.

Cooperative Formation of Chiral Patterns during Growth of Bacterial Colonies

Bacterial colonies can develop chiral morphology in which the colony consists of twisted branches... more Bacterial colonies can develop chiral morphology in which the colony consists of twisted branches, all with the same handedness. Microscopic observations of the chiral growth are presented. We propose that the observed (macroscopic) chirality results from the microscopic chirality of the flagella (via handedness in tumbling) together with orientation interaction between the bacteria. The above assumptions are tested using a generalized version of the communicating walkers model.

Experiments on Laplacian growth

Cluster-Cluster Aggregation

Fractal Measures

chapter in Fractal Growth Phenomena

Novel type of phase transition in a system of self-propelled particles

A simple model with a novel type of dynamics is introduced in order to investigate the emergence ... more A simple model with a novel type of dynamics is introduced in order to investigate the emergence
of self-ordered motion in systems of particles with biologically motivated interaction. In our model
particles are driven with a constant absolute velocity and at each time step assume the average direction
of motion of the particles in their neighborhood with some random perturbation (g) added. We present
numerical evidence that this model results in a kinetic phase transition from no transport (zero average
velocity, ~v, ( = 0) to finite net transport through spontaneous symmetry breaking of the rotational
symmetry. The transition is continuous, since ~v, ~ is found to scale as (71, —g)t with p = 0.45.

Collective Motion

by Tamás Vicsek and Anna Lazar

Wereview the observations and the basic laws describing the essential aspects of collective motio... more Wereview the observations and the basic laws describing the essential aspects of collective
motion — being one of the most common and spectacular manifestation of coordinated
behavior. Our aim is to provide a balanced discussion of the various facets of this highly
multidisciplinary field, including experiments, mathematical methods and models for
simulations, so that readers with a variety of background could get both the basics and
a broader, more detailed picture of the field. The observations we report on include
systems consisting of units ranging from macromolecules through metallic rods and
robots to groups of animals and people. Some emphasis is put on models that are simple
and realistic enough to reproduce the numerous related observations and are useful for
developing concepts for a better understanding of the complexity of systems consisting
of many simultaneously moving entities. As such, these models allow the establishing of
a few fundamental principles of flocking. In particular, it is demonstrated, that in spite of
considerable differences, a number of deep analogies exist between equilibrium statistical
physics systems and those made of self-propelled (in most cases living) units. In both cases
only a few well defined macroscopic/collective states occur and the transitions between

Fluctuations and scaling in biology (book, Oxford)

Contents Preface ... more Contents
Preface
5
Introduction
7
1 Basic concepts (T. Vicsek)
11
1.1 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.1 Noise versus ﬂuctuations . . . . . . . . . . . . . . . . . . . . . 13
1.1.2 Molecular motors driven by noise and ﬂuctuations . . . . . . . 14
1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 Critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 Scaling of event sizes: Avalanches . . . . . . . . . . . . . . . . 17
1.2.3 Scaling of patterns and sequences: Fractals . . . . . . . . . . . 19
1.2.4 Scaling in group motion: Flocks . . . . . . . . . . . . . . . . . 22
2 Introduction to complex patterns, ﬂuctuations and scaling 25
2.1 Fractal geometry (T. Vicsek) . . . . . . . . . . . . . . . . . . . . . .
. 26
2.1.1 Fractals as mathematical and biological objects . . . . . . . . 27
2.1.2 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Useful rules . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 31
2.1.4 Self-similar and self-aﬃne fractals . . . . . . . . . . . . . . . . 33
2.1.5 Multifractals . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.6 Methods for determining fractal dimensions . . . . . . . . . . 36
2.2 Stochastic processes (I. Der´´enyi) . . . . . . . . . . . . . . . . . . . . 39
2.2.1 The physics of microscopic objects . . . . . . . . . . . . . . . 39
2.2.2 Kramers formula and Arrhenius law . . . . . . . . . . . . . . 41
2.3 Continuous phase transitions (Z. Csah´´ok) . . . . . . . . . . . . . . . . 43
2.3.1 The Potts model . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.2 Mean-ﬁeld approximation . . . . . . . . . . . . . . . . . . . . 46
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1

2
CONTENTS
3 Self-organised criticality (SOC) (Z. Csah´ok) 53
3.1 SOC model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 54
3.2 Applications in biology . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 SOC model of evolution . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 SOC in lung inﬂation . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Patterns and correlations
67
4.1 Bacterial colonies (A. Czir´ok) . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.2 Bacteria in colonies . . . . . . . . . . . . . . . . . . . . . . . .
68
4.1.3 Compact morphology . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.4 Branching morphology . . . . . . . . . . . . . . . . . . . . . . 86
4.1.5 Chiral and rotating colonies . . . . . . . . . . . . . . . . . . . 104
4.2 Statistical analysis of DNA sequences (T. Vicsek) . . . . . . . . . . . 112
4.2.1 DNA walk . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 113
4.2.2 Word frequency analysis . . . . . . . . . . . . . . . . . . . . . 117
4.2.3 Vector space techniques . . . . . . . . . . . . . . . . . . . . . 117
4.3 Analysis of brain electrical activity: the dimensional complexity of the electroencephalogram
(M. Moln´ar) . . . . . . . . . . . . . . . . . . . 120
4.3.1 Neurophysiological basis of the electroencephalogram and event- related potentials . . .
. . . . . . . . . . . . . . . . . . . . . . 120
4.3.2 Linear and non-linear methods for the analysis of the EEG . . 122
4.3.3 Examples for the application of PD2 to EEG and ERP analysis 125
4.3.4 Dimensional analysis of ERPs . . . . . . . . . . . . . . . . . . 128
4.3.5 Clinical applications . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 135
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5 Microscopic mechanisms of biological motion (I. Der´enyi, T. Vicsek) 147
5.1 Characterisation of motor proteins . . . . . . . . . . . . . . . . . . . 147
5.1.1 Cytoskeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.2 Muscle contraction . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.3 Rotary motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1.4 Motility assay . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Fluctuation driven transport . . . . . . . . . . . . . . . . . . . . . . . 160
5.2.1 Basic ratchet models . . . . . . . . . . . . . . . . . . . . . . . 162
5.2.2 Brief overview of the models . . . . . . . . . . . . . . . . . . . 163
5.2.3 Illustration of the second law of thermodynamics . . . . . . . 164

CONTENTS
3
5.3 Realistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.1 Kinesin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.2 Myosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.3.3 ATP synthase . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.4 Collective eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4.1 Finite sized particles in a “rocking ratchet” . . . . . . . . . . . 192
5.4.2 Finite sized particles in a “ﬂashing ratchet” . . . . . . . . . . 199
5.4.3 Collective behaviour of rigidly attached particles . . . . . . . . 205
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6 Collective motion
217
6.1 Flocking: collective motion of self-propelled particles (A. Czir´ok, T. Vicsek) . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.1.1 Models and simulations . . . . . . . . . . . . . . . . . . . . . . 218
6.1.2 Scaling properties . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1.3 Further variants of SPP models . . . . . . . . . . . . . . . . . 228
6.1.4 Mean-ﬁeld theory for lattice models . . . . . . . . . . . . . . . 234
6.1.5 Continuum equations for the 1d system . . . . . . . . . . . . . 240
6.1.6 Hydrodynamic formulation for 2D . . . . . . . . . . . . . . . . 247
6.1.7 The existence of long-range order . . . . . . . . . . . . . . . . 251
6.2 Correlated Motion of Pedestrians (D. Helbing, P. Molna´r) . . . . . . 254
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6.2.2 Pedestrian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 255
6.2.3 Trail Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Segregation mechanisms of tissue cells: from experimental data to models

Complex Adaptive Systems Modeling, 2013

Considerable advance has been made in recent years in the research field of pattern formation by ... more Considerable advance has been made in recent years in the research field of pattern formation by segregation of tissue cells. Research has become more quantitative partly due to more in-depth analysis of experimental data and the emergence modeling approaches. In this review we present experimental observations, including some of our new results, on various aspects of two and three dimensional segregation events and then summarize the computational modeling approaches.

Community structure and ethnic preferences in school friendship networks

Recently developed,concepts,and,techniques,of analyzing,complex,systems,provide new,insight into ... more Recently developed,concepts,and,techniques,of analyzing,complex,systems,provide new,insight into the structure of social networks. Uncovering,recurrent preferences,and,organizational,principles in such networks,is a key issue to characterize them. We investigate school friendship networks from the Add Health database. Applying threshold analysis, we find that the friendship networks,do not form a single connected,component,through,mutual,strong nominations,within a school, while under weaker conditions such interconnectedness is present. We extract the

Modeling the Emergence of Modular Leadership Hierarchy During the Collective Motion of Herds Made of Harems

Journal of Statistical Physics, 2014

ABSTRACT Gregarious animals need to make collective decisions in order to keep their cohesiveness... more ABSTRACT Gregarious animals need to make collective decisions in order to keep their cohesiveness. Several species of them live in multilevel societies, and form herds composed of smaller communities. We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected sub-groups (e.g. harems) by combining self organization and social dynamics. It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision. Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortob\&#39;agy, Hungary). We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal.

Universal Patterns of Collective Motion from Minimal Models of Flocking

2008 Second IEEE International Conference on Self-Adaptive and Self-Organizing Systems, 2008

This paper is concerned with the basic laws describing the essential aspects of collective motion... more This paper is concerned with the basic laws describing the essential aspects of collective motion being one of the most common and spectacular manifestation of coordinated actions. Our purpose is to discuss models that are both simple and realistic enough to reproduce the observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many organisms as well as such non-living objects as interacting robots. Understanding the interrelation of these systems has the potential of improving the interpretation of collective behavioral patterns in both living and nonliving systems by learning about similar phenomena in the two domains of nature.

DYNAMICS OF GROWING SELF-AFFINE SURFACES

Topics in Modern Statistical Physics, 1992

The Fractal Nature of Common Patterns

NATO ASI Series, 1993

Chemomodulation of Cellular Movement and Collective Formation of Vortices by Swarming Bacteria

Bacterial colonies have developed sophisticated modes of cooperative behavior which enable them t... more Bacterial colonies have developed sophisticated modes of cooperative behavior which enable them to respond to adverse growth conditions. It has been shown that such behavior can be manifested in formation of complex colonial patterns. Certain Bacillus species exhibit collective migration, "turbulent like" flow and emergence of whirlpools during colonial development. Here we present experimental observations of collective behavior and a generic model to explain such behavior. The model incorporates self-propelled and interacting "particles" (swarmers). We show that velocity interaction between the particles can lead to a synchronized movement. To explain vortices formation, we propose a plausible mechanism involving a special chemotactic response (rotational chemotaxis) which is based on speed modulations according to the concentration of a chemoattractant. This mechanism differs from that exhibited by swimming bacteria. We show that the chemomodulation of swarmers' speed together with the velocity interactions impose a torque on the collective motion and can lead to formation of vortices. The inclusion of both attractive and repulsive rotational chemotaxis in the model captures the salient features of the observed growth patterns~

Collective Dynamics of Two-Mode Stochastic Oscillators

Physical Review Letters, 2001

We study a system of two-mode stochastic oscillators coupled through their collective output. As ... more We study a system of two-mode stochastic oscillators coupled through their collective output. As a function of a relevant parameter four qualitatively distinct regimes of collective behavior are observed. In an extended region of the parameter space the periodicity of the collective output is enhanced by the considered coupling. This system can be used as a new model to describe synchronizationlike phenomena in systems of units with two or more oscillation modes. The model can also explain how periodic dynamics can be generated by coupling largely stochastic units. Similar systems could be responsible for the emergence of rhythmic behavior in complex biological or sociological systems.

Anomalous segregation dynamics of self-propelled particles

New Journal of Physics, 2015

A number of novel experimental and theoretical results have recently been obtained on active soft... more A number of novel experimental and theoretical results have recently been obtained on active soft matter, demonstrating the various interesting universal and anomalous features of this kind of driven systems. Here we consider the adhesion difference-driven segregation of actively moving units, a fundamental but still poorly explored aspect of collective motility. In particular, we propose a model in which particles have a tendency to adhere through a mechanism which makes them both stay in touch and synchronize their direction of motion - but the interaction is limited to particles of the same kind. The calculations corresponding to the related differential equations can be made in parallel, thus a powerful GPU card allows large scale simulations. We find that in a very large system of particles, interacting without explicit alignment rule, three basic segregation regimes seem to exist as a function of time: i) at the beginning the time dependence of the correlation length is analogous to that predicted by the Cahn-Hillard theory, ii) next rapid segregation occurs characterized with a separation of the different kinds of units being faster than any previously suggested speed, finally, iii) the growth of the characteristic sizes in the system slows down due to a new regime in which self-confined, rotating, splitting and re-joining clusters appear. Our results can explain recent observations of segregating tissue cells in vitro.

METHODS FOR DETERMINING FRACTAL DIMENSIONS

Fractal Growth Phenomena, 1989

Social Group Dynamics in Networks

Understanding Complex Systems, 2009

The rich set of interactions between individuals in the society results in complex community stru... more The rich set of interactions between individuals in the society results in complex community structure, capturing highly connected circles of friends, families, or professional cliques in a social network. Due to the frequent changes in the activity and communication patterns of individuals, the associated social and communication network is subject to constant evolution. The cohesive groups of people in such networks can grow by recruiting new members, or contract by loosing members; two (or more) groups may merge into a single community, while a large enough social group can split into several smaller ones; new communities are born and old ones may disappear. We discuss a new algorithm based on a clique percolation technique, that allows to investigate in detail the time dependence of communities on a large scale and as such, to uncover basic relationships of the statistical features of community evolution. According to the results, the behaviour of smaller collaborative or friendship circles and larger communities, e.g., institutions show significant differences. Social groups containing only a few members persist longer on average when the fluctuations of the members is small. In contrast, we find that the condition for stability for large communities is continuous changes in their membership, allowing for the possibility that after some time practically all members are exchanged.

Evolution of the social network of scientific collaborations

Physica A: Statistical Mechanics and its Applications, 2002

The co-authorship network of scientists represents a prototype of complex evolving networks. In a... more The co-authorship network of scientists represents a prototype of complex evolving networks. In addition, it o ers one of the most extensive database to date on social networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an 8-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. Three complementary approaches allow us to obtain a detailed characterization. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, a ecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution.

Networks in life: scaling properties and eigenvalue spectra

Physica A: Statistical Mechanics and its Applications, 2002

We analyze growing networks ranging from collaboration graphs of scientists to the network of sim... more We analyze growing networks ranging from collaboration graphs of scientists to the network of similarities deÿned among the various transcriptional proÿles of living cells. For the explicit demonstration of the scale-free nature and hierarchical organization of these graphs, a deterministic construction is also used. We demonstrate the use of determining the eigenvalue spectra of sparse random graph models for the categorization of small measured networks.