Introduction NETWORKS as a framework for analyzing large scale data sets in molecular cell biology and neurobiology and at the same time a challenge for mathemat- ical theory, that was our motivation for organizing the conference and for editing the present volume of lecture notes from that conference. So, why is this of relevance, and what can we contribute here? It is the basic paradigm of physics that, on the basis of deep and fun- damental mathematical structures, general and universal laws can be formu- lated and experiments can then be devised for unambiguously con¯rming or rejecting those laws. In contrast to this, in modern biology, we are usually confronted with large data sets in which we need to discern some structure. These structures are not self-evident, nor can they be deduced from general principles. Biological structures are aggregate structures of physico-chemical constituents, and their function emerges from the interaction of many ele- ments. Therefore, we need some guidance for which structures to look and to understand how the biological functions are achieved. Networks provide such a paradigm. They constitute an organizational prin- ciple for biological data sets. We need to identify the basic elements in a given framework. Depending on the biological context, these may be certain types of molecules like speci¯c proteins or RNAs, or whole cells, like neurons. We also need to describe the interactions between them, forming bonds, catalyz- ing reactions, or exchanging spikes through synapses. On one hand, we need to ¯nd out which elements interacts with which speci¯c other ones. On the other hand, we need to understand the dynamics of those interactions. The ¯rst aspect leads us to the mathematical structure of a graph. The second aspect brings in the theory of dynamical systems. A dynamical network thus is a graph whose elements dynamically interact according to the structural pattern of the graph. From these local interactions, a collective dynamical pattern emerges that describes the system as a whole. Networks thus link structural and dynamical aspects. In mathematics, often the richest and most interesting patterns emerge from new connections between di®erent theories. The present link between graph theory and dynamical systems in our opinion con¯rms this and leads to challenging research questions. Its ultimate aim is to contribute to the understanding of the formation of structures in biological and other systems. Networks as such, however, are too general for reaching profound theories. There are simply too many types of graphs and dynamical systems, and ar- bitrary combinations of them seem of little use. Therefore, the theory needs VI the concrete biological data to identify the structures and dynamical patterns of interest and substance. In contrast to many areas of applied mathematics that simply provide tool boxes for data analysis, like statistics packages or nu- merical analysis software, here the theory itself is strongly driven by concrete biological ¯ndings. Thus, these proceedings aim at the same time at displaying a mathematical framework for organizing and understanding biological data, at providing a survey of such biological data, and showing the fruitful synthesis, the applica- tion of mathematical principles to those data, and the mathematical research stimulated by those data. We hope that these proceedings will be useful for an introduction to the theoretical aspects, as a survey of important biological networks (see for example the contribution by Prohaska, Mosig, and Stadler on regulatory networks in eukaryotic cells), and as a sample of more concrete case studies of molecular and neural networks Since biological systems are aggregate and composite ones, in each case, we need to decide which are the relevant and crucial aspects, and which ones can be neglected, and perhaps should even be neglected because they obstruct the identi¯cation of the principles. In biological systems, often details at some lower level average out at some higher level. This then leads to the question which details should be included in a model and which ones omitted. While network theory agrees that the models are based on interactions between dis- crete entities, one then has to decide whether only discrete states are possible or whether we should rather admit continuous state values. In the ¯rst case, naturally also the update is carried out at discrete time steps only. In the sim- plest case, there then are only two possible state values, labeled by 0 (rest) and 1 (activity). We then have a so-called Boolean network. Often such a network, even though it obviously represents a very simpli¯ed model, can still capture crucial qualitative aspects. Also, in a situation of only sparse data, it is advantageous to have a very simple with as few parameters as possible, to avoid having to estimate such parameters without su±cient data. Some of the contributions in these proceedings show the advantages of this approach. In other situations, one may know more about the temporal dynamics and can build a corresponding model. A prime example are networks of spiking neurons that have a much better correspondence with neurobiological reality than simple spin systems like the Hop¯eld network. Also, continuous state dy- namics even with discrete temporal updating, like the so-called coupled map lattices, show new features like synchronization that are not so meaningful in discrete state systems. VII A beautiful example where the question of which details to include in a model can be analyzed are transmission delays in networks. In many for- mal models they are, and can be, neglected because they do not a®ect the emerging collective dynamical patterns. In many cases, however, they can also lead to new dynamical phenomena. Atay presents three pertinent case stud- ies, motivated by models in biology and neurobiology. In one of them, it is found that, surprisingly, transmission delays can facilitate synchronization of the individual dynamics in a global network of coupled oscillators. This pro- vides a direct link to synchronization as a proposed mechanism for feature binding in neural systems. The idea, as originally proposed by von der Mals- burg, is that a complex percept can be neurally represented by the coincident ¯ring of the neurons responding to its speci¯c features. Since there does not seem to exist a global temporal coordinator in neural systems, such a syn- chronization must emerge from local interactions between neurons, and the synchronization studies for dynamical systems provide novel insights. Nev- ertheless, synchronization is only the simplest type of collective dynamics, and non-linear dynamical systems can also ¯nd richer global patterns. This is explored from an experimentally driven perspective in the contribution of Eckhorn and his collaborators who identify global dynamical patterns cor- responding to speci¯c sensory inputs. The same issue is approached from a theoretical perspective in the contribution of Ritter and his collaborators who construct neural networks that develop global dynamical patterns in response to their input from an interaction between neurons and a competition between neuron layers. This is then connected with the principles of gestalt formation in cognitive psychology. This reflects an important approach for elucidating the relationship between neural processes and cognitive phenomena. It is a basic insight from the theory of networks, that such a dynami- cal pattern formation also depends on underlying structural aspects, and the contribution of Jost lays some foundations for the analysis of the interplay between structure and dynamics. A basic set of structural parameters of a graph is provided by its Laplacian spectrum, and this in turn characterizes the synchronizability of so-called coupled map lattice dynamics. This is an important example of the fact that one needs to be careful in isolating the correct parameters that determine the global dynamical features of the net- work supported by a graph. Since there far too many di®erent graphs even of moderate size to describe all them exhaustively, we need to identify some graph classes that display fundamental qualitative features as observed in real data. At the same time, we should also be attentive to the speci¯c features that distinguish graphs in some concrete biological domain from other ones. Huang et al. investigate protein-protein interaction networks from that per- spective, and they found that a hierarchical model captures the qualitative features of some species, but not of others. Also in this regard, Li et al. demon- strate in their contribution that real biological networks (their examples are VIII the cell-cycle and life-cycle networks of protein-protein and protein-DNA in- teractions in budding yeast) can have structural and dynamical properties that are profoundly di®erent from the ones of some general graph paradigms like random graphs studied in the literature. In particular, their examples show a dynamical attractor with a large basin of attraction that is reliably at- tained on a speci¯c pathway. This secures the robust function of the network under perturbations, a crucial aspect for any biological system. While in the work of Li et al., structural and dynamical stability of the important cycles is shown, Zhang and Qian study the same system from a stochastic point of view, and they show that the system is also stable against random perturba- tions. Finally, Huang et al. examine still another concept of stability, namely the one against experimental errors, and they reach the fortunate conclusion that protein-protein interaction networks are quite stable in that sense as well. One speci¯c class of graphs that have found important biological applica- tions are trees. They are used for reconstructing and displaying evolutionary histories on the basis of observed similarities between present species. They are also used in other ¯elds, like in linguistics for the reconstruction of histor- ical divergences from common ancestral languages.