# Celso Grebogi

King's College, University of Aberdeen

UK

Freiburg Institute for Advanced Studies

School of Life Sciences - LifeNet

Albertstr. 19

79104 Freiburg im Breisgau

Investigations of the behaviour of dynamical networks using a direct as well as an inverse approach

Complex systems consist of many interacting components, characterised by the presence of emergent behaviour, and whose dynamics are ubiquitous in the life sciences. The applications of fundamental concepts from the theory of nonlinear complex dynamical systems are not only central for understanding cell biology oriented systems biology but it is also essential for the understanding of a range of problems from neurosciences to environmental systems. First principle modelling, inference, as well as control of such complex systems have become increasingly important. Networks of interacting nodes, each with their own dynamics, are one of the main mathematical tools for the description of complex systems. Depending on the particular application, the dynamical behaviour of the nodes can be anything from binary, such as in the Kauffman networks for the description of genetic or metabolic pathways, to deterministic or stochastic and even chaotic networks such as the ones in some neuronal models.

This project proposes to understand the behaviour of dynamical networks from two complementary directions, namely, by using a direct as well as an inverse approach.

The “direct approach” refers to the explicit modelling of the dynamics of the individual nodes forming the network, as well as the network topology, i.e., the coupling among the nodes. As guiding examples, the fundamental aspects of two main types of biological networks will be considered: neuronal networks and biochemical signaling pathways.

A central aspect of neuronal networks that has received little consideration in the literature so far is the interplay among the multiple time scales of oscillators, and how it can lead to novel types of emergent behaviour. Typically, neuronal dynamics exhibits two main time scales: spikes and bursts of spikes, reproducible by well established models, such as the Hindmarsh-Rose and Hodgkin-Huxley models. Even though the characterisation of synchronisation and information transfer in such systems is of central importance for the understanding of fundamental neuronal processes, it is still poorly understood. Therefore, the study will focus on oscillators with two main time scales and how the onset of synchronisation and the increase of information transfer in networks of such oscillators are influenced by the strength and symmetry of the coupling of different time scales, as well as by the coupling among the different nodes of the network.

Depending on the nature of biochemical networks, the approach that will be adopted to study them will be quite different. These networks, in particular the signaling networks, will be considered as input-output systems, and the dynamics on the single nodes will be simplified and represented by binary states (on/off, active/inactive). This simplification arises naturally, e.g., when one considers the states of proteins in phosphorylation cascades that relay a signal further downstream like the MAP-kinase pathway. The phosphorylated or active protein acts as a kinase for the next phosphorylation reaction. The focus of the analysis will be on how the structure of the network influences its functioning. Control strategies will be developed, that will allow the system to be driven into states or types of behaviour that are desirable. The development of these mathematical methods will have a large impact for a broad range of vital biological processes, from stress responses to sensing of nutrients.

The “inverse approach”, also known as the inverse problem, consists of inferring the architecture of the network based on observed data. This problem arises frequently when analysing data from real neuronal systems by using actual data from, say, functional Magnetic Resonance Imaging (fMRI) or electroencephalography (EEG), including single neuron recordings. In this case, it is possible to acquire data about different nodes of the system but without direct access to the underlying topology of the network. In order to gain a deeper understanding of the full complex system under study, such knowledge is of fundamental relevance. For the general case of nonlinear dynamics on the nodes, several methods exist to infer the topology of a network consisting of three or at most four nodes. The extension of existing methods for the analysis of larger networks is a challenging one due to many inherent difficulties. Linear techniques are currently applied to systems consisting of up to a dozen nodes. The increasing of the network size to hundreds of nodes, which can be observed in fMRI experiments, has not been thoroughly addressed so far. One of the strategies of this project is to analyse such large dynamic networks by breaking it up into smaller subnetworks. Another strategy is to extend the kind of analysis already being carried out by my group. Preliminary results show that, for nonlinear systems, a combination of concepts based on phase space behaviour, well-established for the analysis of nonlinear dynamical systems, with concepts based on causality, conventionally used for the analysis of linear systems, is very promising to tackle the inference of such large networks. Based on these results, mathematical strategies will be further developed to infer the architecture of such networks.

- M.S. Baptista, R.M. Rubinger, E.R. Viana, J.C. Sartorelli, U. Parlitz, C. Grebogi: Mutual information rate and bounds for it.
*Plos One*, 2012; 7 (10): e46745. http://dx.doi.org/10.1371/journal.pone.0046745 - "Inference of Granger causal time-dependent influences in noisy multivariate time series", L. Sommerlade, M. Thiel, B. Platt, A. Plano, G. Riedel, C. Grebogi, J. Timmer, B. Schelter,
*Journal of Neuroscience Methods*, 2011 (online); 2012 (print), 203 (1): 173-185 - "A Matter of Life or Death: Modeling DNA Damage and Repair in Bacteria", J. Karschau, C. de Almeida, S. Miller, C. Grebogi, A.P.S. de Moura,
*Biophys J*, 2011; 100 (4): 814-821 - "Characteristics of level-spacing statistics in chaotic graphene billiards" L.A. Huang, Y.C. Lai YC, C. Grebogi,
*Chaos*, 2011; 21 (1): 013102 - "Quantum chaotic scattering in graphene systems", R. Yang R, L. Huang, Y.C. Lai, C. Grebogi,
*Epl-europhys Lett*, 2011; 94 (4): 40004 - "Predicting Catastrophes in Nonlinear Dynamical Systems by Compressive Sensing", W.X. Wang, R. Yang R, V. Kovanis, C. Grebogi,
*Phys Rev Lett*, 2011; 106 (15): 154101 - "Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games", W.X. Wang, X.A. Ni, Y.C. Lai, C. Grebogi,
*Phys Rev E*, 2011; 83 (1): 011917 - "Finite-size effects on open chaotic advection,” R.D. Vilela, A.P.S. Moura, and C. Grebogi, Phys. Rev. E 73, 026302 (2006).
- "Bubbling bifurcation: loss of synchronization and shadowing breakdown in complex systems," R.L. Viana, C. Grebogi, S. Pinto, S. Lopes, A. Batista and J. Kurths, Physica D 206, 94 (2005).
- "Effective dynamics in Hamiltonian systems with mixed phase space," A.E. Motter, A.P.S. Moura, C. Grebogi, and H. Kantz, Phys. Rev. E 71, 036215 (2005).
- "Chemical and biological activity in open flows: a dynamical system approach,” T. Tel, A.P.S. Moura, C. Grebogi, and G. Karolyi, Physics Reports 413, 91 (2005).
- "Poincare recurrence and measure of hyperbolic and nonhyperbolic chaotic attractors," M.S. Baptista, S. Kraut, and C. Grebogi, Phys. Rev. Lett. 95, 094101 (2005).
- "Two-dimensional map for impact oscillator with drift," E. Pavlovskaia and M. Wiercigroch, Phys. Rev. E 70, 036201 (2004).
- "Chemical and biological activity in 3-dimensional flows," A.P.S. Moura and C. Grebogi, Phys. Rev. E 70, 026218 (2004).
- "Unstable dimension variability and codimension-one bifurcations of two-dimensional maps," R.L. Viana, R. Barbosa and C. Grebogi, Phys. Lett. A 321, 244 (2004).
- "Nonhyperbolic chaotic systems and synchronization," E.E.N. Macau and C. Grebogi, Adv. Space Dynamics 4, 123 (2004).
- "Stability properties of nonhyperbolic chaotic attractors with respect to noise," S. Kraut and C. Grebogi, Phys. Rev. Lett. 93, 250603 (2004).
- "Reactive particles in random flows," G. Karolyi, T. Tel, A.P.S. Moura, and C. Grebogi, Phys. Rev. Lett. 92, 174101 (2004).
- "Escaping from nonhyperbolic chaotic attractors," S. Kraut and C. Grebogi, Phys. Rev. Lett. 92, 234101 (2004).
- "Reactions in flows with nonhyperbolic dynamics," A.P.S. Moura and C. Grebogi, Phys. Rev. E 70, 036216 (2004).
- "Universality in active chaos," T. Tel, T. Nishikawa, A.E. Motter, C. Grebogi and Z. Toroczkai, Chaos 14, 72 (2004).