Abstract: Proper theoretical network models of the brain are in need of well-characterized brain areas that are preferably described at a multi-scale level. The suprachiasmatic nucleus (SCN) is the master clock in the mammalian brain and consists of 20,000 individually oscillating cells. Each cell contains a molecular feedback loop that produces an endogenous rhythm with its own intrinsic frequency. In order to obtain a robust and coherent 24-h rhythm that can drive other circadian rhythms in our body, the SCN cells synchronize to each other as a result of neural coupling. In addition to the internal synchronization, the SCN synchronizes to external cycles, such as to the 24-h light-dark cycle and to seasonal cycles. The network structure of the SCN results in a system that shows a balance between robustness on the one hand and flexibility on the other hand. In our lab we perform electrophysiological recordings from single neurons and from populations of about 100 neurons. Furthermore transgenic luciferase expressing mice are used to simultaneously measure the rhythms in gene expression at single cell level. Finally we record electrical activity from populations of neurons with implanted electrodes in freely moving animals. In this preparation, the recorded neurons of the central clock are interacting with other brain areas. We have observed that temporal behavioral patterns and the central clock show scale invariant behavior. With disease and aging, scale invariance is lost, and also in a brain slice preparation when the clock is not communicating with other brain areas, scale invariance is absent. We conclude that scale invariance emerges at the integrated network level. Understanding how neurons and brain regions communicate, coordinate, synchronize, and collectively respond to signals and perturbations is one of the most intriguing, yet unsolved problems in neuroscience. As the output of the SCN is unambiguously measurable in terms of phase and period, the measurements from the different levels of organization, i.e., the molecular level, the cellular level, the organ level and the behavioral level, can be compared. Current studies are aimed at bridging scales, from the micro to the macro level and vice versa, thereby understanding how properties emerge at each of these levels.

Abstract: We consider networks of identical single-input-single-output systems that interact via linear, time-delay coupling functions. The systems itself are inert, that is, their solutions converge to a globally stable equilibrium. However, in the presence of coupling, the network of systems exhibits ongoing oscillatory activity. We study the emergence of oscillations by deriving conditions for 1. The solutions of the time-delay coupled systems to be bounded, 2. the network equilibrium to be unique, and 3. the network equilibrium to be unstable. The network of time-delay coupled inert systems is oscillatory provided that the aforementioned three conditions are satisfied. In addition, using recent results on the existence of partial synchronization manifolds, we identify the patterns of synchronized oscillation that may emerge.

Abstract: Functional connectivity measured during the resting-state in BOLD fMRI has become a popular approach for assessing network interactions in the brain. The study of patterns of connectivity between multiple distributed regions and their associated functional dynamics has been recognised as a powerful tool in cognitive and clinical imaging neurosciences. This talk will provide a basic overview of tools and techniques with a view to estimating directed brain network interactions.

Abstract: We propose and study a random graph model on the hypercubic lattice that interpolates between models of scale-free random graphs and long-range percolation. In our model, each vertex x has a weight Wx, where the weights of different vertices are independent and identically distributed random variables. Given the weights, the edge between x and y is, independently of all other edges, occupied with probability 1?exp(?WxWy/|x?y|?), where (a) ? is the percolation parameter, (b) |x ? y| is the Euclidean distance between x and y, and (c) ? is a long-range parameter.This model gives rise to geometric random graph models that have a high amount of inhomogeneity due to the high variability present in the vertex weights. The model interpolates nicely between long-range percolation models, obtained when the weights are all equal, and scale-free random graph models, obtained when considering the model on the complete graph and letting |x ? y| ? 1 for every x ?= y, so there is no geometry. The most interesting behavior can be observed when the random weights have a power-law distribution, i.e., when P(Wx > w) ? w1?? for some ? > 1. In this case, we see that the degrees are infinite a.s. when ? = ?(? ? 1)/d < 1, while the degrees have a power-law distribution with exponent ? when ? > 1. Our main results describe phase transitions in the positivity of the critical value and in the small-world nature in the percolation cluster as ? varies. Let ?c denote the critical value of the model. Then,we show that ?c =0 when ?<2, while ?c >0 when ?>2. Further, conditionally on 0 and x being connected, the minimal number of edges needed to hop between 0 and x is of order loglog|x| when ? < 2 and at least of order log|x| when ? > 2. These results are similar to the ones in inhomogeneous random graphs, where a wealth of further results is known. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., Wx = 1 for every x), and on inhomogeneous random graphs (i.e., the model on the complete graph of size n and where |x ? y| = n for every x ?= y).Acknowledgements. This work is supported by the Netherlands Organisation for Scientific Re- search (NWO) through VICI grant 639.033.806 and the Gravitation Networks grant 024.002.003. Joint work with Mia Deijfen and Gerard Hooghiemstra.

Abstract: ?The brain has no knowledge until connections are made between neurons. All that we are ? comes from the way our neurons are connected? (Berners Lee). This motto underlies the field of ?connectonics? which over the past decade has gained momentum in neuroscience and neuro-imaging. For the human brain, however, no methods exist to estimate connectivity at the neuronal level from in-vivo recordings. Instead the structural connectivity between brain regions can be estimated from diffusion tensor MR images, or the functional or effective connectivity between brain regions can be estimated from functional MR recordings, or from recordings of the electrical activity in the brain, using EEG or MEG recordings. The functional connectivity at neuronal level changes at different time scales (from maturation and growth to learning and behavior), and so presumably does the functional connectivity of brain regions. To estimate functional connectivity from EEG or MEG it is necessary to estimate from the sensor level data, at the scalp surface, the ?source? level signals, i.e. mean field estimates of the activity representing a cortical area. In a second step the statistical dependencies between these source level signals can be estimated, and it is commonly assumed that such statistical dependencies reflect the connectivity between brain regions (?the functional connectivity?). For both these steps leading to a network description of the brain, there exists many different methods. The perfect method would be multi-variate, time-resolved, unique, would reveal non-linear coupling, and would not suffer from mixing, or from volume conduction and would not depend on the choice of reference in the case of EEG. Clearly such a method does not exist. Moreover the methods used in the two steps of the analysis (source estimation and connectivity estimation) cannot be independently chosen. In a short overview the state of the art techniques in EEG or MEG studies of the neural networks in the brain will be summarized and the short-comings of each of the most commonly used methods will be discussed.

Abstract: Recent advances in functional magnetic resonance imaging (fMRI), diffusion tensor imaging (DTI), electroencephalography (EEG) and magnetoencephalography (MEG) in combination with graph theoretical network analysis have allowed for investigating structural and functional connectivity in the brain. Brain networks show small-world properties, a modular structure and highly connected hub-regions. Understanding of the network is important for understanding the organizational properties of the brain and will become increasingly important for clinical research. The fMRI, DTI, EEG and MEG techniques are able to study large parts of the brain, but they can only map a network of brain regions and are unable to investigate the network of single cells within the inspected brain regions. In order to obtain a complete picture of healthy brain function, the connection between cellular mechanisms and brain regions must be made. Our group investigates the cellular brain network of one such brain region, the circadian clock. The circadian clock, which is located in the suprachiasmatic nuclei (SCN) and drives the daily 24-hour rhythms in our body, is functionally dependent on emergent network properties. While the ability of individual SCN neurons to produce 24-hour rhythms is a cell-autonomous property, the ability of the SCN to respond to light, to adjust to the seasons and to synchronize after a jet-lag is critically dependent upon the state of the neuronal network. The synchronized network output regulates all daily rhythms in our body and is heavily dependent on the interactions between the neurons and the network topology of the clock. We investigated methods to assess network properties on our data, such as small-worldness and scale-freeness for this cellular network. Complicating factors include that this functional network is not static during its oscillation cycle: for example, the network appears to be different between day and night time. This dynamical nature of the network is normally not taken into account in network studies. Furthermore, the SCN network operates at different timescales: the network communication between the neurons in the SCN takes place in the order of milliseconds, while the synchronized network output takes place on a 24-h timescale. How do these timescale differences influence the network properties of the SCN? These dynamical issues open up new aspects for studying brain networks. Finally, to connect brain regions and cellular networks, we do not only look at the SCN network in isolation, but also regard it as being part of a bigger network of brain regions. We currently investigate, using network techniques on our experimental data, how the interactions between the SCN brain region and the rest of the brain is regulated, so that we gain better understanding of its place in the higher order network of the brain. This will in the end result in a hierarchical multi-level network model that operates at different timescales and takes into account the dynamical nature of the biological clock.

Abstract: Networks of coupled nonlinear dynamical systems often display unexpected phenomena. They may for example synchronise. This form of collective behaviour occurs when the agents of the network behave in unison. An example is the simultaneous firing of neurons. It has also been observed that synchrony often emerges or breaks through quite unusual bifurcation scenarios. In this talk I will show how this can be understood with the help of a new geometric framework for network dynamics that we developed. This framework allows for the problem of synchrony breaking to be reduced to its essential dimensions, which in turn enables us to compute and classify a large class of synchrony breaking bifurcations in networks. This is joint work with Jan Sanders and Eddie Nijholt.

Abstract: It is widely accepted in the neuroscience community that the network organization of the brain is scale free and has small world structure. This is considered to be true on multiple levels, ranging from the level of brain regions down to the level of cortical minicolumns. Although high level networks of the brain can be examined using fMRI or other means of measurement, all these techniques are limited in their resolution and they are not expected to provide low level connectivity information about the brain. One possible way to overcome the resolution barrier can be realized as follows. A family of low level (high resolution) networks is used where the nodes of the network have some simple dynamics. The behavior of these nodes can be simulated considering their interconnection network. Simultaneously with this, a virtual measurement can be performed to capture the overall global behavior of the network. This resembles a hypothetical brain with the simulated structure and dynamics placed in an fMRI machine. These virtual measurements can be compared to each other and to real measurement. This comparison can serve as a proxy to assess certain network properties of the low level brain network. All this is done under the premise that the global behavior of large networks of interconnected units with simple dynamics is largely influenced by the underlying network structure. Our goal is to find out to what extent is the scale free and small world nature present in the brain network. In order to achieve this goal we have to specify a family of graphs that is plausible for a low level brain network topology and on top of this the node dynamics also needs to be chosen. Since the cortex can be considered as a folded 2D structure with long range connections, the Scale-Free Percolation random graph model on Z2 with power law node weights is a suitable choice for the graph family in the previously outlined agenda. We can obtain a coupled family of graphs by appropriately changing the parameters alpha and lambda of the graph model simultaneously. This family of graphs contains the Z2 grid as one limiting case while an almost tree-like scale free graph is at the other end of the spectrum. The chosen coupling allows to resemble a certain evolution from the plain grid to a more and more efficient network in terms of information transfer. The chosen node dynamics is given by the paradigm Ising model with its single spin flip Glauber dynamics. We simulate excitation of a brain region coming from another region by considering different Scale-Free Percolation models as the structure of the excited brain region where the external stimulus manifests itself as inhomogeneous external field acting on the Ising model. By comparing globally measured quantities of the excited brain region we can see to what extent is deviation from the Z2 base structure is still beneficial in terms of the simulated brain behavior. This should allow us to gain deeper insight into how the low level network structure of a brain is organized.