RESEARCH PROJECTS 2014 Collective behavior in networks of biological neurons: mathematical modeling and software development Ioanna Chitzanidi, Postdoctoral Researcher National Center for Scientific Research Demokritos Nikos E. Kouvaris, Postdoctoral Research Associate University of Barcelona Christos Antonopoulos, Postdoctoral Research Fellow University of Aberdeen Vasilis Kanas, PhD student University of Patras Filippos Vogiatzian, Master's student University of Amsterdam December 2014
Summary In the current research project we have focused on the characteristic aspects of excitability and synchronization in network-organized neuronal systems. In such networks, the nodes represent excitable neurons while the links connecting the nodes correspond to the synaptic coupling between the neurons. In particular, the effects of network topology on the excitation threshold and the propagation of excitation pulses over the networks has been analyzed. Detailed numerical simulations and analytical calculations have been carried out for scale-free, Erdös-Rényi and tree networks. Moreover, the fascinating effect of chimera states has been investigated for irregular network topologies and inhomogeneous node dynamics. In particular, chimera-like states have been identified in the C.elegans network where the interaction between dynamics and community-based topology plays a significant role. The robustness of chimera states has also been addressed in regular topologies with perturbations in the local dynamics, as well as in hierarchical networks and asymmetrical rings with gaps in the connectivity matrix. For the purpose of these studies a software toolbox was developed in Python2.7 for the manipulation and visualization of complex networks and the corresponding excitable dynamics. Aims and Methodology of the Project The two main objectives of the current project have involved the effects of network topology on the propagation of excitation pulses and the phenomenon of chimera states. In particular, we have focused on the feature of neuronal excitability from a network perspective, i. e. how the propagation of neuronal excitations through a network depends on the network topology, as well as how chimera states are affected by network heterogeneities. Excitation waves For the dynamics of a single neuron (dynamics of a node) we have considered a two-component excitable system, where only the activator can diffuse and the inhibitor varies slowly. This is described by the FitzHugh- Nagumo (FHN) model [1] where u represents fast changes of the electrical potential across the membrane of a neural cell, while the inhibitor variable v is related to the gating mechanism of the membrane channels. Here we have realized a hierarchical organisation of the system on a regular tree with branching ratio k 1, where k is the degree of the networks, 1
Figure 1: Left: (a) Snapshot of an excitation wave which propagates from the root towards the periphery in a regular tree. (b) It can also be represented as a pulse by grouping the nodes with the same distance from the root into a single shell. (c) The time evolution of a single node (or shell) is shown. Right: Evolution of the activator density for an Erdös-Rényi random network. Excitation is applied to the hub (a), it consequently propagates to the neighbouring nodes with degrees k < 6 (b), (c), and finally dies out before the nodes with degrees k > 6 (d). The propagation path is shown with thick green colour. Node labels denote node degrees. i. e. the number of connections of each node. In trees, all nodes with the same distance r from the root can be grouped into a single shell. The activator of a node which belongs to the shell r can be diffusively transported to k 1 nodes in the next shell r + 1 and to just one node in the previous shell r 1. By means of this approximation, instead of investigating propagation of excitation waves directly on a tree network [Figure 1 (a), left panel], we have used the sequence of coupled shells [Figure 1 (b), left panel] which reads, u r = u r u3 r 3 v r + 0.04[u r 1 ku r + (k 1)u r+1 ], v r = 0.02(u r + 1.1). (1) As has been found in Kouvaris et al. [2], the propagation velocity of excitation waves in hierarchical trees decreases with larger degrees until a critical value. At this critical degree waves are still stable and propagate, however, with their minimum velocity. Once this threshold is exceeded, undamped propagation is not possible. Excitation waves loose stability through a saddle-node bifurcation and die out. This behaviour has been 2
revealed in numerical simulations while we were able to predict it by means of a kinematical theory proposed in [2]. The results of this analysis are also relevant for understanding the early stage of excitation spreading in random networks. When activation is applied to a node and we look locally at its vicinity with its first neighbours, activation propagates only to the nodes with degrees smaller than the critical degree. Depending on the system parameters, excitation may propagate through some nodes or disappear before nodes with larger degrees. This degree heterogeneity in random gives rise to the appearance of some preferred paths where can propagate (see green links in Figure 1, right panel). Chimera states Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology [3]. Within this project, our aim has been to investigate the phenomenon of chimera states in networks with irregular topologies and inhomogeneous node dynamics. Since 2012, C.elegans is the only organism to have its connectome completely mapped out: It consists of 302 neurons and about 7000 neuronal synapses. Based on the C.elegans connectivity (adjacency) matrix [4], we have detected 6 communities of neurons using the walktrap and Louvain community detection methods [5,6]. The left panel of Figure 2 shows these 6 communities and their intra-connections. Black links correspond to local electrical synapses between nodes within each community, whereas grey links denote chemical synapses connecting nodes of different communities. We equipped each node of the network with three-variable Hindmarsh- Rose (HR) neural dynamics [7]. The advantage of this model is that it is capable of reproducing several important dynamical behaviors observed in the activity of real neurons studied in vitro, such as rapid firing or regular and chaotic bursting. The equations that govern the dynamics of the neural activity of the whole brain network are the following: ṗ i = q i p 3 i + 3p 2 i n i + 3.2 g n (p i 2) q i = 1 5p 2 i q i, ṅ i = 0.005[4(p i + 1.6) n i ], N B ij S(p j ) g l j=1 N j=1 G ij H(p j ), φ i = q i p i ṗ i q i, i = 1,..., N. (2) p 2 i + q 2 i 3
Figure 2: Left: Communities and connections in the C.elegans network given by the walktrap detection method. Right: Snapshot of variables p and φ (top), and timeseries of node 52 and global order parameter R for the entire network, (bottom). Coupling constants are g n = 0.4 and g l = 0.01. Node indices are reordered by community. Here p i and q i have the same physical meaning as the FHN variables (Eq. 1), while n i represents a slowly varying current. The oscillations of each node are characterized by a geometric phase φ i which is calculated from the first two variables. H(p j ) = p j is the linear electrical coupling function and 1 S(p j ) = is the nonlinear chemical coupling term. B 1+e 10(p j +0.25) ij is the adjacency matrix for chemical connections, G ij is the Laplacian matrix for electrical connections, and the corresponding coupling strengths are gl (electrical) and g n (chemical). The synchronization within each community is calculated using the Kuramoto order parameter [3]: R c = 1 N c e iφc j N c, where N c is the size of the community. Various sychronization patterns may be found by spanning the parameter space (g l, g n ). In this project, we are interested in studying chimera-like states in which syncrhonized neuronal activity of community members coexist and alternate in time with incoherent ones. To detect coupling pairs (g l, g n ) that give rise to such behaviors in the C.elegans brain dynamical network, we have used tools from [8] such as the metastability index and the chimera-like index, which are measures that quantify the metastability and prevalence of chimera states in dynamical networks. Based on our extensive studies for the C.elegans brain dynamical network, we have been able to identify such regions of chimera-like behavior as shown in the right panel of Figure 2, where time snapshots of variables j=1 4
p and φ are displayed in the upper part. We see that while neurons of communities 2 and 4 are almost fully syncrhonized, thus forming a coherent pattern, the same does not hold for the rest of the communities where neurons oscillate in an incoherent manner [10]. Our next task has been to discuss the robustness of chimera states in networks of FHN oscillators (Eq. 1). Considering networks of inhomogeneous elements with regular coupling topology and networks of identical elements with irregular coupling topologies, we have demonstrated that chimera states are robust with respect to these perturbations, and we have analyzed their properties as the inhomogeneities increase. We have found that modifications of coupling topologies cause qualitative changes of chimera states: Additional random links induce a shift of the stability regions in the system parameter plane, whereas gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state. Finally, inspired by recent calculations of fractal dimensions from human brain MRI scans [9], a hierarchical network derived from a Cantor set with fractal dimension < 1 was constructed, and chimera states with nested coherent and incoherent regions were obtained. For further details and corresponding figures refer to [11]. Deliverables The current project has led to 3 publications, 2 conference contributions and the development of 1 software tool. Moreover, it has opened many perspectives for future research and collaboration. The deliverables of this project are the following: Publications 1. Propagation failure of excitation waves on trees and random networks, Nikos E. Kouvaris, Thomas M. Isele, Alexander S. Mikhailov, and Eckehard Schöll, EPL, 106, 68001 (2014). 2. Robustness of chimera states for coupled FitzHugh-Nagumo oscillators: Irregular network topologies and inhomogeneous nodes, Astero Provata, Iryna Omelchenko, Johanne Hizanidis, Eckehard Schöll, and Philipp Hövel, arxiv:1411.5481v1 [nlin.ao] (2014). 3. Chimera-like dynamics and the role of topology in the C.elegans brain network, Johanne Hizanidis, Nikos E. Kouvaris, Gorka Zamorra-López, and Christos Antonopoulos. in preparation (2014). 5
Conference Contributions 1. Nikos Kouvaris: Turing patterns in multiplex networks, Fourth International Workshop on Statistical Mechanics and Dynamical Systems, July 17-19 (Athens). 2. Christos Antonopoulos: Do Brain Networks Evolve by Maximizing Flow of Information?, NUMAN 2014: Recent Approaches to Numerical Analysis: Theory, Methods and Applications, September 2-5 (Chania). Software Tool A Python toolbox has been developed and used for adapting the functional interface of NetworkX and SciPy to our needing. It was also used for manipulating the characteristic properties of networks, such as node degrees and attributes, links weight, spectral properties of the Laplacian matrix etc. Furthermore, we use the SciPy library for handling sparse matrices (more efficient for large networks) and solving systems of ODEs/PDEs (our models) based on our networks. Combining SciPy and NetworkX, and integrating them into our new python module, we have created a flexible library that allows us to solve numerically and visualize the dynamics of excitable systems in various network topologies. This software was designed to be very easily used by people with small or even no special programming skills. All figures presented in this report have been generated with this toolbox. References [1] E. M. Izhikevich, Int. J. Bifurcat. Chaos 10, 1171 (2000). [2] N. E. Kouvaris, T. M. Isele, A. S. Mikhailov, and E. Schöll, Eur. Phys. Lett. 106, 68001 (2014). [3] Y. Kuramoto and D. Battogtokh, Nonlin. Phen. in Complex Sys. 5, 380 (2002). [4] http://cmtk.org/viewer/datasets [Online; accessed 27-February- 2014]. [5] S. Fortunato, Phys. Rep. 486, 75 (2010). [6] V. D Blondel, J.-L. Guillaume, R. Lambiotte, and R. Lefebvre, J. Stat. Mech.: Theory and Experiment, 10, P10008 (2008). 6
[7] J. Hizanidis, V. Kanas, A. Bezerianos, and T. Bountis, Int. J. Bifurcat. Chaos 24, 1450030 (2014). [8] M. Shanahan, Chaos 20, 013108 (2010). [9] P. Katsaloulis, A. Ghosh, A. C. Philippe, A. Provata, and R. Deriche, Eur. Phys. J. B 85, 1 (2012). [10] J. Hizanidis, N. E. Kouvaris, Gorka Zamorra-López, and C. Antonopoulos, in preparation (2014). [11] I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, and P. Hövel, arxiv:1411.5481v1 [nlin.ao] (2014). 7