Organization of excitable dynamics in hierarchical networks - Theoretical analysis, numerical simulation and application to neuroscience

• Applicant: Professor Dr. Marc-Thorsten Hütt
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2009 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 156244915

• Final Report Year: 2016

Final Report Abstract

With our project we established and qualitatively understood several fundamental relationships between network architecture and excitable dynamics. We were able to take advantage of several recent developments in Computational Neuroscience – the interest in correlations between structural connectivity (network) and functional connectivity (excitation patterns, the need for an understanding of sustained activity (i.e. the network mechanisms responsible for amplifying and maintaining excitations) – in order to shape our project and define our research agenda. Our work has been built around a simple three-state, time-discrete stochastic cellular automaton model of excitable dynamics, the SER model. Making use of the simplicity of the model we could establish a few ’stylized facts’ of how network topology shapes excitable dynamics. A remarkably clear and non-trivial picture emerges: In general, simultaneous activation of linked nodes is suppressed (leading to an anti-correlation of the coactivation matrix and the graph’s adjacency matrix). But this suppression can be ’overwritten’ in graphs with high clustering (like encountered within the modules of a modular graph). The (one step) time-delayed co-activation matrix provides information on the propagation of excitations. The strong asymmetry between the upper and lower triangles of this matrix in the case of scalefree graphs is an indicator of the waves described in Müller-Linow et al. (2008): Excitations propagate towards the hub(s) individually, while they emerge from the hub in a coherent fashion. The interplay of several topological scales that is inherent in this picture has prompted two research directions we actively pursued in our most recent work: (1) the role of cycles of different length and barriers along paths, (2) the large-scale organization of excitable dynamics on graphs from the perspective of spatiotemporal pattern formation. Regarding cycles, we show that a network’s inventory of long cycles is a good predictor for transient lengths of excitable dynamics and, relatedly, that self-sustained activity is facilitated by an interplay of long and short cycles. For a variant of the three-state CA model that requires κ percent of active neighbors (relative excitation threshold) we analyze how a single excitation propagates through a random network as a function of the excitation threshold. We show that the onset of sustained activity can be understood as an interplay between topological cycle statistics and path statistics. The effect of modularity on SC/FC correlations has been validated in a more realistic model of excitable dynamics. The results obtained during the funding period go beyond the work program envisioned in the original proposal. We did not expect to be able to quantitatively show the relevance of topological cycles for sustained activity. Also, the analysis of the SER model with a relative excitation threshold has been much richer and more productive than anticipated.

Publications

• (2011). Asymmetric transition and timescale separation in interlinked positive feedback loops. International Journal of Bifurcation and Chaos, 21(07), 1895-1905
  Yordanov, P., Tyanova, S., Hütt, M.-Th. and Lesne, A.
  
• (2012) Building blocks of self-sustained activity in a simple deterministic model of excitable neural networks. Frontiers in Computational Neuroscience 6, 50
  Garcia, G.C., Lesne, A., Hütt, M.-Th. and Hilgetag, C.C.
  
• (2012) Stochastic resonance in discrete excitable dynamics on graphs. Chaos, Solitons & Fractals 45, 611-618
  Hütt, M.-Th., Jain, M., Hilgetag, C. and Lesne, A.
  
• (2013) Model complexity in the study of neural network phenomena. In: Y. Yamaguchi (ed.), Advances in Cognitive Neurodynamics (III), Springer-Verlag, 77-81
  Hilgetag. C.C. , Hütt, M.-Th. and Zhou C.
  
• (2013). Trade-off between multiple constraints enables simultaneous formation of modules and hubs in neural systems. PLoS Comput Biol, 9(3), e1002937
  Chen, Y., Wang, S., Hilgetag, C. C. and Zhou, C.
  
• (2014) Hierarchical modular brain connectivity is a stretch for criticality. Trends in Cognitive Sciences 18, 114-115
  Hilgetag, C.C. and Hütt, M.-Th.
  
• (2014) Network-guided pattern formation of neural dynamics. Phil. Trans. Roy. Soc. B 369, 20130522
  Hütt, M.-Th., Kaiser, M. and Hilgetag, C.C.
  
• (2014) Role of long cycles in excitable dynamics on graphs. Phys. Rev. E 90, 052805
  Garcia, G.C., Lesne, A., Hilgetag C.C. and Hütt, M.-Th.
  
• (2015) A closer look at the apparent correlation of structural and functional connectivity in excitable neural networks. Scientific Reports. 5, 7870
  Messé, A., Hütt, M.-Th., König, P. and Hilgetag, C.