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Dynamical entropy production in cortical circuits with different network topologies

The prevailing explanation for the irregularity of spike sequences in the cerebral cortex is a dynamic balance of excitatory and inhibitory synaptic inputs - the socalled balanced state [1].

Nevertheless its statistical properties are well described by a mean field theory that is independent of the single neuron dynamics, its dynamics is far from being understood. Recently it was found that the stability of the balanced state dynamics depends strongly on the detailed underlying dynamics of individual neurons. Inhibitory networks of leaky integrate-and-fire neurons show stable chaos [2, 3], while a balanced network of neurons with an active spike generation mechanism exhibits deterministic extensive chaos [4].

Previous studies of the dynamics of the balanced state used random (ErdŐs-Rényi) networks. We extended this analysis to arbitrary network topologies and analyzed the entropy production in small world topologies [5], ring networks [6], clustered networks [7], multi-layered networks [8] and topologies with different frequencies of certain network motifs [9]. We derived an analytical expression for the single spike Jacobian containing elements of the coupling matrix, which enabled us to calculate the full Lyapunov spectrum for any desired topology. Using a single neuron model in which action potential onset rapidness [10] and synaptic time constant are adjustable, we simulated the dynamics in numerically exact event-based simulations and calculated Lyapunov spectra, entropy production rate and attractor dimension for a variety of connectivities. We found that the importance of the internal single neuron dynamics for the network stability persists in different topologies. While the Entropy production and Attractor Dimension in clustered [7] and ring networks was very similar to random networks, these dynamical properties were changed substantially when introducing second order network motifs or a small world topology.


  1. van Vreeswijk C, Sompolinsky H: Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science. 1996, 274: 1724-1726. 10.1126/science.274.5293.1724.

    Article  CAS  PubMed  Google Scholar 

  2. Jahnke S, Memmesheimer R-M, Timme M: How Chaotic is the Balanced State?. Frontiers in Computational Neuroscience. 2009, 3: 13-

    Article  PubMed Central  PubMed  Google Scholar 

  3. Zillmer R, Brunel N, Hansel D: Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons. Physical Review E. 2009, 79: 031909-

    Article  Google Scholar 

  4. Monteforte M, Wolf F: Dynamical entropy production in spiking neuron networks in the balanced state. Physical Review Letters. 2010, 105: 268104-

    Article  PubMed  Google Scholar 

  5. Watts , Duncan J, Strogatz , Steven H: Collective dynamics of 'small-world' networks. Nature. 1998, 393 (6684): 440-442. 10.1038/30918.

    Article  CAS  PubMed  Google Scholar 

  6. van Vreeswijk C, Sompolinsky H: Irregular activity in large networks of neurons. Edited by: Chow C, Gutkin B, Hansel D, abd J Dalibard CM. 2005, Les Houches Lectures LXXX on Methods and models in neurophysics. London: Elsevier, 341-402.

    Google Scholar 

  7. Ashok Litwin-Kumar, Brent Doiron: Slow dynamics and high variability in balanced cortical networks with clustered connections. Nature Neuroscience. 2012, DOI: 10.1038/nn.3220

    Google Scholar 

  8. Potjans TC, Diesmann M: 2011, arXiv:1106.5678v1 [q-bio.NC]

  9. Zhao L, Beverlin B, Netoff T, Nykamp DQ: Synchronization from second order network connectivity statistics. Frontiers in Computational Neuroscience. 2011, 5: 28-

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  10. Monteforte M, Wolf F: Single cell dynamics determine strength of chaos in collective network dynamics. Twentieth Annual Computational Neuroscience Meeting: CNS2011. 2011

    Google Scholar 

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Correspondence to Rainer Engelken.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Engelken, R., Monteforte, M. & Wolf, F. Dynamical entropy production in cortical circuits with different network topologies. BMC Neurosci 14 (Suppl 1), P421 (2013).

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