Finding Quasi-Optimal Network Topologies for Information Transmission in Active Networks

PLOS ONE, Oct 2008

This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.

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Finding Quasi-Optimal Network Topologies for Information Transmission in Active Networks

Hussein MS (2008) Finding Quasi-Optimal Network Topologies for Information Transmission in Active Networks. PLoS ONE 3(10): e3479. doi:10.1371/journal.pone.0003479 Finding Quasi-Optimal Network Topologies for Information Transmission in Active Networks Murilo S. Baptista 0 Josue´ X. de Carvalho 0 Mahir S. Hussein 0 Raya Khanin, University of Glasgow, United Kingdom 0 Max-Planck-Institut f u ̈r Physik komplexer Systeme, Dresden, Deutschland, 2 Centro de Matema ́tica da Universidade do Porto , Porto , Portugal , 3 Institute of Physics, University of Sa ̃o Paulo , Sa ̃o Paulo , Brasil This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons. - Funding: This study was financed by the Max Planck Institute for the Physics of Complex Systems, FCT, FAPESP, CNPq, and the Martin Gutzwiller prize 2007/2008 (MSH). Competing Interests: The authors have declared that no competing interests exist. Introduction Given an arbitrary time dependent stimulus that externally excites an active network formed by systems that have some intrinsic dynamics (e.g. neurons and oscillators), how much information from such stimulus can be realized by measuring the time evolution of one of the elements of the network ? Determining how and how much information flows along anatomical brain paths is an important requirement for the understanding of how animals perceive their environment, learn and behave [ 1,2,3 ]. Even though the approaches of Ref. [ 1,2,3,4,5,6 ] have brought considerable understanding on how and how much information from a stimulus is transmitted in a neural network, the relation between network circuits (topology) and information transmission in a neural as well as an active network is still awaiting a more quantitative description [7]. And that is the main thrust of the present manuscript, namely, to present a quantitative way to relate network topology with information in active networks. Since information might not always be easy to be measured or quantified in experiments, we endeavour to clarify the relation between information and synchronization, a phenomenon which is often not only possible to observe but also relatively easy to characterize. We initially proceed along the same line as in Refs. [ 8,9 ], and study the information transfer in autonomous systems. However, instead of treating the information transfer between dynamical systems components, we treat the transfer of information per unit time exchanged between two elements in an autonomous chaotic active network. Thus, we neglect the complex relation between external stimulus and the network and show how to calculate an upper bound value for the mutual information rate (MIR) exchanged between two elements (a communication channel) in an autonomous network. Ultimately, we discuss how to extend this formula to non-chaotic networks suffering the influence of a time-dependent stimulus. Most of this work is directed to ensure the plausibility and validity of the proposed formula for the upper bound of MIR (Sec. Results) and also to study its applications in order to clarify the relation among network topology, information, and synchronization. We do not rely only on results provided by this formula, but we also calculate the MIR by the methods in Refs. [ 10,11 ] and by symbolic encoding the trajectory of the elements forming the network and then measuring the mutual information provided by this discrete sequence of symbols. To illustrate the power of the proposed formula, we applied it to study the exchange of information in networks of coupled chaotic maps (Sec. Methods) and in Hindmarsh-Rose neural networks bidirectionally electrically coupled (Sec. Results). Our formula can be used to a larger class of active networks than the ones here considered. As the networks formed by elements coupled both electrically and chemically (see Ref. [ 12 ]). Still, the studied network topologies are much simpler than the ones found in the brain [ 13,14 ]. Nevertheless, we do believe our approaches can be used to better understand how information is transfered in more realistic networks as the scale-free networks [15], the small-world netw (...truncated)


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Murilo S. Baptista, Josué X. de Carvalho, Mahir S. Hussein. Finding Quasi-Optimal Network Topologies for Information Transmission in Active Networks, PLOS ONE, 2008, 10, DOI: 10.1371/journal.pone.0003479