Experiments on autonomous Boolean networks.

We realize autonomous Boolean networks by using logic gates in their autonomous mode of operation on a field-programmable gate array. This allows us to implement time-continuous systems with complex dynamical behaviors that can be conveniently interconnected into large-scale networks with flexible topologies that consist of time-delay links and a large number of nodes. We demonstrate how we realize networks with periodic, chaotic, and excitable dynamics and study their properties. Field-programmable gate arrays define a new experimental paradigm that holds great potential to test a large body of theoretical results on the dynamics of complex networks, which has been beyond reach of traditional experimental approaches.

[1]  Leonardo L. Gollo,et al.  Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays , 2008, Proceedings of the National Academy of Sciences.

[2]  Paczuski,et al.  Self-organized networks of competing boolean agents , 2000, Physical review letters.

[3]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[4]  J. Danckaert,et al.  Synchronization properties of network motifs: influence of coupling delay and symmetry. , 2008, Chaos.

[5]  Michael Rosenblum,et al.  Experiments on oscillator ensembles with global nonlinear coupling. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Ido Kanter,et al.  Controlling synchronization in large laser networks. , 2012, Physical review letters.

[7]  Lizhong Sun,et al.  A 1.25-GHz 0.35-μm monolithic CMOS PLL based on a multiphase ring oscillator , 2001, IEEE J. Solid State Circuits.

[8]  W. Kinzel,et al.  Nonlocal mechanism for cluster synchronization in neural circuits , 2011, 1103.3634.

[9]  Daniel J. Gauthier,et al.  On the origin of chaos in autonomous Boolean networks , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  A-L Barabási,et al.  Structure and tie strengths in mobile communication networks , 2006, Proceedings of the National Academy of Sciences.

[11]  Eckehard Schöll,et al.  Control of synchronization patterns in neural-like Boolean networks. , 2012, Physical review letters.

[12]  Florian Greil,et al.  Dynamics of critical Kauffman networks under asynchronous stochastic update. , 2005, Physical review letters.

[13]  S. Bornholdt,et al.  Topological evolution of dynamical networks: global criticality from local dynamics. , 2000, Physical review letters.

[14]  R. Roy,et al.  Experimental observation of chimeras in coupled-map lattices , 2012, Nature Physics.

[15]  Harald Haas,et al.  Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication , 2004, Science.

[16]  Yoshiki Kuramoto,et al.  Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Eckehard Schöll,et al.  Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke. , 2008, Chaos.

[18]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[19]  S. Strogatz,et al.  Chimera states for coupled oscillators. , 2004, Physical review letters.

[20]  Q. Ouyang,et al.  The yeast cell-cycle network is robustly designed. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Michael Ghil,et al.  Boolean delay equations. II. Periodic and aperiodic solutions , 1985 .

[22]  P. Hövel,et al.  Loss of coherence in dynamical networks: spatial chaos and chimera states. , 2011, Physical review letters.

[23]  Zvonko G. Vranesic,et al.  Fundamentals of Digital Logic with Verilog Design , 1999 .

[24]  Jia-Ming Liu,et al.  Chaotic radar using nonlinear laser dynamics , 2004 .

[25]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

[26]  S. Kauffman,et al.  Activities and sensitivities in boolean network models. , 2004, Physical review letters.

[27]  S. Strogatz Exploring complex networks , 2001, Nature.

[28]  H. Kato A dynamic formulation of ring oscillator as solitary-wave propagator , 1998 .

[29]  E. G. Jones,et al.  Thalamic circuitry and thalamocortical synchrony. , 2002, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[30]  S Shipp,et al.  The functional logic of cortico-pulvinar connections. , 2003, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[31]  Gert Cauwenberghs,et al.  Neuromorphic Silicon Neuron Circuits , 2011, Front. Neurosci.

[32]  J. Monod,et al.  Genetic regulatory mechanisms in the synthesis of proteins. , 1961, Journal of molecular biology.

[33]  Ido Kanter,et al.  Synthetic reverberating activity patterns embedded in networks of cortical neurons , 2012, 1201.0339.

[34]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[35]  G. Gerosa,et al.  A wide-bandwidth low-voltage PLL for PowerPC microprocessors , 1995 .

[36]  E. Ott,et al.  The effect of network topology on the stability of discrete state models of genetic control , 2009, Proceedings of the National Academy of Sciences.

[37]  Stuart A. Kauffman,et al.  The origins of order , 1993 .

[38]  Ali Hajimiri,et al.  A general theory of phase noise in electrical oscillators , 1998 .

[39]  L. Glass,et al.  Ordered and disordered dynamics in random networks , 1998 .

[40]  David P Rosin,et al.  Ultrafast physical generation of random numbers using hybrid Boolean networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  L. Glass,et al.  Common Chaos in Arbitrarily Complex Feedback Networks , 1997 .

[42]  Daniel J. Gauthier,et al.  Excitability in autonomous Boolean networks , 2012, 1208.6181.

[43]  Zheng Gao,et al.  Boolean chaos. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  L. Glass,et al.  Chaotic Dynamics in an Electronic Model of a Genetic Network , 2005 .

[45]  L. Glass,et al.  Chaos in high-dimensional neural and gene networks , 1996 .

[46]  E Kopelowitz,et al.  Sensitivity of global network dynamics to local parameters versus motif structure in a cortexlike neuronal model. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Carsten Peterson,et al.  Random Boolean network models and the yeast transcriptional network , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[48]  M. Ghil,et al.  Boolean delay equations: A simple way of looking at complex systems , 2006, nlin/0612047.

[49]  K. Showalter,et al.  Chimera and phase-cluster states in populations of coupled chemical oscillators , 2012, Nature Physics.

[50]  Debashis Sahoo,et al.  MiDReG: A method of mining developmentally regulated genes using Boolean implications , 2010, Proceedings of the National Academy of Sciences.

[51]  R. Solé,et al.  Ecological networks and their fragility , 2006, Nature.

[52]  O. Sporns,et al.  Complex brain networks: graph theoretical analysis of structural and functional systems , 2009, Nature Reviews Neuroscience.

[53]  Arkady Pikovsky,et al.  A universal concept in nonlinear sciences , 2006 .