Correspondence between neural threshold networks and Kauffman Boolean cellular automata

Kürten, K. E.

doi:10.1088/0305-4470/21/11/009

Cited by 76 publications

(58 citation statements)

References 5 publications

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“…We approximate the activity of each element by a simple two-state model 22 (σ i = 1 or σ i = 0). The sum of all interactions, activating or inhibitory, determines the state of each node 7,23 (Fig. 1a).…”

Section: (Supplementary Information)mentioning

confidence: 99%

Effects of topology on network evolution

Oikonomou

Cluzel

2006

Nature Phys

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T he ubiquity of scale-free topology in nature raises the question of whether this particular network design confers an evolutionary advantage 1 . A series of studies has identified key principles controlling the growth and the dynamics of scale-free networks 2-4 . Here, we use neuron-based networks of boolean components as a framework for modelling a large class of dynamical behaviours in both natural and artificial systems [5][6][7] . Applying a training algorithm, we characterize how networks with distinct topologies evolve towards a preestablished target function through a process of random mutations and selection [8][9][10] . We find that homogeneous random networks and scale-free networks exhibit drastically different evolutionary paths. Whereas homogeneous random networks accumulate neutral mutations and evolve by sparse punctuated steps 11,12 , scale-free networks evolve rapidly and continuously. Remarkably, this latter property is robust to variations of the degree exponent. In contrast, homogeneous random networks require a specific tuning of their connectivity to optimize their ability to evolve. These results highlight an organizing principle that governs the evolution of complex networks and that can improve the design of engineered systems.In recent years, studies of the architecture of large complex networks have unveiled a topology, known as scale-free, in which the connectivity between elements is power-law distributed 3 . In biology, the intricate interactions of genes and proteins can be viewed as neuron-based networks that control biological signals 13 . It is also common to model large technological and social systems using similar neuronal networks. For example, electronic devices that carry out computational tasks are built using large networks of interconnected logic gates, whereas collective social behaviours emerge from a complex structure of social relations and the dynamics of personal influences 14,15 . In such real-world networks, the topology is important because it mediates the effect of modifications in local interaction that can sometimes affect the networks dynamical behaviour. Topology could thus be a determining factor to modify or evolve the function of networks 16 . In this picture, new dynamical behaviours emerge from 'tinkering' the local interactions from old systems. In living organisms, genetic and protein networks have evolved through a process of mutations and selection to carry out specific functions under specific environmental conditions. This process has inspired physical scientists to apply a similar evolutionary approach to explore solutions to difficult optimization problems and to search for novel designs of artificial systems 17 . For example, evolutionary algorithms have been used to train software and even reconfigurable electronic chips to carry out pre-determined tasks 18 . Also known as 'evolvable hardware' , these electronic devices are composed of large numbers of neuron-like elements whose interactions are programmable 19 . To exploit the power of th...

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Section: (Supplementary Information)mentioning

confidence: 99%

Effects of topology on network evolution

Oikonomou

Cluzel

2006

Nature Phys

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“…function. Another possibility is to specify the value of the function in order to simulate the additive properties of neurons (Bornholdt and Rohlf [2000]; Genoud and Metraux [1999]; Cheng and Titterington [1994]; Wang, Pichler, and Ross [1990]; Kürten [1988a]; Derrida, Gardner, and Zippelius [1987]). …”

Section: Coupling Functionsmentioning

confidence: 99%

Boolean Dynamics with Random Couplings

Aldana¹,

Coppersmith²,

Kadanoff³

2003

Perspectives and Problems in Nolinear Science

204

149

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This paper reviews a class of generic dissipative dynamical systems called N -K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N , there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.

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“…These neural networks have a phase transition similar to RBN [12,13]. Using a fixed connectivity initialization the network evolves to an attractor.…”

Section: Introductionmentioning

confidence: 99%

Self-organized critical random Boolean networks

2001

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Standard Random Boolean Networks display an order-disorder phase transition. We add to the standard Random Boolean Networks a disconnection rule which couples the control and order parameters. By this way, the system is driven to the critical line transition. Under the influence of perturbations the system point out self-organized critical behavior. Several numerical simulations have been done and compared with a proposed analytical treatment.

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Correspondence between neural threshold networks and Kauffman Boolean cellular automata

Cited by 76 publications

References 5 publications

Effects of topology on network evolution

Effects of topology on network evolution

Boolean Dynamics with Random Couplings

Self-organized critical random Boolean networks

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