Critical phenomena in model neural networks

Kürten, K. E.

doi:10.1016/0375-9601(88)90135-1

Cited by 89 publications

(63 citation statements)

References 8 publications

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“…Let us, as an example, consider a particularly simple subset of Boolean networks, the so-called threshold networks (or threshold Boolean networks ;Derrida 1987;Kürten 1988a;Rohlf & Bornholdt 2002). They are a subset of all Boolean networks with the Boolean function of each node depending on the sum of its input signals only.…”

Section: Discrete Network As Models For Cellular Computationmentioning

confidence: 99%

Boolean network models of cellular regulation: prospects and limitations

Bornholdt

2008

J. R. Soc. Interface.

329

261

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Computer models are valuable tools towards an understanding of the cell's biochemical regulatory machinery. Possible levels of description of such models range from modelling the underlying biochemical details to top-down approaches, using tools from the theory of complex networks. The latter, coarse-grained approach is taken where regulatory circuits are classified in graph-theoretical terms, with the elements of the regulatory networks being reduced to simply nodes and links, in order to obtain architectural information about the network. Further, considering dynamics on networks at such an abstract level seems rather unlikely to match dynamical regulatory activity of biological cells. Therefore, it came as a surprise when recently examples of discrete dynamical network models based on very simplistic dynamical elements emerged which in fact do match sequences of regulatory patterns of their biological counterparts. Here I will review such discrete dynamical network models, or Boolean networks, of biological regulatory networks. Further, we will take a look at such models extended with stochastic noise, which allow studying the role of network topology in providing robustness against noise. In the end, we will discuss the interesting question of why at all such simple models can describe aspects of biology despite their simplicity. Finally, prospects of Boolean models in exploratory dynamical models for biological circuits and their mutants will be discussed.

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Section: Discrete Network As Models For Cellular Computationmentioning

confidence: 99%

Boolean network models of cellular regulation: prospects and limitations

Bornholdt

2008

J. R. Soc. Interface.

329

261

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“…It is worth noting that, the dynamics of the overlap becomes deterministic in the limit of large n according to the central limit theorem, which is the so called distance law [22][23][24][25][26], q t+1,s+1 = ϕ(q ts ). In this equation, the equality holds only at q = 0 and q = ±1, and otherwise |ϕ(q)| < |q|.…”

Section: Mean-field Analysismentioning

confidence: 99%

Structure of attractors in randomly connected networks

Toyoizumi

Huang

2015

Phys. Rev. E

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The deterministic dynamics of randomly connected neural networks are studied, where a state of binary neurons evolves according to a discrete-time synchronous update rule. We give a theoretical support that the overlap of systems' states between the current and a previous time develops in time according to a Markovian stochastic process in large networks. This Markovian process predicts how often a network revisits one of previously visited states, depending on the system size. The state concentration probability, i.e., the probability that two distinct states co-evolve to the same state, is utilized to analytically derive various characteristics that quantify attractors' structure. The analytical predictions about the total number of attractors, the typical cycle length, and the number of states belonging to all attractive cycles match well with numerical simulations for relatively large system sizes.

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“…There exist several variants of threshold models. In other variants, the c i j can be continuous quantities chosen at random from some probability distribution; the spin values may be ±1 instead of 1 and 0 (see for instance [4]- [6]), or the update rule in the case that the sum of the inputs is exactly at the threshold can be different.…”

Section: Introductionmentioning

confidence: 99%

The phase diagram of random threshold networks

Szejka¹,

Mihaljev

Drossel

2008

New J. Phys.

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Abstract. Threshold networks are used as models for neural or gene regulatory networks. They show rich dynamical behaviour with a transition between a frozen phase and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae (Li et al 2004 Proc. Natl Acad. Sci. USA 101 4781-6). Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the 'on' and 'off' states, and for the sensitivity λ. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule. Contents

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Critical phenomena in model neural networks

Cited by 89 publications

References 8 publications

Boolean network models of cellular regulation: prospects and limitations

Boolean network models of cellular regulation: prospects and limitations

Structure of attractors in randomly connected networks

The phase diagram of random threshold networks

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