Samuel Johnson | University of Warwick (original) (raw)
Papers by Samuel Johnson
Physical review. E, 2018
We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an ... more We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics-the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erdős-Rényi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erdős-Rényi phenomenology.
Proceedings of the National Academy of Sciences of the United States of America, May 16, 2017
Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on th... more Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on their elements. When described as networks these exhibit few or no cycles, and associated matrices have small leading eigenvalues. It has been suggested that this architecture can confer advantages to the system as a whole, such as "qualitative stability," but this observation does not in itself explain how a loopless structure might arise. We show here that the number of feedback loops in a network, as well as the eigenvalues of associated matrices, is determined by a structural property called trophic coherence, a measure of how neatly nodes fall into distinct levels. Our theory correctly classifies a variety of networks-including those derived from genes, metabolites, species, neurons, words, computers, and trading nations-into two distinct regimes of high and low feedback and provides a null model to gauge the significance of related magnitudes. Because trophic coherence supp...
Chaos (Woodbury, N.Y.), 2016
Trophic coherence, a measure of the extent to which the nodes of a directed network are organised... more Trophic coherence, a measure of the extent to which the nodes of a directed network are organised in levels, has recently been shown to be closely related to many structural and dynamical aspects of complex systems, including graph eigenspectra, the prevalence or absence of feedback cycles, and linear stability. Furthermore, non-trivial trophic structures have been observed in networks of neurons, species, genes, metabolites, cellular signalling, concatenated words, P2P users, and world trade. Here, we consider two simple yet apparently quite different dynamical models-one a susceptible-infected-susceptible epidemic model adapted to include complex contagion and the other an Amari-Hopfield neural network-and show that in both cases the related spreading processes are modulated in similar ways by the trophic coherence of the underlying networks. To do this, we propose a network assembly model which can generate structures with tunable trophic coherence, limiting in either perfectly s...
AIP Conference Proceedings, 2009
Abstract We study excitable networks in which the connection weights vary rapidly with local fiel... more Abstract We study excitable networks in which the connection weights vary rapidly with local fields in a way that mimics resistance or facilitation. The control parameter, Phi, sets the extent to which``static''weights are modified. Considering generic random network topologies, it ensues that the transition to a chaotic regime, or``edge of chaos,''depends crucially on the degree of heterogeneity of the connectivity distribution. In a mean-field approximation and at relatively low temperatures, the critical value is found to be phi c~= 1 ...
Royal Society open science, 2015
Failures of cooperation cause many of society's gravest problems. It is well known that coope... more Failures of cooperation cause many of society's gravest problems. It is well known that cooperation among many players faced with a social dilemma can be maintained thanks to the possibility of punishment, but achieving the initial state of widespread cooperation is often much more difficult. We show here that there exist strategies of 'targeted punishment' whereby a small number of punishers can shift a population of defectors into a state of global cooperation. We conclude by outlining how the international community could use a strategy of this kind to combat climate change.
Non-Equilibrium Statistical Physics Today, 2011
Abstract The behaviour of many complex dynamical systems has been found to depend crucially on th... more Abstract The behaviour of many complex dynamical systems has been found to depend crucially on the structure of the underlying networks of interactions. An intriguing feature of empirical networks is their assortativity--ie, the extent to which the degrees of neighbouring nodes are correlated. However, until very recently it was difficult to take this property into account analytically, most work being exclusively numerical. We get round this problem by considering ensembles of equally correlated graphs [1] and apply this novel technique to ...
Advances in Cognitive Neurodynamics (II), 2010
We present a general theory which allows one to study the effects on emergent, cooperative behavi... more We present a general theory which allows one to study the effects on emergent, cooperative behavior of a complex interplay between different dynamic processes that take place in actual systems at the neuron, synapse and network levels. We consider synaptic changes at different time scales from less than the millisecond to the scale of learning, and the possibility to find a fraction of silent neurons. For some limits of interest, the fixed-point solutions or memories then loose stability and the system shows enhancement of his response to changing external stimuli for particular network topologies. We observe at the edge of chaos that the network activity becomes critical in the sense that the relevant quantities show non-trivial, power-law distributions. We also describe the effect of activity-dependent synaptic processes on the network storage capacity.
Proceedings of the National Academy of Sciences of the United States of America, Jan 16, 2014
Why are large, complex ecosystems stable? Both theory and simulations of current models predict t... more Why are large, complex ecosystems stable? Both theory and simulations of current models predict the onset of instability with growing size and complexity, so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome. We show that trophic coherence-a hitherto ignored feature of food webs that current structural models fail to reproduce-is a better statistical predictor of linear stability than size or complexity. Furthermore, we prove that a maximally coherent network with constant interaction strengths will always be linearly stable. We also propose a simple model that, by correctly capturing the trophic coherence of food webs, accurately reproduces their stability and other basic structural features. Most remarkably, our model shows that stability can increase with size and complexity. This suggests a key to May's paradox, and a range of opportunities and concerns for biodiversity conservation.
Lecture Notes in Computer Science, 2009
We study the dynamics of a simple bistable system driven by multiplicative correlated noise. Such... more We study the dynamics of a simple bistable system driven by multiplicative correlated noise. Such system mimics the dynamics of classical attractor neural networks with an additional source of noise associated, for instance, with the stochasticity of synaptic transmission. We found that the multiplicative noise, which performs as a fluctuating barrier separating the stable solutions, strongly influences the behaviour of the system, giving rise to complex time series and scale-free distributions for the escape times of the system. This finding may be of interest to understand nonlinear phenomena observed in real neural systems and to design bio-inspired artificial neural networks with convenient complex characteristics.
Lecture Notes in Computer Science, 2009
We present an evolving neural network model in which synapses appear and disappear stochastically... more We present an evolving neural network model in which synapses appear and disappear stochastically according to bio-inspired probabilities. These are in general nonlinear functions of the local fields felt by neurons-akin to electrical stimulation-and of the global average field-representing total energy consumption. We find that initial degree distributions then evolve towards stationary states which can either be fairly homogeneous or highly heterogeneous, depending on parameters. The critical cases-which can result in scale-free distributions-are shown to correspond, under a mean-field approximation, to nonlinear drift-diffusion equations. We show how appropriate choices of parameters yield good quantitative agreement with published experimental data concerning synaptic densities during brain development (synaptic pruning).
Physical Review E, 2009
We present an evolving network model in which the total numbers of nodes and edges are conserved,... more We present an evolving network model in which the total numbers of nodes and edges are conserved, but in which edges are continuously rewired according to nonlinear preferential detachment and reattachment. Assuming power-law kernels with exponents α and β, the stationary states the degree distributions evolve towards exhibit a second order phase transition-from relatively homogeneous to highly heterogeneous (with the emergence of starlike structures) at α = β. Temporal evolution of the distribution in this critical regime is shown to follow a nonlinear diffusion equation, arriving at either pure or mixed power-laws, of exponents −α and 1 − α.
PLoS ONE, 2013
Short-term memory in the brain cannot in general be explained the way long-term memory can-as a g... more Short-term memory in the brain cannot in general be explained the way long-term memory can-as a gradual modification of synaptic weights-since it takes place too quickly. Theories based on some form of cellular bistability, however, do not seem able to account for the fact that noisy neurons can collectively store information in a robust manner. We show how a sufficiently clustered network of simple model neurons can be instantly induced into metastable states capable of retaining information for a short time (a few seconds). The mechanism is robust to different network topologies and kinds of neural model. This could constitute a viable means available to the brain for sensory and/or short-term memory with no need of synaptic learning. Relevant phenomena described by neurobiology and psychology, such as local synchronization of synaptic inputs and power-law statistics of forgetting avalanches, emerge naturally from this mechanism, and we suggest possible experiments to test its viability in more biological settings.
PLoS ONE, 2013
Understanding the causes and effects of network structural features is a key task in deciphering ... more Understanding the causes and effects of network structural features is a key task in deciphering complex systems. In this context, the property of network nestedness has aroused a fair amount of interest as regards ecological networks. Indeed, Bastolla et al. introduced a simple measure of network nestedness which opened the door to analytical understanding, allowing them to conclude that biodiversity is strongly enhanced in highly nested mutualistic networks. Here, we suggest a slightly refined version of such a measure of nestedness and study how it is influenced by the most basic structural properties of networks, such as degree distribution and degree-degree correlations (i.e. assortativity). We find that most of the empirically found nestedness stems from heterogeneity in the degree distribution. Once such an influence has been discounted-as a second factor-we find that nestedness is strongly correlated with disassortativity and hence-as random networks have been recently found to be naturally disassortative-they also tend to be naturally nested just as the result of chance.
Physical Review Letters, 2010
Journal of Statistical Mechanics: Theory and Experiment, 2010
It is now generally assumed that the heterogeneity of most networks in nature probably arises via... more It is now generally assumed that the heterogeneity of most networks in nature probably arises via preferential attachment of some sort. However, the origin of various other topological features, such as degree-degree correlations and related characteristics, is often not clear and attributed to specific functional requirements. We show how it is possible to analyse a very general scenario in which nodes gain or lose edges according to any (e.g., nonlinear) functions of local and/or global degree information. Applying our method to two rather different examples of brain development-synaptic pruning in humans and the neural network of the worm C. Elegans-we find that simple biologically motivated assumptions lead to very good agreement with experimental data. In particular, many nontrivial topological features of the worm's brain arise naturally at a critical point.
EPL (Europhysics Letters), 2008
We study the effect of varying wiring in excitable random networks in which connection weights ch... more We study the effect of varying wiring in excitable random networks in which connection weights change with activity to mold local resistance or facilitation due to fatigue. Dynamic attractors, corresponding to patterns of activity, are then easily destabilized according to three main modes, including one in which the activity shows chaotic hopping among the patterns. We describe phase transitions to this regime, and show a monotonous dependence of critical parameters on the heterogeneity of the wiring distribution. Such correlation between topology and functionality implies, in particular, that tasks which require unstable behavior-such as pattern recognition, family discrimination and categorization-can be most efficiently performed on highly heterogeneous networks. It also follows a possible explanation for the abundance in nature of scale-free network topologies.
New Trends and Tools in Complex Networks
We study the effect of topology in an attractor neural network in which a noise parameter meant t... more We study the effect of topology in an attractor neural network in which a noise parameter meant to mimic synaptic depression causes instabilities of the memory patterns leading to complex behaviour, including the possibility of chaos. Investigation of the system shows that three distinct phases can emerge: a ferromagnetic (memory) phase, a phase of chaotic hopping among the attractors, and a phase of periodic patternantipattern switching. In a mean-field approach and for a single pattern, the dynamics of the network is well ...
A wide range of empirical networks���whether biological, technological, information���related or ... more A wide range of empirical networks���whether biological, technological, information���related or linguistic���generically exhibit important degree���degree anticorrelations (ie, they are disassortative), the only exceptions being social ones, which tend to be positively correlated (assortative). Using an information���theory approach, we show that the equilibrium state of highly heterogeneous (scale���free) random networks is disassortative. This not only gives a parsimonious explanation to a long���standing question, but also provides a neutral model ...
Physical Review E, 2011
The performance of attractor neural networks has been shown to depend crucially on the heterogene... more The performance of attractor neural networks has been shown to depend crucially on the heterogeneity of the underlying topology. We take this analysis a step further by examining the effect of degree-degree correlations-or assortativity-on neural-network behavior. We make use of a method recently put forward for studying correlated networks and dynamics thereon, both analytically and computationally, which is independent of how the topology may have evolved. We show how the robustness to noise is greatly enhanced in assortative (positively correlated) neural networks, especially if it is the hub neurons that store the information.
International Journal of Bifurcation and Chaos, 2010
Excitable media may be modeled as simple extensions of the Amari–Hopfield network with dynamic at... more Excitable media may be modeled as simple extensions of the Amari–Hopfield network with dynamic attractors. Some nodes chosen at random remain temporarily quiet, and some of the edges are switched off to adjust the network connectivity, while the weights of the other edges vary with activity. We conclude on the optimum wiring topology and describe nonequilibrium phases and criticality at the edge of irregular behavior.
Physical review. E, 2018
We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an ... more We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics-the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erdős-Rényi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erdős-Rényi phenomenology.
Proceedings of the National Academy of Sciences of the United States of America, May 16, 2017
Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on th... more Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on their elements. When described as networks these exhibit few or no cycles, and associated matrices have small leading eigenvalues. It has been suggested that this architecture can confer advantages to the system as a whole, such as "qualitative stability," but this observation does not in itself explain how a loopless structure might arise. We show here that the number of feedback loops in a network, as well as the eigenvalues of associated matrices, is determined by a structural property called trophic coherence, a measure of how neatly nodes fall into distinct levels. Our theory correctly classifies a variety of networks-including those derived from genes, metabolites, species, neurons, words, computers, and trading nations-into two distinct regimes of high and low feedback and provides a null model to gauge the significance of related magnitudes. Because trophic coherence supp...
Chaos (Woodbury, N.Y.), 2016
Trophic coherence, a measure of the extent to which the nodes of a directed network are organised... more Trophic coherence, a measure of the extent to which the nodes of a directed network are organised in levels, has recently been shown to be closely related to many structural and dynamical aspects of complex systems, including graph eigenspectra, the prevalence or absence of feedback cycles, and linear stability. Furthermore, non-trivial trophic structures have been observed in networks of neurons, species, genes, metabolites, cellular signalling, concatenated words, P2P users, and world trade. Here, we consider two simple yet apparently quite different dynamical models-one a susceptible-infected-susceptible epidemic model adapted to include complex contagion and the other an Amari-Hopfield neural network-and show that in both cases the related spreading processes are modulated in similar ways by the trophic coherence of the underlying networks. To do this, we propose a network assembly model which can generate structures with tunable trophic coherence, limiting in either perfectly s...
AIP Conference Proceedings, 2009
Abstract We study excitable networks in which the connection weights vary rapidly with local fiel... more Abstract We study excitable networks in which the connection weights vary rapidly with local fields in a way that mimics resistance or facilitation. The control parameter, Phi, sets the extent to which``static''weights are modified. Considering generic random network topologies, it ensues that the transition to a chaotic regime, or``edge of chaos,''depends crucially on the degree of heterogeneity of the connectivity distribution. In a mean-field approximation and at relatively low temperatures, the critical value is found to be phi c~= 1 ...
Royal Society open science, 2015
Failures of cooperation cause many of society's gravest problems. It is well known that coope... more Failures of cooperation cause many of society's gravest problems. It is well known that cooperation among many players faced with a social dilemma can be maintained thanks to the possibility of punishment, but achieving the initial state of widespread cooperation is often much more difficult. We show here that there exist strategies of 'targeted punishment' whereby a small number of punishers can shift a population of defectors into a state of global cooperation. We conclude by outlining how the international community could use a strategy of this kind to combat climate change.
Non-Equilibrium Statistical Physics Today, 2011
Abstract The behaviour of many complex dynamical systems has been found to depend crucially on th... more Abstract The behaviour of many complex dynamical systems has been found to depend crucially on the structure of the underlying networks of interactions. An intriguing feature of empirical networks is their assortativity--ie, the extent to which the degrees of neighbouring nodes are correlated. However, until very recently it was difficult to take this property into account analytically, most work being exclusively numerical. We get round this problem by considering ensembles of equally correlated graphs [1] and apply this novel technique to ...
Advances in Cognitive Neurodynamics (II), 2010
We present a general theory which allows one to study the effects on emergent, cooperative behavi... more We present a general theory which allows one to study the effects on emergent, cooperative behavior of a complex interplay between different dynamic processes that take place in actual systems at the neuron, synapse and network levels. We consider synaptic changes at different time scales from less than the millisecond to the scale of learning, and the possibility to find a fraction of silent neurons. For some limits of interest, the fixed-point solutions or memories then loose stability and the system shows enhancement of his response to changing external stimuli for particular network topologies. We observe at the edge of chaos that the network activity becomes critical in the sense that the relevant quantities show non-trivial, power-law distributions. We also describe the effect of activity-dependent synaptic processes on the network storage capacity.
Proceedings of the National Academy of Sciences of the United States of America, Jan 16, 2014
Why are large, complex ecosystems stable? Both theory and simulations of current models predict t... more Why are large, complex ecosystems stable? Both theory and simulations of current models predict the onset of instability with growing size and complexity, so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome. We show that trophic coherence-a hitherto ignored feature of food webs that current structural models fail to reproduce-is a better statistical predictor of linear stability than size or complexity. Furthermore, we prove that a maximally coherent network with constant interaction strengths will always be linearly stable. We also propose a simple model that, by correctly capturing the trophic coherence of food webs, accurately reproduces their stability and other basic structural features. Most remarkably, our model shows that stability can increase with size and complexity. This suggests a key to May's paradox, and a range of opportunities and concerns for biodiversity conservation.
Lecture Notes in Computer Science, 2009
We study the dynamics of a simple bistable system driven by multiplicative correlated noise. Such... more We study the dynamics of a simple bistable system driven by multiplicative correlated noise. Such system mimics the dynamics of classical attractor neural networks with an additional source of noise associated, for instance, with the stochasticity of synaptic transmission. We found that the multiplicative noise, which performs as a fluctuating barrier separating the stable solutions, strongly influences the behaviour of the system, giving rise to complex time series and scale-free distributions for the escape times of the system. This finding may be of interest to understand nonlinear phenomena observed in real neural systems and to design bio-inspired artificial neural networks with convenient complex characteristics.
Lecture Notes in Computer Science, 2009
We present an evolving neural network model in which synapses appear and disappear stochastically... more We present an evolving neural network model in which synapses appear and disappear stochastically according to bio-inspired probabilities. These are in general nonlinear functions of the local fields felt by neurons-akin to electrical stimulation-and of the global average field-representing total energy consumption. We find that initial degree distributions then evolve towards stationary states which can either be fairly homogeneous or highly heterogeneous, depending on parameters. The critical cases-which can result in scale-free distributions-are shown to correspond, under a mean-field approximation, to nonlinear drift-diffusion equations. We show how appropriate choices of parameters yield good quantitative agreement with published experimental data concerning synaptic densities during brain development (synaptic pruning).
Physical Review E, 2009
We present an evolving network model in which the total numbers of nodes and edges are conserved,... more We present an evolving network model in which the total numbers of nodes and edges are conserved, but in which edges are continuously rewired according to nonlinear preferential detachment and reattachment. Assuming power-law kernels with exponents α and β, the stationary states the degree distributions evolve towards exhibit a second order phase transition-from relatively homogeneous to highly heterogeneous (with the emergence of starlike structures) at α = β. Temporal evolution of the distribution in this critical regime is shown to follow a nonlinear diffusion equation, arriving at either pure or mixed power-laws, of exponents −α and 1 − α.
PLoS ONE, 2013
Short-term memory in the brain cannot in general be explained the way long-term memory can-as a g... more Short-term memory in the brain cannot in general be explained the way long-term memory can-as a gradual modification of synaptic weights-since it takes place too quickly. Theories based on some form of cellular bistability, however, do not seem able to account for the fact that noisy neurons can collectively store information in a robust manner. We show how a sufficiently clustered network of simple model neurons can be instantly induced into metastable states capable of retaining information for a short time (a few seconds). The mechanism is robust to different network topologies and kinds of neural model. This could constitute a viable means available to the brain for sensory and/or short-term memory with no need of synaptic learning. Relevant phenomena described by neurobiology and psychology, such as local synchronization of synaptic inputs and power-law statistics of forgetting avalanches, emerge naturally from this mechanism, and we suggest possible experiments to test its viability in more biological settings.
PLoS ONE, 2013
Understanding the causes and effects of network structural features is a key task in deciphering ... more Understanding the causes and effects of network structural features is a key task in deciphering complex systems. In this context, the property of network nestedness has aroused a fair amount of interest as regards ecological networks. Indeed, Bastolla et al. introduced a simple measure of network nestedness which opened the door to analytical understanding, allowing them to conclude that biodiversity is strongly enhanced in highly nested mutualistic networks. Here, we suggest a slightly refined version of such a measure of nestedness and study how it is influenced by the most basic structural properties of networks, such as degree distribution and degree-degree correlations (i.e. assortativity). We find that most of the empirically found nestedness stems from heterogeneity in the degree distribution. Once such an influence has been discounted-as a second factor-we find that nestedness is strongly correlated with disassortativity and hence-as random networks have been recently found to be naturally disassortative-they also tend to be naturally nested just as the result of chance.
Physical Review Letters, 2010
Journal of Statistical Mechanics: Theory and Experiment, 2010
It is now generally assumed that the heterogeneity of most networks in nature probably arises via... more It is now generally assumed that the heterogeneity of most networks in nature probably arises via preferential attachment of some sort. However, the origin of various other topological features, such as degree-degree correlations and related characteristics, is often not clear and attributed to specific functional requirements. We show how it is possible to analyse a very general scenario in which nodes gain or lose edges according to any (e.g., nonlinear) functions of local and/or global degree information. Applying our method to two rather different examples of brain development-synaptic pruning in humans and the neural network of the worm C. Elegans-we find that simple biologically motivated assumptions lead to very good agreement with experimental data. In particular, many nontrivial topological features of the worm's brain arise naturally at a critical point.
EPL (Europhysics Letters), 2008
We study the effect of varying wiring in excitable random networks in which connection weights ch... more We study the effect of varying wiring in excitable random networks in which connection weights change with activity to mold local resistance or facilitation due to fatigue. Dynamic attractors, corresponding to patterns of activity, are then easily destabilized according to three main modes, including one in which the activity shows chaotic hopping among the patterns. We describe phase transitions to this regime, and show a monotonous dependence of critical parameters on the heterogeneity of the wiring distribution. Such correlation between topology and functionality implies, in particular, that tasks which require unstable behavior-such as pattern recognition, family discrimination and categorization-can be most efficiently performed on highly heterogeneous networks. It also follows a possible explanation for the abundance in nature of scale-free network topologies.
New Trends and Tools in Complex Networks
We study the effect of topology in an attractor neural network in which a noise parameter meant t... more We study the effect of topology in an attractor neural network in which a noise parameter meant to mimic synaptic depression causes instabilities of the memory patterns leading to complex behaviour, including the possibility of chaos. Investigation of the system shows that three distinct phases can emerge: a ferromagnetic (memory) phase, a phase of chaotic hopping among the attractors, and a phase of periodic patternantipattern switching. In a mean-field approach and for a single pattern, the dynamics of the network is well ...
A wide range of empirical networks���whether biological, technological, information���related or ... more A wide range of empirical networks���whether biological, technological, information���related or linguistic���generically exhibit important degree���degree anticorrelations (ie, they are disassortative), the only exceptions being social ones, which tend to be positively correlated (assortative). Using an information���theory approach, we show that the equilibrium state of highly heterogeneous (scale���free) random networks is disassortative. This not only gives a parsimonious explanation to a long���standing question, but also provides a neutral model ...
Physical Review E, 2011
The performance of attractor neural networks has been shown to depend crucially on the heterogene... more The performance of attractor neural networks has been shown to depend crucially on the heterogeneity of the underlying topology. We take this analysis a step further by examining the effect of degree-degree correlations-or assortativity-on neural-network behavior. We make use of a method recently put forward for studying correlated networks and dynamics thereon, both analytically and computationally, which is independent of how the topology may have evolved. We show how the robustness to noise is greatly enhanced in assortative (positively correlated) neural networks, especially if it is the hub neurons that store the information.
International Journal of Bifurcation and Chaos, 2010
Excitable media may be modeled as simple extensions of the Amari–Hopfield network with dynamic at... more Excitable media may be modeled as simple extensions of the Amari–Hopfield network with dynamic attractors. Some nodes chosen at random remain temporarily quiet, and some of the edges are switched off to adjust the network connectivity, while the weights of the other edges vary with activity. We conclude on the optimum wiring topology and describe nonequilibrium phases and criticality at the edge of irregular behavior.