Is there switching for replicator dynamics and bimatrix games?

We consider heteroclinic networks for replicator dynamics and bimatrix games, that is, in a simplex or product of simplices, with equilibria at the vertices and connections at the edges-edge networks. Switching dynamics near a heteroclinic network occurs whenever every (infinite) sequence of connections in the network is shadowed by at least one trajectory in its neighborhood. Aguiar and Castro [M.A.D. Aguiar, S.B.S.D. Castro Chaotic switching in a two-person game, Physica D 239 (16), 1598-1609] prove switching near an edge network for the dynamics of the rock-scissors-paper game. Here we give conditions for switching dynamics in general bimatrix games and show that switching near an edge network can never occur for replicator dynamics.     

Mechanism for propagation of rate signals through a 10-layer feedforward neuronal network

Using numerical simulations, we explore the mechanism for propagation of rate signals through a 10-layer feedforward network composed of Hodgkin--Huxley (HH) neurons with sparse connectivity. When white noise is afferent to the input layer, neuronal firing becomes progressively more synchronous in successive layers and synchrony is well developed in deeper layers owing to the feedforward connections between neighboring layers. The synchrony ensures the successful propagation of rate signals through the network when the synaptic conductance is weak. As the synaptic time constant τ_syn varies, coherence resonance is observed in the network activity due to the intrinsic property of HH neurons. This makes the output firing rate single-peaked as a function of τ_syn, suggesting that the signal propagation can be modulated by the synaptic time constant. These results are consistent with experimental results and advance our understanding of how information is processed in feedforward networks.     

The growth of cognition: Free energy minimization and the embryogenesis of cortical computation

The assumption that during cortical embryogenesis neurons and synaptic connections are selected to form an ensemble maximising synchronous oscillation explains mesoscopic cortical development, and a mechanism for cortical information processing is implied by consistency with the Free Energy Principle and Dynamic Logic. A heteroclinic network emerges, with stable and unstable fixed points of oscillation corresponding to activity in symmetrically connected, versus asymmetrically connected, sets of neurons. Simulations of growth explain a wide range of anatomical observations for columnar and non-columnar cortex, superficial patch connections, and the organization and dynamic interactions of neurone response properties. An antenatal scaffold is created, upon which postnatal learning can establish continuously ordered neuronal representations, permitting matching of co-synchronous fields in multiple cortical areas to solve optimization problems as in Dynamic Logic. Fast synaptic competition partitions equilibria, minimizing “the curse of dimensionality”, while perturbations between imperfectly partitioned synchronous fields, under internal reinforcement, enable the cortex to become adaptively self-directed. As learning progresses variational free energy is minimized and entropy bounded.     

Dynamics of simple electronic neural networks

We examine systems of one and two nonlinear threshold switching elements (“neurons”), of the kind used in electronic neural networks. Characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics. When the neurons possess a finite frequency response or a transfer characteristic with a time delay, underdamped transients and instability leading to oscillation can occur. Inertia in the neuron connections is found to cause ringing about fixed points, convoluted basin boundaries, instability and spontaneous oscillation, and chaotic behavior when driven. Furthermore, the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped. These results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks.     

A new local-world evolving network model

In some real complex networks, only a few nodes can obtain the global information about the entire networks, but most of the nodes own only local connections therefore own only local information of the networks. A new local-world evolving network model is proposed in this paper. In the model, not all the nodes obtain local network information, which is different from the local world network model proposed by Li and Chen (LC model). In the LC model, each node has only the local connections therefore owns only local information about the entire networks. Theoretical analysis and numerical simulation show that adjusting the ratio of the number of nodes obtaining the global information of the network to the total number of nodes can effectively control the valuing range for the power-law exponent of the new network. Therefore, if the topological structure of a complex network, especially its exponent of power-law degree distribution, needs controlling, we just add or take away a few nodes which own the global information of the network.     

Architectural issues related to the optical implementation of an EGS network based on embedded control

Free-space optical implementations of switching networks have been proposed to circumvent many of the system-level problems that may be encountered in systems that require many high-density, high-bandwidth connections. The details of a new class of switching network (the EGS network), that is well-suited to free-space implementations, is described. The common control injection problem that plagues most free-space photonic networks, i.e. how can control information from an electronic source be injected into the network for applications that require relatively high network reconfiguration rates, is examined. A new technique for control injection, called embedded control, which permits network operation even with relatively high network reconfiguration rates is also proposed.     

Heteroclinic dynamics in a model of Faraday waves in a square container

We study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross-section and single-frequency forcing. These periodic orbits correspond to quasiperiodic surface waves in the physical system. The heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable Hamiltonian limit, some of which persist in the presence of small damping. The dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate Poincaré maps, with predictions that agree with numerical computations. The results suggest a great richness of possible dynamics of Faraday waves even in simple geometries and with single-frequency forcing.     

Fisher information for spike-based population decoding

We evaluate the Fisher information of a population of model neurons that receive dynamical input and interact via spikes. With spatially independent threshold noise, the spike-based Fisher information that summarizes the information carried by individual spike timings has a particularly simple analytical form. We calculate the loss of information caused by abandoning spike timing and study the effect of synaptic connections on the Fisher information. For a simple spatiotemporal input, we derive the optimal recurrent connectivity that has a local excitation and global inhibition structure. The optimal synaptic connections depend on the spatial or temporal feature of the input that the system is designed to code.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Assortative mixing in networks

A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. Here we measure mixing patterns in a variety of networks and find that social networks are mostly assortatively mixed, but that technological and biological networks tend to be disassortative. We propose a model of an assortatively mixed network, which we study both analytically and numerically. Within this model we find that networks percolate more easily if they are assortative and that they are also more robust to vertex removal.     

Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling

We study synchronization transitions and pattern formation on small-world networks consisting of Morris-Lecar excitable neurons in dependence on the information transmission delay and the rewiring probability. In addition, networks formed via gap junctional connections and coupling via chemical synapses are considered separately. For gap-junctionally coupled networks we show that short delays can induce zigzag fronts of excitations, whereas long delays can further detriment synchronization due to a dynamic clustering anti-phase synchronization transition. For the synaptically coupled networks, on the other hand, we find that the clustering anti-phase synchronization can appear as a direct consequence of the prolongation of information transmission delay, without being accompanied by zigzag excitatory fronts. Irrespective of the coupling type, however, we show that an appropriate small-world topology can always restore synchronized activity if only the information transmission delays are short or moderate at most. Long information transmission delays always evoke anti-phase synchronization and clustering, in which case the fine-tuning of the network topology fails to restore the synchronization of neuronal activity.     

Firing activity of complex space-clamped FitzHugh-Nagumo neural networks

We investigate how firing activity of complex neural networks depends on the random long-range connections and coupling strength. Network elements are described by excitable space-clamped FitzHugh-Nagumo (SCFHN) neurons with the values of parameters at which no firing activity occurs. It is found that for a given appropriate coupling strength C, there exists a critical fraction of random connections (or randomness) p^*, such that if p > p^* the firing neurons, which are absent in the nearest-neighbor network, occur. The firing activity becomes more frequent as randomness p is further increased. On the other hand, when the p is smaller, there are no active neurons in network, no matter what the value of C is. For a given larger p, there exist optimal coupling strength levels, where firing activity reaches its maximum. To the best of our knowledge, this is a novel mechanism for the emergence of firing activity in neurons.     

Amplified Signal Response by Neuronal Diversity on Complex Networks

The effect of diversity on dynamics of coupled FitzHugh-Nagumo neurons on complex networks is numerically investigated, where each neuron is subjected to an external subthreshold signal. With the diversity the network is a mixture of excitable and oscillatory neurons, and the diversity is determined by the variance of the system＇s parameter. The complex network is constructed by randomly adding long-range connections （shortcuts） on a nearest-neighbouring coupled one-dimensional chain. Numerical results show that external signals are maximally magnified at an intermediate value of the diversity, as in the case of well-known stochastic resonance, burthermore, the effects of the number of shortcuts and coupled strength on the diversity-induced phenomena are also discussed. These findings exhibit that the diversity may play a constructive role in response to external signal, and highlight the importance of the diversity on such complex networks.     

Spiking sychronization regulated by noise in three types of Hodgkin—Huxley neuronal networks

In this paper,we study spiking synchronization in three different types of Hodgkin-Huxley neuronal networks,which are the small-world,regular,and random neuronal networks.All the neurons are subjected to subthreshold stimulus and external noise.It is found that in each of all the neuronal networks there is an optimal strength of noise to induce the maximal spiking synchronization.We further demonstrate that in each of the neuronal networks there is a range of synaptic conductance to induce the effect that an optimal strength of noise maximizes the spiking synchronization.Only when the magnitude of the synaptic conductance is moderate,will the effect be considerable.However,if the synaptic conductance is small or large,the effect vanishes.As the connections between neurons increase,the synaptic conductance to maximize the effect decreases.Therefore,we show quantitatively that the noise-induced maximal synchronization in the Hodgkin-Huxley neuronal network is a general effect,regardless of the specific type of neuronal network.     

Lateral predictive coding revisited: internal model,symmetry breaking,and response time

Predictive coding is a promising theoretical framework in neuroscience for understanding information transmission and perception. It posits that the brain perceives the external world through internal models and updates these models under the guidance of prediction errors. Previous studies on predictive coding emphasized top-down feedback interactions in hierarchical multilayered networks but largely ignored lateral recurrent interactions. We perform analytical and numerical investigations in this work on the effects of single-layer lateral interactions. We consider a simple predictive response dynamics and run it on the MNIST dataset of hand-written digits. We find that learning will generally break the interaction symmetry between peer neurons, and that high input correlation between two neurons does not necessarily bring strong direct interactions between them. The optimized network responds to familiar input signals much faster than to novel or random inputs, and it significantly reduces the correlations between the output states of pairs of neurons.     

Spreading processes on dynamically changing contact networks

We develop and analyze an agent-based model for the study of information propagation in dynamic contact networks. We represent information as a state of a node in a network that can be probabilistically transferred to an adjacent node within a single time step. The model is based on a closed (yet sufficiently large) population that can support processes of link generation and annihilation using different contact regimes. Our study is confined to the case of homogeneous contacts, where each agent establishes and breaks contacts in the same way. We consider information to be available for spreading in a fixed time window (i.e. finite memory). We find, surprisingly, that information transmission (measured as the proportion of informed nodes after a fixed number of time steps) is identical for dynamic preferential and random networks, but radically different for the associate mixing contact regime. We also find that the probability of transmission is, similarly counterintuitively, not a main driver of the process as opposed the the main network par maters determining contact lifetime and the turnover rate on connections. We discuss the explanation and the significance of these results in the light of the fundamental difference between dynamic and static (cumulative) networks.     

FitzHugh-Nagumo神经元网络的联想记忆与分割

以广泛讨论的Fitz Hugh-Nagumo神经元节点组成脉动神经元网络,从神经系统空时模式编码理论研究网络的记忆(或模式)存储与时间分割问题.给定一个输入模式,它是几种模式的叠加,网络能够以一部分神经元同步发放的形式一个接一个地分割出每一种模式.如果输入的模式有缺损,系统能够把它们恢复成原型,即神经网络的联想记忆功能.模拟需要调节耦合强度和噪声强度等参数使得网络在特定的参数值和中等强度噪声达到最优的时间分割,与广泛讨论的随机共振现象一致.     

Sparse coding for layered neural networks

We investigate storage capacity of two types of fully connected layered neural networks with sparse coding when binary patterns are embedded into the networks by a Hebbian learning rule. One of them is a layered network, in which a transfer function of even layers is different from that of odd layers. The other is a layered network with intra-layer connections, in which the transfer function of inter-layer is different from that of intra-layer, and inter-layered neurons and intra-layered neurons are updated alternately. We derive recursion relations for order parameters by means of the signal-to-noise ratio method, and then apply the self-control threshold method proposed by Dominguez and Bollé to both layered networks with monotonic transfer functions. We find that a critical value _C of storage capacity is about 0.11|a ln a|⁻¹ (a1) for both layered networks, where a is a neuronal activity. It turns out that the basin of attraction is larger for both layered networks when the self-control threshold method is applied.     

Disorder and decision cost in spatial networks

We introduce the concept of decision cost of a spatial graph, which measures the disorder of a given network taking into account not only the connections between nodes but their position in a two-dimensional map. The influence of the network size is evaluated and we show that normalization of the decision cost allows us to compare the degree of disorder of networks of different sizes. Under this framework, we measure the disorder of the connections between airports of two different countries and obtain some conclusions about which of them is more disordered. The introduced concepts (decision cost and disorder of spatial networks) can easily be extended to Euclidean networks of higher dimensions, and also to networks whose nodes have a certain fitness property (i.e., one-dimensional).