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Register a new userStructural Characterization of Oscillations in Brain Networks with Rate Dynamics
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Among the versatile forms of dynamical patterns of activity exhibited by the brain, oscillations are one of the most salient and extensively studied, yet are still far from being well understood. In this paper, we provide various structural characterizations of the existence of oscillatory behavior in neural networks using a classical neural mass model of mesoscale brain activity called linear-threshold dynamics. Exploiting the switched-affine nature of this dynamics, we obtain various necessary and/or sufficient conditions on the network structure and its external input for the existence of oscillations in (i) two-dimensional excitatory-inhibitory networks (E-I pairs), (ii) networks with one inhibitory but arbitrary number of excitatory nodes, (iii) purely inhibitory networks with an arbitrary number of nodes, and (iv) networks of E-I pairs. Throughout our treatment, and given the arbitrary dimensionality of the considered dynamics, we rely on the lack of stable equilibria as a system-based proxy for the existence of oscillations, and provide extensive numerical results to support its tight relationship with the more standard, signal-based definition of oscillations in computational neuroscience.
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The fast-slow dynamics of an eco-evolutionary system are studied, where we consider the feedback actions of environmental resources that are classified into those that are self-renewing and those externally supplied. We show although these two types of resources are drastically different, the resulting closed-loop systems bear close resemblances, which include the same equilibria and their stability conditions on the boundary of the phase space, and the similar appearances of equilibria in the interior. After closer examination of specific choices of parameter values, we disclose that the global dynamical behaviors of the two types of closed-loop systems can be fundamentally different in terms of limit cycles: the system with self-renewing resources undergoes a generalized Hopf bifurcation such that one stable limit cycle and one unstable limit cycle can coexist; the system with externally supplied resources can only have the stable limit cycle induced by a supercritical Hopf bifurcation. Finally, the explorative analysis is carried out to show the discovered dynamic behaviors are robust in even larger parameter space.
We study the problem of estimating the parameters (i.e., infection rate and recovery rate) governing the spread of epidemics in networks. Such parameters are typically estimated by measuring various characteristics (such as the number of infected and recovered individuals) of the infected populations over time. However, these measurements also incur certain costs, depending on the population being tested and the times at which the tests are administered. We thus formulate the epidemic parameter estimation problem as an optimization problem, where the goal is to either minimize the total cost spent on collecting measurements, or to optimize the parameter estimates while remaining within a measurement budget. We show that these problems are NP-hard to solve in general, and then propose approximation algorithms with performance guarantees. We validate our algorithms using numerical examples.
Trust and distrust are common in the opinion interactions among agents in social networks, and they are described by the edges with positive and negative weights in the signed digraph, respectively. It has been shown in social psychology that although the opinions of most agents (followers) tend to prevail, sometimes one agent (leader) with a firm stand and strong influence can impact or even overthrow the preferences of followers. This paper aims to analyze how the leader influences the formation of followers opinions in signed social networks. In addition, this paper considers an asynchronous evolution mechanism of trust/distrust level based on opinion difference, in which the trust/distrust level between neighboring agents is portrayed as a nonlinear weight function of their opinion difference, and each agent interacts with the neighbors to update the trust/distrust level and opinion at the times determined by its own will. Based on the related properties of sub-stochastic and super-stochastic matrices, the inequality conditions about positive and negative weights to achieve opinion consensus and polarization are established. Some numerical simulations based on two well-known networks called the ``12 Angry Men network and the Karate Club network are provided to verify the correctness of the theoretical results.
Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the structure of the NDS is identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.
We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We show that the distance-based lower bound is usually better than the zero-forcing-based bound when the leaders do not constitute a zero-forcing set. On the other hand, we also show that any set of leaders that can be shown to achieve complete SSC via the distance-based bound is necessarily a zero-forcing set. These results indicate that while the zero-forcing based approach may be preferable when the focus is only on verifying complete SSC, the distance-based approach is usually more informative when partial SSC is also of interest. Furthermore, we also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two bounds. We support our analysis with numerical results for various graphs and leader sets.
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