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If we had to synthesize the approaches that have been proposed so far to explain the presence of criticality spreading over such a wide range of natural systems, we could broadly consider two general positions. On the one hand, we find approaches assuming that biological systems are self-tuned, either by learning or evolution, to regions of the parameter space displaying optimal fitness, and that these optimal points are often placed near critical points due to the functional advantages of critical behaviour3,12. In other words, criticality is understood as a by-product derived from the adaptive or survival processes of living systems. On the other hand, we find different views focused on the idea of self-organized criticality systems (SOC), in which criticality emerges spontaneously from simple local interactions, without fine tuning of the parameters of the model. Typically, SOC models exploit clever local rules (e.g. in cellular automata) that produce critical behaviour in a specific context13–15.
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study
| 85.75 |
Nevertheless both approaches present some weaknesses. Understanding criticality in biological systems as an indirect consequence of adaptation to external circumstances is not very explanatory and does not provide ways to test alternatives. In general, explanations that simply assume that they are the consequence of the ‘survival of the fittest’ are often unfalsifiable. With regard to the second approach, it is well known that many SOC models are highly idealized and, in many cases, they are not able to capture the basic interactions of living systems, often failing to provide general explanations of the ubiquitous emergence of criticality16. For that reason, although both adaptive self-tuning and SOC models present interesting insights, they are generally applicable in a relatively narrow set of contexts or under highly idealized conditions.
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review
| 99.56 |
In this paper, we explore an alternative approach to examine how a system can display critical activity in a wide variety of situations. We propose a model that, using only general local mechanisms, is aimed to adaptively maintain the behaviour of the system around a critical point of its parameter space by maintaining certain relational invariants. In other words, instead of thinking about criticality as a by-product of adaptation to complex environments or a spontaneous property of certain systems, we inquire into the possibility that biological systems might are equipped with adaptive mechanisms aimed to preserve an internal equilibrium near critical points while they interact with their environment. Among other outcomes, the existence of cheap learning mechanisms maintaining the parameters of a system around regions of criticality could drastically reduce the cost of searching large parametric spaces for finding fit solutions, or even generate interesting solutions in an unsupervised manner.
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study
| 100.0 |
The paper is structured as follows. First, we propose a novel method that appears to be capable of driving a system near critical points by maintaining certain relational invariants. In our case, these invariants are extracted from the correlation structure of a well-known model operating at a critical point, where correlations scale with distance according to a power law function. Second, we propose a simple learning rule maintaining this correlation invariance, and hypothesize that it could be used for driving systems in different contexts to operate near critical points. We test the model using two classical examples of learning and control: the Mountain Car and a double pendulum. We show evidence suggesting that the general rule proposed here is able to drive adaptive agents with no free parameters towards critical points of operation. At the same time, the agents themselves are poised at points of behavioural transition, where they are able to exploit a broad span of dynamic possibilities available in their environment, suggesting a link between an internal search of critical points and the exploration of external behavioural points of transition. Finally, we suggest further tests of criticality and discuss the limitations and possible generalization of this synthetic approach as a contribution towards understanding deeper principles governing biological and cognitive systems.
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study
| 99.94 |
Inspired by the ideas described above, we present a novel simple mechanism designed to test whether a general adaptive system is able to drive a neural controller near criticality by imposing certain patterns in the organizational structure of the system. This focus on the system’s organization instead of the mechanistic properties of its components is supported by the existence of well-known universality classes that provide a unified expression for families of systems operating under criticality17.
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other
| 99.7 |
In physics, the concept of universality allows to group a great variety of different critical phenomena into a small number of universality classes in such a way that all systems belonging to a given universality class are essentially identical near the critical point. Thus, systems belonging to the same universality class, even if defined by very different material parameters or physical properties, have the same critical exponents characterizing diverging observables. For example, in different spin and percolation models, we find that the family of all bidimensional lattices (square, triangular, hexagonal and so forth) spatial correlations follow the asymptotic form c(r) ∝ 1/rη near the phase transition point, where η is the same for all lattice structures of dimension 2 in a particular model.
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study
| 99.94 |
This surprising property provides a perspective on criticality in terms of universal relations, suggesting that we could model criticality using simple and non-specific mechanisms independently of the individual parameters of the system. Our hypothesis is the following: if all systems belonging to the same universality class present the same distribution of correlations at criticality, adjusting an arbitrary system to reproduce the same distribution of correlations might drive the system to a similar critical point. If, as we have said, in the neighborhood of critical points, critical exponents assume the same universal values for a particular class, it could be enough to use in our analysis a very simple (but nontrivial) model. Looking for generality, we use the least structured statistical model (i.e. a maximum entropy model) of a network of interacting units, constrained only by pairwise correlations between them. This is known as the Ising model in physics or the Boltzmann Machine in computer science18. The interest of using it is that it is also one of the simplest models of criticality that that can be solved analytically19.
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study
| 100.0 |
An Ising model can be specified as a neural network of N binary variables only constrained by pairwise correlations. Units can have a value of +1 or −1 and are affected by local bias hi and couplings Jij between pairs of units. These parameters take continuous values and we assume couplings to be symmetric with Jii = 0. In order to simulate the behaviour of the model, units are updated sequentially in a random order using Glauber dynamics, by which each unit is activated with a probability that follows a sigmoid function:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P({s}_{i}(t+1))={[1+{e}^{-\beta 2{H}_{i}(t){s}_{i}(t+1)}]}^{-1}$$\end{document}P(si(t+1))=[1+e−β2Hi(t)si(t+1)]−1where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{i}(s)={h}_{i}+{\sum }_{j}{J}_{ij}{s}_{j}(t)$$\end{document}Hi(s)=hi+∑jJijsj(t) is the effective field received by neuron i summing the internal field and inputs from other neurons (and 2Hi(s)si(t + 1) is the energy difference required to flip the sign of unit i). The state of the model will be updated by sequentially applying Glauber dynamics (i.e. Equation 1) each simulation step to all units of the network in a random order. When updated sequentially, an Ising model with symmetric couplings will reach an equilibrium maximum entropy distribution:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(s)=\frac{1}{Z}{\exp }[\beta (\sum _{i}{h}_{i}{s}_{i}+\sum _{i < j}{J}_{ij}{s}_{i}{s}_{j})]$$\end{document}P(s)=1Zexp[β(∑ihisi+∑i<jJijsisj)]where the distribution follows an exponential family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(s)=\frac{1}{Z}{e}^{-\beta E(s)}$$\end{document}P(s)=1Ze−βE(s) and Z is a normalization value. The energy E(s) of each state of size N is defined in terms of the bias hi and couplings Jij between pairs of units, with β = 1/(TkB), being kB the Boltzmann’s constant and T the temperature of the system. Without loss of generality we can set an operating working temperature such that β = 1.
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study
| 100.0 |
Following the intuition introduced above about universality classes, we are looking for a model that preserves certain structure in the correlations of the system. There is some experimental evidence showing that, given an Ising model near a critical point, one could build a family of models by learning correlations drawn at random from the original system, which will be poised near a critical point20. Inspired by this idea, we propose to reproduce and support criticality by maintaining a distribution of correlations of a particular universality class. Interestingly, the Ising model is a well-studied example of a universality class. In the case of bidimensional lattices, pairwise correlations follow the asymptotic form c(r) ∝ 1/rη, where η = 1/42, and r is the distance between units.
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study
| 99.94 |
However, instead of restricting a model to a particular set of mechanisms or a given topology, we decide to design a learning rule that preserves a distribution of correlations of a critical point belonging to a specific universality class. This could capture some of the properties of that universality class without choosing a specific topology or parametrical configuration. The next goal would be to test whether this adaptive mechanism has the capability of driving an arbitrary system to a critical point. And, in case it were so, the system poised near a critical point should display interesting features of adaptive behaviour as maximal sensitivity or a wide dynamic range of behaviours.
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other
| 99.9 |
In order to test this idea we design a simple learning rule to adjust the parameters of an arbitrary Ising model to the desired distribution of correlations. In a nutshell, the learning rule will operate as follows: (1) The distribution of correlations of a finite square lattice Ising model is calculated, (2) the new model is defined by assigning each neuron reference correlation values for its synapses randomly sampled from the previous distribution, (3) during learning, each neuron sorts its synapses by their correlation strength, and adjusts these correlations to the reference values assigned using an inverse Ising learning rule. We proceed now to explain these points in detail.
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study
| 99.94 |
First, since the size of the models employed here is far from the thermodynamic limit, instead of directly using the diverging asymptotic form c(r) ∝ 1/rη, we approximate it by computing the correlation structure of a finite model operating at a known critical temperature. One of the few cases where the Ising model presents an exact solution is a model with zero fields and a bidimensional square lattice connectivity, in which a critical point appears at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J={\rm{l}}{\rm{o}}{\rm{g}}\,(1+\sqrt{2})/(2\beta )$$\end{document}J=log(1+2)/(2β)19. Exploiting this, we build a 20 × 20 square lattice Ising model operating at critical temperature with periodic boundary conditions to generate reference correlations to be used by the learning rule. We simulate the model using Glauber Dynamics, generating 106 samples, after an initial run of 105 updates from a random state. From this simulation, we obtain the distribution of correlations in the system P(cij), where cij = 〈sisj〉, observed in Fig. 1. Since the fields hi of all units are zero, the means mi = 〈si〉 of all units are also zero.Figure 1Distribution of correlation values used for learning, generated from a 20 × 20 lattice Ising model at critical temperature.
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study
| 100.0 |
Second, once the distribution of correlations has been obtained, we generate new models by adjusting their correlations to match the distribution P(cij). For doing so applying only local information, we assign each neuron a set of correlation values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ik}^{\ast },k=\mathrm{1...}\,N$$\end{document}cik∗,k=1...N drawn at random from P(cij). At each step, we will compute the actual correlations between a neuron i and its neighbours j as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm, and generate reference values for learning by sorting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ik}^{\ast }$$\end{document}cik∗ to match the order of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm. We will denote the sorted reference values as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{\ast }$$\end{document}cij∗. All the values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{\ast }$$\end{document}mi⁎ will be set equal to zero. The reason for sorting the values of the correlations is to give more flexibility to the rule, since we are only interested in maintaining a c(r) ∝ 1/rη relation, independently of which connection holds each value.
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study
| 99.94 |
For the third step, the problem is that it is not trivial finding which combination of hi and Jij generates a specific combination of mj and cij. This is known as the ‘inverse Ising problem’, which can be solved by using a simple gradient descent rule18:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}{h}_{i}\leftarrow {h}_{i}+\mu ({m}_{i}^{\ast }-{m}_{i}^{m})\\ {J}_{ji}\leftarrow {J}_{ji}+\mu ({c}_{ij}^{\ast }-{c}_{ij}^{m})\end{array}$$\end{document}hi←hi+μ(mi⁎−mim)Jji←Jji+μ(cij⁎−cijm)where μ is a constant learning rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{\ast }$$\end{document}mi⁎ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{\ast }$$\end{document}cij⁎ are the reference mean and correlations of the learning algorithm, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{m}$$\end{document}mim and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm are the mean and correlations of the model for the current values of hi and Jij. Generally, performing each learning step is computationally expensive, since it requires summing over all possible states of s, although approximate methods such as Monte Carlo sampling are generally used to speed up learning. Similarly, we compute the approximate correlations by simulating our networks using the Glauber dynamics in Equation 1 for a number of steps.
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other
| 99.44 |
As a demonstration of our method, we apply the learning rule to 10 different networks for sizes N = 4, 8, 16, 32, 64 assigning them means and correlations drawn at random from the distribution found for the 20 × 20 lattice Ising model. For each network, we apply Equation 3 for learning \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{\ast },{c}_{ij}^{\ast }$$\end{document}mi∗,cij∗, estimating the actual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{m}$$\end{document}mim and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm with Glauber dynamics.
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study
| 99.94 |
In order to simplify the process, we have made some operative decisions. For instance, since precision of learning is not important (the objective is to capture the overall distribution), we do not wait for convergence of the algorithm and simply update the learning rule 1000 times. We use a learning rate μ = 0.01 and compute 1000N samples for each learning step, being N the size of the system. For simplicity also, instead of the critical temperature of the lattice Ising model, we set an arbitrary inverse temperature of β = 1. Note that the choice of operating temperature is irrelevant, since it only implies a rescaling of the parameters that the algorithm will compensate.
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other
| 99.6 |
For each network, we test if it is near a critical point by computing its heat capacity. A divergence in the heat capacity is a sufficient indicator of a continuous phase transition indicating the presence of a critical point with maximal sensitivity to parametrical changes1. In our model, the heat capacity is represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{\partial H}{\partial \beta }=$$\end{document}C(β)=−β∂H∂β=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta }^{2}(\langle {E}^{2}(s)\rangle -{\langle E(s)\rangle }^{2})$$\end{document}β2(〈E2(s)〉−〈E(s)〉2), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(s)=-\,{\sum }_{i}{h}_{i}{s}_{i}-{\sum }_{i < j}\,{J}_{ij}{s}_{i}{s}_{j}$$\end{document}E(s)=−∑ihisi−∑i<jJijsisj is the energy of the Ising model and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=-\,\sum _{s}P(s)$$\end{document}H=−∑sP(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{log}\,(P(s))$$\end{document}log(P(s)) is the entropy of the system. So we test if a system is at criticality by using the entropy H of the system as an order parameter and looking for continuous phase transitions associated with critical points. We detect a continuous phase transition if the entropy presents a sharp but continuous transition in which the derivative of the entropy (the heat capacity) diverges as the system size increases.
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study
| 100.0 |
We simulate each network for 105 steps for different values of β, and we find that all the heat capacity of the 10 networks diverges at the operating temperature β = 1 (Fig. 2A), showing values similar to those of the original lattice Ising model with periodic boundaries. Let us point out that although the distribution of correlations is similar to the lattice Ising model, the structure of the network is radically changed. Instead of the original ordered structure of a uniform lattice, we now have a disordered distribution of couplings Jij (Fig. 2B), including both positive and negative values. Also, each execution of the learning algorithm yields a completely different arrangement of values of couplings Jij.Figure 2(A) Divergence of the heat capacity in 10 models after learning random correlations sampled from Fig. 1. Maximum and minimum values are shown by the grey area. (B) Distribution of the values of the coupling matrix Jij of a 64 units Ising model after learning correlations sampled from Fig. 1. (the order of the nodes in the coupling matrix has been arranged using hierarchical clustering). Values of Jij correspond to the values in the colour bar.
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| 100.0 |
(A) Divergence of the heat capacity in 10 models after learning random correlations sampled from Fig. 1. Maximum and minimum values are shown by the grey area. (B) Distribution of the values of the coupling matrix Jij of a 64 units Ising model after learning correlations sampled from Fig. 1. (the order of the nodes in the coupling matrix has been arranged using hierarchical clustering). Values of Jij correspond to the values in the colour bar.
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study
| 99.94 |
In the following section, we test the capacity of this learning rule for driving the neural controller of an embodied agent towards a critical point. In order to do so, we need to take into account the environment during learning. If we consider two interconnected Ising models (one being the neural controller and other being the environment) Equation 3 holds perfectly if we only apply it to the values of i and j corresponding to units of the neural controller. In our case, we do not use an Ising model as an environment but instead we use two classical examples in reinforcement learning with the goal of testing a more realist scenario. Therefore, our learning rule will be valid as long as the statistics of the environment can be approximated by an Ising model with an arbitrary number of units. Luckily, Ising models in the form of Boltzmann machines are universal approximators21 and the stationary distribution of any arbitrary environment can be approximated by an equivalent Ising model.
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study
| 97.2 |
In order to evaluate the behaviour of the proposed learning rule, we test it in two embodied situations using the OpenAI Gym toolkit22. We define a neural network consisting of an Ising model defined as in Equation 2, describing a network of N = 6 + Hh units, with Nh hidden units, 2 motor units and 4 sensor units. Motor units define the actions performed by the agents. In sensor units, the magnetic field of the unit is not a fixed parameter but it is be updated with the value of an external input hi = Ii. Sensor units and motor units are only connected to hidden neurons, while hidden neurons are connected to all other neurons (Fig. 3A). We choose this configuration because it is widely used in neural networks and allowed recurrence in the connectivity of hidden units, although the architecture choice is not restrictive. All units are assigned reference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ik}^{\ast }$$\end{document}cik∗ value (selected at random from distribution P(cij) shown in Fig. 1), and all units except sensor units are assigned an objective mean value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{\ast }=0$$\end{document}mi∗=0. During learning, the agent applies the rule in Equation 3 for adjusting its means and correlations to the assigned values. Each simulation step, the units of the Ising model are updated in a sequential random order using Glauber dynamics (Equation 1).Figure 3(A) Structure of the embodied neural controller for a model with N = 12 units and Nh = 6 hidden units. (B) Mountain Car environment: an under-powered car that must drive up a steep hill by balancing itself to gain momentum. (C) Acrobot environment: an agent has to balance a double pendulum to reach the high part of the environment.
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| 100.0 |
(A) Structure of the embodied neural controller for a model with N = 12 units and Nh = 6 hidden units. (B) Mountain Car environment: an under-powered car that must drive up a steep hill by balancing itself to gain momentum. (C) Acrobot environment: an agent has to balance a double pendulum to reach the high part of the environment.
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other
| 99.9 |
The first embodiment of the network consists of the Mountain Car environment23. This environment is a classical testbed in reinforcement learning depicting an under-powered car that must drive up a steep hill (Fig. 3B). Since gravity is stronger than the car’s engine, the vehicle must learn to gain potential energy by driving to one hill before the car is able to make it to the goal at the top of the opposite hill (see Methods). The neural network receives the speed of the system as an input and controls the force of the car’s engine as an output. The second embodiment consists of a double pendulum or ‘Acrobot’24, which has to coordinate the movements of two connected links to lift its weight (Fig. 3C, see Methods). The neural network receives the speed of the first pendulum and controls the torque applied on the joint between the two pendulums.
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| 99.8 |
For sizes Nh = 1, 2, 4, 8, 16, 32, 64, we train 10 agents in both environments, applying the learning rule from Equation 3, with η = 0.01. Note that agents during learning have no other explicit goal other than adjusting the correlations of the system to a random sample extracted from the probability distribution in Fig. 1. In Supplementary Videos S1 and S2 we can observe an example of the behaviour of agents with N = 64 after training. In Fig. S2C,D we can see the distribution of correlations of an agent with Nh = 64 (the result is similar for all agents and sizes) and Fig. S2E,F shows the error between the distribution in Fig. S2C,D and Fig. 1 for this agent. In Fig. S2C,D we can observe how agents after learning display a correlation structure consistent with the correlation distribution c(r) ∝ 1/rη of the lattice Ising model at criticality.
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study
| 100.0 |
In this section, we analyse the behaviour of the neural controllers and the behavioural patterns of the agents with respect to the possibilities of their parameter space. The goal is to test if the learning rule proposed here is effective for driving the agent near critical points. As we observe below, the 10 agents display quite similar behaviour for each environment, despite the fact that each one has learned different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ik}^{\ast }$$\end{document}cik∗ and Jij. We are interested in analysing if there is anything special about the configuration reached by the agents after learning.
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study
| 99.94 |
In order to compare the agents with other behavioural possibilities, we explore the parameter space by changing the parameter β of the agents. Modifying the value of β is equivalent to a global rescaling of the parameters of the agent transforming hi ← β · hi and Jij ← β · Jij, thus exploring the parameter space along one specific direction. That is, changing β is just a way of testing one dimension in the parameter space of possible models. First, we assess the presence of criticality in the neural controller of the agent. Specifically, we look for the presence of a continuous phase transition in which an order parameter of the system (the entropy) presents a sharp transition displaying a divergence of its derivative (the heat capacity). Second, we analyze not only the behaviour of the neural controller but the agent as a whole, looking for behavioural transitions of the agents and divergences of the susceptibility of the agent’s behaviour to parametrical changes.
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study
| 100.0 |
In order to test whether the agents are being driven towards a critical point, we analyse signatures of critical behaviour in the neural controller of the agent. As we mentioned before, a sufficient indicator for criticality is the presence of a divergence of the heat capacity of the system (as in Fig. 2A). A divergence in the heat capacity indicates the presence of a continuous phase transition presenting a critical point in which the system is maximally sensitive to parametrical changes. Unfortunately, when the neural controller is embodied in the Mountain Car and Acrobot environments, we can no longer access a formal description of the probability distribution of the agent-environment system, thus we cannot directly compute from the energy of the system values as the entropy or heat capacity of the system. Nevertheless, we can still directly compute the entropy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x)=-\,{\sum }_{x}P(x)\,\mathrm{log}\,P(x)$$\end{document}H(x)=−∑xP(x)logP(x) of any variable of the system by estimating its probability distribution P(x) through simulations. In order to compute the entropy and the heat capacity of different variables, we simulate the agent’s behaviour for 101 values of β, logarithmically distributed in the interval [10−1, 101]. We run the 10 agents for each embodiment during 106 simulation steps, reseting the agent’s position and state every 5 · 104 simulation steps.
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| 100.0 |
First, we compute the entropy of the probability function of hidden neurons in the controller. Due to computational constraints (the number of states of the probability distribution increases with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2}^{{N}_{h}}$$\end{document}2Nh) we only compute the entropy in agents up to Nh = 16 hidden units. Displaying the entropy of the neurons for different values of β we observe that the agents are near an order-disorder transition (Fig. 4A,B). Larger sizes make the transition sharper and closer to the operating temperature β = 1. From the entropy of a variable, its heat capacity can be computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{{\rm{\partial }}H(\beta )}{{\rm{\partial }}\beta }$$\end{document}C(β)=−β∂H(β)∂β. From the computed 101 values of entropy, H(β) is estimated by fitting a curve using cubic B-splines25 as indicated in the Methods section. In Fig. 4C,D we observe how the system displays a similar divergence of the heat capacity of the neural network as the Ising model, suggesting that the robot’s neural controller is at a critical point.Figure 4(A,B) Entropy of the agent’s neural controller for different sizes up to N = 22 (Nh = 16), for 10 different agents and 101 values of β. (C,D) Heat capacity of the agent’s neurons. Heat capacity computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{\partial H(\beta )}{\partial \beta }$$\end{document}C(β)=−β∂H(β)∂β, where H(β) is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the model presents a divergence of its heat capacity as the number of neurons N increases, suggesting that the neural controller of the system is near a continuous phase transition.
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| 100.0 |
(A,B) Entropy of the agent’s neural controller for different sizes up to N = 22 (Nh = 16), for 10 different agents and 101 values of β. (C,D) Heat capacity of the agent’s neurons. Heat capacity computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{\partial H(\beta )}{\partial \beta }$$\end{document}C(β)=−β∂H(β)∂β, where H(β) is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the model presents a divergence of its heat capacity as the number of neurons N increases, suggesting that the neural controller of the system is near a continuous phase transition.
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| 99.94 |
In order to confirm a divergence in the observed transition, larger systems should be evaluated. Since evaluating the entropy of the hidden neurons is computationally infeasible for large sizes, we repeat the analysis for the state of the 4 sensor units of the network for a larger amount of hidden units. In Fig. 5 we show the entropy and heat capacity of the sensor units for sizes up to Nh = 64 hidden neurons, where we can observe a similar picture than in Fig. 4, suggesting that the heat capacity diverges, and a second order transition takes place in the neural controller of the agent. These results suggest that the agent’s neural controller is operating near a point of criticality, resembling a continuous phase transition, suggesting that the neural controller self-organizes to present maximal sensitivity to changes in its parameter space.Figure 5(A,B) Entropy of the sensor units for different sizes up to N = 70 (Nh = 64), for 10 different agents and 101 values of β. (C,D) Heat capacity of the agent’s neurons. Heat capacity computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{{\rm{\partial }}H(\beta )}{{\rm{\partial }}\beta }$$\end{document}C(β)=−β∂H(β)∂β, where H(β) is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the system presents a divergence of its heat capacity as the number of neurons N increases, suggesting that the neural controller of the system is near a second order phase transition.
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| 100.0 |
(A,B) Entropy of the sensor units for different sizes up to N = 70 (Nh = 64), for 10 different agents and 101 values of β. (C,D) Heat capacity of the agent’s neurons. Heat capacity computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\beta )=-\,\beta \frac{{\rm{\partial }}H(\beta )}{{\rm{\partial }}\beta }$$\end{document}C(β)=−β∂H(β)∂β, where H(β) is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the system presents a divergence of its heat capacity as the number of neurons N increases, suggesting that the neural controller of the system is near a second order phase transition.
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study
| 99.7 |
In addition to the presence of a continuous phase transition, a classical signature of criticality is the presence of power law distributions in the statistical descriptions of the states of a system. Unlike an isolated Ising model, our neural network is connected to an environment and the probability distribution of the Ising neural controller is no longer described by Equation 2. Thus, we compute the probability distribution of the system by counting the occurrence of each state of the units s to compute the probability distribution of the Ising model P(s). We simulate the system at β = 1 for 108 simulation steps. As in training, the agent’s position and state are reset every 5 · 104 simulation steps. We observe that all agents approximately follow Zipf’s law for the Mountain Car (Fig. 6A) and Acrobot embodiments (Fig. 6B), with error bars in a very narrow range. The power-law distribution of neural activation patterns suggests that the neural controller of the agents is operating near a critical point. We have to note that the sole occurrence of a power law is generally insufficient to assess the presence of criticality and may arise naturally in some non-equilibrium conditions. Nevertheless, together with the apparent divergence of the heat capacity it supports the idea that the neural controller of the agents might be poised near a critical state.Figure 6Ranked probability distribution function of the neural network for 10 different agents and different sizes in (A) the Mountain Car and (B) the Acrobot embodiments. The real distribution is compared with a distribution following Zipf’s law, (i.e. P(s) = 1/rank, dash-dotted line). We observe a good agreement between the model and Zipf’s law, suggesting critical scaling.
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| 100.0 |
Ranked probability distribution function of the neural network for 10 different agents and different sizes in (A) the Mountain Car and (B) the Acrobot embodiments. The real distribution is compared with a distribution following Zipf’s law, (i.e. P(s) = 1/rank, dash-dotted line). We observe a good agreement between the model and Zipf’s law, suggesting critical scaling.
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other
| 99.7 |
What does it imply for the agent to adapt to be poised near a critical point? It should be remarked here that agents are given no explicit goal. They only tend to adapt to behavioural patterns maintaining a distribution of correlations randomly sampled from the distribution shown in Fig. 1. To explore this issue, we examine the different behavioural modes of the agent while exploring its parameter space by changing the value of β. The behaviour of the Mountain Car can be described just by its horizontal position x and speed v at different moments of time. As well, the horizontal and vertical positions of the tip of the Acrobot’s links is a good description of its behaviour.
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other
| 99.75 |
In Fig. 7A–C we can observe the behaviour of a Mountain Car agent with Nh = 64 for β = {0.75, 1, 1.2}, respectively. We observe that for values of β lower than the operating temperature, the agents are not able to reach the top of the mountain. On the other hand, when β is higher, the agents present more ‘rigid’ trajectories going from one mountain peak to the other. At β = 1 the agent is able to reach the top of the mountain (note that the peaks of the mountain are located at x = −π/2 and x = π/6) while displaying larger behavioural diversity. Similarly, in Fig. 7D–F, we observe that the Acrobot agent with Nh = 64 at β = 1 displays a diverse range of behaviours, being able to reach the top of the plane while, when β is lowered or increased, it drifts to other behavioural modes in less diverse regimes. Although only one agent is represented for each environment, the results of Fig. 7 are similar in all agents and sizes.Figure 7Transition in the behavioural regime of the agents with Nh = 64. We show the behaviour of two individual agents with different values of β for the Mountain Car (A–C) and Acrobot (D–F) embodiments. We observe that β = 1 is a transition point between two modes of behaviour in both agents.
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study
| 100.0 |
Transition in the behavioural regime of the agents with Nh = 64. We show the behaviour of two individual agents with different values of β for the Mountain Car (A–C) and Acrobot (D–F) embodiments. We observe that β = 1 is a transition point between two modes of behaviour in both agents.
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| 99.3 |
To get a more general picture of different agents and sizes, we can analyse the behavioural transitions in relevant variables of the agent-environment systems. Furthermore, we analyse whether the behaviour of the agent, and not only its neural controller, is near a critical point. For inspecting this, we calculate the mean height of agents 〈y〉 in our simulations (Fig. 8A,B), and compute the susceptibility of this height value as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }_{y}(\beta )=\beta \frac{\partial \langle y\rangle }{\partial \beta }$$\end{document}χy(β)=β∂〈y〉∂β (Fig. 8C,D). In this case, the susceptibility appears to increase monotonically with size when size is doubled, even if these increases are not as uniform as in Figs 4 and 5. Although further tests of criticality could validate if criticality is also found in the whole agent-environment system, the figures suggest that the continuous phase transition of the neural controller corresponds with a sharp transition in the agent’s behaviour.Figure 8(A,B) Mean height 〈y〉 of the agents for different sizes up to N = 70 (Nh = 64), for 10 different agents and 101 values of β. (C,D) Susceptibility of the agent’s behaviour, computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }_{y}(\beta )=\beta \frac{\partial \langle y\rangle }{\partial \beta }$$\end{document}χy(β)=β∂〈y〉∂β, where 〈y〉 is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the behaviour of the agent presents a sharp transition around β = 1.
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study
| 100.0 |
(A,B) Mean height 〈y〉 of the agents for different sizes up to N = 70 (Nh = 64), for 10 different agents and 101 values of β. (C,D) Susceptibility of the agent’s behaviour, computed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }_{y}(\beta )=\beta \frac{\partial \langle y\rangle }{\partial \beta }$$\end{document}χy(β)=β∂〈y〉∂β, where 〈y〉 is estimated using B-splines. Maximum and minimum values are shown by the grey area. The figures suggest that in both the Mountain Car (left) and Acrobot (right) embodiments the behaviour of the agent presents a sharp transition around β = 1.
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study
| 99.94 |
Recapitulating the main ideas presented so far, we have tested how, by taking a set of correlations chosen at random from a distribution generated by a lattice Ising model at a critical point, we can construct a new model that appears to be also near a critical point in its parameter space. Moreover, if an embodied agent maintains these correlations using a simple learning rule while interacting with its environment–as a sort of organizational homeostasis–the agent neural controller seems to be driven to a critical point, which coincides with behavioural transitions in the agent-environment parameter space. Due to computational limitations for estimating the probability distribution of the embodied Ising networks, we only calculate the entropy and heat capacity of neural controllers up to Nh = 16 hidden neurons (and N = 22 total neurons) and approximate larger sizes computing the entropy and heat capacity of Ns = 4 sensor neurons for neural controllers up to N = 70 total neurons. These results are still far from the thermodynamic limit and further tests should confirm the results presented here for larger sizes. Nevertheless, in all cases we observe a clear diverging tendency of the heat capacity every time the size of the system is doubled. Tests for larger sizes could be performed by designing environments that can be described by a Gibbs distribution, avoiding the combinatorial explosion in computing the heat capacity of the system by calculating it directly from the energy of the whole agent-environment system.
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study
| 100.0 |
These results suggest the possibility that criticality could be diagnosed and induced directly from the maintenance of a given distribution of correlations rather than modelling a precise mechanistic structure. Also, our results show that criticality could be generated by quite simple mechanisms only relying on local information, maintaining specific correlations around a given value. Here we have implemented the mechanism as a simple Boltzmann Learning process, but other rules could have the same effect, such as the combination of Hebbian and anti-Hebbian tendencies in specific ratios or other simple mechanisms. In our model, we only require the system to maintain a distribution of relations between the components of the system. This connects with systemic approaches to biology interested not in specific or intrinsic components of biological systems but in the networks of relations and processes26–28. It is also in line with notions of relational invariance in Piaget’s approach to functional invariants in cognitive development29 or Maturana and Varela’s ideas of autopoietic machines, defined as homeostatic systems that maintain their own organization constant as a network of relations between components30.
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study
| 99.94 |
Assuming a similar systemic perspective, we have derived learning rule intended to drive a system towards critical points by maintaining an invariant structure of correlations roughly defined by a critical exponent 1/rη. Our approach assumes a different point of view on self-organized criticality, in which the distribution of correlations is not the consequence of criticality in a specific topology but the cause driving an indeterminate topology to what appears to be a critical point. The question now is whether imposing connections derived from a 1/rη function is a strong assumption or implies particularly exigent circumstances. We do not think so, since power law functions can be naturally generated by simple rules of preferential attachment favouring ‘rich-get-richer’ cumulative inequalities31, or directly as a natural consequence of certain geometries of space (e.g. gravitation laws32).
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study
| 97.56 |
Our model only assumes that a system is going to adapt in order to preserve an internal network of relations. It emphasizes the maintenance of organizational structures capable of reproducing the behaviour of living systems without relying on internal models of the external source of sensory input. This contrasts with other approaches which have focused on understanding criticality as a strategy to effectively represent a complex and variable external world3, for example, studying criticality in predictive coding or deep learning architectures dealing with complex inputs33,34. In those cases, an internalist view is assumed, where the neural controller represents structures of an external world, whose complexity may be the cause of critical activity in the controller. Instead, our approach is agnostic in terms of the inputs or the external world of an organism, and deals only with how an agent rearranges its internal structures facing different environments.
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other
| 72.75 |
The agents presented here are not specifically designed for a particular problem. In simple terms, our agents generate (preserving the same internal neural organization) a wide variability and richness of behaviours (avoiding both disorder and explosive and indiscriminate propagation) that permits them to explore the space of parameters and eventually achieve solutions that they were not designed to find. The empirical evidence of experiments shown here supports this idea. A parallel could be established with the concept of play, which can be understood as a ‘rule-breaker’ activity of the constraints of a stable and self-equilibrating regime of behaviours which has no concrete goals35. A model as the one presented here could be used for exploring life-like autonomous behaviour without the need for explicit internal representations, goals, or rule-based behaviour. Conceptual models of critical activity based on the maintenance of a system’s relational invariants could help in the development of a synthetic route towards the exploration of adaptive and embodied criticality.
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| 98.7 |
This environment consists in a car with mass m moving along a one-dimensional environment. In this environment, the agent moves its position in an horizontal axis x, limited to an interval of [−1.5π, 0.5π]. Each horizontal position represents a point in an environment with two mountains, whose height is defined as y = 0.55 + 0.45 sin(3x). The velocity in the horizontal axis is updated each time step as v(t + 1) = v(t) + 0.001a − 0.0025 cos(3x), where a is the action of the motor which can be either −1, 0, 1, impulsing the car with a force F = ma. The inputs Ii fed to the sensor units are defined as an array of 4 units, which are assigned the instantaneous velocity of the car discretized into an array of 4 bits. Each input Ii is assigned a value of 1 if its corresponding bit is active and −1 otherwise.
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other
| 99.9 |
The Acrobot is a two-link planar robot composed of two pendulums joined at their tip, with a motor applying a torque in clockwise or counterclockwise directions on the joint between the two links. The position of the system is defined by the angles of both pendulums θ1 and θ2, whose behaviour is defined by the following system of differential equations:4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}{\ddot{\theta }}_{1} & = & -({d}_{2}{\ddot{\theta }}_{2}+{\varphi }_{1})/{d}_{1}\\ {\ddot{\theta }}_{2} & = & {({m}_{2}{l}_{c2}^{2}+{I}_{2}-\frac{{d}_{2}^{2}}{{d}_{1}})}^{-1}(\tau +\frac{{d}_{2}}{{d}_{1}}{\varphi }_{1}-{\varphi }_{2})\\ {d}_{1} & = & {m}_{1}{l}_{c1}^{2}+{m}_{2}({l}_{1}^{2}+{l}_{c2}^{2}+2{l}_{1}{l}_{c2}\,\cos \,({\theta }_{2}))+{I}_{1}+{I}_{2}\\ {d}_{2} & = & {m}_{2}({l}_{c2}^{2}+{l}_{1}{l}_{c2}\,\cos \,({\theta }_{2}))+{I}_{2}\\ {\varphi }_{2} & = & {m}_{2}{l}_{c2}g\,\cos \,({\theta }_{1}+{\theta }_{2}-\pi /2)\\ {\varphi }_{1} & = & -{m}_{2}{l}_{1}{l}_{c2}{\dot{\theta }}_{2}^{2}\,\sin \,({\theta }_{2})-2{m}_{2}{l}_{1}{l}_{c2}{\dot{\theta }}_{2}{\dot{\theta }}_{1}\,\sin \,({\theta }_{2})\\ & & +({m}_{1}{l}_{c1}+{m}_{2}{l}_{1})g\,\cos \,({\theta }_{1}-\pi /2)+{\varphi }_{2}\end{array}$$\end{document}θ¨1=−(d2θ¨2+φ1)/d1θ¨2=(m2lc22+I2−d22d1)−1(τ+d2d1φ1−φ2)d1=m1lc12+m2(l12+lc22+2l1lc2cos(θ2))+I1+I2d2=m2(lc22+l1lc2cos(θ2))+I2φ2=m2lc2gcos(θ1+θ2−π/2)φ1=−m2l1lc2θ˙22sin(θ2)−2m2l1lc2θ˙2θ˙1sin(θ2)+(m1lc1+m2l1)gcos(θ1−π/2)+φ2where τ is the torque applied to the system which can be either −1, 0, 1, m1 = m2 = m is the mass of the links, l1 = l2 = 1 is the length of the links and lc1 = lc2 = 0.5 are the lengths to the center of mass of the links, I1 = I2 = 1 are the moments of inertia of the links and g = 9.8 is the gravity. As well, variables d1 and d2 are the total moments of inertia of each link, and ϕ1 and ϕ2 are linked to the potential energy of the system.
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other
| 99.8 |
Similarly to the Mountain Car, the inputs fed to the sensor units in the Acrobot embodiment are defined as an array defined with 4 sensor units, encoding the angular speed of the first link with binary values (encoding active and inactive bits as +1 and −1).
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other
| 99.9 |
In order to make the tasks challenging, we set the maximum velocity allowed to the Mountain car to ±0.045 (typically is set to 0.07) and the mass of the Acrobot links to m = 1.75 (typically m = 1). These parameters are designed to make it difficult for agents controlled by neural networks with random parameters solve the task (reaching the top of the environment), having success rates of 6.1% for the Mountain Car and 3.1% for the Acrobot. Success rates were evaluated by simulating 1000 neural controllers with random parameters (sampled from a uniform distribution in the range [−2, 2]). The Mountain Car was simulated for 1000 simulation steps starting from a random position in the valley between [0.4, 0.6], and was considered successful when reached the maximum position at least once. The Acrobot was simulated for 5000 simulation steps from the bottom position (angles and angular speeds between [−0.1, 01]) and was consider successful if reached a vertical position higher than 1.8.
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study
| 100.0 |
During training, agents are initialized in the starting random positions in the bottom of their environments (x ∈ [0.4, 0.6] and v = 0 for the Mountain Car and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{1},{\theta }_{2},{\dot{\theta }}_{1},{\dot{\theta }}_{2}\,\in \,[\,-\,0.1,01]$$\end{document}θ1,θ2,θ˙1,θ˙2∈[−0.1,01] for the Acrobot). The state of the neural network is randomized and the initial parameters hi and Jij are set to zero. Agents are simulated for 1000 trials of 5000 steps, computing each trial the values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{m}$$\end{document}mim and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm and applying Equation 3 at the end of the trial. The agent’s position and state are reset every 5 · 104 simulation steps.
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study
| 96.6 |
Acute suppurative thyroiditis (AST) is a rare but potentially lethal endocrine condition, representing just 0.1–0.7% of all thyroid disease . The rarity of AST is thought to be influenced by several protective trophic factors and anatomic barriers specific to the thyroid gland, including the presence of iodine, hydrogen peroxide, lymphatic drainage, and a thyroid capsule. These mechanisms have been suggested to provide a shield against the development of thyroid suppuration [2–4].
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review
| 99.9 |
AST is most commonly encountered in the pediatric and adolescent populations, usually in those with congenital anomalies such as a pyriform sinus fistula [5, 6]. Only 8% of cases occur in adulthood; in many cases, the source of infection is not obvious [2, 7]. It may present similarly to subacute thyroiditis, with painful thyroid swelling, fever, and leukocytosis following an upper respiratory tract infection (URTI). AST exhibits a wide range of illness severity, from complete resolution back to a euthyroid state to a greater than 12% mortality rate if no interventions are initiated [2, 3].
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review
| 99.9 |
Given the high mortality rate without intervention, early diagnosis and treatment of AST remain paramount in order to prevent poor outcomes. There are limited recommendations in the current published literature for a streamlined diagnostic approach and management strategy to provide guidance to clinicians [2, 8, 9]. In this case report, we highlight a diagnostic approach and management of AST by use of targeted intravenous (IV) antibiotics followed by transition to oral antibiotics in an adult patient with a history of IV drug use and a preexisting goiter.
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clinical case
| 99.94 |
The patient is a 26-year-old female with a past medical history significant for IV drug use and a right-sided multinodular goiter who presented with five days of anterior neck swelling, predominantly on the right side, and associated tenderness to palpation. In the week prior to presentation, the patient had developed a sore throat, fatigue, and rhinorrhea, along with an abscess on the posterior aspect of her left forearm where she injects heroin. She denied ever injecting drugs into her neck or shoulders. Her review of systems was negative for fevers, chills, rigors, shortness of breath, wheezing, difficulty breathing, dysphagia, and chest pain. She further denied weakness, muscle twitching, diarrhea, constipation, weight loss, weight gain, and palpitations. Her multinodular goiter had been discovered about one year prior. At the time, she was asymptomatic and thyroid function tests were within normal limits, but no fine-needle aspiration (FNA) biopsy was performed.
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clinical case
| 100.0 |
On admission, the patient was febrile to 100.9°F and tachycardic to 130 beats/minute, with blood pressure of 134/88 mm·Hg, respiratory rate of 18 breaths/minute, and oxygen saturation of 99% on room air. Bedside incision and drainage of the left forearm abscess was performed, and cultures grew methicillin-sensitive Staphylococcus aureus (MSSA). Laboratory workup was significant for an elevated white blood cell count of 23.8/μL with differential showing 84.6% neutrophils. Thyroid function tests showed a TSH of 0.02 U/mL, free T4 of 3.89 ng/dL, and free T3 of 5.1 pg/mL, consistent with thyrotoxicosis. The patient was HIV negative and hepatitis C antibody positive. Within the first 24 hours, one out of two blood cultures was positive for MSSA. Computed tomography (CT) scan of the neck with contrast showed a 6.4 × 6.1 × 7.1 cm mass regional to the right thyroid lobe, heterogeneous in appearance with multiple septations and with solid enhancing and cystic components (Figure 1). Mild-to-moderate edema/inflammatory changes were evident along the retropharyngeal space. Additionally, mild luminal narrowing and posterior displacement of the right common carotid artery, lateral displacement and marked narrowing of the right internal jugular vein, and leftward deviation of the trachea were apparent.
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clinical case
| 100.0 |
The patient was started on IV ampicillin-sulbactam, as her clinical picture was concerning for AST. Additionally, she was started on naproxen 500 mg twice daily and propranolol 20 mg three times daily for inflammation and thyrotoxicosis, respectively. Over the first four days of hospitalization, the patient remained afebrile, and the leukocytosis and neck swelling improved. A thyroid ultrasound was performed on day 4 to further evaluate for a drainable abscess given the concern for AST. The ultrasound showed a heterogeneous, necrotic, and hypervascular mass in the right lobe measuring 6.8 × 4.8 × 4.5 cm, without any drainable abscess (Figure 2). The patient subsequently underwent ultrasound-guided FNA; cultures grew MSSA, and cytology was consistent with follicular lesion of undetermined significance. A transesophageal echocardiogram ruled out endocarditis. On day 8 of hospitalization, the patient was transitioned from IV ampicillin-sulbactam to oral amoxicillin-clavulanate and discharged home to complete a total of 14 days of antibiotics. She was also discharged on the same regimen of propranolol and naproxen. The patient was instructed to follow up at the endocrinology clinic in one month and receive a repeat FNA in three to six months.
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clinical case
| 100.0 |
Here, we present a case of AST that could have been misdiagnosed and improperly treated as subacute thyroiditis. Both AST and subacute thyroiditis may present with painful thyroid swelling, fever, and leukocytosis following a URTI. In this case, atypical features for AST include the patient's age, presence of thyrotoxicosis, lack of dysphagia, dysphonia, and airway compromise, and marked clinical improvement within four days of admission . Thyrotoxicosis only occurs in 5–10% of patients with AST and is more often associated with subacute thyroiditis. Subacute thyroiditis, while far more common, is self-limiting; AST, while rare, can be life-threatening. Nevertheless, the patient had several risk factors for AST that increased clinical suspicion, allowing for an appropriate diagnostic workup and course of antibiotics to prevent complications. Misdiagnosis is particularly dangerous as administration of prednisone, used for subacute thyroiditis, may lead to deterioration of AST .
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clinical case
| 99.94 |
In adults, proposed routes of infection for AST include lymphatic or hematogenous spread, direct inoculation of the thyroid or surrounding anatomy, direct extension of an abscess, and spread through a pyriform sinus fistula, usually in the setting of preexisting thyroid disease or an immunocompromised state [2, 11]. Based on the patient's history, several mechanisms may explain her AST. She was an IV drug user who presented with an MSSA abscess in her forearm, along with MSSA bacteremia. Thus, there is the strong possibility for hematogenous spread of MSSA from the forearm abscess to the thyroid. Although the patient denied injecting heroin into her neck or shoulders, there is also the possibility for direct inoculation of the thyroid. Furthermore, the patient had a preexisting multinodular goiter of one-year duration that showed follicular lesion of undetermined significance on cytology. Abnormal thyroid structures, such as multinodular goiters, nodules, or malignancies, have been postulated to make the thyroid gland more susceptible to suppuration . In particular, Erdamar et al. found that the enzymatic free radical defense system was impaired in 41 patients with multinodular goiters and papillary carcinomas . A weakened antioxidant defense mechanism may predispose the thyroid to bacterial infections. Few other cases in the literature report AST in the context of a preexisting multinodular goiter [13–15].
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clinical case
| 99.9 |
Given the potentially life-threatening sequelae of AST, the threshold for performing ultrasound-guided FNA to rule in/out AST should be relatively low, especially in a patient with one or more predisposing factors. Ideally, thyroid FNA cultures should be obtained prior to antibiotic administration to prevent unnecessary treatment and reduce the risk of a false-negative thyroid culture. Nevertheless, the patient in this case was treated empirically with ampicillin-sulbactam prior to obtaining a thyroid FNA a few days into her hospital stay. Had ampicillin-sulbactam not been started early on in her hospital stay, the thyroid mass may have further compressed the trachea, leading to significant narrowing and airway compromise.
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clinical case
| 99.8 |
Because AST is rare and may present with varying levels of severity, there are few high-level evidence studies that clearly define optimal management strategies. AST was traditionally managed with a combination of surgical and medical treatment modalities involving partial or total thyroidectomy or surgical drainage and targeted antibiotic therapy. In more recent nonrandomized clinical practice and case reports, there has been a greater emphasis on an upfront medical approach with targeted IV antibiotic therapy [8, 16]. The optimal duration of IV antibiotic therapy is unclear, with expert opinion recommending 14 days . Isolated case reports have continued IV antibiotics for up to 37 days, even in patients who demonstrated early response . As our patient lacked signs of airway compromise and a drainable thyroid abscess, and improved rapidly over the first four days, the decision was made by day 8 to transition her to oral antibiotics and discharge on a 14-day total course. Further studies are needed to refine therapeutic approaches based on the severity of disease presentation.
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clinical case
| 99.9 |
This case demonstrates the importance of assessing patient risk factors for AST, even when clinical presentation may seem more typical of subacute thyroiditis. Prompt diagnosis with ultrasound-guided FNA and cultures may allow for targeted antibiotic therapy and prevent complications associated with AST. There is a lack of high-quality studies assessing the optimal management strategy for AST; as such, treatment should be guided by the severity of disease presentation. Further investigation is needed to assess the effectiveness of shorter courses of therapy and/or transition to oral antibiotics in less severe presentations showing early clinical improvement.
|
clinical case
| 99.9 |
The soybean genome has been sequenced and expression patterns of most soybean genes are known [1, 2]. Although putative genes have been predicted based on the DNA sequence and when possible annotated for inferred function based on protein homology, many soybean genes remain uncharacterized even at this level of annotation. Rapid identification of biological functions of soybean genes will require an indexed insertional mutant collection suitable for reverse genetics studies.
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study
| 99.94 |
Gene silencing through RNAi has been successfully used in the functional characterization of plant genes . However, the incomplete inactivation of target genes is a common problem with such gene silencing approaches and makes the data interpretation difficult . T-DNA insertion mutagenesis, in which insertion of T-DNA into the coding or promoter sequence of a gene can disrupt its function, has been effectively utilized in Arabidopsis [5–7]. Gene editing has been shown to be a powerful approach for the functional analyses of genes in plant species including maize [8, 9]. In soybean, the main bottleneck associated with the application of RNAi, T-DNA insertion mutagenesis or gene editing approaches for functional analyses of a large number of genes is the lack of a high-throughput transformation procedure and availability of large greenhouse spaces for growing transformants.
|
review
| 99.8 |
Transposable elements are a major component of the genomes of higher eukaryotes and are widely distributed among plant species . Functional characterization of thousands of soybean genes could be facilitated by knockout mutants induced by transposons [11–14]. Active endogenous transposable elements have been identified in several plant species and have been effectively used in functional characterization of plant genes [15, 16].
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review
| 98.4 |
The utility of active transposons from maize (Ac/Ds), rice (mPing) and tobacco (Tnt1) in tagging soybean genes has been demonstrated [11–13]. However, this approach requires genetic transformation. Handling thousands of transgenic lines in the field prior to deregulation is impractical and cumbersome. Furthermore, products generated through genetic transformation are often not well received by a significant proportion of the end users worldwide and the deregulation step could be lengthy and expensive. Tagging of soybean genes using an endogenous transposable element is therefore an attractive solution not only for functional analyses of tens of thousands of soybean genes, but also in generating desirable mutants for rapid genetic improvement of important traits in the commercial soybean cultivars. Since mutants created by an endogenous transposon are not GMOs, the products produced from incorporation of such mutants should be accepted by all consumers.
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study
| 99.9 |
Five genes, W1, W3, W4, Wm, and Wp, regulate pigmentation in flowers and hypocotyls of soybean . The mutant w4-m allele at the W4 locus is characterized by altered pigment accumulation patterns in flowers and hypocotyls . The soybean line with w4-m allele was added to soybean collection and assigned the genetic type collection number T322 . W4 contains the DFR2 gene encoding a functional dihydroflavonol-4-reductase 2 enzyme. Somatic excision of a 20,548-bp CACTA-type transposable element, Tgm9, from Intron 2 of DFR2 causes variegated flowers and hypocotyls with dark brown sectors . Since Tgm9 is located in an intron, both precise and imprecise excisions of the element lead to restoration of the wild-type phenotype. Excision of the element from the germ tissues of the parent plant results in at least some progenies that carry only purple flowers and are termed germinal revertants. The mutation rates in several genetic loci among the revertants with purple flowers were much higher than the rate of spontaneous mutation . Therefore, it was hypothesized that following excision of Tgm9 from DFR2, the element inserts into new genetic loci.
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study
| 100.0 |
Tgm9 excises at a high rate with a germinal reversion frequency of about 6% per generation . Several mutant genes, unlinked to the W4 locus, were isolated by screening germinal revertants for morphological mutant phenotypes [20, 22, 23]. For example, male-sterility, female-sterility and root necrotic root mutants were identified by screening thousands of germinal revertants carrying purple flowers [24, 25]. The present study was undertaken to determine if Tgm9 is suitable for creating knockout mutants for conducting large-scale functional analyses in soybean.
|
study
| 100.0 |
Seeds of chlorophyll deficient mutants (Genetic type collection numbers T323, T325 and T346), male-sterile, female-fertile mutant (T359) and partial female sterile mutants (T364, T365 and T367) were obtained from Dr. Randall Nelson, USDA-Agricultural Research Services.
|
other
| 99.9 |
For generating germinal revertants for this study, we identified T322 (w4-m) plants that showed variegated flowers and their progenies were grown to identify germinal revertants. One germinal revertant per progeny row was selected for this study (S1 Fig). Leaf tissues from each of the selected germinal revertants was harvested for DNA preparation according to a previously described method .
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study
| 100.0 |
A GenomeWalker Universal kit (Clontech Laboratories, Inc., Mountain View, CA, USA) was used to find the unknown genomic DNA sequences adjacent to a transposable element using the manufacturer’s instructions. DNA (2.5 μg) was independently digested with four restriction enzymes (DraI, EcoRV, PvuII, and StuI) to generate blunt ended fragments (S2 Fig). After phenol:chloroform::1:1 purification, digested genomic DNA was ligated to the GenomeWalker adaptor to generate genomic DNA libraries. The four genomic libraries, generated through digestion of DNA with four restriction endonucleases for each germinal revertant were used for the first PCR (PCR 1) using the outer adaptor primer (AP1) and an outer Tgm9-specific primer (Trans R1) (S2 Fig; S1 Table). The PCR 1 mixture was then diluted to 100 times and used as a template for a second or “nested” PCR (PCR 2). PCR 2 reactions were conducted using the nested adaptor primer (AP2) and a nested Tgm9-specific primer (Trans R2) (S2 Fig). For visualization, the resulting PCR products were separated on a 1.5% agarose gel at 100 V for 1 hour. As there are some residual/truncated copies of Tgm9 in T322, only unique bands of individual mutants were sequenced (S2 Fig). Residual/truncated copies were displayed as common bands among the selected mutants (S3 Fig). Unique PCR products were sequenced and evaluated for presence of over 100 bp Tgm9-end specific sequences.
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study
| 100.0 |
Long range PCR was performed to amplify a region spanning 2,523 base pairs of the 5’-end of the transposon in two steps using two nested primers from Tgm9 and two from the mutant MER3 gene of mer3 (S1 Table). The PCR products were amplified using Phusion high fidelity DNA polymerase (Thermo Fisher, USA). Bands of correct size were extracted from a 0.8% agarose gel using IBI gel extraction kit (IBI Scientific, USA). The samples were sequenced using the inner Tgm9 and inner MER3 primers at the DNA Facility at Iowa State University. The resulting sequence was aligned with the Tgm9 (GQ344503.1) and Tgmt* (EU190440.1) sequences.
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study
| 100.0 |
Tgm9-insertion mutants were named using an initial 'T9' with a nomenclature similar to that used for other mutant types in SoyBase. The location of each Tgm9 insertion was identified by using BLAST to find the location of the element's flanking sequence. The location was determined in both the Wm82.a1 and Wm82.a2 soybean genome assemblies (http://soybase.org/aboutgenomenomenclature.php). The proportion of exons and introns in the soybean genome sequences that were mapped to chromosomes were calculated by considering the predicted exonic and intronic sequences in the Wm82.a2 genome assembly (1).
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study
| 100.0 |
The data in the GFF file used at SoyBase for the Wm82.a2 genome assembly was parsed into the component parts of the genes. For each gene we used the longest splice variant and determined (i) the full length of the gene, i.e. the region transcribed; (ii) the length of the CDS, i.e. the translated mRNA from ATG to TAG with introns removed; (iii) the length of the 5’ UTR (un-translated region) in the translated mRNA, i.e. with any introns removed; (iv) the length of the 3’ UTR in the translated mRNA, i.e. with any introns removed; (v) the combined length of all introns, i.e. those sequences spliced out of the 5’ UTR and/or the translated region and/or the 3’ UTR. These values were summed for all gene models in the Wm82.a2 genome assembly. We used the sum of the 5’ UTR, CDS and 3’ UTR lengths as the total exon length in the genome. The proportions of exon (11.6%) and intron (10.5%) sequences (S2 Table) are used as the expected values of random Tgm9 insertion into the soybean genome.
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study
| 100.0 |
Most of the data are presented in the manuscript or as supplemental data files. The data for the Tgm9 insertion sites have been deposited to SoyBase and are available at http://soybase.org/gb2/gbrowse/gmax2.0x/?start=1;stop=56831624;ref=Gm01;width=1024;version=100;flip=0;grid=1;id=28f86f22137273c155fe3e1756259be0;l=tgm9-gene_models_wm82_a2_v1-pericentromere%3Aoverview.
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other
| 99.9 |
We investigated the applicability of Tgm9 in generating knockout mutants for functional analyses of soybean genes as follows. First, we determined if the insertion following excision of Tgm9 from the mutable w4-m allele resulted in mutation among the mutants previously identified from screening germinal revertants, generated from the T322 line [20, 23, 27–31] (Table 1). Second, we investigated 124 random germinal revertants to determine the properties of Tgm9 transposition.
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study
| 100.0 |
Several morphological mutants were previously identified from screening thousands of germinal revertants generated from the T322 (w4-m) mutable line. These mutants include chlorophyll deficient mutants (T323, T325 and T346) [23, 27], male-sterile, female-fertile mutant (T359) and partial female sterile (fsp) mutants (T364, T365 and T367) (Table 1). The genes responsible for the mutant phenotypes in these mutants have been previously genetically mapped [29–31]. To determine if the mutations in these mutants were caused by Tgm9-insertion, we identified the Tgm9-insertion sites using a transposon display approach for each of the mutants (S2 Fig).
|
study
| 100.0 |
The Tgm9-insertion sites were mapped to single genes in each of the five mutants investigated (Tables 1 and 2; Fig 1). The five Tgm9-induced mutant genes identified in these five mutants are from previously mapped mutant loci or regions. For example, in three chlorophyll deficient mutants (y20 mdh1-n) a Tgm9 insertion was identified in the first intron of Glyma.12G159300 encoding lactate/malate dehydrogenase (Table 2). The y20 Mdh1-n locus was originally mapped to Chromosome 12 and shown to be flanked by microsatellite markers Satt253 and Satt302 (Table 1). Glyma.12G159300 is located in this y20 Mdh1-n genomic region. Similarly, the Tgm9-insertion site in the ms9 mutant was located in the first intron of Glyma.03G152300 that has no functional annotation (Table 2). The gene is in the ms9 region flanked by Satt521 and Satt237 (https://soybase.org) [30, 32] (Tables 1 and 2; Fig 1). The Tgm9-insertion sites in Fsp2 and Fsp3 were detected in the 7th intron and 2nd exon of Glyma.06G174200 and Glyma.08G359000, respectively (Table 2). Glyma.06G174200 encodes a haloacid dehalogenase-like hydrolase; Glyma.08G359000 encodes the embryo-specific protein 3, (ATS3) (Table 2) (https://soybase.org). The Fsp2 locus is flanked by Satt170 and Satt277 on Chromosome 6 . Glyma.06G174200 is located in this Fsp2 region (Tables 1 and 2; Fig 1). Likewise, the Tgm9 insertion sites in the fsp3 and fsp5 mutants were also localized to the genomic regions to which Fsp3 and Fsp5 were genetically mapped (Fig 1; Table 2).
|
study
| 100.0 |
To better understand the transposition patterns of Tgm9, we identified 124 germinal revertants bearing only purple flowers, each from individual T322 (w4-m) mutable plants (S1 Fig). Therefore, each revertant was unique. We were able to determine insertion sites from 105 mutants (S3 Table). Whether Tgm9 failed to insert into new genetic loci or we failed to recover the insertion sites among the remaining 19 mutants is unknown.
|
study
| 100.0 |
Physical mapping of the Tgm9 insertion sites among 105 mutants revealed that the element transposes to all 20 chromosomes (Fig 2). Insertion sites per chromosome ranged from 1 to 16 per chromosome (Fig 2). These vast differences in the distribution of insertions among chromosomes could be due to the small sample size of the population studied or due to a preference of the element to transpose into some genomic regions. There were six genomic regions, <1 Mb in size, which were enriched in Tgm9 insertions. Of the 105 identified mutants, 24 were mapped to these six regions: (i) Chromosome 1 (six Tgm9 insertions between 50.06 and 50.50 Mb region (http://www.soybase.org/SequenceIntro.php); (ii) Chromosome 9 (four between 4.07 and 5.07 Mb); (iii) Chromosome 9 (five between 8.77 and 9.06 Mb), (iv) Chromosome 15 (three between 2.58 and 2.87 Mb); (v) Chromosome 17 (three between 40.65 and 40.89 Mb); and (vi) Chromosome 18 (three between 54.31 and 54.51 Mb) (Fig 2). In soybean the euchromatic region constitutes 43% of the genome . In this study, Tgm9 transposed to euchromatic regions 77.1% of the time. The remaining 22.9% of insertions were mapped to the pericentromeric regions (Fig 2).
|
study
| 100.0 |
Green arrows represent locations of Tgm9 in genes (exons and introns) and red arrows show Tgm9 insertions in other genomic regions. A purple arrow shows the location of Tgm9 in W4 on Chromosome 17. Centromeres (black rectangles) and heterochromatic regions (grey areas) are shown on individual chromosomes. Scale is represented in million base pairs (Mb) of DNA.
|
other
| 99.75 |
To study the distribution of transposition events of Tgm9 across the chromosomes, the physical locations of the Tgm9 insertion sites in the soybean genome were investigated. Although Tgm9-insertion in promoters can alter phenotypes, due to the limitation in precisely predicting the promoter sequences we calculated the number of insertions that were localized to only exon- and intron-sequences. These mutations are expected to knock-out gene function, and therefore are desired mutations for functional analyses of soybean genes. Of the 105 insertions studied, 16.2% and 9.5% of the Tgm9-insertions were localized to exons and introns, respectively (Fig 3, S3 Table). Thus, 25.7% of the insertions were generated in gene-sequences. In the soybean genome, gene-sequences cover 22.1% of the genome (S3 Table), which is very comparable to the observed insertion rate of 25.7% in the gene-sequences. The calculated χ2 value (1.92 at df = 1) suggests that the observed frequencies of Tgm9 insertions in exons and introns (Fig 3) is statistically not significant at p = 0.05 from the expected values resulting from random insertion of the element in exon and intron sequences. We therefore conclude based on the study of 105 mutants that the Tgm9 element appears to transpose randomly in the soybean genome (Fig 3, S3 Table).
|
study
| 100.0 |
The observed frequency of Tgm9 insertions in exonic and intronic sequences were calculated from 105 Tgm9 insertion loci (S3 Table). The expected frequency of Tgm9 is based on random insertion of the element in the exonic and intronic sequences calculated from the soybean genome sequences mapped to chromosomes (Soybase). The χ2 value (1.7752 at df = 1) calculated for observed Tgm9 insertion frequencies and expected frequencies of random Tgm9 insertions in exon and intron sequences is statistically non-significant at p = 0.05.
|
study
| 100.0 |
The availability of the genome sequence has greatly expedited molecular and genomic research in soybean . Reverse genetic approaches such as TILLING and deletion mutants induced by fast neutron irradiation are being exploited for functional characterization of the soybean genome [33, 34]. Heterologous transposable elements have also been successfully applied in functional characterization of soybean genes [11–13].
|
study
| 71.3 |
The Ac/Ds transposon system in maize causes insertion mutations in closely linked locations. It has been applied to target specific chromosomal regions in several plant species [35, 36]. In soybean, the feasibility of Ac/Ds was tested by sequencing flanking regions of 200 individual mutants and was validated by isolating a gene involved in male-fertility . Similarly, mPing, a miniature inverted repeat transposable element (MITE) from rice was evaluated in soybean . mPing was shown to transpose to unlinked regions with a strong preference for gene-containing regions . Tnt1, a tobacco retrotransposon, was effectively transformed into soybean and was shown to be re-activated by tissue culture .
|
study
| 100.0 |
Transposon-induced mutants are preferred over mutants generated using irradiation or chemical mutagenesis approaches because they typically contain a very few mutations per mutant and therefore are easier to analyze and utilize in breeding programs. However, heterologous transposable elements have to be genetically transformed and at least for the maize Ac/Ds system requires the generation of a large collection of transgenic soybean lines because the element moves only to linked genomic regions . The requirement of tissue culture for the activation of Tnt1 limits its uses in functional analyses of soybean genes because tissue culture itself can generate new genetic and epigenetic mutations, broadly known as somaclonal variation. Furthermore, functional analyses of tens of thousands of soybean genes will require growing a huge number of transgenic soybean lines in the field with proper care not to release transgenes to the environment. Once a product is developed from such a huge effort, the seed industry must then wait to get approval for deregulating the transgenic soybean lines. Then, there is also the issue of non-acceptance of transgenic soybeans by a significant proportion of soybean consumers. Given the complexities associated with the use of heterologous transposons summarized above, endogenous transposable elements are better suited for functional analyses of soybean genes and the generation of useful mutants for breeding desirable cultivars.
|
review
| 99.9 |
Several endogenous transposable elements such as Tgm1, Tgm2, Tgm3, Tgm4, Tgm5, Tgm6, Tgm7, Tgm8, Tgm9, Tgmt* and TgmR* have been identified in soybean [14, 38–40]. Of these, Tgm9 has been shown to be active and has recently been used to clone a male-sterile, female-sterile gene (GmMER3) that encodes an ATP dependent DNA helicase [14, 41, 42]. It is possible, although unlikely, to have another Tgm sequence(s) causing mutation in GmMER3 or other genes investigated in this study. To determine if the mutation in the male-sterile, female-sterile (mer3) mutant was caused by Tgm9, and not by any other active elements such as Tgmt*, the 3’-end 2,523 bp region of the transposon in the GmMER3 gene of the mer3 mutant was amplified and sequenced. The sequence matched perfectly with the Tgm9 sequence suggesting that the mer3 mutation was caused by Tgm9 insertion, not by Tgmt* (S4 Fig). Furthermore, sequencing of the T322 genome revealed that Tgm9 is the only intact CACTA-type element identified in this cultivar. In addition to the expected location of Tgm9 in W4, an additional copy of the element was identified in a locus on Chromosome 19. This Tgm9 element was hemizygous in T322 and absent among 15 selected Tgm9-induced mutants. This suggests that most likely the Tgm9 copy on Chromosome 19 was generated from a recent transposition event (J. Baumbach, S. Srivastava and M.K. Bhattacharya, unpublished).
|
study
| 100.0 |
Here we have demonstrated that following excision from dfr2, Tgm9 induces mutations in unlinked soybean genes (Fig 1). We have also demonstrated from analysis of 105 independent Tgm9-induced mutants that the element transposes to unlinked loci on all 20 soybean chromosomes (Fig 2). Since Tgm9 is the only intact CACTA-type element in T322, the mutant genes or sequences identified in this study were resulted from insertion of Tgm9. To our knowledge, Tgm9 is only reported active transposable element in soybean .
|
study
| 100.0 |
In this study we have observed that Tgm9 transposes into unlinked loci following excision from the w4-m mutant allele and that such transpositions could cause functional mutations in over 25% of the mutants. There are 15,166 single copy genes in the soybean genome and Tgm9 should be a suitable tool for generating mutations in most of these genes . However, Tgm9 has preference for certain genomic regions (Fig 2). This will reduce the efficiency of Tgm9 in inducing mutations in such a study for functional analyses of soybean genes. Utilizing Tgm9 to advance soybean genetics and biology will therefore require a large indexed Tgm9-induced mutant population. These mutants can be easily identified by taking the advantage of next generation sequencing platforms [7, 43] and displayed in the SoyBase genome browser (Fig 4). The ability of Tgm9 to precisely excise from mutant loci for reconstituting the wild-type function may eliminate the need of complementation analyses of mutants through genetic transformation for identification of novel genes .
|
study
| 100.0 |
The track for Tgm9-insertion lines shows the location of two Tgm9-induced insertions (T91200307 and T91200364) on Chromosome 15. Note that in the T91200307 mutant, the Tgm9 insertion site is located in the inter-genic region, and in T91200364, the insertion site is in the 2nd exon of Glyma.15G033300. (http://soybase.org/gb2/gbrowse/gmax2.0x/?start=1;stop=56831624;ref=Gm01;width=1024;version=100;flip=0;grid=1;id=28f86f22137273c155fe3e1756259be0;l=tgm9-gene_models_wm82_a2_v1-pericentromere%3Aoverview {Please use user name: Bhatt; password: tgm9#}).
|
study
| 99.9 |
The 20,548-bp Tgm9 element has been reported to be fractured during transposition, leading to the generation of stable mutants which could be used in soybean breeding programs . Furthermore, cultivars generated by using such Tgm9-induced mutants can reach consumers rapidly since they do not require the lengthy deregulation process of mutants generated from the use of heterologous transposon systems.
|
other
| 76.25 |
In this investigation we have shown that Tgm9 is an active transposable element that can induce mutations in unlinked genes on all 20 soybean chromosomes. It appears based on the study of 105 mutants that the Tgm9 element randomly transposes into genes and over 25% of the mutants are expected to be knockout mutants, suitable for functional analyses of soybean genes. As there is a single active copy of this element in the mutable T322 line, identification of the insertion sites in a large collection of Tgm9-induced mutants is feasible through application of a next-generation sequencing platform. In soybean, the functions of the majority of the genes are still unknown. The generation of an indexed transposon-induced mutant population using Tgm9 and their display in the SoyBase genome browser in the context of the other information available at SoyBase (Fig 4) will likely facilitate the functional characterization of most of the 15,166 single copy soybean genes . Desirable Tgm9-induced mutant genes carrying inactive, fractured Tgm9-elements can also be identified and incorporated into elite breeding programs leading to relatively rapid release of genetically improved cultivars for commercial cultivation.
|
study
| 100.0 |
Adaptor has 5’ extended strand with no binding site for primers AP1 (Adaptor Primer 1) or AP2 (Adaptor Primer 2). Binding site for AP1 or AP2 can only be generated by transposon specific primers (TransR1 or TransR2). Exposed 3’ end of the adaptor is blocked by amino group to prevent extension. Unique bands from different lanes are excised and sequenced. The bands that are common to all lanes are most likely ancient transposition events.
|
other
| 99.5 |
A single progeny row was grown from each of the six independent mutable plants harvested in 2014. From each row, two plants were selected for transposon display: (i) one plant with only green stem; (ii) the other plant with only purple stem (germinal revertant). The PCR fingerprints of each of the six plants with only green stem are shown on lanes 1 through 6; and those for six plants with purple stem on lanes 7 through 12. Note that two plants, Plant # 1 under green stem heading and Plant # 1 under purple stem heading, were descended from the same mutable plant harvested in 2014. Two restriction endonucleases, EcoRV and PvuII, were used in digesting the genomic DNA for generating the transposon displays. White arrows show the amplification of some of the residual insertions; whereas, red arrows indicate the progeny-specific amplification presumably from new Tgm9 insertion sites in distinct loci. DFR2-specific amplification was observed for the plants with purple stems (germinal revertants). Note that in Plant # 5 with purple stem failed to amplify the Tgm9 insertion site at the DFR2 intron II presumably because of simultaneous Tgm9 excision from both DFR2 copies. The sibling plants with green stems failed to amplify DFR2 because of the presence of Tgm9 in both DFR2 copies. Sub-PCR of the two strong PCR amplified ~750 bp fragments in Plant # 3 and 6 with green stem using DFR2 F and TransR2 primers (Supplemental Table 1) indicated that the intense amplified PCR products were from the DFR2 locus (data are not shown).
|
study
| 100.0 |
Transposon insertion sequence from the GmMER3 gene was amplified by conducting long-range PCR and compared to both Tgm9 and Tgmt*, the two highly similar transposons characterized from soybean. A) The orientation of the Tgm9 insertion in the MER3 gene and primers used for nested PCR and sequencing are shown. B) First PCR product was amplified using primers 1 and 2. The amplified PCR product was 2,823 bp. C) The nested PCR product was amplified using primers 3 and 4. The amplified PCR product was 2,758 bp. Primer 3 was used for sequencing the PCR product represented by the dashed line X. Primer 4 was used to sequence the PCR product and produced sequence represented by the dashed line Y. D) Sequence from primer 3 matched the MER3 gene and the start of the 5’ end of the Tgm9 transposon sequence shown with the red font. E) Sequence from primer 4 aligned to the Tgm9 and Tgmt*. The PCR product matches the Tgm9 sequence perfectly. Polymorphic nucleotides between the insertion sequence and Tgmt* sequence are shown in red font.
|
study
| 100.0 |
Since many intra/intermolecular vibration modes of biomaterials such as protein1, 2 and DNA3–6 are located within THz spectral range, there has been great interest in research with THz spectroscopic system for biomaterials. Specifically, in contrast to other optical techniques including ultraviolet or X-rays, its non-invasive and non-ionizing properties allow THz technique to be utilized as spectroscopy for even more complex structural biomaterials comprising cells7 and tissues8 without worrying about thermal fluctuations or other nonlinear side effects. In protein, especially, the conformational information plays a major role in molecular interactions and binding with ligands, which can be analyzed by the THz spectroscopy9, since the protein-ligand binding energy is in the THz range as well. However, the detection of such small change of optical property could be limited using THz spectroscopic system as it is, due to its too small absorption cross-section. Recently, metamaterial sensing chip based THz detection techniques have been developed for highly sensitive and selective detections of carbohydrates10, chemical compounds11, 12, thin sample layer13, 14 and microorganisms15 and overcoming the limit of sensitivity with typical THz spectroscopic systems. Here, we report THz optical characteristics of three different AI viruses investigated using a highly sensitive THz spectroscopy system assisted by nanoscale metamaterial sensing chips. The sensing chip induces huge THz field enhancement16 as much as 50 times, leading to increase the detection sensitivity and selectivity at the same time. Therefore, the three different AI virus samples could be clearly discriminated with respect to their optical parameters. The quantification for one of the virus subtypes was also performed verifying the excellence of the THz nano-metamaterial sensing chip, finally.
|
study
| 99.94 |
Since several subtypes of Avian Influenza (AI) viruses, frequently causing worldwide outbreaks, are extremely pathogenic, a development of quick and accurate diagnostic methods has been highly demanded (see details in Supplementary Information). In order to investigate fundamental optical characteristics of targeted virus samples, THz transmission measurements without any assistive sensing chip were performed firstly. The targeted three virus samples are: A/NWS/33 (H1N1), A/wild bird/Korea/K09-652/2009 (H5N2), and A/Korean native chicken/Korea/K040110/2010 (H9N2), as shown in Table 1. Among three different virus samples, we chose the H9N2 subtype virus and control (without inoculation of any virus) in pellet forms after freeze-drying process. An absorbance was extracted from the transmission measurement through a pallet type sample in a same way described in ref. 10. The H9N2 and control samples have no recognizable absorption features in the THz spectrum (Fig. 1a) due to superposition of many protein vibration modes and inhomogeneous broadening of absorption features17. In contrast to earlier works10–12, it is not easy to define fundamental resonance frequency to design nano-antenna based sensing chip for the target bio sample in this case without unique spectral features from their intrinsic modes. In this experiment, therefore, multi-resonance nano-antenna was suggested which is based on the concept of THz nano-antennas with a log-periodic alignment18. For an optically unknown target molecules, the multi-resonance nano-antenna is very useful, since it maintains very high sensitivity in ultrabroadband THz regime.Table 1The strain names with subtype and total protein concentration of each virus samples are represented.Subtype-strain nameTotal protein concentration (mg/ml)A/NWS/33 (H1N1)0.54A/wild bird/Korea/K09-652/2009 (H5N2)0.2A/Korean native chicken/Korea/k040110/2010 (H9N2)0.28 Figure 1THz detection of virus samples. (a) Absorption spectra for pallet types of virus included a protein sample (H9N2) and a control sample without virus in it. (b) A conceptual schematic of THz detection of virus samples in liquid state using a nano slot-antenna array based sensing chip. (c) Optical images of dropped virus solutions onto the multi-resonance nano-antenna array before (top) and after (down) THz excitation. (d) Transmittance spectra through multi-resonance nano-antenna that have three resonance peaks, with and without H9N2 sample onto the antenna. (e) The difference in transmitted intensity (ΔT) and shifted resonance frequency from each fundamental resonance peak of multi-resonance nano-antenna (Δf) for H9N2 sample are represented.
|
study
| 99.94 |
THz detection of virus samples. (a) Absorption spectra for pallet types of virus included a protein sample (H9N2) and a control sample without virus in it. (b) A conceptual schematic of THz detection of virus samples in liquid state using a nano slot-antenna array based sensing chip. (c) Optical images of dropped virus solutions onto the multi-resonance nano-antenna array before (top) and after (down) THz excitation. (d) Transmittance spectra through multi-resonance nano-antenna that have three resonance peaks, with and without H9N2 sample onto the antenna. (e) The difference in transmitted intensity (ΔT) and shifted resonance frequency from each fundamental resonance peak of multi-resonance nano-antenna (Δf) for H9N2 sample are represented.
|
study
| 100.0 |
Figure 1b shows a schematic of THz transmittance measurement through the virus sample dropped sensing chip. The multi-resonance nano-antenna sensing chip was fabricated with two-dimensional punctured rectangular slots with a width of w = 500 nm, lengths of l = 200, 100, 67, and 50 μm, and spacings between adjacent two antennas are 15, 7.5, and 3.75 μm in the horizontal direction, respectively. The second type of nano-antenna sensing chip, used in later measurements, was fabricated with a length of l = 40 um. The periods between slots are 40 um in the horizontal direction and 50 um in the vertical direction. These nano-antennas are patterned in a 150-nm-thick gold on the top of double-side-polished 500-μm-thick silicon wafer using an e-beam lithography technique. The total slot numbers more than 1000 within a whole area of 2 mm × 2 mm are designed to minimize possible errors from a random distribution of protein sample during the liquid drop casting. Each slot has a fundamental resonance (f res); for example, the multi-resonance nano-antenna sensing chip has three resonances at 0.62, 0.93, and 1.31 THz, respectively, and the single-resonance nano-antenna sensing chip has a single fundamental resonance at 1.4 THz according to the effective refractive index of substrate19. For the measurements of virus samples, we dropped 2 uL of virus solution in liquid state and dried it for 1 hour in controlled laboratory atmosphere (30% of relative humidity and 20 degree of Celsius) to avoid any possible signal error from the water absorption at THz frequency20. The dropped virus sample builds a thin film on the multi-resonance sensing chip as shown in the top of Fig. 1c. After 20 minutes of THz field excitation, diffraction pattern appeared in the middle of film as shown in the bottom of Fig. 1c. Without THz field, the pattern disappeared within 5 minutes. Protein aggregation and crystallization effect under high electric field have been reported previously21–23. Even though our case is not exactly matched to the conditions for the reported protein crystallization, such as concentration of protein, precipitants, pH, and temperature of protein solution, it can be assumed that the changed surface is related to the temporal aggregation as a beginning step of protein crystallization under strongly enhanced THz electric field. The further study on this interesting effect of protein aggregation and induced crystallization using nano-antenna assisted THz field is needed, but it will not be discussed here in depth (out of scope of this report).
|
study
| 100.0 |
THz optical response for H9N2 virus sample was characterized with transmission measurement through multi-resonance nano-antenna (Fig. 1d). The THz optical responses in terms of the changed values at three maximum transmittances (ΔT) and shifted resonance frequencies (Δf) are respectively plotted in Fig. 1e. As a meaningful frequency for further measurements, 1.4 THz was selected owing to its biggest changes in both ΔT and Δf values, implying the highest sensitivity. Further results with 1.4 THz single-resonance nano-antenna in Fig. 2 present that the viruses can be classified in terms of resonance frequency and decreased transmission ratio.Figure 2THz optical properties for various types of virus samples. (a) Normalized THz transmission spectra measured using the single-resonance nano-antenna without virus sample (black), and with selected three viruses, H1N1 (blue), H5N2 (cyan), and H9N2 (magenta). (b) FDTD simulation results of transmittances for the single-resonance nano-antenna and the three different model samples with various composition of dielectric constants (n and κ) are shown. Inset indicates a used geometry for simulation.
|
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THz optical properties for various types of virus samples. (a) Normalized THz transmission spectra measured using the single-resonance nano-antenna without virus sample (black), and with selected three viruses, H1N1 (blue), H5N2 (cyan), and H9N2 (magenta). (b) FDTD simulation results of transmittances for the single-resonance nano-antenna and the three different model samples with various composition of dielectric constants (n and κ) are shown. Inset indicates a used geometry for simulation.
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Measured transmission spectra via single-resonance nano-antenna sensing chip with a fundamental resonance frequency at 1.4 THz with and without three AI virus samples are shown in Fig. 2a. Each virus sample shows a distinct transmission change and a shift in the resonance frequency related to their different surfaces and inherent strains. Further analysis and comparison of THz optical characteristics for different subtype viruses were investigated through numerical simulations using the Finite-difference time-domain (FDTD) method (Fig. 2b) (details in Supplementary Information). In FDTD simulations calculating transmission spectra, virus samples are considered as 5 μm-thick, homogeneous dielectric clads possessing different complex refractive indices as illustrated in inset (see the methods). In Fig. 2, experimental results on transmittance characteristics of various samples are compared with numerical FDTD results showing a good agreement. The transmission value change is affected by the molecular absorption accompanying with the change of the imaginary part of the refractive index, κ of each sample. Also, the change of the real part of the refractive index, n is related to the composition of the virus samples. The spectral changes in transmission, therefore, can be confident evidences to define and identify the virus samples, as well presented in both experiments and simulations.
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In order to get more insight into detection mechanism using THz sensing chip, concentration dependence was examined for one subtype of the viruses. We measured transmission spectra for H9N2 virus in various concentrations via nano-antenna sensing chip (f res = 1.4 THz) as shown in Fig. 3a–d. The various concentrations of virus samples were prepared by diluting virus samples (total protein concentration: 0.28 mg/dl) with buffer liquid (total protein concentration: 0 mg/dl) by 1:1 and 2:1 volume ratio, producing four samples: 0, 1, 0.14, 0.28 mg/ml, respectively. The buffer is prepared without inoculation of any virus, as same as the control sample used in pellet experiment. As the concentration of virus increases, the maximum transmittance value decreases, due to its absorption change as shown in Fig. 3e (magenta). It is noted that the clear linearity can be extracted from the data, allowing a promising quantification tool of such protein samples which is one of the critical issues in protein studies. On the one hand, the resonance frequency is maintained, whereas the maximum value of the transmittance was obviously changed in terms of the concentration. The virus-concentration-dependent refractive index was assumed with an effective medium approximation that treats heterogeneous media as homogeneous. The film-likely coated sample on the metamaterial is assumed as 5 μm thick cladding, same as previous FDTD simulations in Fig. 2b, and the densest concentration of protein among three virus samples is 0.54 mg/ml. According to ref. 22, the density of the influenza virus is calculated as 1.104 g/ml24, the volume fraction in the coated virus sample on the sensing chip can be estimated as about 0.05%. The used volume fraction is too small to evidently affect the whole refractive index of the sample. Therefore, the resonance frequency is not significantly changed in terms of the concentration increase, besides initial change Δf = 0.18 THz was observed from bare antenna to first dropping. The FDTD simulation results (Fig. 3e) also confirm that the transmittance value is mainly changed, whereas the resonance shift is negligible with increasing of κ. To exclude the concentration dependency in classifying and subtyping various viruses, considering the normalization per unit mass of virus is essential.Figure 3Virus quantification by nano-metamaterial based THz detection. (a–d) Normalized THz spectra for various concentrations (0, 1, 0.14, 0.28 mg/ml) of H9N2 virus in the buffer solution. (e) The changes in the maximum values of the normalized transmittances (ΔT, magenta closed circle) and shifted resonance frequency (Δf, green closed triangle) are plotted for H9N2 virus in different concentrations as a function of concentration level. The red bar is error bar of buffer solution measurement. Black line and gray dashed line are linear fitting of the transmittance change and frequency shift data, respectively. (f) FDTD simulation results of transmittances for three different model samples with various composition of dielectric constants (n and κ) are shown.
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