X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure r and the target distortion d. For mean-square error distortion, the difference is at most d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of the Gaussian channel with the same noise power is at most X with possibly dependent coordinates. Our proof technique leverages tools from convex geometry.We derive a lower bound on the differential entropy of a log-concave random variable  It was es, thenN(X+Y)≤4Nes, thenN(X+Y)≤4Nquality (N(X−Y)≤e2gated in . For furgated in . In Sectrelated,N(X+Y)≤πed, the rate-distortion function r-th moment distortion measure is given byd, explicitly linking the storage requirements for X to the information content of X (measured by d. As shown in [The bound  on the der bound  states tiven by1RXd=infPXr bound [RXd≥RXd≜h (r=2), (RXd≥h(X)−shown in ,17 underd is linked to the optimality of simple lattice quantizers [d, where The speed of convergence in  and its antizers , an insiantizers  showed td↓0. Ziv  proved td, provided that the source density is log-concave: for mean-square error distortion , we havBesides leading to the reverse entropy power inequality and the reverse Shannon lower bound, the new bounds on the differential entropy allow us to bound the capacity of additive noise memoryless channels, provided that the noise follows a log-concave distribution.Z is given by (21)CZ(P)The rest of the paper is organized as follows. log-concave if f is log-concave if for every A function X in We say that a random vector Our first result is a lower bound on the differential entropy of symmetric log-concave random variable in terms of its moments.Theorem 1.Let X be a symmetric log-concave random variable. Then, for every  holds with equality for uniform distribution in the limit Moreover, As we will see in Theorem 3, for he bound  tightensp) as e) ≈ 1.4 bits at The difference between the upper bound in  and the The next theorem, due to Karlin, Proschan and Barlow , shows tTheorem 2.Let X be a symmetric log-concave random variable. Then, for every  with equality [Moreover, the Laplace distribution satisfies equality .p, we deduce that for every symmetric log-concave random variable X, for every Combining Theorem 2 with the well-known fact that X and a Gaussian X.Using Theorem 1 and , we immeCorollary 1.Let X be a symmetric log-concave random variable. Then, for every where Remark 1. in the limitThe uniform distribution achieves equality in Indeed, if U is uniformly distributed on a symmetric interval, thenand so, in the limit We next provide a lower bound for the differential entropy of log-concave random variables that are not necessarily symmetric.Theorem 3.Let X be a log-concave random variable. Then, for every  tightens asMoreover, for The next proposition is an analog of Theorem 2 for log-concave random variables that are not necessarily symmetric.Proposition 1.Let X be a log-concave random variable. Then, for every Remark 2..Contrary to Theorem 2, we do not know whether there exists a distribution that realizes equality in X and a Gaussian X. Recall the definition of Using Theorem 3, we immediately obtain the following upper bound for the relative entropy of Δp in .Corollary 2.Let X be a zero-mean, log-concave random variable. Then, for every where X, one can optimize over p to further tighten x+As γ→0, , and thuudied in ,27.Since for fixed For example, extended Cauchy distributions, that is, distributions of the formX in We say that a random vector Theorem 6.Let We derive the following lower bound on the differential entropy for one-dimensional symmetric Corollary 4.Let Notice that  reduces Theorem 7.Let X and Y be uncorrelated log-concave random variables. Then,As an application of Theorems 3 and 4, we establish in Theorems 7 and 8 below a reverse form of the entropy power inequality X in Rn,  side of . Here,∥he bound  on D=12,Fuce thatF(−1)=fX81). In the context of information theory, Lemma 1 has been previously applied to obtain reverse entropy power inequalities [Actually, the powerful and versatile result of Lemma 1, which implies ualities , as wellualities . In thisProof of Theorem 2.The result follows by applying Lemma 1 to the values The proof leverages the ideas from .Proof of Theorem 3.Y be an independent copy of X. Jensen’s inequality yieldsLet equality  to X−Y ty yieldsh(X)=−∫fXo obtain1fX−Y(0)≥o obtain1fX−Y(0)≥Proof of Proposition 1.Y be an independent copy of X. Since Let isotropic if X is symmetric and for all unit vectors X is isotropic if its covariance matrix isotropic constant of X.We say that a random vector hyperplane conjecture, asks whether the isotropic constant of an isotropic log-concave random vector is also bounded from above by a universal constant (independent of the dimension). This conjecture holds under additional assumptions, but, in full generality, Lemma 2.Let X be an isotropic log-concave random vector in It is well known that imension . A long-eader to . We will[ℓX  Let equality . Upper-b Let equa2πe,and  follows.If Z is independent of X and realizes the maximum differential entropy under the r-th moment constraint, Z is given by(2) Let fined in . Hence,If en, from , we obtaProof. X is a symmetric log-concave random variable.Denote for brevity X. This choice of Assume that If en, from , we obtaRemark 7.1) Let us explain the strategy in the proof of Theorems 9 and 10. By definition, To estimate or one can apply Jensen’s inequality directly to get the bound provides a tighter bound over . This justifies choosing  in the proof of .A simple study shows that  , which leads the tightening of  for  , we bound r-th norm by second norm, and we note that by the independence of X and Z,To justify the choice of  provides a tighter bound over .A simple study shows that 2) Using Corollary 2, if , and wherewhere is the rate-distortion function of a Gaussian source with the same variance ee e.g.,  ≤, using (RXcov(d)−If Recall that the capacity of such a channel isWe compare the capacity Proof. X such that The lower bound is well known, as mentioned in . To obtaombining , 141) a a(141)h−h−hγ-concaveables in . Finallyables in  and 66)γ-concaveUnder the Gaussian assumption, information-theoretic limits in many communication scenarios admit simple closed-form expressions. Our work demonstrates that, at least in three such scenarios , the information-theoretic limits admit a closed-form approximation with at most 1 bit of error if the Gaussian assumption is relaxed to the log-concave one. We hope that the approach will be useful in gaining insights into those communication and data processing scenarios in which the Gaussianity of the observed distributions is violated but the log-concavity is preserved.


Nature Communications 10.1038/s41467-020-18739-4, published online 2 October 2020.Correction to: The original version of this Article contained an error in Fig. 3, in which the types of tegmino-tegminal stridulation in Gryllidea and in Tettigonioidea shown on the right side of the phylogeny were reversed. The correct version of Fig. 3 is:which replaces the previous incorrect version.The original version of this Article contained an error in the last sentence of the Fig. 3 legend, which incorrectly read ‘The common ancestor of Gryllidea evolved “left-over-right” stridulation, the common ancestor of Hagloidea evolved “ambidextrous” stridulation, and the common ancestor of Tettigonioidea evolved “right-over-left” stridulation’. The correct version replaces this sentence with ‘The common ancestor of Gryllidea evolved “right-over-left” stridulation, the common ancestor of Hagloidea evolved “ambidextrous” stridulation, and the common ancestor of Tettigonioidea evolved “left-over-right” stridulation’.43’. The correct version replaces this sentence with ‘Crickets and mole crickets stridulate by moving the right forewing over the left, and katydids stridulate in the opposite way by moving the left forewing over the right43’.The original version of this Article contained an error in the Discussion, which incorrectly read ‘Crickets and mole crickets stridulate by moving the left forewing over the right, and katydids stridulate in the opposite way by moving the right forewing over the leftThis has been corrected in both the PDF and HTML versions of the Article.


We then argue that, unlike IB and PF, these problems lead to easily interpretable guarantees in a variety of inference tasks with statistical constraints on accuracy and privacy. While the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to a binary case, we derive closed form expressions for several bottleneck problems.Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems , and strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding, and dependence dilution. Leveraging these connections, we prove a new cardinality bound on the auxiliary variable in IB, making its computation more tractable for discrete random variables. In the second part, we introduce a general family of optimization problems, termed “bottleneck problems”, by replacing mutual information in IB and PF with other notions of mutual information, namely  Y is a target variable and X is an observable correlated variable with joint distribution T of X that is maximally “informative” for inferring Y. The bottleneck variable T is assumed to be generated from X by applying a random function F to X, i.e., Y given X, that we denote byT using the mutual information T that solves the constrained optimization problemYXT.Optimization formulations that involve information-theoretic quantities  have been instrumental in a variety of learning problems found in machine learning. A notable example is the information bottleneck  method . SupposeX and Y. Wyner and Ziv )Once ion The result in  was provIn this section, we illustrate several information-theoretic settings which shed light on the operational interpretation of both tings in ,30. In p problem ,89 under problem . Here, w problem  and sour problem .X of a source of interest Y. She wishes to transmit a rate-constrained description from her observation  to Bob such that he can recover Y with small average distortion. More precisely, let n i.i.d. samples of n probability measures on T)Suppose Alice has access only to a noisy version Tsybakov , who shofined in . For morfunctionIt can bY^ as T)where R=erred to ,93. Notiuch thatlim supn→essed in  for discently in  for any n i.i.d. samples Q, we wish to test whether Q is a product distribution. This task is formulated by the following hypothesis test:Y observations UΦ,Ψ and LΦ,Ψ as4LΦ,ΨU(β)≔We will apply this approach to analytically compute YXT, generates unique Let Definition 1.Let U be a discrete and V be an arbitrary random variables supported on andMoreover, the multiplicative gain of the observation V in guessing U is defined as (the reason for ∞ in the notation becomes clear later)U with or without the observation V, respectively. Intuitively, V is in estimating U: If it is small, then it means it is nearly as hard for an adversary observing V to guess U as it is without V. This observation motivates the use of As the names suggest, T can only improve, we have Y and T; instead, it means T is useless in estimating Y. As an example, consider Y and T are clearly dependent; i.e., Y is uniform  is given in In the following theorem, we apply the technique delineated in Theorem 11.For andwhere As described in T satisfying YXT and Y. Similarly, to prove  (see ) correspfined in , with reTheorem 12.Let where Moreover,where X and Y connected through T forming YXT, we haveX with probability at lest Y) is to obfuscate the higher-likely event T such that By letting nnel see (62)PT∗| we havePc(X|T)≥λ we havePc(X|T)≥λComputing Lemma 9.For any pair of random variables andwhere The previous lemma can be directly applied to derive upper and lower bounds for In this section, we describe another instantiation of the general framework introduced in terms of functions Definition 4.Let where vergence  between f, we have f-information is always non-negative. If, furthermore, f is strictly convex at 1, then equality holds if and only f-divergence in [f-divergence and f-information can be found in [f with the property f-information. Popular examples include T-information [f-information in the binary case for a certain function f, see [Due to convexity of gence in  and applfound in  and the ormation , and alsormation  for. It milar to ), then tn f, see  (Theoremf-information can be utilized in (Let lized in  and 41)∼PXYIn light of the discussion in f-divergence is sometimes called Hellinger divergence of order Y given T is lower bounded by Y can be reconstructed with small MMSE given an observation of T. This result serves a natural justification for Consider the function ee e.g., . Note thn et al.  and Asoon et al.  showed tUnfortunately, our approach described in fined in , for the)−1  when X is a Gaussian perturbation of Y. We also utilized the celebrated I-MMSE relationship [T is considered to be a Gaussian perturbation of X.Following the recent surge in the use of information bottleneck , we equality  and stroequality , we obtationship  to derivf-information [In the second part of the paper, we argue that the choice of (Shannon’s) mutual information in both quations  that canquations . Neverthormation  or f-information . Invokinormation ,34, we a


Scientific reports, 10.1038/s41598-019-41819-5 published online 01 April 2019Correction to: The original version of this Article contained an error in the title of the paper, where the words “domain swapping” were incorrectly presented without a space between the words, as “domainswapping”. This has now been corrected in the PDF and HTML versions of the Article and in the accompanying Supplementary Information file.


One characteristic of epilepsy is the variety of mechanisms leading to the epileptic state, which are still largely unknown. Refractory status epilepticus (RSE) and depolarization block (DB) are other pathological brain activities linked to epilepsy, whose patterns are different and whose mechanisms remain poorly understood. In epileptogenic network modeling, the Epileptor is a generic phenomenological model that has been recently developed to describe the dynamics of seizures. Here, we performed a detailed qualitative analysis of the Epileptor model based on dynamical systems theory and bifurcation analysis, and investigate the dynamic evolution of “normal” activity toward seizures and to the pathological RSE and DB states. The mechanisms of the transition between states are called bifurcations. Our detailed analysis demonstrates that the generic model undergoes different bifurcation types at seizure offset, when varying some selected parameters. We show that the pathological and normal activities can coexist within the same model under some conditions, and demonstrate that there are many pathways leading to and away from these activities. We here archive systematically all behaviors and dynamic regimes of the Epileptor model to serve as a resource in the development of patient-specific brain network models, and more generally in epilepsy research. Epilepsy is characterized by patient-specific electrophysiological discharges. The range of mechanisms and pathways leading to the same type of seizure, however, is large. The Epileptor model has found many applications in epilepsy research and clinical applications, because it allows the classification and dynamic modeling of seizure types independent of the knowledge of its underlying biophysical mechanisms. It is based purely on the dynamic features of the seizure. We provide here a complete functional atlas of all possible behaviors of the Epileptor model, which serves as a useful resource in modeling brain networks in epilepsy. More, we explore the contribution of the Epileptor model to better understand the dynamics of the refractory status epilepticus and depolarization block phenomena, which are linked to epilepsy.Epilepsy is a condition of the nervous system in which neuronal populations manifest as repeated epileptic seizures lasting a few minutes. These seizures are spontaneous and commonly accompanied by fast abnormal discharges , after which the brain activity slowly returns to normal. Epileptic seizures may present in different forms, as well as transitions from and to these pathological states. Some seizures are controlled by medication, particularly antiepileptic drugs (AEDs). However, there are seizures that last >1 h without returning to baseline, and do not respond to AEDs (31-43%), resulting then in so-called refractory status epilepticus (RSE) . PatientOther neuroelectric phenomena are linked to epilepsy, among which spreading depression (SD) is the most prominent. SD is characterized by a slowly propagating depolarization wave [or depolarization block (DB)] in neuronal networks, followed by a shutdown of brain activity . DB is aThe mechanisms underlying the genesis of epilepsy, DB, and RSE are still largely unknown. In this article, we take an integrative approach toward the understanding of these neuroelectric phenomena. Our thinking is inspired by the wish to understand the underlying dynamic underpinnings rather than the biophysiological basis.In this context, a large number of experimental and computational models have been proposed to clarify the basic mechanisms of seizures and DB. Most computational models rely on biophysically realistic parameters, trying to reproduce experimental data . Althougwww.epinov.com; z, evolving on a very slow timescale called the permittivity variable  written in the following form:In what follows, unless otherwise stated, the values of the parameters are as follows:a = 1; b = 3; c = 1; d = 5; m =  0Subsystem 1: γ = 0.01Subsystem 2: z dynamics: r  =  0.00,035; s  =  4; x0 = −1.6.Slow x1 and y1 comprise the subsystem 1 responsible for fast discharges, and x2 and y2 the subsystem 2 involved in spike-wave events . e events . The thiAll stochastic simulations are performed with linear additive Gaussian white noise with a zero mean and a variance of 0.0025 using the Euler–Maruyama method. Deterministic simulations are performed using the Runge-Kutta method with a maximal time step of 0.01.m and x0 parameters. The solutions are the vector equilibrium points. We first determine the equilibrium points of the whole system when the parameter x0 varies, for two values of the parameter m (m  =  0 and m  =  0.5). We determine the stability of the equilibrium points E by evaluating the eigenvalues of the Jacobian matrix J at the equilibrium point E, which is stable if all the real parts of the eigenvalues of J are negative  parameter space for equilibrium points, which consists of various areas, each of which is characterized by coexisting equilibrium points with different stability. Moreover, when the parameter We explore the stability of the coexisting equilibrium points for a range of z, x1) bifurcation diagram of the Epileptor for constant m, here m  =  0.5. The  bifurcation diagram comprises the equilibrium points of the Epileptor, where x1 is the first coordinate of the vector of equilibrium points that we determined with respect to z. All equilibrium point solutions were obtained using the Matlab solve symbolic solver, where only the  curve is divided into different branches, of which each is numerically continued separately representing a different stability type of the equilibrium points solutions. Bifurcations were identified at stability changes for given z-values and are indicated by dots in the corresponding visualizations. As the main parameters m and m and One important step in the analysis of a mathematical model is the geometrical analysis of bifurcations, which here was performed on the deterministic Epileptor model. First, we draw a  phase space below the SLE attractor shown in z equation of the Epileptor model  bifurcation diagram. Thus, we modify the z equation to stabilize the final state. We interpret graphically the divergence of the Epileptor by introducing the averaging method, which is used to locate the periodic orbits in the phase space. The periodic orbits are intersections of the z-nullcline and the z-nullcline is given by x1 for each z constant written in the following form:When using this We also determine the periodic orbits by using the Pontryagin’s averaging technique . To illux0 varies in a  bifurcation diagram of periodic orbits. We determine the bifurcation that results in the appearance or disappearance of the periodic orbits. Second, we explore two parameter spaces of periodic orbits for where periodic orbits are the zeros of the x0 can change the stability of equilibrium points and then can control the Epileptor behavior. Using the fact that a trajectory moves around the equilibrium point when it is stable, we use the bifurcation analysis to find the values of x0 for which an equilibrium point is stable. There is a qualitative distinction between the stable equilibrium points, which we discuss in more detail later.The parameter m and x0.The Epileptor behavior depends on the stability of its equilibrium points. More, there is a pre-existing limit cycle (LC) attractor, which is a stable periodic orbit. In this case, the trajectory can have a behavior that is controlled by the stability of the equilibrium points or can jump to the limit cycle attractor. Thus, there is a coexistence of two attractors. We identify all coexisting attractors according to the values of the parameters We provide a detailed analysis of the subsystem 1 dynamics without coupling .z is constant  by solving the following equations:We analytically find the equilibrium points  between them. When z further increases, the stable limit cycle disappears through a homoclinic bifurcation, HB  bifurcation diagram and the z-nullcline, which is related to the parameter x0. We show how the z-nullcline moves in the bifurcation diagram when varying x0, and use the bifurcation diagram to find the equilibrium points of the fast-slow subsystem for each value of x0.We analytically find the equilibrium points by solving  of z˙=0 . We detenegative . We grapAs follows from the Epileptor analysis, we determine the periodic orbits of the fast-slow subsystem as well as their stability by using the averaging method and the Pontryagin’s averaging technique. The slow averaged nullcline is given by the following:z constant.x0 varies and determine the bifurcation leading to their appearance or disappearance.Here, we plot the graph of x0 for which an equilibrium point is stable. We show how stable equilibrium points can correspond to different states.Our approach to stabilizing the equilibrium points of the fast-slow subsystem is the same as that of the Epileptor model: analyze bifurcation diagrams of the fast-slow subsystem and then find the values of m and x0, which control the stability of the equilibrium point. We discuss the coexisting attractors when m and x0 have different values.The fast-slow subsystem exhibits a bistability of two attractors as the Epileptor model. The first attractor is the fast-slow limit cycle. The behavior of the second attractor depends on the parameters In this section, we analyze the dynamics of subsystem 2 (without coupling), which generates SWEs. We explore the equilibrium points, evaluating their stability and determining the bifurcation types.The subsystem 2 equations are given by the following:x2, y2) by solving the following equations:We analytically find the equilibrium points . State variables x2 and y2 are responsible for generating the SWEs with an intermediate timescale (ψ2). We plot the Epileptor trajectory in a  phase space  diagram for m  =  0.5  parameter space of equilibrium points in The stability of equilibrium points changes when we vary m and x0 A,B. TrajThe parameter space comprises nine areas. The equilibrium point is an unstable focus in area 1, a stable node in area 5, and a saddle in area 4. The Epileptor model has three equilibrium points in areas 2, 3, 6, 7, 8, and 9. In area 2, a stable node, a saddle and an unstable focus coexist. In area 3, a stable focus, a saddle, and an unstable focus coexist. In area 6, three saddles coexist. In area 7, two saddles and one unstable focus coexist. In area 9, two unstable foci and one saddle coexist. In area 8, an unstable focus, an unstable node, and a saddle coexist.m, x0) parameter space of equilibrium points for We plot a  bifurcation diagram of the Epileptor model in m  =  0.5. x1 is the first component of the equilibrium points E. z acts as a control parameter (r  =  0). Z-shaped curve (Z-curve). Z-lower and Z-upper branches consist of stable nodes and unstable foci, respectively. The Z-middle branch separates Z-lower and Z-upper branches and consists of saddles. The Z-middle and Z-upper branches collide as z increases in a saddle-node bifurcation SN3  consists of unstable foci and another (dash-dotted) consists of saddles. The two sub-branches collide as z increases in a saddle-node bifurcation SN2. The equilibrium points E of the Epileptor model lie at the intersection of the z-nullcline and the curve of equilibrium points (z-nullcline (x0. The z-nullcline moves downward (x0 decreases) or upward (x0 increases) in the bifurcation diagram; hence, it intersects the curve of equilibrium points at different sites. When x0 = −1.6, then the z-nullcline intersects the Z-middle branch, which consists of saddles  occurs through a saddle-node bifurcation, SN1. The transition from ictal to normal states (second bifurcation) occurs though a homoclinic bifurcation, HB. The homoclinic bifurcation z(HB) corresponds to an offset threshold and the saddle-node bifurcation 1, HB].Moreover, there are two branches above the ed curve , inset. m points . The z-n saddles . Consistm in the dynamics of the Epileptor, we plot a  bifurcation diagram, when m decreases. Let m  =  0  bifurcation with Z-curve, the lower branch consists of saddles and the upper branch comprises two sub-branches (z decreases in a SNIC bifurcation. The second sub-branch (dash-dotted) consists of saddles. The two sub-branches collide as z increases in a saddle-node bifurcation SN2. We plot a trajectory in Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1 and the trajectory switches to the Z-upper branch exhibiting an oscillatory solution, which terminates as z increases in a SNIC bifurcation (see inset). Then the trajectory is at the upper branch above the Z-curve, which consists of stable foci. The trajectory exhibits a nonoscillatory solution, which terminates as z increases in a saddle-node bifurcation SN2 and the trajectory switches to the Z-lower branch. The Z-lower branch corresponds to the normal state. The Z-upper branch and the upper branch above the Z-curve correspond to the ictal state (see inset). The transition from normal to ictal states (first bifurcation) occurs through a saddle-node bifurcation, SN1. The transition from ictal to normal states (second bifurcation) occurs through a saddle-node bifurcation, SN2. Fast discharges characterizing the ictal state correspond to the oscillatory solution, which disappears through a saddle-node on invariant circle bifurcation SNIC. Therefore, the transitions between ictal and normal states are said to be of a fold/circle bifurcation. The system is bistable on .The branches , inset. Z-shaped curve. Above, there are the following three branches: two branches consisting of saddles, and one branch consisting of stable foci for m ≤ 0 and unstable foci for m > 0. The Epileptor model undergoes a SNIC bifurcation for m ≤ 0 and an HB bifurcation for m > 0. The transitions between ictal and normal states occur through a fold/circle bifurcation for m ≤ 0 and through a fold/homoclinic bifurcation for m  >  0. The system is bistable on  for m  ≤  0 and on  for m > 0.We conclude that the bifurcation diagrams of the Epileptor model is a z, x1) bifurcation diagram for m  =  0 in Z-shaped curve. Z-lower, Z-middle, and Z-upper branches consist of stable nodes, saddles, and unstable foci, respectively. Decreasing z, Z-lower and Z-middle branches collide in a saddle-node bifurcation, SN1. Increasing z, Z-upper and Z-middle branches collide in another saddle-node bifurcation, SN2. The Z-upper branch terminates as z decreases in a saddle-node bifurcation SN4. Moreover, two branches appear as z decreases. The lower branch (dashed) consists of unstable foci, and the upper branch (dash-dotted) consists of saddles. Increasing z, lower and upper branches collide in a saddle-node bifurcation SN3. Decreasing z, the upper branch terminates in a saddle-node bifurcation SN4.We now analyze the bifurcation diagram when x0 = −1.6, the z-nullcline is at the Z-middle branch, then the equilibrium point is a saddle. We plot a trajectory in Z-lower and Z-upper branches. When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation, SN1, and the trajectory switches to the Z-upper branch, which consists of unstable foci. An unstable focus is surrounded by a stable periodic orbit  and Z-upper (dashed) branches correspond to normal and ictal states, respectively. The transition from normal to ictal states occurs through a saddle-node bifurcation SN1. The transition from ictal to normal states occurs through an HB. Thus, the transitions between ictal and normal states occur through a fold/homoclinic bifurcation. The system is bistable on .Let m = −0.5, we plot a  bifurcation diagram in Z-upper branch is divided into two sub-branches separated by a Hopf (H) bifurcation (z increases in a saddle-node bifurcation SN2. The second sub-branch (dashed) consists of unstable foci and terminates as z decreases in a saddle-node bifurcation SN4. An unstable focus is surrounded by a stable periodic orbit  exhibiting an oscillatory solution, which terminates as z increases in an H bifurcation. Then, the trajectory is at the Z-upper (solid) sub-branch exhibiting a nonoscillatory solution, which terminates as z increases in a saddle-node bifurcation SN2. Z-lower (solid) and Z-upper branches correspond to normal and ictal states, respectively. The oscillatory solution corresponds to fast discharges. The transition from normal to ictal states occurs through a saddle-node bifurcation, SN1. The transition from ictal to normal states occurs through a saddle-node bifurcation SN2. The oscillatory solution (fast discharges) terminates in an H bifurcation. The transitions between ictal and normal states are said to be of a fold/Hopf bifurcation. The system is bistable on .Let urcation . The firm = −1, we plot a  bifurcation diagram in Z-shaped curve. The Hopf bifurcation point H disappears, then the Z-upper branch consists of stable foci only, which is the equilibrium point of nonoscillatory state. Increasing z, the Z-upper (solid) branch terminates in a SNIC bifurcation  branch consists of saddles, and the lower (dashed) one consists of unstable foci. An unstable focus is surrounded by a stable periodic orbit  branch, which consists of stable foci, thereby exhibiting a nonoscillatory solution. Then, the stable focus disappears as z increases through a saddle-node bifurcation, SN2, and the trajectory switches to the Z-lower branch. The transition from Z-lower to Z-upper branches occurs through a saddle-node bifurcation, SN1, and the transition from Z-upper to Z-lower branches occurs through a saddle-node bifurcation, SN2. The transitions between Z-lower and Z-upper branches occur through a fold/fold bifurcation. The system is bistable on . Here, the SLE attractor reduces to a periodic switch between a nonoscillatory state and a NS.Let urcation . Below tZ-lower and Z-upper branches occur through a fold/homoclinic bifurcation, m = −0.5; and a fold/fold bifurcation, 1, SN2] for m <  0 and on  for m  ≥  0. Moreover, the transitions between ictal and normal states occur through a fold/circle bifurcation when We conclude that when y1, ψ, z) phase space below the SLE attractor shown in z equation (i2 are on the left of the separatrix (S), which means that i2 are below the separatrix (S) and the SLE attractor in the phase space. To explain this divergence, we plot the z-nullcline and the z, x1) bifurcation diagram for m  =  0.5  branch. Then three equilibrium points coexist: two unstable foci and one saddle. The z-nullcline intersects the We consider  diagram . The z-nz, trajectories converge to LC and the amplitude of periodic orbits increases, leading to LC with a large amplitude. The intersection of the z-nullcline and the We plot the olutions –8. Decrex0 = −1.6, two periodic orbits exist for m  =  0.5 (m  =  0 (Let   =  0.5  and m  =(m  =  0 . We showz-nullcline moves downward (x0 decreases) or upward (x0 increases) in the bifurcation diagram, hence the existence of periodic orbits depends on x0. We find S and LC in a  bifurcation diagram of periodic orbits  markers and black dots correspond to LC and S, respectively. Only LC exists for large x0. Decreasing x0, LC and S coexist, and collide as x0 is further decreased in a saddle-node of periodic orbits bifurcation (SNPO), then fades.The c orbits A. x0 actm = −16) shows that for some x0, a third periodic orbit appears, which is labeled as blue top (+) markers. This periodic orbit is stable, and is localized above LC and S . SLC and LC are two stable limit cycles that coexist for some x0, and are separated by S. SLC exists after a SNPO bifurcation occurs when m = −16 and disappears when x0 is further decreased.LC and S B. Since m, x0) parameter spaces of periodic orbits: for We explore two  bifurcation diagram for m = 1 in z-nullcline is at the middle branch and the equilibrium point is a saddle. The z-nullcline intersects the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1 and the trajectory switches to the Z-upper branch, which consists of unstable foci. Then the trajectory exhibits an oscillatory solution on the Z-upper branch and continues to SLC at which z stabilizes  bifurcation diagram for m = −2 in x0 = 0, the equilibrium point is an unstable focus. The z-nullcline intersects the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1, and the trajectory switches to the Z-upper (solid) branch, which consists of stable foci. Increasing z, the trajectory exhibits a nonoscillatory solution on the Z-upper branch (solid), which terminates in a SNIC bifurcation  curve. Let Z-upper branch consists of unstable foci when m  =  0  bifurcation diagram for m = −8 (Z-shaped curve. Z-lower (solid), Z-middle (dash-dotted), and Z-upper (solid) branches consist of stable nodes, saddles, and stable nodes, respectively. The Z-upper branch terminates as z decreases in a SNIC bifurcation (Z-upper branch (m decreases to the Z-upper (solid) branch. The lower branch consists of unstable foci, and the upper branch consists of saddles. Increasing z, the lower and upper branches collide in a saddle-node bifurcation, SN3. Note that x0 = −1.6, the equilibrium point is a saddle. When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation, SN1, and the trajectory switches to the Z-upper branch, which consists of stable nodes. The trajectory exhibits a silent activity on the Z-upper branch, which terminates as z increases in a saddle-node bifurcation SN2. Then, the trajectory switches to the Z-lower branch. The transition between Z-lower and Z-upper branches occur through a fold/fold bifurcation. The system is bistable on . Time series of the Epileptor model ψ, subsystem 1 ψ1, and subsystem 2 ψ2 are plotted in m underlying a slow wave. This scenario corresponds to a depolarization block  bifurcation diagram for m = −8 (Z-shaped curve. Z-lower (solid), Z-middle (dash-dotted), and Z-upper (dashed) branches consist of stable nodes, saddles, and unstable foci, respectively. Above the Z-curve, lower branch consists of saddles and upper branch consists of stable nodes , and hence does not enter into a DB.The DB occurs when ion, SN2 A, and th1, SN2] interval, on which the Epileptor model is bistable. We determine the branches of the  curve existing on this interval . When x0 increases, the Epileptor stabilizes in the ictal (nonoscillatory) state, and when x0 decreases, the Epileptor model stabilizes in the normal state.The transitions between ictal and normal states of the SLE attractor occur when the equilibrium point is a saddle (m and x0 = −1.6), the Epileptor switches differently to the normal state according to the bifurcation type: homoclinic, SNIC, or Hopf. The Epileptor switches to the normal state just after a homoclinic bifurcation, and remains in the nonoscillatory state after SNIC and Hopf bifurcations, thereby exhibiting a nonoscillatory solution. The latter disappears as z increases through a saddle-node bifurcation, SN2, and then the Epileptor switches to the normal state. When increasing x0, the stability of the equilibrium point changes, and the Epileptor remains in the nonoscillatory state. We illustrate this scenario in During the ictal state, epileptiform fast discharges (oscillatory state) disappear through three bifurcation types depending on urcation , 6, a SNurcation , or a Hourcation . When thm  =  0 and x0 = −0.9  curve: lower and upper (solid) branches above the Z-curve, and the Z-upper (dashed) branch. The z-nullcline intersects the upper branch above Z at C. Then three equilibrium points coexist: an unstable focus, a saddle, and a stable focus (C). When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation, SN1, and the trajectory switches to the Z-upper branch exhibiting an oscillatory solution (fast discharges), which terminates as z increases in a SNIC bifurcation. Then, the trajectory exhibits a nonoscillatory solution at the upper (solid) branch above the Z-curve, and continues to C at which z stabilizes. Therefore, the transition to the Z-lower branch does not occur and the Epileptor remains in the nonoscillatory state. We plot this scenario in a  phase space  sub-branch at C which is a stable focus. When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1 and the trajectory switches to the Z-upper (dashed) sub-branch exhibiting an oscillatory solution (fast discharges), which terminates as z increases in a Hopf bifurcation, H. Then, the trajectory exhibits a nonoscillatory solution at the Z-upper (solid) sub-branch, which consists of stable foci, and continues to C at which z stabilizes. The transition to the Z-lower branch does not occur, and the Epileptor remains in the nonoscillatory state. We plot this scenario in a  phase space  phase space (see C is responsible for the oscillations (2) around the stable focus. After a transient normal state, the Epileptor spirals into the equilibrium point of the nonoscillatory state exhibiting damped oscillations, and does not re-enter the normal state.Let urcation . When inm (x0 = −1.6). When increasing x0, the Epileptor model stabilizes its stable node, which corresponds to the equilibrium point of DB. To see this, we plot a  bifurcation diagram when x0 = −0.8 in z-nullcline intersects the Z-upper (solid) branch at C, which is a stable node. When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1 and the trajectory switches to the Z-upper branch and enters into a DB. The trajectory continues as z increases to C at which z stabilizes. Thus, the transition to the Z-lower branch does not occur and the Epileptor remains in DB. We plot this scenario in a  phase space (see x0 = −1.6), the normal state of the SLE attractor disappears through a saddle-node bifurcation SN1. When decreasing x0, the Epileptor stabilizes its stable node, which is the equilibrium point of the normal state. To see this, we plot a  bifurcation diagram when x0 = −2.5 in z-nullcline intersects the Z-lower branch at C, which corresponds to the equilibrium point of the normal state. The ictal state (Z-upper branch) exists but not the equilibrium point, hence the Epileptor remains in the normal state, at C which is a stable node.When the equilibrium point is a saddle (m.m = 0.5 (Z-upper branch (dashed), the oscillatory solution terminates as z increases in an HB bifurcation and the trajectory switches to the Z-lower branch.m = 0.5 A: When am = 0 (Z-upper branch (dashed), the oscillatory solution terminates as z increases in a SNIC bifurcation, then the trajectory exhibits a nonoscillatory solution on the upper branch above the Z-curve, which terminates as z increases in a saddle-node bifurcation SN2. Then the trajectory switches to the Z-lower branch.m = 0 B: When aWhen m.m ≥ 0 (Z-upper (dashed) branch, which terminates as z increases in a homoclinic bifurcation HB, and then it switches to the Z-lower branch.For m ≥ 0 C: The trm = −0.5 (Z-upper sub-(dashed) branch, which terminates as z increases in a Hopf bifurcation H. Then the trajectory exhibits a nonoscillatory solution on the Z-upper (solid) sub-branch, which terminates as z increases in a saddle-node bifurcation SN2, and the trajectory switches to the Z-lower branch.For m = −0.5 D: The trm ≤ −1, when a trajectory starts at i with Z-upper branch, which terminates as z increases in a saddle-node bifurcation, SN2. Then the trajectory switches to the Z-lower branch  bifurcation diagram for m  =  0.5 (x0 = −1.6). The behavior of the first trajectory (right) corresponds to SLEs, which occur through a fold/homoclinic bifurcation. The Z-middle branch acts as a separatrix between ictal and normal states. The second trajectory (left) converges to LC. The saddle periodic orbit (S) separates LC and the SLE attractor. We plot LC, SLEs, and S in a  phase space  phase space  bifurcation diagram (x0 = −1.6). The transitions between ictal and normal states occur through a fold/circle bifurcation. Then SLEs and LC coexist and are separated by a saddle periodic orbit (S). We plot LC, SLEs, and S in a  phase space ( diagram . The firse space A. SLEs aA fold/Hopf bifurcation:x0 = −1.6). Parameter settings in Y, X, z) phase space (se space A. SLEs am = −1 (x0 = −1.6). Parameter settings in Y, X, z) phase space  phase space  bifurcation diagram is plotted in m = −8  phase space . Let m  =  0, the equilibrium point is a stable node when x0 is below −2.1  phase space . Parameter settings in Y, X, z) phase space  is the equilibrium point of the nonoscillatory state. Y, X, z) phase space . LC (below) and the nonoscillatory state (above) coexist. The nonoscillatory state shown in A, A, and A is the same as that shown in space A, A, 30A.m, x0 and We identified seven types of coexisting attractors: a coexistence of SLE and LC –21, a courcation , 19, a furcation , or a fourcation .m, x0) parameter space of periodic orbits and equilibrium points in To characterize the different types of the coexisting attractors, we explore a  by using 32, which shows 18 (1–18) areas:For each area, the Epileptor model has different equilibrium points, depending on (I) LC: Area I in z-nullcline intersects the z-nullcline intersects the  curve at different equilibrium points. The equilibrium point is an unstable focus in area 1 and a saddle in area 14. The equilibrium points are one saddle and two unstable foci in area 7. The equilibrium points are one unstable focus and two saddles in area 9. The equilibrium points are three saddles in area 10. The equilibrium points are a saddle, an unstable focus, and an unstable node in area 8. Trajectories exhibit only a fast-slow cyclic behavior (LC), which is plotted in (II) LC and SLC: Area II in z-nullcline intersects the z-nullcline intersects the  curve at one equilibrium point, which is an unstable focus. LC and SLC coexist. The stable manifold of the saddle periodic orbit (S) separates the basin of attraction of LC and the basin of attraction of SLC. Trajectories exhibit two periodic solutions depending on initial conditions. The first solution corresponds to a fast-slow cyclic behavior (LC). The second solution corresponds to a periodic solution with a small amplitude (SLC).(III) SLC: Area III in z-nullcline intersects the z-nullcline intersects the  curve at one equilibrium point, which is an unstable focus. The trajectories exhibit only a periodic solution with a small amplitude (SLC).(IV) Nonoscillatory state: Area IV in z-nullcline and the z-nullcline intersects the  curve at different equilibrium points. The equilibrium points are a stable node, an unstable focus, and a saddle in area 4. Trajectories remain in the nonoscillatory state.(V) LC and a nonoscillatory state: Area V in z-nullcline intersects the z-nullcline intersects the  curve at different equilibrium points. The equilibrium points are a stable node, an unstable focus, and a saddle in area 5. The equilibrium points are a stable focus, an unstable focus, and a saddle in area 6. Trajectories either exhibit a fast-slow cyclic behavior (LC) or remain in the nonoscillatory state. The coexistence of LC and the nonoscillatory state is plotted in (VI) SLE with a fold/circle bifurcation: Area VI in z-nullcline and the z-nullcline intersects the Z-middle branch, which consists of saddles. Thus, a unique saddle equilibrium point exists. Since m ≤  0, an SLE occurs through a fold/circle bifurcation.(VII) LC and SLE with a fold/circle bifurcation: Area VII in z-nullcline intersects the (VIII) LC and SLE with a fold/homoclinic bifurcation: Area VIII in z-nullcline intersects the (IX) NS: Area IX in z-nullcline and the z-nullcline intersects the  curve at the Z-lower branch, and then the equilibrium point is a stable node in area 17. The Epileptor remains in NS after a transient period, which is shown in (X) LC and NS: Area X in z-nullcline intersects the z-nullcline intersects the  curve at the Z-lower branch, which is a stable node in area 18. Then LC and NS coexist and are separated by a saddle periodic orbit (S). This coexistence is plotted in  LC and a chaotic state: Areas XI and XII in z-nullcline intersects the z-nullcline intersects the z-nullcline intersects the Z-middle branch which consists of saddles. An SLE occurs through a fold/homoclinic bifurcation in area 16. Then LC and SLE coexist in area 16. S acts as a separatrix between the basin of attraction of LC and the basin of attraction of SLE in area 16.x0 in m  =  1 and in m  =  1.5. The system is chaotic when the ISI is irregular. Therefore, LC and the chaotic state coexist in areas 15 and 16 and are separated by a saddle periodic orbit (S).We calculate the times between two successive spikes, which are referred to as the interspike intervals (ISIs) with respect to m, x0) parameter space of periodic orbits and equilibrium points in The parameter space of the Epileptor model includes attractors as LC, SLE, and normal state, but neither DB nor SLE with a fold/Hopf bifurcation. In fact, using bifurcation analysis, we demonstrated that DB and Hopf bifurcation appeared when decreasing The parameter space is divided into two parts, which are separated by a bold line (SNPO bifurcation) . Above tm and x0. To analyze this further, we determine the equilibrium points and the periodic orbits of each area (I-XIII) by using 35, which shows 20 (1–20) areas:For each area, the Epileptor model has different equilibrium points depending on (I) LC: Area (I) in z-nullcline intersects the z-nullcline intersects the  curve at different equilibrium points. The equilibrium point is an unstable focus in area 1 and a saddle in area 17. The equilibrium points are one saddle and two unstable foci in area 7. Trajectories exhibit only a fast-slow cyclic behavior (LC).(II) LC and SLC: Area (II) in z-nullcline intersects the z-nullcline intersects the  curve at one equilibrium point, which is an unstable focus in area 2 and a saddle in area 18. The trajectories exhibit two periodic solutions depending on the initial conditions. The first solution corresponds to a fast-slow cyclic behavior (LC). The second solution corresponds to a periodic solution with a small amplitude (SLC). Moreover, the SLC behavior depends on the stability of the equilibrium point. The SLC behavior is plotted in (III) SLC: Area III in z-nullcline intersects the z-nullcline intersects the  curve at one equilibrium point, which is an unstable focus. Trajectories exhibit only a periodic solution with a small amplitude (SLC). The SLC behavior is plotted in (IV) DB: Area IV in z-nullcline and the z-nullcline intersects the  curve at different equilibrium points. The equilibrium point is a stable node in area 8. The equilibrium points are a stable node, an unstable focus, and a saddle in area 4. The Epileptor remains in DB after a transient period, which is plotted in (V) LC and a DB: Area V in z-nullcline intersects the z-nullcline intersects the  curve at different equilibrium points. The equilibrium points are a stable node, an unstable focus, and a saddle in area 5. The equilibrium point is a stable node in area 9. Trajectories either exhibit a fast-slow cyclic behavior (LC) or enter into a DB. The coexistence of LC and DB is plotted in (VI) LC and a nonoscillatory state: Area VI in z-nullcline intersects the z-nullcline intersects the  curve at different equilibrium points. The equilibrium points are a stable focus, an unstable focus, and a saddle in area 6. The equilibrium point is a stable focus in area 10 and a stable node in area 11. Trajectories either exhibit a fast-slow cyclic behavior (LC) or remain in a nonoscillatory state. The coexistence of LC and the nonoscillatory state is plotted in (VII) Periodic switch between DB and NS: Area VII in z-nullcline and the z-nullcline intersects the Z-middle branch which consists of saddles. Then only a saddle equilibrium point exists in area 12. The ictal state of SLE reduces to DB with decreasing m. Periodic switch between DB and NS occurs through a fold/fold bifurcation, and is plotted in (VIII) LC and a periodic switch between DB and NS: Area VIII in z-nullcline intersects the m. Periodic switch between DB and NS occurs through a fold/fold bifurcation, and coexists with LC. Both are separated by a saddle periodic orbit (S). The coexistence of LC and periodic switch between DB and NS is plotted in (IX) LC and a periodic switch between nonoscillatory state and NS: Area IX in z-nullcline intersects the (X) LC and SLE with a fold/Hopf bifurcation: Area X in z-nullcline intersects the urcation . Then LC(XI) LC and SLE with a fold/homoclinic bifurcation: Area XI in z-nullcline intersects the  area 16 . Then LC(XII) A NS: Area XII in z-nullcline and the z-nullcline intersects the  curve at the Z-lower branch, and then the equilibrium point is a stable node in area 19. The Epileptor remains in NS after a transient period, which is shown in (XIII) LC and a NS: Area XIII in z-nullcline intersects the z-nullcline intersects the  curve at the Z-lower branch, and hence the equilibrium point is a stable node in area 20. Then LC and NS coexist and are separated by a saddle periodic orbit (S). This coexistence is plotted in m and x0 vary  by solving the following equations:x1 is a solution of:We analytically find the equilibrium points  exists if:X is a solution of We find J:To determine the equilibrium points stability, we analyze the eigenvalues of the following Jacobian matrix J is defined at the equilibrium point . To find the eigenvalues of J, we solve the characteristic equation:Tr(J) and Det(J) are the trace and the determinant of the matrix J, respectively.J are given by:The trace and the determinant of J (Tr(J) and Det(J) are negative and positive.An equilibrium point is stable if all the real parts of the eigenvalues of J are negative . We can J . We deteTr(J) are as follows:The roots of Det(J) are as follows:X is a solution of The roots of x1 (x1 ≥ 0) exists (see We conclude that an equilibrium point ists see  if:(49)m.We find the stability of equilibrium points x1- and y1-nullclines. The x1-nullcline . Trajectories i1 and i2 converge to one equilibrium point, which is a stable node. Trajectories i3 and i4 converge to one equilibrium point, which is a stable focus when m  =  0 (m  =  0). By increasing (m  =  1 and m  =  1.5), the stable focus loses its stability, and the equilibrium point is an unstable focus. Trajectories i3 and i4 converge to a stable limit cycle, which surrounds the unstable focus. The radius of the limit cycle increases as m increases (m  =  1 and m  =  1.5).When −Iext1=0 A. We ploz  =  3.1, three equilibrium points coexist. The first one is a stable node and the second one is a stable or an unstable focus, depending on m. The third equilibrium point is a saddle. The stable manifold of the saddle corresponds to a separatrix between the first and second equilibrium point. The subsystem 1 undergoes a supercritical Andronov–Hopf bifurcation, H, as m increases at m(H)  =  0.514  for large x0. Decreasing x0, S (black dotted squares) appears. Stable (LC) and saddle (S) periodic orbits approach each other as x0 decreases, collide in a saddle-node periodic orbit bifurcation, and then fades. LC and S disappear through a SNPO bifurcation  markers] appear. Hence, LC and SLC1 coexist and are separated by S. Decreasing x0, SLC1 disappears. Further decreasing x0, LC and S disappear through an SNPO bifurcation  bifurcation diagram the equilibrium points, labeled as dots  bifurcation diagram when m = −8 in Z-lower branch consists of stable nodes, which are the equilibrium points of the NS. The Z-upper branch consists of stable nodes (z-nullcline is at the Z-middle branch which consists of saddles (x0 = −1.6). When a trajectory is at the Z-lower branch, the stable node disappears as z decreases through a saddle-node bifurcation SN1 and then the trajectory enters into a depolarization block, which terminates as z increases in a saddle-node bifurcation, SN2. Thus, the transitions between the Z-lower and Z-upper branches are reduced to a periodic switch between a NS and a DB. Time series are plotted in When Z-lower and Z-upper branches occur when the equilibrium point is a saddle (x0 = −1.6). When x0 is increased, the fast-slow subsystem stabilizes the equilibrium point of the Z-upper branch, and when x0 is decreased, the fast-slow subsystem stabilizes the equilibrium point of the Z-lower branch .The transitions between x0 = −0.5. We plot a  bifurcation diagram when m  =  0 in z-nullcline is at the Z-upper branch which consists of a stable focus. Decreasing z, the stable node disappears through a saddle-node bifurcation SN1 and the fast-slow subsystem switches to the Z-upper branch (dashed) exhibiting an oscillatory solution, which terminates in a Hopf bifurcation, H. The fast-slow subsystem exhibits a nonoscillatory solution at the Z-upper branch (solid) and continues to C, at which z stabilizes. The imaginary part of the eigenvalues of C is responsible for the oscillations around the stable focus, which is the equilibrium point of the nonoscillatory state. Time series are plotted in Let x0 = −0.5. We plot a  bifurcation diagram when m = −8 in z-nullcline is at the Z-upper branch which consists of stable nodes. Decreasing z, the stable node disappears through a saddle-node bifurcation, SN1, then the fast-slow subsystem switches to the Z-upper branch (solid) exhibiting DB and continues to C, at which z stabilizes. The imaginary part of the eigenvalues of C goes to zero, and therefore the oscillations around C disappear. Here, the stable node (C) is the equilibrium point of DB. Time series are plotted in Let x0 = −2.4. We plot a  bifurcation diagram when m  =  0 in z-nullcline is at the Z-lower branch which consists of stable nodes. When a trajectory is at the Z-upper branch (dashed), it exhibits an oscillatory solution, which terminates as z increases in a Hopf bifurcation, H, and then a nonoscillatory solution on the Z-upper branch (solid), which terminates as z increases in a saddle-node bifurcation SN2. The fast-slow subsystem continues to C, at which z stabilizes. The stable node (C) is the equilibrium point of the NS. Time series are plotted in Let Case 1: Only LC exists. m  =  2 and x0 = −1.6, which is LC  acts as a separatrix between LC and SLE  parameter space of the fast-slow subsystem in m and x0 (area I) only LC exists. The adjacent area II shows bistability of LC and a stable focus. Both attractors are separated by a saddle periodic orbit. In the middle, LC coexists with the SLE attractor separated by a saddle periodic orbit. SLE occurs via a saddle-node/saddle-node bifurcation (area VI) and a saddle-node/homoclinic bifurcation (area VII). In area VIII, the homoclinic bifurcation HB is not completed and gives rise to another (coexisting) stable periodic orbit with a small amplitude, SLC1. In area V, DB coexists with LC. In area IX, normal brain activity and LC coexist. In area X, only normal activity exists.To characterize the coexisting attractors, we explore a ((I) LC: The z-nullcline intersects the z, x1) curve at different equilibrium points. The equilibrium point can be a stable focus, an unstable focus, or a saddle. All trajectories converge to LC. Time series are plotted in (II) Coexistence of LC and a stable focus: The z-nullcline intersects the  curve at one equilibrium point, which is a stable focus, and intersects the Z-lower branch, the stable node disappears through a saddle-node bifurcation SN1 and the trajectory switches to the Z-upper branch, which comprises a Hopf bifurcation point, H. If z decreases in a Hopf bifurcation, H. Then, the trajectory continues to the stable focus at which z stabilizes. If z decreases to the stable focus at which z stabilizes. The stable focus is the equilibrium point of the nonoscillatory state. The system is bistable on .If a trajectory is in the basin of attraction of the stable focus, then it exhibits a nonoscillatory solution following two scenarios. When the trajectory is at the m decreases to a stable node . Here, the stable node is the equilibrium point of the DB, and then DB is present in area II. LC and DB coexist as m decreases.Using analytic techniques, we observe that the stable focus reduces as (V) Coexistence of LC and a periodic switch between DB and NS: The z-nullcline intersects the  curve at one equilibrium point, which is a saddle, and intersects the H, exists for m ≤ 1  Coexistence of LC and SLE with a fold/fold bifurcation: Using numerical techniques, there is a saddle-node bifurcation at both offset and onset seizures. We can deduce that SLE occurs with a fold/fold bifurcation in area VI. The z-nullcline intersects the Z-upper branch consists of a stable focus, which is the equilibrium point of the nonoscillatory state. The SLE attractor reduces then to a periodic switch between nonoscillatory state and NS, which occurs through a fold/fold bifurcation, coexisting with LC. Time series are plotted in m increases, m increases through a fold/Hopf bifurcation in area VI, coexisting with LC. Time series are plotted in F1for LC.Using analytic techniques , we can (VII) Coexistence of LC and SLE with a fold/homoclinic bifurcation: The z-nullcline intersects the  curve at one equilibrium point, which is a saddle, and intersects the m > 1, then the transitions between Z-upper and Z-lower branches occur through a fold/homoclinic bifurcation. For m  =  1, a Hopf bifurcation point, H, exists such that Z-upper branch, then it exhibits an oscillatory solution, which touches the Z-middle branch before the Hopf bifurcation, H. Then the transitions between Z-upper and Z-lower branches occur through a fold/homoclinic bifurcation for m  =  1. LC and a fold/homoclinic bifurcation coexist and are separated by S. Time series are plotted in )>z(SN1) . When a 1:(VIII) Coexistence of LC and SLC The z-nullcline intersects the  curve at one equilibrium point, which is a saddle, and intersects the 1, and S. Then LC and SLC1 coexist and are separated by S. If a trajectory is in the basin of attraction of SLC1, then it exhibits a periodic solution with a small amplitude. When the trajectory is at the Z-lower branch, the stable node disappears through a saddle-node bifurcation, SN1, and the trajectory switches to the Z-upper branch. Then, the trajectory exhibits, as z increases, an oscillatory solution, which stabilizes at SLC1. Time series are plotted in 1 and in (IX) Coexistence of LC and a NS: The z-nullcline intersects the  curve at one equilibrium point which is a stable node, and intersects the m. We discuss these scenarios below in area X.In the following areas, LC and S disappear through a SNPO bifurcation.(III) DB: The z-nullcline intersects the  curve at one equilibrium point, which is a stable node <z(SN1) . When a (IV) Periodic switch between DB and NS: This area exhibits a periodic switch between DB and NS, like area V. In contrast, LC and S disappear through a SNPO bifurcation.Therefore, the fast-slow subsystem only switches between DB and NS.(X) NS: The z-nullcline intersects the  curve at one equilibrium point, which is a stable node. Time series of this solution are plotted in m.m < 1 (first scenario), a Hopf bifurcation point, H, exists. If a trajectory is at the Z-upper sub-branch, which consists of unstable foci, it exhibits an oscillatory solution that terminates as z increases in a Hopf bifurcation, H. Then, the trajectory exhibits a nonoscillatory solution on the Z-upper sub-branch, which consists of stable foci. The nonoscillatory solution terminates as z decreases in a saddle-node bifurcation SN2 and the trajectory switches to the Z-lower branch.For m = 1 (second scenario), a Hopf bifurcation point H exists. If a trajectory is at the Z-upper sub-branch, which consists of unstable foci, it exhibits an oscillatory solution, which touches the Z-middle branch before the Hopf bifurcation, H, as z increases. Then the trajectory switches to the Z-lower branch through an HB.For m > 1 (third scenario), a Hopf bifurcation point, H, does not exist. If a trajectory is at the Z-upper branch, which consists of unstable foci, then it exhibits an oscillatory solution, which terminates as z increases in a homoclinic bifurcation, and the trajectory switches to the Z-lower branch.For Z-lower branch consists of stable nodes, which are the equilibrium points of the normal state. At the Z-lower branch, the trajectory continues to a stable node, at which z stabilizes. Therefore, the fast-slow subsystem only remains in the normal state. Time series are plotted in The m increases.Note that the SNPO curve (bold line) decreases as m and x0. When LC coexists with an attractor and both are separated by S, the trajectories converge to LC or the attractor, depending on initial conditions. When S does not exist  bifurcation diagram for x0 = −0.1  by solving the following equations:We analytically determine the equilibrium points  by using the x2- and y2-nullclines, where the x2-nullcline corresponds to a cubic curve given by:y2-nullcline corresponds to two straight lines given by:We graphically determine the equilibrium points  bifurcation diagram of the subsystem 2 in x2) curve is an S-shaped curve, which comprises four branches. S-lower and S-upper branches consist of stable nodes and unstable foci, respectively. The S-middle branch consists of saddles and acts as a separatrix between the S-lower and S-upper branches. The S-upper branch is divided into two sub-branches: a stable limit cycle surrounds one sub-branch (red) but does not surround the other sub-branch (green). We call the red sub-branch a limit cycle branch, which is limited by S-middle and S-upper branches collide in a saddle-node bifurcation, SN. Increasing further S-middle and S-lower branches collide in a SNIC bifurcation  and z change the input z in z increases, then z decreases, then ψ2; g(x1) and z.We now link the dynamics of subsystem 2 to its behavior  bursters as proposed by m, controls the Epileptor dynamics during the ictal period, and the second, x0, controls the equilibrium points of the Epileptor. Bifurcations that define how the Epileptor qualitatively changes its behavior were identified using the scaling behavior of frequency and amplitude during seizure onset and offset. A taxonomy of seizures (SLEs) assembles the bifurcations into 16 different classes , RSE, DB, and normal brain activity (NS) in the brain, under a variety of conditions. Furthermore, the model predicts the existence of different paths between these neuroelectric activities which may help to explain the mechanisms underlying their genesis, making progress in clinical and brain research.


Since, in Einstein gravity, a massless scalar field with lightlike gradient behaves as a null dust, one could expect that it can act as the matter source of Vaidya geometries. We show that this is impossible because the Klein–Gordon equation forces the null geodesic congruence tangent to the scalar field gradient to have zero expansion, contradicting a basic property of Vaidya solutions. By contrast, exact plane waves travelling at light speed and sourced by a scalar field acting as a null dust are possible. R is its trace, and e.g., [The Vaidya solutions of the Einstein equations , 21\docuor)e.g., , 5). Thepp-waves [A null dust appears frequently in classical and quantum gravity, in Vaidya spacetimes , 2, pp-wpp-waves –8, Robinpp-waves , twistinpp-waves –11, in spp-waves –21, blacpp-waves –24, and pp-waves , 25. Colpp-waves , 27. A nu is a retarded null coordinate, m(u) is a regular mass function, and u. The stress-energy tensor of the null dust sourcing the Vaidya geometry isAn outgoing Vaidya solution , 2 It is well known –44 that  of view . However of view  has prop of view  for a diThere remains the case affinely-parametrized null geodesics.As noted in the previous section, since a null vector is defined up to a multiplicative (positive) factor, one could choose a different parametrization of the null dust in which its stress-energy tensor is instead written as ab=ρℓaℓb . For a gNow consider the congruence of null geodesics tangent to the null vector geometry , or ingogeometry ). Let usgeometry  first: bgeometry , their ev is the advanced null coordinate) sourced by a scalar field, that would have to be The same argument, adapted to the replacement f(u) is an arbitrary (but regular) function of u, k are constants, and [It is possible to obtain analytic solutions of the Einstein equations  travelliy2,where 17\documents, and 21\documeVaidya geometries sourced by a massless scalar field mentclass2pt{minimane wave  can be s


The Editors-in-Chief have retracted this article (Antoine et al. All authors agree to this retraction.


This method provides an effective alternative to metasurface bandwidth extension and can also be extended to higher bands such as THz and infrared frequencies.In this paper, we propose a method of designing ultra-wideband single-layer metasurfaces for cross-polarization conversion, via the introduction of Fano resonances. By adding sub-branches onto the unit cell structure, the induced surface currents are disturbed, leading to coexistence of both bright and dark modes at higher frequencies. Due to the strong interaction between the two modes, Fano resonance can be produced. In this way, five resonances in all are produced by the single-layer metasurface. The first four are conventional and are generated by electric and magnetic resonances, whereas the fifth one is caused by Fano resonance, which further extends the bandwidth. A prototype was designed, fabricated and measured to verify this method. Both the simulated and measured results show that a 1:4.4 bandwidth can be achieved for both  Metamaterials can achieve a mass of exotic physical effects, such as polarization conversion7, negative refraction8, perfect absorber10 and so on. In order to reduce bulky volume and be more suitable for application, a two-dimensional form of metamaterial is generated, called as metasurface, which is a planar array composed of sub-wavelength metamaterial units. In the past decade, study of polarization converter is quite familiar to scientific community, but it is still a key issue in electromagnetic (EM) wave. Many meaningful and excellent works have been proposed such as linear-to-linear, linear-to-circular and dual-band polarization15. If a facile structure can achieve more wideband and higher-efficiency simultaneously, it will further promote the development of various polarization manipulating devices such as high-efficient circularly polarized antennas17, more reduction of radar cross section (RCS)19, and even to construct more complex EM waves20.Metamaterials refer to artificial objects exhibiting intriguing properties which do not exist in nature22 and chiral metasurfaces24 have been accounted. However, these polarizers suffer from either the narrow bandwidth or the low-efficiency, which extremely limit their practical applications. One method to extend the bandwidth is to stack multilayers25, which will lead to bulky thickness and difficult preparation. The other method is to use multiple resonances. Chen et al.6 have indicated that an ultra-wideband metasurface can achieve a 1/4 3 dB bandwidth through four electric and magnetic Lorentzian resonances but the polarization conversion ratio (PCR) is only above 50%. Therefore, it is a priority for us to expand bandwidth under the condition of maintaining the high-efficiency.Many polarization converters based on anisotropic metasurfaces28. Fano resonances is viewed as a quantum interference between a discrete state and a continuum state in atomic physics30. The shape of Fano resonance is distinctively different from the most common observed spectral line shapes known as symmetric Lorentzian line shapes. It is characterized by an asymmetric spectral shape and have been achieved in various electromagnetic metamaterial structures31.Fano resonances have been research hotspots recently, due to their capability of providing ultra-narrow spectral linewidth and hence various potential applications such as sensing, slow light, and nano-lasing33. In this paper, we aim to further extend the bandwidth via the introduction of Fano resonance. To this end, we first give an optimized double-head arrow  that consist of one long straight cut-wire and two oblique V-shaped wires, according to multiple plasmon resonances of Chen et al.6. Next, two sub-branches are added to the unit 1, generating a composited structure . When combine them together, the canceling current will be formed between the branch and sub-branch, which can be regarded as dark mode, as shown in Fig. x- and y-polarized waves, with almost all polarization conversion ratio (PCR) above 90%.At present, most wideband polarization rotators are created by increasing the number of Lorentzian resonances but seldom involve Fano resonances6 have indicated that the bandwidth can be broadened through multiple plasmon resonances. On the basis of it, we give the unit 1 shown in Fig. e = 13.7 mm, f = 6.95 mm, g = 0.70 mm, m = 0.30 mm, r = 0.5 mm. Then, full-wave simulations are performed using the frequency domain solver in CST Microwave Studio with periodic boundary conditions in x- and y-directions and open conditions along the z-direction. The polarization conversion properties are studied by the reflection coefficients. y- (x-) polarized wave. As shown in Fig. Chen et al.entclass1pt{minimax- and y-directions with a periodicity of p = 12.2 mm. Top layer is a 0.018 mm metal pattern, whose geometrical parameters are illustrated as follows: a = 6.95 mm, b = 2.82 mm, c = 13.7 mm, d = 0.5 mm, k = 0.7 mm, l = 2.51 mm, r = 0.5 mm. The second layer is F4B dielectric spacer with a thickness y-polarized and x-polarized incident waves. Compared with Fig. On the other hand, hoping to get Fano resonance to broaden bandwidth of polarization conversion, two sub-branches are added to the unit 1, generating the unit 2 depicted in Fig. The PCR is used to evaluate the performance of the polarization conversion, which is defined as entclass1pt{minimau- and v-coordinate system is introduced. The v-axis is along 45° direction with respect to the y-axis and u-axis is perpendicular to v-axis, as shown in Fig. y-axis. The electric field can be decomposed into two orthogonal components (directions u and v). Hence, the electric field of the incident EM wave can be expressed as u- and v-axes, respectively. The electric field of the reflected wave can be written as u- and v-axes, respectively. Due to the anisotropic characteristic of the metasurface, a different phase change x-direction. The incident polarization is rotated by 90°. To verify the polarization conversion performance of the unit 2, we carried out numerical simulations of reflection phase and amplitude characteristics. As displayed in Fig. u- and v-polarized waves are basically equal and we get 180°−39°To better understand the principle of polarization conversion, entclass1pt{minimaentclass1pt{minimau- and v-axes can excite resonant eigen-modes. Figure u-polarized EM waves while the other two eigen-modes (ii) and (iv) are excited by v-polarized EM waves. Thus, when the y-polarized waves are incident, five resonances can be excited simultaneously because the y-polarized waves have both u- and v-components. However, there is a special resonance, the eigen-mode (v). We find that co-polarization reflection in (v) is far less than others. It means the resonance in (v) is fundamentally different from the other resonances.The curve of Fano resonance is asymmetry, which is a typical feature different from that of conventional symmetric resonance curves. As shown in Fig. In order to judge what type of each resonance, it is better to analyze surface current distributions. Therefore, we monitor the surface currents at each eigen-frequency as shown in Fig. 35.Magnetic resonances are generated by anti-parallel currents coupling between top metal pattern and the background sheet. Electric resonances are generated by parallel currents. Fano resonance can be introduced by the strong interaction between bright and dark modes. More intuitively, Fano resonances can be caused by the interaction between the electric dipolesv-direction. Therefore, in Fig. Aiming to make it more intuitive, the partial surface current is shown by the arrows painting in the Fig. Besides, we also simulate the surface current distributions of unit 1 shown in Fig. 37. From the Fig. Fano resonances suggest a good approximation of constructive and destructive interference of discrete states and continuum states, which are viewed as dark modes and bright modes40. It can be seen from the Fig. u- and v-direction of the surface current components in the ground sheet. In the v-direction of Fig. u-direction, it is different for the surface currents between both sides of the cut-wire structure. From the color we can realize that the redder the color, the greater the current. Main surface currents are centralized on one side while there's almost nothing on the other side. Therefore, according to the vector synthesis principle, the main surface currents in v-direction are the two red arrows, which are parallel to v-direction surface current component in the ground sheet. Then, electrical resonances in the u-direction are formed. It is the interaction between the electric dipole and the quadrupole that causes Fano resonance with an asymmetrical spectrum. In conclusion, the first four resonances result in expansion of the operating frequency range because they are generated by electric and magnetic resonances. The fifth resonance is Fano resonance caused by interaction between dark modes and bright modes, which leads to further expansion of bandwidth.On the other hand, Fano resonances also can be intuitively explained by the coupled-mode theory of two classical Lorentzian oscillators with electromagnetic waves as driving force. In a case of two coupled oscillators, there are effectively two driving forces acting on the first oscillator, which are out of phase and cancel each otherb) is a critical factor affecting the bandwidth and the performance of Fano resonance. As exhibited in Fig. b is 0, 1.5, 2.0, 2.3, 2.8, 3.3. Compared b = 1.5 with b = 0 in Fig. b can almost independently change the frequency of Fano resonance, and the resonant frequency will decrease with the increase of parameter b while have almost no effect on the first four resonances. Although the Fano resonance has a characteristic of narrow band, it still can be regarded as a method to further expand bandwidth on the basis of an ultra-wideband polarization conversion metasurface. Hence, in the principle of ensuring high efficiency while maximizing bandwidth, we successfully connect the narrow band of the Fano resonance with the wide band caused by the first four Lorentz resonances by adjusting parameter b to 2.8, as shown in the green lines of Fig. b = 2.8 does increase by a peak around 13.3 GHz compared with b = 0 and b = 1.5. Meanwhile, in the process of changing parameter b as shown in Fig. It is worth mentioning that the length of oblique V-shaped wires  reduction by combining with its 90°, 180° and 270° rotated ones to create the destructive interferences cancellation. Meanwhile, the proposed structure is a potential candidate for many polarization control devices due to its ultra-wideband and high efficiency performance or ultrasensitive sensors because the energy of Fano resonances is strongly confined which is suitable for switching devices, sensors, isolators, etc. This work also provides a notable avenue to metasurface bandwidth extension using Fano resonances and can also be extended to higher bands such as THz and infrared frequencies.


Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing q-congruences for truncated basic hypergeometric series with the base being an odd power of q. We also give a number of related conjectures including q-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a prime greater than 3.We provide several new  In his first letter to Hardy from 1913, Ramanujan announced that function . During example, , 27, 31)example, , q-superq-notation: For an indeterminate q, letq-shifted factorial. For convenience, we compactly writeq-shifted factorials. Moreover,q-integer, which can be defined by n, including negative n, which in particular gives nth cyclotomic polynomial in q, which may be defined asnth primitive root of unity.Recently, the authors , the following identity:q-Gamma functionIndeed, by replacing ain from , after mq by q-analogue of Euler’s identityq-series identities.Now, since 1- limit  becomes51 we get  while font}q2 in  and putt2/6. See  for receoddWe turn to discussing whether some of the results obtained in the paper can be further strengthened. We have proved Theorems  side of  with oddNevertheless, we would like to give the following result being similar to Theorem r be an even integer with Let The proof of Theorem We can also prove the following refinement of  and 1.81.8. HoweLet We would like to propose the following three conjectures which are similar to Corollary r be a positive integer and let p be a prime with Let r a positive integer, thenLet r a positive integer, thenLet q-analogues of (Conjectures ogues of  as follon be an integer greater than 1. ThenLet p be a prime. ThenLet p) modulo It is clear that the  case of  reduces  case of  modulo \ize that , while sq-analogue of (Finally, we give a partial and a complete logue of  as follon be an integer greater than 1. ThenLet p be a prime. ThenLet It is clear that the  case of  reduces uence in .


Q-neutrosophic soft sets are essentially neutrosophic soft sets characterized by three independent two-dimensional membership functions which stand for uncertainty, indeterminacy and falsity. Thus, it can be applied to two-dimensional imprecise, indeterminate and inconsistent data which appear in most real life problems. Relations are a suitable tool for describing correspondences between objects. In this study we introduce and discuss Q-neutrosophic soft relations, which can be discussed as a generalization of fuzzy soft relations, intuitionistic fuzzy soft relations, and neutrosophic soft relations. Q-neutrosophic soft relation is a sub Q-neutrosophic soft set of the Cartesian product of the Q-neutrosophic soft sets, in other words Q-neutrosophic soft relation is Q-neutrosophic soft sets in a Cartesian product of universes. We also present the notions of inverse, composition of Q-neutrosophic soft relations and functions along with some related theorems and properties. Reflexivity, symmetry, transitivity as well as equivalence relations and equivalence classes of Q-neutrosophic soft relations are also defined. Some properties of these concepts are presented and supported by real life examples. Finally, an algorithm to solve decision making problems using Q-neutrosophic soft relations is developed and verified by an example to show the efficiency of this method. The theory of neutrosophic set was firstly proposed by Smarandache  as a genAnother commonly used method in handling uncertainties and representing incomplete and unreliable data is soft set theory which was established by Molodtsov  as a genQ-fuzzy soft sets was established by Adam and Hassan ,21. The Since fuzzy set theory was introduced by Zadeh , it has The utilization of fuzzy relations  derived The relations between fuzzy sets, soft sets and their extensions have been widely studied. However, these relations do not encompass indeterminate information which appears in two dimensional universal sets. To overcome this, we introduce the Q-neutrosophic soft relation (Q-NSR), which represents the degree of presence, absence or indeterminacy of interaction between the elements of the Q-neutrosophic soft sets (Q-NSSs). Thus, it serves the indeterminacy and two-dimensionality of a data set at the same time, which cannot be served by fuzzy sets, soft sets and their extensions models. We present the concepts of inverse, functions and composition of Q-neutrosophic soft relations (Q-NSRs), some related theorems and properties. We define reflexivity, symmetry, transitivity as well as equivalence relations and equivalence classes of Q-neutrosophic soft relations (Q-NSRs). To show the ability of this model to solve decision making problems with two-dimensional indeterminate information, we developed an algorithm to solve decision making problems using Q-neutrosophic soft relations (Q-NSRs) and illustrate it by an example.In this section, we review the notions of soft sets, neutrosophic sets, neutrosophic soft sets with some of their properties which are pertinent to this work. The Q-neutrosophic soft sets (Q-NSSs) is also introduced.X.Soft set theory was first introduced by Molodtsov  as a parDefinition 1A pair . A pairNeutrosophic set was established by Smarandache  as a genDefinition 2Γ on the universe X is defined asA neutrosophic set and. A neutDefinition 3Γ and Ψ be two neutrosophic sets, then we say that Γ is a subset of Ψ denoted by Let . Let Γ Maji  presenteDefinition 4Let X be an initial universe set and E be a set of parameters. Consider . Let X We will now introduce the concept of Q-neutrosophic set to provide a way to deal with uncertain, indeterminate and inconsistent two-dimensional information. We also extend this concept to multi Q-neutrosophic set and Q-neutrosophic soft set (Q-NSS).Definition 5.Let X be a universal set and Q be a nonempty set. A Q-neutrosophic set where Note that the set of all Q-neutrosophic sets over X will be denoted by Definition 6.Let X be a universal set, Q be any nonempty set, l be any positive integer and I be a unit interval where where l is called the dimension of The set of all multi Q-neutrosophic sets of dimension l in X and Q is denoted by Definition 7.Let X be a universal set, E be a set of parameters, and Q be a nonempty set. Let such that The set of all Q-neutrosophic soft sets (Q-NSSs) in X and Q is denoted by Definition 8.Let The following example illustrates the above definition of the Q-NSS.Example 1.Suppose we want to examine the attractiveness of houses that a person is considering purchasing. Suppose there are three houses in the universe The Q-NSS In this section, after introducing the Cartesian product of two Q-NSSs, we will characterize the idea of Q-NSR, and present two fundamental operations of Q-NSRs, namely inverse and composition with some essential properties.In the following we define the Cartesian product of two Q-NSSs followed by an illustrative example.Definition 9.If X is an initial universal set, Q is a nonempty set, E is a set of parameters, where Example 2.Suppose we have a set of students The Cartesian product of where elements will look likeNow, we introduce the relation between two Q-NSSs, followed by the definitions of the domain and the range of a Q-NSR with some illustrative examples.Definition 10.If X is an initial universal set, Q is a nonempty set, E is a set of parameters, where for all If If Definition 11.Let R be a Q-NSR from Example 3.Reconsider Example 2 with a relation R from Then Definition 12.The identity relation Example 4.In Example 2, the relation Now, we introduce the operations of inverse and composition of two Q-NSRs followed by examples and relevant theorems.Definition 13.If It is to be noted that the inverse of R is defined by reversing the order of every pair belonging to R.Example 5.Reconsider R as in Example 3, where we would then have Theorem 1.Suppose 1. 2. If Proof. If Next, we will propose the definition of the composition of Q-NSRs along with an illustrative example, followed by related theorem.Definition 14.If where for all Example 6.Let Suppose that Define the Q-NSRs The relation The composition between the Q-NSRs The components Theorem 2.If Proof. If Similar results follow for the rest of the terms. This completes the proof. ☐In this section, we will introduce various types of Q-NSRs, partitions and equivalence classes of Q-NSSs with some related theorems.Definition 15.Let R be a relation on 1. 2. 3 4 Example 7.Consider a Q-NSS Consider a relation Definition 16.Let Example 8.Reconsider R as in Example 3, we would then have Lemma 1.Let Proof. Suppose Conversely, suppose Now, we define the partition of a Q-NSS followed by some related theorems.Definition 17.A collection of nonempty Q-NS subsets 1. 2. Example 9.Let Suppose Remark 1.Elements of the partition are called a block of Corresponding to a partition Theorem 3.Let Proof. Reflexive: Let Symmetric: If Transitive: If Remark 2.The equivalence Q-NSR defined in the above theorem is called an equivalence relation determined by the partition P. In the previous example the equivalence relation determined by the partition Theorem 4.Corresponding to every equivalence relation defined on a Q-NSS Proof. A corresponding to P of (i)(ii)If Let Since Now for the second part, let Remark 3.The partition constructed in the above theorem consisting of all equivalence classes of In this section, we present the concept of Q-NS function and some special types of Q-neutrosophic soft functions with related theorems.Definition 18.Let Example 10.Reconsider Example 2. The Q-NSR f which consists of a science student with excellent GPA forms a Q-NS function from Definition 19.Let 1. f is injective (one to one) if 2. f is surjective (onto) if 3. f is bijective if it is both injective and surjective.4. f is a constant function if all elements in 5. f is an identity function if the identity Q-NS function I on a QNSS Theorem 5.Let 1. 2. 3. Proof. 1. Let 2. Let This implies, Now, clearly 3. Let f is one to one. Conversely, suppose Definition 20.Let Definition 21.Let Now we will define and propose a few theorems on the composition of Q-neutrosophic soft functions.Theorem 6.If Proof. f is one to one, Let f is surjective, there exists a unique element Now Theorem 7.Let Proof. f and g are one to one we have Let g is onto. Again since f is onto there exists Let In this section, we present an application of Q-NSR in a decision-making problem.The problem we consider is as below.E be a set of parameters where Let Suppose the Q-NSS Step 1.X, Construct two Q-NSSs over Step 2.Construct a Q-NSR Step 3.Compute the comparison table using the formula: Step 4.Select the highest numerical grades for each column in the comparison table.Step 5.Find the score table which shows the objects with highest numerical grades corresponding to every pair of parameters.Step 6.Compute the score Step 7.M where The decision is any one of the elements in Then we can select a car on the basis of their parameters using Q-NSR, by applying the following algorithm.To execute the above steps, we will use the Q-NSSs We obtain the Q-NSR We compute the comparison table as:The highest numerical grade for each column in The score of each object by taking the sum of these numerical grades are: d” of color “q”, as the appropriate solution for selecting the most suitable car according to the basis of parameters Hence, This work established the notion of relations between Q-NSSs. It improved the concept of relations between fuzzy sets, soft sets and hybrid models of these two types of sets because the relations have been constructed based on the Q-NSSs model which has three independent two-dimensional membership functions for uncertainty, indeterminacy and falsity. It provided an adequate parametrization tool to deal with the aspects of two-dimensional imprecise, indeterminate and inconsistent data which appear in most real life problems. The notions of inverse, composition of Q-NSRs and functions were also presented along with some related theorems and properties. The ideas of reflexive, symmetric, transitive and equivalence Q-NSRs were defined followed by the introduction of the associated properties with illustrative examples. An algorithm using Q-NSR was constructed and applied to a decision making problem. Q-NSS facilitates the way to numerous prospects for future research since it serves the two-dimensionality and indeterminacy simultaneously. Thus, it can be extended using the refined neutrosophic set , soft ex


Scientific Reports 10.1038/s41598-020-62298-z, published online 23 April 2020Correction to: The original version of this Article contained errors.All instances of ‘San Juan Islands’ now read ‘Gulf Islands’.rs,NIR in Eq. (1), where the first instance of a minus should have been a plus:Additionally, as a result of clerical error the wrong sign was used in the definition of R′now reads:This has now been corrected in both PDF and HTML version of the Article.


Scientific Reports 10.1038/s41598-019-41318-7, published online 20 March 2019Correction to: Nematus oligospilus” was incorrectly given as “Nematus Oligospilus”.The original version of this Article contained an error in the title, where “In addition, in Fig. 2, panel (b) was a duplication of part of panel (a).These errors have now been corrected in the PDF and HTML versions of the Article.