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This paper extends the second-order three-wave flow of Fordy and Kulish to propose a generalized second-order flow of the three-wave hierarchy. Then the Lax pair of the generalized second-order thr...

A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

Based on Hirota bilinear method, the N -soliton solution of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is derived explicitly, from which some localized waves such as soliton, breather, lump and their interactions are obtained by the approach of long wave limit. Especially, by selecting particular parameter constraints in the N -soliton solutions, the one breather or one lump can be obtained from two-soliton; the elastic interaction solutions between one bell-shaped soliton and one breather or between one bell-shaped soliton and one lump can be obtained from three-soliton; the elastic interaction solutions among two bell-shaped solitons and one breather, among two bell-shaped solitons and one lump, between two breathers or between two lumps can be obtained from four-soliton; the elastic interactions solutions among one bell-shaped soliton and two breathers, among one breather and three bell-shaped solitons, among one lump and three bell-shaped solitons, among one bell-shaped soliton and two breathers or among one breather, one lump and one bell-shaped soliton can be obtained from five-soliton. Detailed behaviors of such interaction phenomena are illustrated analytically and graphically. The results obtained in this paper may be helpful for understanding the evolution of nonlinear localized waves in shallow water.

A new lattice hierarchy is constructed from a discrete matrix spectral problem. By the Tu scheme technique, the associated Hamiltonian structures and infinitely many conservation laws of this hierarchy are derived. Then a symplectic map is proposed based on the Lax pair and the adjoint Lax pair. Furthermore, the N-fold Darboux transformation and explicitly exact solutions of the first two equations in the hierarchy are investigated. Finally, the density profiles of these exact solutions are presented to illustrate the overtaking collisions of solitary waves.

In this paper, the N -soliton solution is constructed for the ( 2 + 1 )-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.

The Riemann problem of the defocusing AB system is explored by Whitham modulation theory. As a result, the modulation periodic wave solutions along with the corresponding Whitham equations are derived. The basic wave structures that act as building blocks in the classification of solution to the Riemann Problem are proposed graphically. Finally, a discontinuous initial value problem is solved and a exotic wave pattern is found, whose soliton front is just the dark soliton in the defocusing AB system.

The Lax pair of the Merola–Ragnisco–Tu (MRT) equation is derived by the prolongation technique. Then an integrable lattice hierarchy and the associated Hamiltonian structures of the hierarchy are constructed. Furthermore, the N-fold Douboux transformations for the MRT and higher-order MRT equations are established respectively, some explicit solutions of the two equations are obtained and the graphs are shown to illustrate the inelastic overtaking interactions of these soliton solutions. Finally, the infinitely many conservation laws for the MRT and higher-order MRT equations are listed.

In this paper, a high-efficient, split-step, and implicit corrected parallel smoothed particle hydrodynamics (SS-ICPSPH) method is developed to simulate the dynamic systems of several nonlinear Schrodinger/Gross–Pitaevskii equations (NLSE/GPE). The proposed method is motivated by the split-step for the equation, the corrected symmetric kernel gradient for the traditional SPH and the implicit scheme for time, respectively. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. Firstly, the numerical accuracy and the merits of the proposed method are tested by solving 2D NLSE, and compared with the analytical results. Secondly, the new method is extended to simulate the 2D/3D two-component GPE, compared with high accuracy finite difference results. Thirdly, the proposed method is extended to investigate the sheet-like vortices in rotating Bose–Einstein condensate . Finally, the implicit corrected SPH scheme is tentatively extended to capture the propagation process of free surface wave in a rectangular pool with initial perturbation. All the numerical results show the ability and the reliability of the proposed method.

Some two-component Korteweg–de Vries systems are studied by prolongation technique and Painleve analysis. Especially, the two-component KdV system conjectured to be integrable by Foursov is proved to be both Lax integrable and P-integrable. Its conservation laws are investigated based on the obtained Lax pair. Furthermore, it is shown that the three two-component Korteweg–de Vries systems are identical under certain invertible linear transformations. Finally, the auto-Backlund transformation and some exact solutions for the two-component Korteweg–de Vries system are derived explicitly.

An integrable lattice hierarchy is constructed from a discrete matrix spectral problem, in which one of the Suris systems is the first member of this hierarchy. Some related properties such as Hamiltonian structure of this lattice hierarchy are discussed. The Suris system is solved by the N-fold Darboux transformation. As a result, the multi-soliton solutions are derived and the soliton structures along with the interaction behaviors among solitons are shown graphically. Finally, the infinitely many conservation laws of the Suris system are given.

The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions., Published version of the article. 42 pages, 13 figures

In this paper, we study the application of a version of the method of simplest equation for obtaining exact traveling wave solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms. The Duffing-type equation is used as simplest auxiliary equation. In the meantime, the proposed method is proved to be a powerful mathematical tool for obtaining exact solutions of nonlinear partial differential equations in mathematical physics.

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painleve integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665-669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth-order Lax pairs instead of the fifth-order ones. Moreover, the Painleve integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painleve integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto-Backlund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, the symmetry group and similarity reductions of the two-dimensional generalized Benney system are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Firstly, the vector field associated with the Lie group of transformation is obtained. Then the point transformations are proposed, which keep the solutions of the generalized Benney system invariant. Finally, the symmetry reductions and explicitly exact solutions of the generalized Benney system are derived by solving the corresponding symmetry equations.

In the present paper, the prolongation technique and Painleve analysis are performed to a two-component Korteweg–de Vries system. It is proved that this system is both Lax integrable and P-integrable. By embedding the prolongation algebra in the s l ( 3 ; C ) algebra, the 3×3 Lax representation of the system is derived. Moreover, the auto-Backlund transformation and some exact solutions for the two-component Korteweg–de Vries system are proposed explicitly, and it is shown that this system owns solitary wave solutions which demonstrate fission and fusion behaviors.

In this paper, based on the zero curvature equation, an arbitrary order matrix spectral problem is studied and its associated multi-component cubic–quintic nonlinear Schrodinger integrable hierarchy is derived. In order to solve the multi-component cubic–quintic nonlinear Schrodinger system, a class of Riemann–Hilbert problem is proposed with appropriate transformation. Through the special Riemann–Hilbert problem, where the jump matrix is considered to be an identity matrix, the soliton solutions of all integrable equations are explicitly calculated. The specific examples of one-soliton, two-soliton and N -soliton solutions are explicitly presented.

We propose a method to compute objective contour feature. Conventional objective chain algorithms correlate the motion with area characteristics of the objectives to reach the goals. Nevertheless, they may not be efficient under certain circumstances of complicated tracking. In this paper, we incorporate the contour characteristic of the objective into conventional methods and thus construct a function to estimate the similarity of the objective. Experiments show that more reliable results are obtained when rigid objects are tracked.

This paper presents a multi-frame super-resolution (SR) reconstruction algorithm based on diffusion tensor regularization term. Firstly, L1-norm structure is used as data fidelity term, anisotropic diffusion equation with directional smooth characteristics is introduced as a prior knowledge to optimize reconstruction result. Secondly, combined with shock filter, SR reconstruction energy functional is established. Finally, Euler-Lagrange equation based on nonlinear diffusion model is exported. Simulation results validate that the proposed algorithm enhances image edges and suppresses noise effectively, which proves better robustness.

In this paper, we study the integrability and exact bright soliton solutions to the coupled nonlinear Schrodinger equation with higher-order effects arising from nonlinear fiber medium. Firstly, the Lax integrability of this equation is investigated by prolongation technique and an integrable generalized coupled higher-order nonlinear Schrodinger equation is proposed. Then the general N bright-bright soliton solutions of this integrable equation are obtained by Riemann-Hilbert formulation and the collision dynamics between two solitons is analyzed. Finally, an integrable generalized n-coupled higher-order nonlinear Schrodinger equation together with its linear spectral problem are given.

In this paper, the Lie group analysis method is applied to the geometric average Asian option pricing equation in financial problems. Firstly, the complete Lie symmetry group and infinitesimal generators of this equation are derived. Then the optimal system with one parameter for the Lie symmetry algebra are obtained, which gives the possibility to describe a complete set of invariant solutions to the pricing equation. Finally, based on the optimal system the symmetry reductions and corresponding closed form solutions for the pricing equation are proposed.

Nonlinear Schrodinger (NLS)-type equations can describe some physical phenomena in nonlinear optics, fluids, plasmas, etc. Under consideration in this paper is a discretized second-order in time nonlinear Schrodinger equation. Conservation laws and N-fold Darboux transformation (DT) are constructed by means of symbolic computations and its Lax representation. N-soliton solutions in terms of determinant are derived with the obtained DT. Structures of these solutions are shown graphically. Inelastic interaction phenomena between/among the two-, three-and four-soliton solutions are discussed, they might be helpful for understanding some physical phenomena.

The classical bilateral filtering algorithm is a non-linear and non-iterative image denoising method in spatial domain which utilizes the spatial information and the intensity information between a point and its neighbors to smooth the noisy images while preserving edges well. To further improve the image denoising performance, a spatially adaptive bilateral filtering image deonoising algorithm with low computational complexity is proposed. The proposed algorithm takes advantage of the local statistics characteristic of the image signal to better preserve the edges or textures while suppressing the noise. Experiment results show that the proposed image denoising algorithm achieves better performance than the classical bilateral filtering image denoising method.

In this paper, we investigate the exact solutions and conservation laws of a general Hirota equation. Firstly, the N-fold Darboux transformation of this equation is proposed. Then by choosing three kinds of seed solutions, the multisoliton solutions, breather solutions, and rogue wave solutions of the general Hirota equation are obtained based on the Darboux transformation. Finally, the conservation laws of this equation are derived by using its linear spectral problem. The results in this paper may be useful in the study of ultrashort optical solitons in optical fibers.

In this work, the option pricing Black–Scholes model with dividend yield is investigated via Lie symmetry analysis. As a result, the complete Lie symmetry group and infinitesimal generators of the one-dimensional Black–Scholes equation are derived. On the basis of these infinitesimal generators, the similarity variables and newly explicit solutions of the Black–Scholes equation are obtained by solving the corresponding characteristic equations. Finally, figures for an explicit solution with different dividend yields are presented to demonstrate the novel properties.

The Lax integrability of the coupled KdV equations derived from two-layer fluids [S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids, J. Phys. A: Math. Gen. 39 (2006) 513–527] is investigated by means of prolongation technique. As a result, the Lax pairs of some Painleve integrable coupled KdV equations and several new coupled KdV equations are obtained. Finally, the Miura transformations and some coupled modified KdV equations associated with the Lax integrable coupled KdV equations are derived by an easy way.

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived.

The integrability of a coupled KdV system is studied by prolongation technique and singularity analysis. As a result, Backlund transformation and linear spectral problem associated with this system are obtained. Some special solutions of the system are also proposed.

Hirota’s bilinear method is a powerful tool for obtaining a wide class of exact solutions of soliton equations. The crucial step of this method is to find a suitable dependent variable transformation, i.e. Hirota’s transformation, which transforms a soliton equation into a Hirota bilinear equation. In this paper, a systematic method to construct Hirota’s transformations of continuous soliton equations is proposed. And some examples are given to illuminate the availability of this method. In addition, a new two-soliton solution of a coupled nonlinear Schrödinger equation is obtained.

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1+1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.

In this paper, the multiple Riccati equations rational expansion method is further extended to construct non-travelling wave solutions of the (2 + 1)-dimensional Painleve integrable Burgers equation and the (2 + 1)-dimensional Breaking soliton equation, as a result, some double solitary-like wave solutions and complexiton solutions of the two equations are obtained. The extended method can also be applied to solve some other nonlinear evolution equations.

The present paper deals with families of non-trivial novel solutions of the general elliptic equation ϕ ′ 2 ( ξ ) = d d ξ ϕ 2 = a 0 + a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 ϕ 4 . Based on these novel solutions, a direct and generalized algebraic algorithm is described to construct the new non-travelling wave solutions of systems of nonlinear partial differential equations (NLPDEs). Subsequently, a series of important non-travelling wave solutions of the (2 + 1)-dimensional Burgers equations are obtained.

Some special types of solutions with arbitrary functions for the (N+1)-dimensional sine-cosine-Gordon equation, the (N+1)-dimensional double sinh-Gordon equation, and the (N+1)-dimensional sinh-cosinh-Gordon equations are constructed by means of the separation transformation approach. As an illustrative sample, the properties of some solutions are shown by their figures.

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of first-order elliptic equation ϕ ′ 2 = a 0 + a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 ϕ 4 (where ϕ ′ = d d x ϕ ) are obtained. To our knowledge, these nontrivial solutions can not be found in [Chaos Solitons Fract. 26 (2005) 785–794] and [Phys. Lett. A 336 (2005) 463–476] by Yomba and other existent papers until now. By using these nontrivial solutions, a direct algebraic method is described to construct several kinds of exact non-travelling wave solutions for the (2 + 1)-dimensional Breaking soliton equations and the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equation. By using this method, many other physically important nonlinear partial differential equations (NLPDEs) can be investigated and new non-travelling wave solutions can be explicitly obtained with the aid of symbolic computation system Maple.

In this paper, by using symbolic and algebra computation, Chen and Wang's multiple Riccati equations rational expansion method was further extended. Many double soliton-like and other novel combined forms of exact solutions of the (2+1)-dimensional Breaking soliton equation are derived by using the extended multiple Riccati equations expansion method.

In this paper, with the aid of the symbolic computation we have improved the extended F-expansion method in [Chaos, Solitons and Fractals 22 (2004) 111] and proposed the further improved F-expansion method. By this method, we have gotten many new exact solutions which we have never seen before within our knowledge of the (2+1)-dimensional Nizhnik-Novikov-Veselov equations. In addition,the solutions we get are more general than the solutions the extended F-expansion method gets. At the same time, our method is more convenient than the method in [Chaos, Solitons and Fractals 18 (2003) 299-309] and the solutions we get are more abundant than the solutions in [H-q Zh. Appl. Math. E-Notes 1 (2001) 139]. In other word, the solutions in [H-q Zh. Appl. Math. E-Notes 1 (2001) 139] are part of our solutions. Our method can also apply to other partial differential equations and also get many new exact solutions.

In this paper, with the aid of the symbolic computation we improve the extended F-expansion method in [Chaos, Solitons & Fractals 2004; 22:111] and propose the further improved F-expansion method. Using this method, we have gotten many new exact solutions which we have never seen before within our knowledge of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. In addition,the solutions we get are more general than the solutions that the extended F-expansion method gets.The solutions we get include Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions and so on. Our method can also apply to other partial differential equations and can also get many new exact solutions.

In this article, we present dynamical approaches and multi-quadratic integer programming techniques to study the problem of seizure prediction. The data used in our studies consist of continuous intracranial electroencephalograms (EEGs) from patients with temporal lobe epilepsy. The results of this study can be used as a criterion to pre-select the critical electrode sites that can be used to predict epileptic seizures.

In this paper, making use of the truncated Laurent series expansion method and symbolic computation we get the auto-Bäcklund transformation of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation. As a result, single soliton solution, single soliton-like solution, multi-soliton solution, multi-soliton-like solution, the rational solution and other exact solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation are found. These solutions may be useful to explain some physical phenomena.

First, the theory of random process is linked with the statistical description of epileptic human brain process. A statistical information approach to the adaptive analysis of the electroencephalogram (EEG) is proposed. Then, the problem of time window recognition of the global stochastic model based upon Bayesian estimation and the use of global optimization for restricted experimental data are proposed. A robust algorithm for estimating unknown parameters of stochastic models is considered. The ability of nonlinear time-series analysis to extract features from brain EEG signal for detecting epileptic seizures is evaluated.

The phenomenon of epilepsy, one of the most common neurological disorders, constitutes a unique opportunity to study the dynamics of spatiotemporal state transitions in real, complex, nonlinear dynamical systems. We previously demonstrated that measures of chaos and angular frequency obtained from electroencephalographic (EEG) signals generated by critical sites in the cerebral cortex converge progressively (dynamical entrainment) from the asymptomatic interictal state to the ictal state (seizure) [L.D. Iasemidis, P. Pardalos, J.C. Sackellares and D.-S. Shiau (2001). Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Combinatorial Optimization, 5, 9–26; L.D. Iasemidis, D.-S. Shiau, P.M. Pardalos and J.C. Sackellares (2002). Phase entrainment and predictability of epileptic seizures. In: P.M. Pardalos and J. Principe (Eds.), Biocomputing, pp. 59–84. Kluwer Academic Publishers]. This observation suggests the possibility of developing algorith...

The soliton interaction is investigated based on solving the nonisospectral generalized Sawada-Kotera (GSK) equation. By using Hirota method, the analytic one-, two-, three-, andN-soliton solutions of this model are obtained. According to those solutions, the relevant properties and features of line-soliton and bright-soliton are illustrated. The results of this paper will be useful to the study of soliton resonance in the inhomogeneous media.

Epilepsy is one of the most common disorders of the nervous system. The progressive entrainment between an epileptogenic focus and normal brain areas results to transitions of the brain from chaotic to less chaotic spatiotemporal states, the epileptic seizures. The entrainment between two brain sites can be quantified by the T-index from the measures of chaos (e.g., Lyapunov exponents) of the electrical activity (EEG) of the brain. By applying the optimization theory, in particular quadratic zero-one programming, we were able to select the most entrained brain sites 10 minutes before seizures and subsequently follow their entrainment over 2 hours before seizures. In five patients with 3–24 seizures, we found that over 90% of the seizures are predictable by the optimal selection of electrode sites. This procedure, which is applied to epilepsy research for the first time, shows the possibility of prediction of epileptic seizures well in advance (19.8 to 42.9 minutes) of their occurrence.

A Darboux transformation is constructed for the generalized Hirota-Satsuma coupled KdV system and the result is compared with the recent work of Geng and his collaborators [X. G. Geng et al., Phys. Rev. E 79, 056602 (2009) and X. G. Geng and G. L. He, J. Math. Phys. 51, 033514 (2010)]. It is shown that our Darboux transformation may be applied to three interesting reductions of the general system. In addition, the iteration of this Darboux transformation is worked out, and some solutions to the associated systems are obtained.

We have previously reported preictal spatiotemporal transitions in human mesial temporal lobe epilepsy (MTLE) using short term Lyapunov exponent (STL max ) and average angular frequency (\( \Omega \) ). These results have prompted us to apply the quantitative nonlinear methods to a limbic epilepsy rat (CLE), as this model has several important features of human MTLE. The present study tests the hypothesis that preictal dynamical changes similar to those seen in human MTLE exist in the CLE model. Forty-two, 2-hr epoch data sets from 4 CLE rats (mean seizure duration 74±20 sec) are analyzed, each containing a focal onset seizure and intracranial data beginning 1 hr before the seizure onset. Three nonlinear measures, correlation integral, short-term largest Lyapunov exponent and average angular frequency are used in the current study. Data analyses show multiple transient drops in STL max values during the preictal period followed by a significant drop during the ictal period. Average angular frequency values demonstrate transient peaks during the preictal period followed by a significant peak during the ictal period. Convergence among electrode sites is also observed in both STL max and \( \Omega \) values before seizure onset. Results suggest that dynamical changes precede and accompany seizures in rat CLE. Thus, it may be possible to use the rat CLE model as a tool to refine and test real-time seizure prediction, and closed-loop intervention techniques.

There is growing evidence that temporal lobe seizures are preceded by a preictal transition, characterized by a gradual dynamical change from asymptomatic interictal state to seizure. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates STLmax, a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites. The optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures. Specifically, a quadratically constrained quadratic 0-1 programming problem is formulated to identify critical electrode sites. The automated seizure warning algorithm was tested in continuous, long-term EEG recordings obtained from 5 patients with temporal lobe epilepsy. For individual patient, we use the first half of seizures to train the parameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve analysis. With the best parameter setting, the algorithm applied to all cases predicted an average of 91.7% of seizures with an average false prediction rate of 0.196 per hour. These results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes.

Seizure occurrences seem to be random and unpredictable. However, recent studies in epileptic patients suggest that seizures are deterministic rather than random. There is growing evidence that seizures develop minutes to hours before clinical onset. Our previous studies have shown that quantitative analysis based on chaos theory of long-term intracranial electroencephalogram (EEG) recordings may enable us to observe the seizure's development in advance before clinical onset. The period of seizure's development is called a preictal transition period, which is characterized by gradual dynamical changes in EEG signals of critical electrode sites from asymptomatic interictal state to seizure. Techniques used to detect a preictal transition include statistical analysis of EEG signals, optimization techniques, and nonlinear dynamics. In this paper, we herein present optimization techniques, specifically multi-quadratic 0-1 programming, for the selection of the cortical sites that are involved with seizure's development during the preictal transition period. The results of this study can be used as a criterion to preselect the critical electrode sites that can be used to predict epileptic seizures.

Epilepsy is a common neurological disorder characterized by recurrent seizures, most of which appear to occur spontaneously. Our research, employing novel signal processing techniques based on the theory of nonlinear dynamics, led us to the hypothesis that seizures represent a spatiotemporal state transition in a complex chaotic system. Through the analysis of long-term intracranial EEG recordings obtained in patients with medically intractable seizures, we discovered that seizures were preceded by a preictal transition that evolves over tens of minutes. This transition is followed by a seizure. Following the seizure, the spatiotemporal dynamics appear to be reset. The study of this process has been hampered by its complexity and variability. A major problem was that the transitions involve a subset of brain sites that vary from seizure to seizure, even in the same patient. However, by combining dynamical analytic techniques with a powerful global optimization algorithm for selecting critical electrode sites, we have been able to elucidate important dynamical characteristics underlying human epilepsy. We illustrate the use of these approaches in confirming our hypothesis regarding postictal resetting of the preictal transition by the seizure. It is anticipated that these observations will lead to a better understanding of the physiological processes involved. From a practical perspective, this study indicates that it may be possible to develop novel therapeutic approaches involving carefully timed interventions and reset the preictal transition of the brain well prior to the onset of the seizure.