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Involvement of memristive term and additive physical variables including magnetic flux and charge can enhance the physical description of the biophysical neurons. Neural circuits coupled with memristors can be built and tamed to mimic the intrinsic biophysical characteristics and dynamical properties of biological neurons, and these memristive oscillator models are effective in predicting the mode transition in neural activities and self-organization in collective electric behaviors of neural networks. Any proposal of memristive map neurons requires reliable physical description. For example, the energy definition and self-adaptive working mechanism are crucial to verify the reliability of memristive maps. A capacitive variable is useful to describe the membrane potential, while the complexity of ion channels requires careful evaluation and description by using inductive variables relative to the electromagnetic field. In this work, a charge-controlled memristor is connected to an inductor in series for building a hybrid ion channel, and then a capacitor and a nonlinear resistor are combined to couple the ion channel. As a result, a simple memristive neural circuit is designed to discern the inner effect of electric field and magnetic field synchronously. The energy function is defined and verified with theoretical proof. Furthermore, a linear transformation is applied to convert this memristive neuron into a memristive map with an exact energy description, in which its dynamics and mode transition will be controlled by an adaptive law when its energy is beyond the threshold. Additive noise is imposed to induce coherence resonance, which can be detected by using the statistical analysis and average value for Hamilton energy function during changes in noise intensity. This scheme provides guidance for energy definition in memristive maps and the intrinsic energy regulation mechanism in neural activities is explained from physical and dynamical aspects.

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

We propose a two layer neural network for computation of an approximate convex-hull of a set of points or a set of circles/ellipses of different sizes. The algorithm is based on a very elegant concept — shrinking of a rubber band surrounding the set of planar objects. Logically, a set of neurons is placed on a circle (rubber band) surrounding the objects. Each neuron has a parameter vector associated with it. This may be viewed as the current position of the neuron. The given set of points/objects exerts a force of attraction on every neuron, which determines how its current position will be updated (as if, the force determines the direction of movement of the neuron lying on the rubber band). As the network evolves, the neurons (parameter vectors) approximate the convex-hull more and more accurately. The scheme can be applied to find the convex-hull of a planar set of circles or ellipses or a mixture of the two. Some properties related to the evolution of the algorithm are also presented.

In this paper, we propose a parallel algorithm for the traffic control problem (an NP-complete problem) on crossbar switch networks. This problem is to find a set of conflict-free paths such that the maximum number of message packets can be transmitted over the network. The problem can be represented by an energy function. Then by applying our parallel algorithm, the state of the energy function is iteratively updated toward a stable state. When the energy function reaches a stable state, the state represents a solution of the problem. The empirical results show that the throughputs of the proposed algorithm are much better than the linear algorithm. We have shown that the time complexity of a parallel algorithm is O(n) by using n² processors. Furthermore, since the traffic control problem can be reduced to the traveling salesman problem, the proposed algorithm can be further applied to some other NP-complete problems.

Graph cuts is an image segmentation method by which the region and boundary information of objects can be revolved comprehensively. Because of the complex spatial characteristics of high-dimensional images, time complexity and segmentation accuracy of graph cuts methods for high-dimensional images need to be improved. This paper proposes a new three-dimensional multilevel banded graph cuts model to increase its accuracy and reduce its complexity. Firstly, three-dimensional image is viewed as a high-dimensional space to construct three-dimensional network graphs. A pyramid image sequence is created by Gaussian pyramid downsampling procedure. Then, a new energy function is built according to the spatial characteristics of the three-dimensional image, in which the adjacent points are expressed by using a 26-connected system. At last, the banded graph is constructed on a narrow band around the object/background. The graph cuts method is performed on the banded graph layer by layer to obtain the object region sequentially. In order to verify the proposed method, we have performed an experiment on a set of three-dimensional colon CT images, and compared the results with local region active contour and Chan–Vese model. The experimental results demonstrate that the proposed method can segment colon tissues from three-dimensional abdominal CT images accurately. The segmentation accuracy can be increased to 95.1% and the time complexity is reduced by about 30% of the other two methods.

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

In this study, a new 3D chaotic system was first transformed into a Kolmogorov-type system to describe the vector field from the viewpoint of torque. In this Kolmogorov-type system, only inertial torque and non-Rayleigh dissipation exist. Thus, this is different from previously reported Kolmogorov-type systems that are generally decomposed into inertial torque, internal torque, dissipation, and external torque. Moreover, by analyzing these two torques, the physical background of the system and the key factors of chaos generation were also determined. That is, the inertial torque and non-Rayleigh dissipation are responsible for chaos generation in the new 3D chaotic system. Then, the Casimir function and Hamiltonian energy function were also analyzed to investigate the cycling of energy in the chaotic system. Finally, both an analog circuit and a Field Programmable Gate Array (FPGA) circuit were designed to implement the chaotic system. All of the experimentally obtained results are consistent with the results of numerical analysis, which did not only indicate the chaotic characteristics of the 3D chaotic system physically, but also provided physical models for engineering applications.

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases.

Physics-based computational approaches to predicting the structure of macromolecules such as proteins are gaining increased use, but there are remaining challenges. In the current work, it is demonstrated that in energy-based prediction methods, the degree of optimization of the sampled structures can influence the prediction results. In particular, discrepancies in the degree of local sampling can bias the predictions in favor of the oversampled structures by shifting the local probability distributions of the minimum sampled energies. In simple systems, it is shown that the magnitude of the errors can be calculated from the energy surface, and for certain model systems, derived analytically. Further, it is shown that for energy wells whose forms differ only by a randomly assigned energy shift, the optimal accuracy of prediction is achieved when the sampling around each structure is equal. Energy correction terms can be used in cases of unequal sampling to reproduce the total probabilities that would occur under equal sampling, but optimal corrections only partially restore the prediction accuracy lost to unequal sampling. For multiwell systems, the determination of the correction terms is a multibody problem; it is shown that the involved cross-correlation multiple integrals can be reduced to simpler integrals. The possible implications of the current analysis for macromolecular structure prediction are discussed.