Probing the attractors in neural networks

I propose tools to probe the nature of the retrieval attractors in neural networks. These include the activity distribution, the evolutions of the state damage, activity damage and temporal correlation damage. They enable us to demonstrate that the retrieval attractors in dillute asymmetric neutral networks are not clouds of attractors, but consists of a single chaotic attractor for each stored pattern. Furthermore, they facilitate the devise of effective freezing procedures, which significantly improve the quality of retrieval in dilute asymmetric neural networks.     

Structures of attractors in periodically forced neural oscillators

Periodically forced oscillations in the membrane potential of squid giant axons are analysed by stroboscopic plots at multi-phases of the forcing function. The analysis clarifies the global structures of the attractors in the phase space R² × S¹.     

A universal scaling theory for complexity of analog computation

We discuss the computational complexity of solving linear programming problems by means of an analog computer. The latter is modeled by a dynamical system which converges to the optimal vertex solution. We analyze various probability ensembles of linear programming problems. For each one of these we obtain numerically the probability distribution functions of certain quantities which measure the complexity. Remarkably, in the asymptotic limit of very large problems, each of these probability distribution functions reduces to a universal scaling function, depending on a single scaling variable and independent of the details of its parent probability ensemble. These functions are reminiscent of the scaling functions familiar in the theory of phase transitions. The results reported here extend analytical and numerical results obtained recently for the Gaussian ensemble.     

Scaling and universality of the complexity of analog computation

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.     

Coding and computation with neural spike trains

We study a simple model for the statistics of neural spike trains as they encode a continuously varying signal. The model is motivated with reference to several recent experiments on sensory neurons, and we show how analogies between the relevant probabilistic issues in neural coding and statistical mechanics can be exploited. Results are given for the information capacity of the code, for the optimal structure of code-reading algorithms, and for the time delays which arise in optimal processing of the coded signal. In addition, we show how simple analog computations can be expressed directly in terms of transformations of the spike train. The rules for reading the code and for optimal analog computation depend on the context for behavioral decision making-the relative weights assigned to different types of errors, the relative importance of different signals. We find that there is a conflict between minimizing this context dependence of the code and maximizing its information capacity; a compromise can be achieved by appropriate preprocessing (filtering) of the encoded signal. Experiments on auditory and visual neurons qualitatively confirm the predicted filtering. Similarly, the structure of the optimal multiplier neuron is shown to depend upon the intensity and spectral content of incoming signals, and these predictions compare favorably with experiments on a movement-sensitive cell in the fly visual system.     

A Hopfield neural network with multiple attractors and its FPGA design

Neural network is important for a wide range of applications. Especially, a small neural network can display various complex behaviors. In this work, the investigations of a Hopfield neural network and its field programmable gate array (FPGA) implementation have been reported. The considered Hopfield neural network is simple because it includes only three neurons. It is interesting that we observed chaos and numerous coexisting attractors in such a network. In addition, the network has been implemented via an FPGA platform to verify its feasibility.     

Strange non-chaotic attractors in a state controlled-cellular neural network-based quasiperiodically forced MLC circuit

The adsorption energies, bond order, atomic charge, optical properties, and electrostatic potential of nitrogen molecules of armchair single-walled carbon nanotubes (SWCNTs) and nitrogen-doped single-walled carbon nanotubes (N-SWCNTs) were investigated using density functional theory (DFT). Our results show that adsorption of the \(\hbox {N}_{2}\) molecules on the external wall of a nanotube is more effective than on the internal wall in SWCNTs. The results show that \(\hbox {N}_{2}\) molecule(s) are weakly bonded to SWCNTs and N-SWCNTs through van der Waals-type interactions. The interaction of \(\hbox {N}_{2}\) molecules with SWCNTs and N-SWCNTs is physisorption as the adsorption energy and charge transfer are small, and adsorption distance is large. The electronic transitions from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) (\(\hbox {H}\rightarrow \hbox {L}\)) have the maximum wavelength and the lowest oscillator strength. The potential sensor on the surface of pristine SWCNTs and N-SWCNTs for the adsorption of \(\hbox {N}_{2}\) molecule(s) is investigated. The N-loaded single-walled carbon nanotube is introduced as a better \(\hbox {N}_{2}\) molecule(s) detector when compared with SWCNTs.     

Numerical analyses and breadboard experiments of twin attractors in two-neuron-based non-autonomous Hopfield neural network

This paper investigates twin attractors in a two-neuron-based non-autonomous Hopfield neural network (HNN) through numerical analyses and hardware experiments. Stability analysis of the DC equilibrium point is executed and an unstable saddle-focus is found in the parameter region of interest. The stimulus-associated dynamical behaviors are numerically explored by bifurcation diagrams and dynamical map in two-dimensional parameter-space, from which coexisting twin attractors behavior can be observed with the variations of two stimulus-associated parameters. Moreover, breadboard experiment investigations are carried out, which effectively verify the numerical simulations.     

Fixed-point quantum search

The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.     

Active noise control in a duct by an analog neural network circuit

Conventional active noise control (ANC) in ducts has been realized with digital signal processing. The physical size of the conventional ANC systems is usually large owing to the signal processing interval, and the cost of the system depends on the price of the digital signal processor (DSP). This paper proposes a new ANC system with an analog neural network circuit, which will process signals in short time periods without DSP. The proposed neural network circuit has a simple structure consisting of analog multipliers and an integrator, and we simulated the performance of the circuit by HSPICE. We also fabricated a circuit connected to a real duct and confirmed operation of the proposed ANC system.     

Coexistence behavior of asymmetric attractors in hyperbolic-type memristive Hopfield neural network and its application in image encryption

The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits. This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network(HNN) with coupled hyperbolic memristors through the modification of a single coupling connection weight. The bistable mode of the hyperbolic memristive HNN(m HNN), characterized by the coexistence of asymmetric chaos and periodic attractors, is effectively demonstrated through the utilization of convent...     

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional(5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.     

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.     

Femtosecond soliton attractors in fibres

It is shown that the attraction of femtosecond solitons in fibres is possible. Soliton attraction is calculated numerically, taking into account the fifth-order non-linearity in the refractive index of the fibre for the varying initial conditions. The role of the non-linear and dispersion higher-order effects and self-frequency shift in soliton attraction is also considered.     

Trapped ion chain as a neural network: error resistant quantum computation

We demonstrate the possibility of realizing a neural network in a chain of trapped ions with induced long range interactions. Such models permit one to store information distributed over the whole system. The storage capacity of such a network, which depends on the phonon spectrum of the system, can be controlled by changing the external trapping potential. We analyze the implementation of error resistant universal quantum information processing in such systems.