A solution method for decomposing vector fields in Hamilton energy

Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the existing systems.This derivation process consists of three steps:step 1,decomposing the vector field;step 2,solving the Hamilton energy function;and step 3,verifying uniqueness.In order to easily choose an appropriate decomposition method,we propose a classification criterion based on the form of system state variables,i.e.,type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation.Moreover,exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition.Finally,we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior.This solution provides a new approach for deducing the Hamilton energy function,especially in high-dimensional systems.

Diffusive field coupling-induced synchronization between neural circuits under energy balance

When charged bodies come up close to each other,the field energy is diffused and their states are regulated under bidirectional field coupling.For biological neurons,the diversity in intrinsic electric and magnetic field energy can create synaptic connection for fast energy balance and synaptic current is passed across the synapse channel;as a result,energy is pumped and exchanged to induce synchronous firing modes.In this paper,a capacitor is used to connect two neural circuits and energy propagation is activated along the coupling channel.The intrinsic field energy in the two neural circuits is exchanged and the coupling intensity is controlled adaptively using the Heaviside function.Some field energy is saved in the coupling channel and is then sent back to the coupled neural circuits to reach energy balance.Therefore the circuits can reach possible energy balance and complete synchronization.It is possible that the diffusive energy of the coupled neurons inspires the synaptic connections to grow stronger for possible energy balance.

Memristors-coupled neuron models with multiple firing patterns and homogeneous and heterogeneous multistability

Memristors are extensively used to estimate the external electromagnetic stimulation and synapses for neurons.In this paper,two distinct scenarios,i.e.,an ideal memristor serves as external electromagnetic stimulation and a locally active memristor serves as a synapse,are formulated to investigate the impact of a memristor on a two-dimensional Hindmarsh-Rose neuron model.Numerical simulations show that the neuronal models in different scenarios have multiple burst firing patterns.The introduction of the memristor makes the neuronal model exhibit complex dynamical behaviors.Finally,the simulation circuit and DSP hardware implementation results validate the physical mechanism,as well as the reliability of the biological neuron model.

Dynamical analysis,geometric control and digital hardware implementation of a complex-valued laser system with a locally active memristor

In order to make the peak and offset of the signal meet the requirements of artificial equipment,dynamical analysis and geometric control of the laser system have become indispensable.In this paper,a locally active memristor with non-volatile memory is introduced into a complex-valued Lorenz laser system.By using numerical measures,complex dynamical behaviors of the memristive laser system are uncovered.It appears the alternating appearance of quasi-periodic and chaotic oscillations.The mechanism of transformation from a quasi-periodic pattern to a chaotic one is revealed from the perspective of Hamilton energy.Interestingly,initial-values-oriented extreme multi-stability patterns are found,where the coexisting attractors have the same Lyapunov exponents.In addition,the introduction of a memristor greatly improves the complexity of the laser system.Moreover,to control the amplitude and offset of the chaotic signal,two kinds of geometric control methods including amplitude control and rotation control are designed.The results show that these two geometric control methods have revised the size and position of the chaotic signal without changing the chaotic dynamics.Finally,a digital hardware device is developed and the experiment outputs agree fairly well with those of the numerical simulations.

Memristive oscillator to memristive map,energy characteristic

Most of nonlinear oscillators composed of capacitive and inductive variables can obtain the Hamilton energy by using the Helmholtz theorem when the models are rewritten in equivalent vector forms.The energy functions for biophysical neurons can be obtained by applying scale transformation on the physical field energy in their equivalent neural circuits.Realistic dynamical systems often have exact energy functions,while some mathematical models just suggest generic Lyapunov functions,and the energy function is effective to predict mode transition.In this paper,a memristive oscillator is approached by two kinds of memristor-based nonlinear circuits,and the energy functions are defined to predict the dependence of oscillatory modes on energy level.In absence of capacitive variable for capacitor,the physical time t and charge q are converted into dimensionless variables by using combination of resistance and inductance(L,R),e.g.,τ=t×R/L.Discrete energy function for each memristive map is proposed by applying the similar weights as energy function for the memristive oscillator.For example,energy function for the map is obtained by replacing the variables and parameters of the memristive oscillator with corresponding variables and parameters for the memristive map.The memristive map prefers to keep lower average energy than the memristive oscillator,and chaos is generated in a discrete system with two variables.The scheme is helpful for energy definition in maps,and it provides possible guidance for verifying the reliability of maps by considering the energy characteristic.

Two simple memristive maps with adaptive energy regulation and digital signal process verification

Mathematical models can produce desired dynamics and statistical properties with the insertion of suitable nonlinear terms,while energy characteristics are crucial for practical application because any hardware realizations of nonlinear systems are relative to energy flow.The involvement of memristive terms relative to memristors enables multistability and initial-dependent property in memristive systems.In this study,two kinds of memristors are used to couple a capacitor or an inductor,along with a nonlinear resistor,to build different neural circuits.The corresponding circuit equations are derived to develop two different types of memristive oscillators,which are further converted into two kinds of memristive maps after linear transformation.The Hamilton energy function for memristive oscillators is obtained by applying the Helmholz theorem or by mapping from the field energy of the memristive circuits.The Hamilton energy functions for both memristive maps are obtained by replacing the gains and discrete variables for the memristive oscillator with the corresponding parameters and variables.The two memristive maps have rich dynamic behaviors including coherence resonance under noisy excitation,and an adaptive growth law for parameters is presented to express the self-adaptive property of the memristive maps.A digital signal process(DSP)platform is used to verify these results.Our scheme will provide a theoretical basis and experimental guidance for oscillator-to-map transformation and discrete map-energy calculation.

An adaptive energy regulation in a memristive map linearized from a circuit with two memristive channels

Nonlinear circuits can show multistability when a magnetic flux-dependent memristor(MFDM) or a charge-sensitive memristor(CSM) is incorporated into a one branch circuit,which helps estimate magnetic or electric field effects.In this paper,two different kinds of memristors are incorporated into two branch circuits composed of a capacitor and a nonlinear resistor,thus a memristive circuit with double memristive channels is designed.The circuit equations are presented,and the dynamics in this oscillator with two memristive terms are discussed.Then,the memristive oscillator is converted into a memristive map by applying linear transformation on the sampled time series for the memristive oscillator.The Hamilton energy function for the memristive oscillator is obtained by using the Helmholtz theorem,and it can be mapped from the field energy of the memristive circuit.An energy function for the dual memristive map is suggested by imposing suitable weights on the discrete energy function.The dynamical behaviors of the new memristive map are investigated,and an adaptive law is proposed to regulate the firing mode in the memristive map.This work will provide a theoretical basis and experimental guidance for oscillator-to-map transformation and discrete map energy calculation.

Energy flow accounts for the adaptive property of functional synapses

In the presence of external stimuli and electromagnetic radiation(EMR),biological neurons can exhibit different firing patterns and switch to appropriate firing modes because of intrinsic self-adaption.Coupling to memristive synapses can discern the EMR effect,and memristive synapses connecting to neurons can be effectively regulated by external physical fields.From a dynamical viewpoint,the appropriate setting for memristive synapse intensity can trigger changes in neural activities;however,the biophysical mechanism of adaptive regulation in the memristive biophysical neuron has not been clarified.Herein,a memristor is used to control a simple neural circuit by generating a memristive current,and an equivalent memristive neuron model is obtained.A single firing mode can be stabilized in the absence of EMR,while multiple firing modes occur in the neuron under EMR.The gain of the memristive synaptic current is dependent on the energy flow,and the shunted energy flow in the memristive channel can control the energy ratio between the electric field and magnetic field.The growth and enhancement of the memristive synapse depend on the energy flow across the memristive channel.The memristive synapse is enhanced when its field energy is below the threshold,and it is suppressed when its field energy is above the threshold.These results explain why and how multiple firing modes are induced and controlled in biological neurons.Furthermore,the self-adaption property of memristive neurons was also clarified.Thus,the control of energy flow in the memristive synapse can effectively regulate the membrane potentials,and neural activities can be effectively controlled to select suitable body gaits.

Phase synchronization and energy balance between neurons

A functional neuron has been developed from a simple neural circuit by incorporating a phototube and a thermistor in different branch circuits.The physical field energy is controlled by the photocurrent across the phototube and the channel current across the thermistor.The firing mode of this neuron is controlled synchronously by external temperature and illumination.There is energy diversity when two functional neurons are exposed to different illumination and temperature conditions.As a result,synapse connections can be created and activated in an adaptive way when field energy is exchanged between neurons.We propose two kinds of criteria to discuss the enhancement of synapse connections to neurons.The energy diversity between neurons determines the increase of the coupling intensity and synaptic current for neurons,and the realization of synchronization is helpful in maintaining energy balance between neurons.The first criterion is similar to the saturation gain scheme in that the coupling intensity is increased with a constant step within a certain period until it reaches energy balance or complete synchronization.The second criterion is that the coupling intensity increases exponentially before reaching energy balance.When two neurons become non-identical,phase synchronization can be controlled during the activation of synapse connections to neurons.For two identical neurons,the second criterion for taming synaptic intensity is effective for reaching complete synchronization and energy balance,even in the presence of noise.This indicates that a synapse connection may prefer to enhance its coupling intensity exponentially.These results are helpful in discovering why synapses are awaken and synaptic current becomes time-varying when any neurons are excited by external stimuli.The potential biophysical mechanism is that energy balance is broken and then synapse connections are activated to maintain an adaptive energy balance between the neurons.These results provide guidance for designing and training intelligent neural networks by taming the coupling channels with gradient energy distribution.