Asymptotic Input-Output Relationship Predicts Electric Field Effect on Sublinear Dendritic Integration of AMPA Synapses.

An extracellular electric field (EF) induces transmembrane polarizations on extremely inhomogeneous spaces. Evidence shows that EF-induced somatic polarization in pyramidal cells can modulate the neuronal input-output (I/O) function. However, it remains unclear whether and how dendritic polarization participates in the dendritic integration and contributes to the neuronal I/O function. To this end, we built a computational model of a simplified pyramidal cell with multi-dendritic tufts, one dendritic trunk, and one soma to describe the interactions among EF, dendritic integration, and somatic output, in which the EFs were modeled by inserting inhomogeneous extracellular potentials. We aimed to establish the underlying relationship between dendritic polarization and dendritic integration by analyzing the dynamics of subthreshold membrane potentials in response to AMPA synapses in the presence of constant EFs. The model-based singular perturbation analysis showed that the equilibrium mapping of a fast subsystem can serve as the asymptotic subthreshold I/O relationship for sublinear dendritic integration. This allows us to predict the tendency of EF-mediated dendritic integration by showing how EF changes modify equilibrium mapping. EF-induced hyperpolarization of distal dendrites receiving synapses inputs was found to play a key role in facilitating the AMPA receptor-evoked excitatory postsynaptic potential (EPSP) by enhancing the driving force of synaptic inputs. A significantly higher efficacy of EF modulation effect on global AMPA-type dendritic integration was found compared with local AMPA-type dendritic integration. During the generation of an action potential (AP), the relative contribution of EF-modulated dendritic integration and EF-induced somatic polarization was determined to show their collaboration in promoting or inhibiting the somatic excitability, depending on the EF polarity. These findings are crucial for understanding the EF modulation effect on neuronal computation, which provides insight into the modulation mechanism of noninvasive brain modulation.

Neural recruitment by ephaptic coupling in epilepsy.

OBJECTIVE: One of the challenges in treating patients with drug-resistant epilepsy is that the mechanisms of seizures are unknown. Most current interventions are based on the assumption that epileptic activity recruits neurons and progresses by synaptic transmission. However, several experimental studies have shown that neural activity in rodent hippocampi can propagate independently of synaptic transmission. Recent studies suggest these waves are self-propagating by electric field (ephaptic) coupling. In this study, we tested the hypothesis that neural recruitment during seizures can occur by electric field coupling. METHODS: 4-Aminopyridine was used in both in vivo and in vitro preparation to trigger seizures or epileptiform activity. A transection was made in the in vivo hippocampus and in vitro hippocampal and cortical slices to study whether the induced seizure activity can recruit neurons across the gap. A computational model was built to test whether ephaptic coupling alone can account for neural recruitment across the transection. The model prediction was further validated by in vitro experiments. RESULTS: Experimental results show that electric fields generated by seizure-like activity in the hippocampus both in vitro and in vivo can recruit neurons locally and through a transection of the tissue. The computational model suggests that the neural recruitment across the transection is mediated by electric field coupling. With in vitro experiments, we show that a dielectric material can block the recruitment of epileptiform activity across a transection, and that the electric fields measured within the gap are similar to those predicted by model simulations. Furthermore, this nonsynaptic neural recruitment is also observed in cortical slices, suggesting that this effect is robust in brain tissue. SIGNIFICANCE: These results indicate that ephaptic coupling, a nonsynaptic mechanism, can underlie neural recruitment by a small electric field generated by seizure activity and could explain the low success rate of surgical transections in epilepsy patients.

Slow periodic activity in the longitudinal hippocampal slice can self-propagate non-synaptically by a mechanism consistent with ephaptic coupling.

KEY POINTS: Slow periodic activity can propagate with speeds around 0.1 m s-1 and be modulated by weak electric fields. Slow periodic activity in the longitudinal hippocampal slice can propagate without chemical synaptic transmission or gap junctions, but can generate electric fields which in turn activate neighbouring cells. Applying local extracellular electric fields with amplitude in the range of endogenous fields is sufficient to modulate or block the propagation of this activity both in the in silico and in the in vitro models. Results support the hypothesis that endogenous electric fields, previously thought to be too small to trigger neural activity, play a significant role in the self-propagation of slow periodic activity in the hippocampus. Experiments indicate that a neural network can give rise to sustained self-propagating waves by ephaptic coupling, suggesting a novel propagation mechanism for neural activity under normal physiological conditions. ABSTRACT: Slow oscillations are a standard feature observed in the cortex and the hippocampus during slow wave sleep. Slow oscillations are characterized by low-frequency periodic activity (<1 Hz) and are thought to be related to memory consolidation. These waves are assumed to be a reflection of the underlying neural activity, but it is not known if they can, by themselves, be self-sustained and propagate. Previous studies have shown that slow periodic activity can be reproduced in the in vitro preparation to mimic in vivo slow oscillations. Slow periodic activity can propagate with speeds around 0.1 m s-1 and be modulated by weak electric fields. In the present study, we show that slow periodic activity in the longitudinal hippocampal slice is a self-regenerating wave which can propagate with and without chemical or electrical synaptic transmission at the same speeds. We also show that applying local extracellular electric fields can modulate or even block the propagation of this wave in both in silico and in vitro models. Our results support the notion that ephaptic coupling plays a significant role in the propagation of the slow hippocampal periodic activity. Moreover, these results indicate that a neural network can give rise to sustained self-propagating waves by ephaptic coupling, suggesting a novel propagation mechanism for neural activity under normal physiological conditions.

Particle swarm optimization algorithm based parameters estimation and control of epileptiform spikes in a neural mass model.

This paper proposes an epilepsy detection and closed-loop control strategy based on Particle Swarm Optimization (PSO) algorithm. The proposed strategy can effectively suppress the epileptic spikes in neural mass models, where the epileptiform spikes are recognized as the biomarkers of transitions from the normal (interictal) activity to the seizure (ictal) activity. In addition, the PSO algorithm shows capabilities of accurate estimation for the time evolution of key model parameters and practical detection for all the epileptic spikes. The estimation effects of unmeasurable parameters are improved significantly compared with unscented Kalman filter. When the estimated excitatory-inhibitory ratio exceeds a threshold value, the epileptiform spikes can be inhibited immediately by adopting the proportion-integration controller. Besides, numerical simulations are carried out to illustrate the effectiveness of the proposed method as well as the potential value for the model-based early seizure detection and closed-loop control treatment design.

Spike-frequency adaptation of a two-compartment neuron modulated by extracellular electric fields.

Spike-frequency adaptation has been shown to play an important role in neural coding. Based on a reduced two-compartment model, here we investigate how two common adaptation currents, i.e., voltage-sensitive potassium current (I(M)) and calcium-sensitive potassium current (I(AHP)), modulate neuronal responses to extracellular electric fields. It is shown that two adaptation mechanisms lead to distinct effects on the dynamical behavior of the neuron to electric fields. These effects depend on a neuronal morphological parameter that characterizes the ratio of soma area to total membrane area and internal coupling conductance. In the case of I(AHP) current, changing the morphological parameter switches spike initiation dynamics between saddle-node on invariant cycle bifurcation and supercritical Hopf bifurcation, whereas it only switches between subcritical and supercritical Hopf bifurcations for I(M) current. Unlike the morphological parameter, internal coupling conductance is unable to alter the bifurcation scenario for both adaptation currents. We also find that the electric field threshold for triggering neuronal steady-state firing is determined by two parameters, especially by the morphological parameter. Furthermore, the neuron with I(AHP) current generates mixed-mode oscillations through the canard phenomenon for some small values of the morphological parameter. All these results suggest that morphological properties play a critical role in field-induced effects on neuronal dynamics, which could qualitatively alter the outcome of adaptation by modulating internal current between soma and dendrite. The findings are readily testable in experiments, which could help to reveal the mechanisms underlying how the neuron responds to electric field stimulus.

Suppression of seizures based on the multi-coupled neural mass model.

Epilepsy is one of the most common serious neurological disorders, which affects approximately 1% of population in the world. In order to effectively control the seizures, we propose a novel control methodology, which combines the feedback linearization control (FLC) with the underlying mechanism of epilepsy, to achieve the suppression of seizures. The three coupled neural mass model is constructed to study the property of the electroencephalographs (EEGs). Meanwhile, with the model we research on the propagation of epileptiform waves and the synchronization of populations, which are taken as the foundation of our control method. Results show that the proposed approach not only yields excellent performances in clamping the pathological spiking patterns to the reference signals derived under the normal state but also achieves the normalization of the pathological parameter, where the parameters are estimated from EEGs with Unscented Kalman Filter. The specific contribution of this paper is to treat the epilepsy from its pathogenesis with the FLC, which provides critical theoretical basis for the clinical treatment of neurological disorders.

Multiple feature extraction and classification of electroencephalograph signal for Alzheimers' with spectrum and bispectrum.

In this paper, we have combined experimental neurophysiologic recording and statistical analysis to investigate the nonlinear characteristic and the cognitive function of the brain. Spectrum and bispectrum analyses are proposed to extract multiple effective features of electroencephalograph (EEG) signals from Alzheimer's disease (AD) patients and further applied to distinguish AD patients from the normal controls. Spectral analysis based on autoregressive Burg method is first used to quantify the power distribution of EEG series in the frequency domain. Compared to the control group, the relative power spectral density of AD group is significantly higher in the theta frequency band, while lower in the alpha frequency bands. In addition, median frequency of spectrum is decreased, and spectral entropy ratio of these two frequency bands undergoes drastic changes at the P3 electrode in the central-parietal brain region, implying that the electrophysiological behavior in AD brain is much slower and less irregular. In order to explore the nonlinear high order information, bispectral analysis which measures the complexity of phase-coupling is further applied to P3 electrode in the whole frequency band. It is demonstrated that less bispectral peaks appear and the amplitudes of peaks fall, suggesting a decrease of non-Gaussianity and nonlinearity of EEG in ADs. Notably, the application of this method to five brain regions shows higher concentration of the weighted center of bispectrum and lower complexity reflecting phase-coupling by bispectral entropy. Based on spectrum and bispectrum analyses, six efficient features are extracted and then applied to discriminate AD from the normal in the five brain regions. The classification results indicate that all these features could differentiate AD patients from the normal controls with a maximum accuracy of 90.2%. Particularly, different brain regions are sensitive to different features. Moreover, the optimal combination of features obtained by discriminant analysis may improve the classification accuracy. These results demonstrate the great promise for scape EEG spectral and bispectral features as a potential effective method for detection of AD, which may facilitate our understanding of the pathological mechanism of the disease.

Endogenous field feedback promotes the detectability for exogenous electric signal in the hybrid coupled population.

This paper presents the endogenous electric field in chemical or electrical synaptic coupled networks, aiming to study the role of endogenous field feedback in the signal propagation in neural systems. It shows that the feedback of endogenous fields to network activities can reduce the required energy of the noise and enhance the transmission of input signals in hybrid coupled populations. As a common and important nonsynaptic interactive method among neurons, particularly, the endogenous filed feedback can not only promote the detectability of exogenous weak signal in hybrid coupled neural population but also enhance the robustness of the detectability against noise. Furthermore, with the increasing of field coupling strengths, the endogenous field feedback is conductive to the stochastic resonance by facilitating the transition of cluster activities from the no spiking to spiking regions. Distinct from synaptic coupling, the endogenous field feedback can play a role as internal driving force to boost the population activities, which is similar to the noise. Thus, it can help to transmit exogenous weak signals within the network in the absence of noise drive via the stochastic-like resonance.

Complexity extraction of electroencephalograms in Alzheimer's disease with weighted-permutation entropy.

In this paper, weighted-permutation entropy (WPE) is applied to investigating the complexity abnormalities of Alzheimer's disease (AD) by analyzing 16-channel electroencephalograph (EEG) signals from 14 severe AD patients and 14 age-matched normal subjects. The WPE values are estimated in the delta, the theta, the alpha, and the beta sub-bands for each channel with an overlapped sliding window. WPE is modified from the permutation entropy (PE), which has been recently suggested as a measurement to extract the complexity of the EEG signals. The advantage of WPE over PE is verified by both the model simulated and the experimental EEG signals. Although the results show that both the average PE and WPE of AD patients are decreased in contrast with the normal group in these four sub-bands, especially in the theta band, WPE can exhibit a better performance in distinguishing the AD patients from the normal controls by the more significant differences in the four sub-bands, which may be attributed to the brain dysfunction. Thus, it suggests that WPE may become a probable useful tool to detect brain dysfunction in AD and it seems to be promising to disclose the abnormalities of brain activity for other neural disease.

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model.

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.

Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks.

The phenomenon of stochastic resonance in Newman-Watts small-world neuronal networks is investigated when the strength of synaptic connections between neurons is adaptively adjusted by spike-time-dependent plasticity (STDP). It is shown that irrespective of the synaptic connectivity is fixed or adaptive, the phenomenon of stochastic resonance occurs. The efficiency of network stochastic resonance can be largely enhanced by STDP in the coupling process. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed one when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, we demonstrate that the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of the adaptive network. Furthermore, the small-world topology can significantly affect stochastic resonance of excitable neuronal networks. It is found that there exists an optimal probability of adding links by which the noise-induced transmission of weak periodic signal peaks.

Decreased coherence and functional connectivity of electroencephalograph in Alzheimer's disease.

In this paper, we investigate the abnormalities of electroencephalograph (EEG) signals in the Alzheimer's disease (AD) by analyzing 16-scalp electrodes EEG signals and make a comparison with the normal controls. Coherence is introduced to measure the pair-wise normalized linear synchrony and functional correlations between two EEG signals in different frequency domains, and graph analysis is further used to investigate the influence of AD on the functional connectivity of human brain. Data analysis results show that, compared with the control group, the pair-wise coherence of AD group is significantly decreased, especially for the theta and alpha frequency bands in the frontal and parieto-occipital regions. Furthermore, functional connectivity among different brain regions is reconstructed based on EEG, which exhibit obvious small-world properties. Graph analysis demonstrates that the local functional connections between regions for AD decrease. In addition, it is found that small-world properties of AD networks are largely weakened, by calculating its average path lengths, clustering coefficients, global efficiency, local efficiency, and small-worldness. The obtained results show that both pair-wise coherence and functional network can be taken as effective measures to distinguish AD patients from the normal, which may benefit our understanding of the disease.

A combined method to estimate parameters of the thalamocortical model from a heavily noise-corrupted time series of action potential.

A combined method composing of the unscented Kalman filter (UKF) and the synchronization-based method is proposed for estimating electrophysiological variables and parameters of a thalamocortical (TC) neuron model, which is commonly used for studying Parkinson's disease for its relay role of connecting the basal ganglia and the cortex. In this work, we take into account the condition when only the time series of action potential with heavy noise are available. Numerical results demonstrate that not only this method can estimate model parameters from the extracted time series of action potential successfully but also the effect of its estimation is much better than the only use of the UKF or synchronization-based method, with a higher accuracy and a better robustness against noise, especially under the severe noise conditions. Considering the rather important role of TC neuron in the normal and pathological brain functions, the exploration of the method to estimate the critical parameters could have important implications for the study of its nonlinear dynamics and further treatment of Parkinson's disease.

Frequency-dependent membrane polarization across neocortical cell types and subcellular elements by transcranial alternating current stimulation.

Objective.Transcranial alternating current stimulation (tACS) is a non-invasive brain stimulation technique that directly interacts with ongoing brain oscillations in a frequency-dependent manner. However, it remains largely unclear how the cellular effects of tACS vary between cell types and subcellular elements.Approach.In this study, we use a set of morphologically realistic models of neocortical neurons to simulate the cellular response to uniform oscillating electric fields (EFs). We systematically characterize the membrane polarization in the soma, axons, and dendrites with varying field directions, intensities, and frequencies.Main results.Pyramidal cells are more sensitive to axial EF that is roughly parallel to the cortical column, while interneurons are sensitive to axial EF and transverse EF that is tangent to the cortical surface. Membrane polarization in each subcellular element increases linearly with EF intensity, and its slope, i.e. polarization length, highly depends on the stimulation frequency. At each frequency, pyramidal cells are more polarized than interneurons. Axons usually experience the highest polarization, followed by the dendrites and soma. Moreover, a visible frequency resonance presents in the apical dendrites of pyramidal cells, while the other subcellular elements primarily exhibit low-pass filtering properties. In contrast, each subcellular element of interneurons exhibits complex frequency-dependent polarization. Polarization phase in each subcellular element of cortical neurons lags that of field and exhibits high-pass filtering properties. These results demonstrate that the membrane polarization is not only frequency-dependent, but also cell type- and subcellular element-specific. Through relating effective length and ion mechanism with polarization, we emphasize the crucial role of cell morphology and biophysics in determining the frequency-dependent membrane polarization.Significance.Our findings highlight the diverse polarization patterns across cell types as well as subcellular elements, which provide some insights into the tACS cellular effects and should be considered when understanding the neural spiking activity by tACS.

Electric field effects on neuronal input-output relationship by regulating NMDA spikes.

Evidence shows that the dendritic polarization induced by weak electrical field (EF) can affect the neuronal input-output function via modulating dendritic integration of AMPA synapses, indicating that the supralinear dendritic integration of NMDA synapses can also be influenced by dendritic polarization. However, it remains unknown how dendritic polarization affects NMDA-type dendritic integration, and then contributes to neuronal input-output relationship. Here, we used a computational model of pyramidal neuron with inhomogeneous extracellular potentials to characterize the relationship among EF, dendritic integration, and somatic output. Basing on singular perturbation we analyzed the subthreshold dynamics of membrane potentials in response to NMDA synapses, and found that the equilibrium mapping of a fast subsystem can characterize the asymptotic subthreshold input-output (sI/O) relationship for EF-regulated supralinear dendritic integration, allowing us to predict the tendency of EF-regulated dendritic integration by showing the variation of equilibrium mapping under EF stimulation. EF-induced depolarization at distal dendrites receiving synapses plays a crucial role in shifting the steep change of sI/O left by facilitating dendritic NMDA spike generation and in decreasing the plateau of sI/O via reducing driving force. And more effective EF modulation appears at sparsely activated NMDA receptors compared with clustered synaptic inputs. During the action potential (AP) generation, the respective contribution of EF-regulated dendritic integration and EF-induced somatic polarization was identified to show their synergetic or antagonistic effect on AP generation, depending on neuronal excitability. These results provided insight in understanding the modulation effect of EF on neuronal computation, which is important for optimizing noninvasive brain stimulation. Supplementary Information: The online version contains supplementary material available at 10.1007/s11571-022-09922-y.

Assessment of impaired consciousness using EEG-based connectivity features and convolutional neural networks.

Growing electroencephalogram (EEG) studies have linked the abnormities of functional brain networks with disorders of consciousness (DOC). However, due to network data's high-dimensional and non-Euclidean properties, it is difficult to exploit the brain connectivity information that can effectively detect the consciousness levels of DOC patients via deep learning. To take maximum advantage of network information in assessing impaired consciousness, we utilized the functional connectivity with convolutional neural network (CNN) and employed three rearrangement schemes to improve the evaluation performance of brain networks. In addition, the gradient-weighted class activation mapping (Grad-CAM) was adopted to visualize the classification contributions of connections among different areas. We demonstrated that the classification performance was significantly enhanced by applying network rearrangement techniques compared to those obtained by the original connectivity matrix (with an accuracy of 75.0%). The highest classification accuracy (87.2%) was achieved by rearranging the alpha network based on the anatomical regions. The inter-region connections (i.e., frontal-parietal and frontal-occipital connectivity) played dominant roles in the classification of patients with different consciousness states. The effectiveness of functional connectivity in revealing individual differences in brain activity was further validated by the correlation between behavioral performance and connections among specific regions. These findings suggest that our proposed assessment model could detect the residual consciousness of patients.

Scalable Multi-Hierarchy Embedded Platform for Neural Population Simulations.

Brain-inspired structured neural circuits are the cornerstones of both computational and perceived intelligence. Real-time simulations of large-scale high-dimensional neural populations with complex nonlinearities pose a significant challenge. Taking advantage of distributed computations using embedded multi-cores, we propose an ARM-based scalable multi-hierarchy parallel computing platform (EmPaas) for neural population simulations. EmPaas is constructed using 340 ARM Cortex-M4 microprocessors to achieve high-speed and high-accuracy parallel computing. The tree-two-dimensional grid-like hybrid topology completes the overall construction, reducing communication strain and power consumption. As an instance of embedded computing, the optimized model for a biologically plausible basal ganglia-thalamus (BG-TH) network is deployed into this platform to verify the performance. At an operating frequency of 168 MHz, the BG-TH network consisting of 4000 Izhikevich neurons is simulated in the platform for 3000 ms with a power consumption of 56.565 mW per core and an actual time of 2748.57 ms, which shows the parallel computing approach significantly improves computational efficiency. EmPaas can meet the requirement of real-time performance with the maximum amount of 2000 Izhikevich neurons loaded in each Extended Community Unit (ECUnit), which provides a new practical method for research in large-scale brain network simulation and brain-inspired computing.

Effects of time delay on the stochastic resonance in small-world neuronal networks.

The effects of time delay on stochastic resonance in small-world neuronal networks are investigated. Without delay, an intermediate intensity of additive noise is able to optimize the temporal response of the neural system to the subthreshold periodic signal imposed on all neurons constituting the network. The time delay in the coupling process can either enhance or destroy stochastic resonance of neuronal activity in the small-world network. In particular, appropriately tuned delays can induce multiple stochastic resonances, which appear intermittently at integer multiples of the oscillation period of weak external forcing. It is found that the delay-induced multiple stochastic resonances are most efficient when the forcing frequency is close to the global-resonance frequency of each individual neuron. Furthermore, the impact of time delay on stochastic resonance is largely independent of the small-world topology, except for resonance peaks. Considering that information transmission delays are inevitable in intra- and inter-neuronal communication, the presented results could have important implications for the weak signal detection and information propagation in neural systems.

The estimation of neurotransmitter release probability in feedforward neuronal network based on adaptive synchronization.

In this paper, we proposed a new approach to estimate unknown parameters and topology of a neuronal network based on the adaptive synchronization control scheme. A virtual neuronal network is constructed as an observer to track the membrane potential of the corresponding neurons in the original network. When they achieve synchronization, the unknown parameters and topology of the original network are obtained. The method is applied to estimate the real-time status of the connection in the feedforward network and the neurotransmitter release probability of unreliable synapses is obtained by statistic computation. Numerical simulations are also performed to demonstrate the effectiveness of the proposed adaptive controller. The obtained results may have important implications in system identification in neural science.

Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization.

This paper presents an adaptive anticipatory synchronization based method for simultaneous identification of topology and parameters of uncertain nonlinearly coupled complex dynamical networks with time delays. An adaptive controller is proposed, based on Lyapunov stability theorem and Barbǎlat's Lemma, to guarantee the stability of the anticipatory synchronization manifold between drive and response networks. Meanwhile, not only the identification criteria of network topology and system parameters are obtained but also the anticipatory time is identified. Numerical simulation results illustrate the effectiveness of the proposed method.