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Chizhevsky VN, Lakhmitski MV. Improvement of signal propagation in the optoelectronic artificial spiking neuron by vibrational resonance. Phys Rev E 2024; 109:014211. [PMID: 38366496 DOI: 10.1103/physreve.109.014211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2023] [Accepted: 12/04/2023] [Indexed: 02/18/2024]
Abstract
Experimental evidence of vibrational resonance (VR) in the optoelectronic artificial spiking neuron based on a single photon avalanche diode and a vertical cavity laser driven by two periodic signals with low and high frequencies is reported. It is shown that a very weak subthreshold low-frequency (LF) periodic signal can be greatly amplified by the additional high-frequency (HF) signal. The phenomenon shows up as a nonmonotonic resonant dependence of the LF response on the amplitude of the HF signal. Simultaneously, a strong resonant rise of the signal-to-noise ratio is also observed. In addition, for the characterization of VR an area under the first LF period in the probability density function of interspike intervals for the LF signal and the maximal amplitude in this area were used, both of which also demonstrate a resonant behavior depending on the amplitude of the HF signal.
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Affiliation(s)
- V N Chizhevsky
- B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, 220072 Minsk, Belarus
| | - M V Lakhmitski
- B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, 220072 Minsk, Belarus
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2
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Martínez N, Deza RR, Montani F. Characterizing the information transmission of inverse stochastic resonance and noise-induced activity amplification in neuronal systems. Phys Rev E 2023; 107:054402. [PMID: 37329070 DOI: 10.1103/physreve.107.054402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 04/13/2023] [Indexed: 06/18/2023]
Abstract
Purkinje cells exhibit a reduction of the mean firing rate at intermediate-noise intensities, which is somewhat reminiscent of the response enhancement known as "stochastic resonance" (SR). Although the comparison with the stochastic resonance ends here, the current phenomenon has been given the name "inverse stochastic resonance" (ISR). Recent research has demonstrated that the ISR effect, like its close relative "nonstandard SR" [or, more correctly, noise-induced activity amplification (NIAA)], has been shown to stem from the weak-noise quenching of the initial distribution, in bistable regimes where the metastable state has a larger attraction basin than the global minimum. To understand the underlying mechanism of the ISR and NIAA phenomena, we study the probability distribution function of a one-dimensional system subjected to a bistable potential that has the property of symmetry, i.e., if we change the sign of one of its parameters, we can obtain both phenomena with the same properties in the depth of the wells and the width of their basins of attraction subjected to Gaussian white noise with variable intensity. Previous work has shown that one can theoretically determine the probability distribution function using the convex sum between the behavior at small and high noise intensities. To determine the probability distribution function more precisely, we resort to the "weighted ensemble Brownian dynamics simulation" model, which provides an accurate estimate of the probability distribution function for both low and high noise intensities and, most importantly, for the transition of both behaviors. In this way, on the one hand, we show that both phenomena emerge from a metastable system where, in the case of ISR, the global minimum of the system is in a state of lower activity, while in the case of NIAA, the global minimum is in a state of increased activity, the importance of which does not depend on the width of the basins of attraction. On the other hand, we see that quantifiers such as Fisher information, statistical complexity, and especially Shannon entropy fail to distinguish them, but they show the existence of the mentioned phenomena. Thus, noise management may well be a mechanism by which Purkinje cells find an efficient way to transmit information in the cerebral cortex.
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Affiliation(s)
- Nataniel Martínez
- IFIMAR (CONICET), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, B7602AYL Mar del Plata, Argentina
| | - Roberto R Deza
- IFIMAR (CONICET), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, B7602AYL Mar del Plata, Argentina
| | - Fernando Montani
- IFLP (CONICET), Facultad de Ciencias Exactas, Universidad Nacional de La Plata, B1900 La Plata, Argentina
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3
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Calim A, Palabas T, Uzuntarla M. Stochastic and vibrational resonance in complex networks of neurons. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200236. [PMID: 33840216 DOI: 10.1098/rsta.2020.0236] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 02/03/2021] [Indexed: 05/22/2023]
Abstract
The concept of resonance in nonlinear systems is crucial and traditionally refers to a specific realization of maximum response provoked by a particular external perturbation. Depending on the system and the nature of perturbation, many different resonance types have been identified in various fields of science. A prominent example is in neuroscience where it has been widely accepted that a neural system may exhibit resonances at microscopic, mesoscopic and macroscopic scales and benefit from such resonances in various tasks. In this context, the two well-known forms are stochastic and vibrational resonance phenomena which manifest that detection and propagation of a feeble information signal in neural structures can be enhanced by additional perturbations via these two resonance mechanisms. Given the importance of network architecture in proper functioning of the nervous system, we here present a review of recent studies on stochastic and vibrational resonance phenomena in neuronal media, focusing mainly on their emergence in complex networks of neurons as well as in simple network structures that represent local behaviours of neuron communities. From this perspective, we aim to provide a secure guide by including theoretical and experimental approaches that analyse in detail possible reasons and necessary conditions for the appearance of stochastic resonance and vibrational resonance in neural systems. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.
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Affiliation(s)
- Ali Calim
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey
| | - Tugba Palabas
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey
| | - Muhammet Uzuntarla
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey
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Calim A, Longtin A, Uzuntarla M. Vibrational resonance in a neuron-astrocyte coupled model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200267. [PMID: 33840211 DOI: 10.1098/rsta.2020.0267] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 10/28/2020] [Indexed: 05/22/2023]
Abstract
Recent findings have revealed that not only neurons but also astrocytes, a special type of glial cells, are major players of neuronal information processing. It is now widely accepted that they contribute to the regulation of their microenvironment by cross-talking with neurons via gliotransmitters. In this context, we here study the phenomenon of vibrational resonance in neurons by considering their interaction with astrocytes. Our analysis of a neuron-astrocyte pair reveals that intracellular dynamics of astrocytes can induce a double vibrational resonance effect in the weak signal detection performance of a neuron, exhibiting two distinct wells centred at different high-frequency driving amplitudes. We also identify the underlying mechanism of this behaviour, showing that the interaction of widely separated time scales of neurons, astrocytes and driving signals is the key factor for the emergence and control of double vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.
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Affiliation(s)
- Ali Calim
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey
| | - Andre Longtin
- Department of Physics, University of Ottawa, Ottawa, Ontario, Canada
| | - Muhammet Uzuntarla
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey
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Zhang G, Liu Y, He L. Research on fault detection of asymmetric piecewise well-posed stochastic resonance system. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2021; 92:025116. [PMID: 33648081 DOI: 10.1063/5.0041204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2020] [Accepted: 01/30/2021] [Indexed: 06/12/2023]
Abstract
Stochastic resonance of an asymmetric piecewise well-posed system driven by a periodic forcing and Gaussian white noise is investigated. Aiming at the problem that the output saturation of the classical stochastic resonance (CSR) system needs to be further improved, the dimensionality of the quartic function is reduced to a quadratic function, and the well position of the function becomes asymmetric. First, the potential function and mean first passage time are analyzed, and then the signal to noise ratio formula of the system is derived through adiabatic approximation theory. Second, the system is simulated and tested. Theoretical analysis and numerical simulation show that the system in a well-posed symmetric case has better performance than the CSR system, but is better in a well-posed asymmetric case. Finally, the bearing fault detection is processed by using the proposed system. The results show that the fault frequency can be more accurately identified by the well-posed asymmetry, and the energy of the characteristic signal can be improved further. The theoretical basis and reference value of the system are provided for further application in practical engineering testing.
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Affiliation(s)
- Gang Zhang
- School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
| | - Yilin Liu
- School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
| | - Lifang He
- School of Optoelectronic Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
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Yang Z, Guo D, Zhang Y, Wu S, Yao D. Visual Evoked Response Modulation Occurs in a Complementary Manner Under Dynamic Circuit Framework. IEEE Trans Neural Syst Rehabil Eng 2019; 27:2005-2014. [DOI: 10.1109/tnsre.2019.2940712] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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8
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Mondal A, Upadhyay RK, Ma J, Yadav BK, Sharma SK, Mondal A. Bifurcation analysis and diverse firing activities of a modified excitable neuron model. Cogn Neurodyn 2019; 13:393-407. [PMID: 31354884 DOI: 10.1007/s11571-019-09526-z] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2018] [Revised: 12/27/2018] [Accepted: 02/20/2019] [Indexed: 11/29/2022] Open
Abstract
Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh-Rose model (H-R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.
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Affiliation(s)
- Argha Mondal
- 1Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand 826004 India.,2Computational Neuroscience Center, University of Washington, Seattle, USA
| | - Ranjit Kumar Upadhyay
- 1Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand 826004 India
| | - Jun Ma
- 3Department of Physics, Lanzhou University of Technology, Lanzhou, 730050 People's Republic of China
| | - Binesh Kumar Yadav
- 1Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand 826004 India
| | - Sanjeev Kumar Sharma
- 1Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand 826004 India
| | - Arnab Mondal
- 1Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand 826004 India
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Doungmo Goufo EF, Tabi CB. On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input. CHAOS (WOODBURY, N.Y.) 2019; 29:023104. [PMID: 30823721 DOI: 10.1063/1.5083180] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2018] [Accepted: 01/13/2019] [Indexed: 06/09/2023]
Abstract
Since the neurologists Hindmarsh and Rose improved the Hodgkin-Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh-Rose neurons with an external current input. Combining with fractional differentiation, the model is generalized with the introduction of an additional parameter, the non-integer order of the derivative σ, and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamics show that in the standard case where the control parameter σ=1, the nerve cell's behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as σ decreases (σ=0.9 and σ=0.85) with the pole of attraction becoming chaotic.
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Affiliation(s)
| | - Conrad Bertrand Tabi
- Botswana International University of Science and Technology, P/Bag 16, Palapye, Botswana
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10
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Agaoglu SN, Calim A, Hövel P, Ozer M, Uzuntarla M. Vibrational resonance in a scale-free network with different coupling schemes. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.09.070] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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11
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Propagation of firing rate by synchronization in a feed-forward multilayer Hindmarsh–Rose neural network. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.09.037] [Citation(s) in RCA: 50] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
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12
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Energy expenditure computation of a single bursting neuron. Cogn Neurodyn 2018; 13:75-87. [PMID: 30728872 PMCID: PMC6339863 DOI: 10.1007/s11571-018-9503-3] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2017] [Revised: 08/22/2018] [Accepted: 08/28/2018] [Indexed: 01/06/2023] Open
Abstract
Brief bursts of high-frequency spikes are a common firing pattern of neurons. The cellular mechanisms of bursting and its biological significance remain a matter of debate. Focusing on the energy aspect, this paper proposes a neural energy calculation method based on the Chay model of bursting. The flow of ions across the membrane of the bursting neuron with or without current stimulation and its power which contributes to the change of the transmembrane electrical potential energy are analyzed here in detail. We find that during the depolarization of spikes in bursting this power becomes negative, which was also discovered in previous research with another energy model. We also find that the neuron’s energy consumption during bursting is minimal. Especially in the spontaneous state without stimulation, the total energy consumption (2.152 × 10−7 J) during 30 s of bursting is very similar to the biological energy consumption (2.468 × 10−7 J) during the generation of a single action potential, as shown in Wang et al. (Neural Plast 2017, 2017a). Our results suggest that this property of low energy consumption could simply be the consequence of the biophysics of generating bursts, which is consistent with the principle of energy minimization. Our results also imply that neural energy plays a critical role in neural coding, which opens a new avenue for research of a central challenge facing neuroscience today.
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13
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Zhu J, Liu X. Measuring spike timing distance in the Hindmarsh-Rose neurons. Cogn Neurodyn 2017; 12:225-234. [PMID: 29564030 DOI: 10.1007/s11571-017-9466-9] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Revised: 11/28/2017] [Accepted: 12/19/2017] [Indexed: 11/28/2022] Open
Abstract
In the present paper, a simple spike timing distance is defined which can be used to measure the degree of synchronization with the information only encoded in the precise timing of the spike trains. Via calculating the spike timing distance defined in this paper, the spike train similarity of uncoupled Hindmarsh-Rose neurons in bursting or spiking states with different initial conditions is investigated and the results are compared with other spike train distance measures. Later, the spike timing distance measure is applied to study the synchronization of coupled or common noise-stimulated neurons. Counterintuitively, the addition of weak coupling or common noise doesn't enhance the degree of synchronization although after critical values, both of them can induce complete synchronizations. More interestingly, the common noise plays opposite roles for weak and strong enough couplings. Finally, it should be noted that the measure defined in this paper can be extended to measure large neuronal ensembles and the lag synchronization.
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Affiliation(s)
- Jinjie Zhu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 YuDao Street, Nanjing, 210016 Jiangsu Province People's Republic of China
| | - Xianbin Liu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 YuDao Street, Nanjing, 210016 Jiangsu Province People's Republic of China
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Yang X, Yu Y, Sun Z. Autapse-induced multiple stochastic resonances in a modular neuronal network. CHAOS (WOODBURY, N.Y.) 2017; 27:083117. [PMID: 28863486 DOI: 10.1063/1.4999100] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This study investigates the nontrivial effects of autapse on stochastic resonance in a modular neuronal network subjected to bounded noise. The resonance effect of autapse is detected by imposing a self-feedback loop with autaptic strength and autaptic time delay to each constituent neuron. Numerical simulations have demonstrated that bounded noise with the proper level of amplitude can induce stochastic resonance; moreover, the noise induced resonance dynamics can be significantly shaped by the autapse. In detail, for a specific range of autaptic strength, multiple stochastic resonances can be induced when the autaptic time delays are appropriately adjusted. These appropriately adjusted delays are detected to nearly approach integer multiples of the period of the external weak signal when the autaptic strength is very near zero; otherwise, they do not match the period of the external weak signal when the autaptic strength is slightly greater than zero. Surprisingly, in both cases, the differences between arbitrary two adjacent adjusted autaptic delays are always approximately equal to the period of the weak signal. The phenomenon of autaptic delay induced multiple stochastic resonances is further confirmed to be robust against the period of the external weak signal and the intramodule probability of subnetwork. These findings could have important implications for weak signal detection and information propagation in realistic neural systems.
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Affiliation(s)
- XiaoLi Yang
- College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - YanHu Yu
- College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - ZhongKui Sun
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
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Nobukawa S, Nishimura H, Yamanishi T. Chaotic Resonance in Typical Routes to Chaos in the Izhikevich Neuron Model. Sci Rep 2017; 7:1331. [PMID: 28465524 PMCID: PMC5430992 DOI: 10.1038/s41598-017-01511-y] [Citation(s) in RCA: 34] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2017] [Accepted: 03/29/2017] [Indexed: 11/09/2022] Open
Abstract
Chaotic resonance (CR), in which a system responds to a weak signal through the effects of chaotic activities, is a known function of chaos in neural systems. The current belief suggests that chaotic states are induced by different routes to chaos in spiking neural systems. However, few studies have compared the efficiency of signal responses in CR across the different chaotic states in spiking neural systems. We focused herein on the Izhikevich neuron model, comparing the characteristics of CR in the chaotic states arising through the period-doubling or tangent bifurcation routes. We found that the signal response in CR had a unimodal maximum with respect to the stability of chaotic orbits in the tested chaotic states. Furthermore, the efficiency of signal responses at the edge of chaos became especially high as a result of synchronization between the input signal and the periodic component in chaotic spiking activity.
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Affiliation(s)
- Sou Nobukawa
- Department of Computer Science, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, 275-0016, Japan.
| | - Haruhiko Nishimura
- Graduate School of Applied Informatics, University of Hyogo, 7-1-28 Chuo-ku, Kobe, 650-8588, Japan
| | - Teruya Yamanishi
- Department of Management Information Science, Fukui University of Technology, 3-6-1 Gakuen, Fukui, 910-8505, Japan
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16
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Guo S, Wang C, Ma J, Jin W. Transmission of blocked electric pulses in a cable neuron model by using an electric field. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2016.08.023] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
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17
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Han R, Wang J, Yu H, Deng B, Wei X, Qin Y, Wang H. Intrinsic excitability state of local neuronal population modulates signal propagation in feed-forward neural networks. CHAOS (WOODBURY, N.Y.) 2015; 25:043108. [PMID: 25933656 DOI: 10.1063/1.4917014] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Reliable signal propagation across distributed brain areas is an essential requirement for cognitive function, and it has been investigated extensively in computational studies where feed-forward network (FFN) is taken as a generic model. But it is still unclear how distinct local network states, which are intrinsically generated by synaptic interactions within each layer, would affect the ability of FFN to transmit information. Here we investigate the impact of such network states on propagating transient synchrony (synfire) and firing rate by a combination of numerical simulations and analytical approach. Specifically, local network dynamics is attributed to the competition between excitatory and inhibitory neurons within each layer. Our results show that concomitant with different local network states, the performance of signal propagation differs dramatically. For both synfire propagation and firing rate propagation, there exists an optimal local excitability state, respectively, that optimizes the performance of signal propagation. Furthermore, we find that long-range connections strongly change the dependence of spiking activity propagation on local network state and propose that these two factors work jointly to determine information transmission across distributed networks. Finally, a simple mean field approach that bridges response properties of long-range connectivity and local subnetworks is utilized to reveal the underlying mechanism.
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Affiliation(s)
- Ruixue Han
- School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People's Republic of China
| | - Jiang Wang
- School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People's Republic of China
| | - Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People's Republic of China
| | - Bin Deng
- School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People's Republic of China
| | - Xilei Wei
- School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, People's Republic of China
| | - Yingmei Qin
- School of Automation and Electrical Engineering, Tianjin University of Technology and Education Tianjin, Tianjin 300222, China
| | - Haixu Wang
- Department of Statistics and Actuarial Science, Simon Fraser University, 507-9188 University Crescent, Burnaby BC V5A 0A5, Canada
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Wei X, Zhang D, Lu M, Wang J, Yu H, Che Y. Endogenous field feedback promotes the detectability for exogenous electric signal in the hybrid coupled population. CHAOS (WOODBURY, N.Y.) 2015; 25:013113. [PMID: 25637924 DOI: 10.1063/1.4906545] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Xile Wei
- Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Danhong Zhang
- Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Meili Lu
- School of Informational Technology and Engineering, Tianjin University of Technology and Education, Tianjin 300222, China
| | - Jiang Wang
- Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Haitao Yu
- Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Yanqiu Che
- School of Automation and Electrical Engineering, Tianjin University of Technology and Education, Tianjin 300222, China
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19
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Tang J, Luo JM, Ma J. Information transmission in a neuron-astrocyte coupled model. PLoS One 2013; 8:e80324. [PMID: 24312211 PMCID: PMC3843665 DOI: 10.1371/journal.pone.0080324] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2013] [Accepted: 10/07/2013] [Indexed: 11/18/2022] Open
Abstract
A coupled model containing two neurons and one astrocyte is constructed by integrating Hodgkin-Huxley neuronal model and Li-Rinzel calcium model. Based on this hybrid model, information transmission between neurons is studied numerically. Our results show that when the successive spikes are produced in neuron 1 (N1), the bursting-like spikes (BLSs) occur in two neurons simultaneously during the spikes being transferred to neuron 2 (N2). The existence of the astrocyte and a higher expression level of mGluRs facilitate the occurrence of BLSs, but the rate of occurrence is not sensitive to the parameters. Furthermore, time delay τ occurs during the information transmission, and τ is almost independent of the effect of the astrocyte. Additionally, we found that low coupling strength may result in the distortion of the information, and this distortion is also proven to be almost independent of the astrocyte.
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Affiliation(s)
- Jun Tang
- College of Science, China University of Mining and Technology, Xuzhou, China
- * E-mail:
| | - Jin-Ming Luo
- College of Science, China University of Mining and Technology, Xuzhou, China
| | - Jun Ma
- Department of Physics, Lanzhou University of Technology, Lanzhou, China
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Jun D, Guang-Jun Z, Yong X, Hong Y, Jue W. Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model. Cogn Neurodyn 2013; 8:167-75. [PMID: 24624236 DOI: 10.1007/s11571-013-9273-x] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2012] [Revised: 10/03/2013] [Accepted: 10/24/2013] [Indexed: 11/25/2022] Open
Abstract
Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.
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Affiliation(s)
- Dong Jun
- College of Science, Air Force Engineering University, Xi'an, 710051 China ; The First Aeronautical Institute of Air Force, Xinyang, 464000 Henan China
| | - Zhang Guang-Jun
- College of Science, Air Force Engineering University, Xi'an, 710051 China ; School of Life Science and technology, Xi'an Jiao tong University, Xi'an, 710049 China
| | - Xie Yong
- School of Aerospace, Xi'an Jiao tong University, Xi'an, 710049 China
| | - Yao Hong
- College of Science, Air Force Engineering University, Xi'an, 710051 China
| | - Wang Jue
- School of Life Science and technology, Xi'an Jiao tong University, Xi'an, 710049 China
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21
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Gong Y, Xu B, Wu Y. Adaptive coupling optimized spiking coherence and synchronization in Newman-Watts neuronal networks. CHAOS (WOODBURY, N.Y.) 2013; 23:033105. [PMID: 24089941 DOI: 10.1063/1.4813224] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
In this paper, we have numerically studied the effect of adaptive coupling on the temporal coherence and synchronization of spiking activity in Newman-Watts Hodgkin-Huxley neuronal networks. It is found that random shortcuts can enhance the spiking synchronization more rapidly when the increment speed of adaptive coupling is increased and can optimize the temporal coherence of spikes only when the increment speed of adaptive coupling is appropriate. It is also found that adaptive coupling strength can enhance the synchronization of spikes and can optimize the temporal coherence of spikes when random shortcuts are appropriate. These results show that adaptive coupling has a big influence on random shortcuts related spiking activity and can enhance and optimize the temporal coherence and synchronization of spiking activity of the network. These findings can help better understand the roles of adaptive coupling for improving the information processing and transmission in neural systems.
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Affiliation(s)
- Yubing Gong
- School of Physics and Optoelectronic Engineering, Ludong University, Yantai, Shandong 264025, China
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22
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Yu H, Wang J, Du J, Deng B, Wei X, Liu C. Effects of time delay and random rewiring on the stochastic resonance in excitable small-world neuronal networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052917. [PMID: 23767608 DOI: 10.1103/physreve.87.052917] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2012] [Indexed: 06/02/2023]
Abstract
The effects of time delay and rewiring probability on stochastic resonance and spatiotemporal order in small-world neuronal networks are studied in this paper. Numerical results show that, irrespective of the pacemaker introduced to one single neuron or all neurons of the network, the phenomenon of stochastic resonance occurs. The time delay in the coupling process can either enhance or destroy stochastic resonance on small-world neuronal networks. In particular, appropriately tuned delays can induce multiple stochastic resonances, which appear intermittently at integer multiples of the oscillation period of the pacemaker. More importantly, it is found that the small-world topology can significantly affect the stochastic resonance on excitable neuronal networks. For small time delays, increasing the rewiring probability can largely enhance the efficiency of pacemaker-driven stochastic resonance. We argue that the time delay and the rewiring probability both play a key role in determining the ability of the small-world neuronal network to improve the noise-induced outreach of the localized subthreshold pacemaker.
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Affiliation(s)
- Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
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23
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Yu H, Wang J, Du J, Deng B, Wei X, Liu C. Effects of time delay on the stochastic resonance in small-world neuronal networks. CHAOS (WOODBURY, N.Y.) 2013; 23:013128. [PMID: 23556965 DOI: 10.1063/1.4790829] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
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.
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Affiliation(s)
- Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
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24
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Yu H, Wang J, Sun J, Yu H. Effects of hybrid synapses on the vibrational resonance in small-world neuronal networks. CHAOS (WOODBURY, N.Y.) 2012; 22:033105. [PMID: 23020444 DOI: 10.1063/1.4729462] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We investigate the effect of vibrational resonance in small-world neuronal networks with hybrid chemical and electrical synapses. It is shown that, irrespective of the probability of chemical synapses, an optimal amplitude of high-frequency component of the signal can optimize the dynamical response of neuron populations to the low-frequency component, which encodes the information. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, which determine the ability of neuronal networks to enhance the outreach of localized subthreshold low-frequency signal. In particular, chemical synaptic coupling is more efficient than the electrical coupling for the transmission of local input signal due to its selective coupling. Moreover, there exists an optimal small-world topology characterized by an optimal value of rewiring probability, warranting the largest peak value of the system response. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for signal processing in neuronal systems.
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Affiliation(s)
- Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
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25
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Guo D, Li C. Stochastic resonance in Hodgkin-Huxley neuron induced by unreliable synaptic transmission. J Theor Biol 2012; 308:105-14. [PMID: 22687443 DOI: 10.1016/j.jtbi.2012.05.034] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2012] [Revised: 05/30/2012] [Accepted: 05/31/2012] [Indexed: 11/25/2022]
Abstract
We systematically investigate the stochastic dynamics of a single Hodgkin-Huxley neuron driven by stochastic excitatory and inhibitory input spikes via unreliable synapses in this paper. Based on the mean-filed theory, a novel intrinsic neuronal noise regulation mechanism stemming from unreliable synapses is presented. Our simulation results show that, under certain conditions, the stochastic resonance phenomenon is able to be induced by the unreliable synaptic transmission, which can be well explained by the theoretical prediction. To a certain degree, the results presented here provide insights into the functional roles of unreliable synapses in neural information processing.
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Affiliation(s)
- Daqing Guo
- School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, People's Republic of China.
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26
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Lin X, Gong Y, Wang L, Ma X. Coherence resonance and bi-resonance by time-periodic coupling strength in Hodgkin-Huxley neuron networks. Sci China Chem 2011. [DOI: 10.1007/s11426-011-4474-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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27
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Yu H, Wang J, Liu C, Deng B, Wei X. Vibrational resonance in excitable neuronal systems. CHAOS (WOODBURY, N.Y.) 2011; 21:043101. [PMID: 22225338 DOI: 10.1063/1.3644390] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
In this paper, we investigate the effect of a high-frequency driving on the dynamical response of excitable neuronal systems to a subthreshold low-frequency signal by numerical simulation. We demonstrate the occurrence of vibrational resonance in spatially extended neuronal networks. Different network topologies from single small-world networks to modular networks of small-world subnetworks are considered. It is shown that an optimal amplitude of high-frequency driving enhances the response of neuron populations to a low-frequency signal. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, such as the coupling strength between neurons, network size, and rewiring probability of single small-world networks, as well as the number of links between different subnetworks and the number of subnetworks in the modular networks. All these parameters play a key role in determining the ability of the network to enhance the outreach of the localized subthreshold low-frequency signal. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for the weak signal detection and information propagation across neuronal systems.
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Affiliation(s)
- Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
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28
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Lin X, Gong Y, Wang L. Multiple coherence resonance induced by time-periodic coupling in stochastic Hodgkin-Huxley neuronal networks. CHAOS (WOODBURY, N.Y.) 2011; 21:043109. [PMID: 22225346 DOI: 10.1063/1.3652847] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
In this paper, we study the effect of time-periodic coupling strength (TPCS) on the spiking coherence of Newman-Watts small-world networks of stochastic Hodgkin-Huxley (HH) neurons and investigate the relations between the coupling strength and channel noise when coherence resonance (CR) occurs. It is found that, when the amplitude of TPCS is varied, the spiking induced by channel noise can exhibit CR and coherence bi-resonance (CBR), and the CR moves to a smaller patch area (bigger channel noise) when the amplitude increases; when the frequency of TPCS is varied, the intrinsic spiking can exhibit CBR and multiple CR, and the CR always occurs when the frequency is equal to or multiple of the spiking period, manifesting as the locking between the frequencies of the intrinsic spiking and the coupling strength. These results show that TPCS can greatly enhance and optimize the intrinsic spiking coherence, and favors the spiking with bigger channel noise to exhibit CR. This implies that, compared to constant coupling strength, TPCS may play a more efficient role for improving the time precision of the information processing in stochastic neuronal networks.
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Affiliation(s)
- Xiu Lin
- School of Physics, Ludong University, Yantai, Shandong 264025, China
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29
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Franović I, Todorović K, Vasović N, Burić N. Stability, bifurcations, and dynamics of global variables of a system of bursting neurons. CHAOS (WOODBURY, N.Y.) 2011; 21:033109. [PMID: 21974644 DOI: 10.1063/1.3619293] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.
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Affiliation(s)
- Igor Franović
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
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30
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Chapeau-Blondeau F, Rousseau D, Delahaies A. Rényi entropy measure of noise-aided information transmission in a binary channel. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:051112. [PMID: 20866190 DOI: 10.1103/physreve.81.051112] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2010] [Indexed: 05/29/2023]
Abstract
This paper analyzes a binary channel by means of information measures based on the Rényi entropy. The analysis extends, and contains as a special case, the classic reference model of binary information transmission based on the Shannon entropy measure. The extended model is used to investigate further possibilities and properties of stochastic resonance or noise-aided information transmission. The results demonstrate that stochastic resonance occurs in the information channel and is registered by the Rényi entropy measures at any finite order, including the Shannon order. Furthermore, in definite conditions, when seeking the Rényi information measures that best exploit stochastic resonance, then nontrivial orders differing from the Shannon case usually emerge. In this way, through binary information transmission, stochastic resonance identifies optimal Rényi measures of information differing from the classic Shannon measure. A confrontation of the quantitative information measures with visual perception is also proposed in an experiment of noise-aided binary image transmission.
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Affiliation(s)
- François Chapeau-Blondeau
- Laboratoire d'Ingénierie des Systèmes Automatisés (LISA), Université d'Angers, 62 Avenue Notre Dame du Lac, 49000 Angers, France
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31
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32
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Wang Q, Perc M, Duan Z, Chen G. Delay-induced multiple stochastic resonances on scale-free neuronal networks. CHAOS (WOODBURY, N.Y.) 2009; 19:023112. [PMID: 19566247 DOI: 10.1063/1.3133126] [Citation(s) in RCA: 48] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We study the effects of periodic subthreshold pacemaker activity and time-delayed coupling on stochastic resonance over scale-free neuronal networks. As the two extreme options, we introduce the pacemaker, respectively, to the neuron with the highest degree and to one of the neurons with the lowest degree within the network, but we also consider the case when all neurons are exposed to the periodic forcing. In the absence of delay, we show that an intermediate intensity of noise is able to optimally assist the pacemaker in imposing its rhythm on the whole ensemble, irrespective to its placing, thus providing evidences for stochastic resonance on the scale-free neuronal networks. Interestingly thereby, if the forcing in form of a periodic pulse train is introduced to all neurons forming the network, the stochastic resonance decreases as compared to the case when only a single neuron is paced. Moreover, we show that finite delays in coupling can significantly affect the stochastic resonance on scale-free neuronal networks. In particular, appropriately tuned delays can induce multiple stochastic resonances independently of the placing of the pacemaker, but they can also altogether destroy stochastic resonance. Delay-induced multiple stochastic resonances manifest as well-expressed maxima of the correlation measure, appearing at every multiple of the pacemaker period. We argue that fine-tuned delays and locally active pacemakers are vital for assuring optimal conditions for stochastic resonance on complex neuronal networks.
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Affiliation(s)
- Qingyun Wang
- Department of Mechanics and Aerospace Engineering, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
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33
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Li X, Wang J, Hu W. Effects of chemical synapses on the enhancement of signal propagation in coupled neurons near the canard regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041902. [PMID: 17995021 DOI: 10.1103/physreve.76.041902] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2007] [Revised: 07/27/2007] [Indexed: 05/25/2023]
Abstract
The response of three coupled FitzHugh-Nagumo neurons, under Gaussian white noise, to a subthreshold periodic signal is studied in this paper. By combining the canard dynamics, chemical coupling, and stochastic resonance together, the information transfer in this neural system is investigated. We find that chemical synaptic coupling is more efficient than the well-known linear coupling (gap junction) for local signal input, i.e., only one of the three neurons is subject to the periodic signal. This weak and local input is common in biological systems for the sake of low energy consumption.
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Affiliation(s)
- Xiumin Li
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China.
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34
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Gosak M, Perc M. Proximity to periodic windows in bifurcation diagrams as a gateway to coherence resonance in chaotic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:037201. [PMID: 17930370 DOI: 10.1103/physreve.76.037201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2007] [Indexed: 05/25/2023]
Abstract
We show that chaotic states situated in the proximity of periodic windows in bifurcation diagrams are eligible for the observation of coherence resonance. In particular, additive Gaussian noise of appropriate intensity can enhance the temporal order in such chaotic states in a resonant manner. Results obtained for the logistic map and the Lorenz equations suggest that the presented mechanism of coherence resonance is valid beyond particularities of individual systems. We attribute the findings to the increasing attraction of imminent periodic orbits and the ability of noise to anticipate their existence and use a modified wavelet analysis to support our arguments.
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Affiliation(s)
- Marko Gosak
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska cesta 160, SI-2000 Maribor, Slovenia
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35
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Perc M, Gosak M, Marhl M. From stochasticity to determinism in the collective dynamics of diffusively coupled cells. Chem Phys Lett 2006. [DOI: 10.1016/j.cplett.2006.01.065] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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