Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control
Spiking neuron models which exhibit rich dynamics are usually defined by hybrid dynamical systems. It is revealed that mathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitative analysis for a quadratic integrate-and-fire mode...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2015/836402 |
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| author | Guangyao Tang Jin Yang Sanyi Tang |
| author_facet | Guangyao Tang Jin Yang Sanyi Tang |
| author_sort | Guangyao Tang |
| collection | DOAJ |
| description | Spiking neuron models which exhibit rich dynamics are usually
defined by hybrid dynamical systems. It is revealed that
mathematical analysis of these models has important significance.
Therefore, in this work, we provide a comprehensively qualitative
analysis for a quadratic integrate-and-fire model by using the
theories of hybrid dynamical system. Firstly, the exact
impulsive and phase sets are defined according to the phase
portraits of the proposed model, and then the
Poincaré map is constructed. Furthermore, the
conditions for the existence and stability of an order 1 periodic
solution are provided. Moreover, the existence and nonexistence
of an order k k≥2 periodic solution have been studied
theoretically and numerically, and the results show
that the system has periodic solutions with any period. Finally,
some biological implications of the mathematical results are
discussed. |
| format | Article |
| id | doaj-art-d5ca18d1a15d4dc486f153a65cce655d |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-d5ca18d1a15d4dc486f153a65cce655d2025-02-03T01:31:22ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2015-01-01201510.1155/2015/836402836402Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback ControlGuangyao Tang0Jin Yang1Sanyi Tang2Key Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, Hubei 445000, ChinaDepartment of Mathematics, Chongqing Jiaotong University, Chongqing 400074, ChinaCollege of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, ChinaSpiking neuron models which exhibit rich dynamics are usually defined by hybrid dynamical systems. It is revealed that mathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitative analysis for a quadratic integrate-and-fire model by using the theories of hybrid dynamical system. Firstly, the exact impulsive and phase sets are defined according to the phase portraits of the proposed model, and then the Poincaré map is constructed. Furthermore, the conditions for the existence and stability of an order 1 periodic solution are provided. Moreover, the existence and nonexistence of an order k k≥2 periodic solution have been studied theoretically and numerically, and the results show that the system has periodic solutions with any period. Finally, some biological implications of the mathematical results are discussed.http://dx.doi.org/10.1155/2015/836402 |
| spellingShingle | Guangyao Tang Jin Yang Sanyi Tang Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control Discrete Dynamics in Nature and Society |
| title | Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control |
| title_full | Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control |
| title_fullStr | Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control |
| title_full_unstemmed | Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control |
| title_short | Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control |
| title_sort | qualitative analysis of a quadratic integrate and fire neuron model with state dependent feedback control |
| url | http://dx.doi.org/10.1155/2015/836402 |
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