scholarly journals Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guangyao Tang ◽  
Jin Yang ◽  
Sanyi Tang
Keyword(s):  
Periodic Solution ◽  
Qualitative Analysis ◽  
Phase Portraits ◽  
Neuron Models ◽  
Integrate And Fire ◽  
State Dependent ◽  
Proposed Model ◽  
Existence And Stability ◽  
Spiking Neuron Models ◽  
Important Significance

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 orderk  k≥2periodic 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.

Interspike interval distributions of spiking neurons driven by fluctuating inputs

2011 ◽  
Vol 106 (1) ◽  
pp. 361-373 ◽  
Author(s):  
Srdjan Ostojic
Keyword(s):  
Coefficient Of Variation ◽  
Cortical Neurons ◽  
Interspike Interval ◽  
Full Range ◽  
Spiking Neurons ◽  
Neuron Models ◽  
Integrate And Fire ◽  
Gamma Distributions ◽  
Spiking Neuron Models ◽  
Analytical Tools

Interspike interval (ISI) distributions of cortical neurons exhibit a range of different shapes. Wide ISI distributions are believed to stem from a balance of excitatory and inhibitory inputs that leads to a strongly fluctuating total drive. An important question is whether the full range of experimentally observed ISI distributions can be reproduced by modulating this balance. To address this issue, we investigate the shape of the ISI distributions of spiking neuron models receiving fluctuating inputs. Using analytical tools to describe the ISI distribution of a leaky integrate-and-fire (LIF) neuron, we identify three key features: 1) the ISI distribution displays an exponential decay at long ISIs independently of the strength of the fluctuating input; 2) as the amplitude of the input fluctuations is increased, the ISI distribution evolves progressively between three types, a narrow distribution (suprathreshold input), an exponential with an effective refractory period (subthreshold but suprareset input), and a bursting exponential (subreset input); 3) the shape of the ISI distribution is approximately independent of the mean ISI and determined only by the coefficient of variation. Numerical simulations show that these features are not specific to the LIF model but are also present in the ISI distributions of the exponential integrate-and-fire model and a Hodgkin-Huxley-like model. Moreover, we observe that for a fixed mean and coefficient of variation of ISIs, the full ISI distributions of the three models are nearly identical. We conclude that the ISI distributions of spiking neurons in the presence of fluctuating inputs are well described by gamma distributions.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

scholarly journals Chaotic States Induced By Resetting Process In Izhikevich Neuron Model

2015 ◽  
Vol 5 (2) ◽  
pp. 109-119 ◽  
Author(s):  
Sou Nobukawa ◽  
Haruhiko Nishimura ◽  
Teruya Yamanishi ◽  
Jian-Qin Liu
Keyword(s):  
Neuron Model ◽  
Large Range ◽  
Neuron Models ◽  
Spike Generation ◽  
Izhikevich Neuron Model ◽  
State Dependent ◽  
Spiking Neuron Models ◽  
Generation Mechanisms ◽  
Spike Patterns ◽  
Transition Scheme

Abstract Several hybrid neuron models, which combine continuous spike-generation mechanisms and discontinuous resetting process after spiking, have been proposed as a simple transition scheme for membrane potential between spike and hyperpolarization. As one of the hybrid spiking neuron models, Izhikevich neuron model can reproduce major spike patterns observed in the cerebral cortex only by tuning a few parameters and also exhibit chaotic states in specific conditions. However, there are a few studies concerning the chaotic states over a large range of parameters due to the difficulty of dealing with the state dependent jump on the resetting process in this model. In this study, we examine the dependence of the system behavior on the resetting parameters by using Lyapunov exponent with saltation matrix and Poincaré section methods, and classify the routes to chaos.

Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops

2007 ◽  
Vol 19 (8) ◽  
pp. 2124-2148 ◽  
Author(s):  
Jianfu Ma ◽  
Jianhong Wu
Keyword(s):  
Periodic Solutions ◽  
Asymptotically Stable ◽  
Spiking Neuron ◽  
Phase Resetting ◽  
Neuron Models ◽  
Absolute Refractory Period ◽  
Integrate And Fire ◽  
Stable Periodic ◽  
Spiking Neuron Models ◽  
Integrate And Fire Neuron

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.

scholarly journals Multi-State Dependent Impulsive Control for Pest Management

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Huidong Cheng ◽  
Fang Wang ◽  
Tongqian Zhang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Qualitative Analysis ◽  
Pest Management ◽  
Pest Control ◽  
Impulsive Control ◽  
Management Strategies ◽  
Control Methods ◽  
State Dependent ◽  
Order One Periodic Solution

According to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution of such system, and further, the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results. Our results show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.

Dynamical Behavior and Bifurcation Analysis of the SIR Model with Continuous Treatment and State-Dependent Impulsive Control

2019 ◽  
Vol 29 (10) ◽  
pp. 1950131 ◽  
Author(s):  
Qian Li ◽  
Yanni Xiao
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Reproduction Number ◽  
Continuous Treatment ◽  
Positive Order ◽  
Susceptible Population ◽  
State Dependent ◽  
Existence And Stability ◽  
Impulsive Model ◽  
Disease Free

In this study, we propose a state-dependent impulsive model describing the susceptible individuals-triggered interventions. We find that the model with susceptible individuals-guided impulsive interventions can exhibit very complex dynamical behaviors with rich biological meanings. We note that this formulated impulsive model has disease-free periodic solution, and we can investigate the threshold dynamics by defining the control reproduction number. We study the existence and stability of the disease-free periodic solution (DFPS) for [Formula: see text]. Our results show that, even if the basic reproduction number [Formula: see text], the DFPS can still be stable when the threshold level of susceptible population [Formula: see text], indicating that with a proper chosen [Formula: see text], the state-dependent impulsive strategy can effectively control the development of the infectious disease and eradicate the disease eventually. By employing the bifurcation theory, we investigate the bifurcation phenomenon near the DFPS with respect to some key parameters, and observe that a positive order-1 periodic solution can bifurcate from the DFPS via a transcritical bifurcation. By utilizing numerical simulation, we further explore the existence and stability of the positive order-[Formula: see text] periodic solutions, and found the feasibility of stable positive order-1, order-2 and order-3 periodic solutions, that imply the existence of chaos. In particular, we find that there can be three positive order-1 periodic solutions simultaneously, in which one is stable and the other two are unstable. Our finding indicates that the comprehensive strategy combining continuous treatment with state-dependent impulsive vaccination and isolation plays a crucial role in controlling the prevalence and further spread of the infectious diseases.

scholarly journals Dynamics and bifurcation analysis of a state-dependent impulsive SIS model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jinyan Wang
Keyword(s):  
Periodic Solution ◽  
Control Measures ◽  
Global Dynamics ◽  
Sis Model ◽  
Susceptible Population ◽  
Ode System ◽  
State Dependent ◽  
Existence And Stability ◽  
Modelling Studies ◽  
The Stability

AbstractRecently, considering the susceptible population size-guided implementations of control measures, several modelling studies investigated the global dynamics and bifurcation phenomena of the state-dependent impulsive SIR models. In this study, we propose a state-dependent impulsive model based on the SIS model. We firstly recall the complicated dynamics of the ODE system with saturated treatment. Based on the dynamics of the ODE system, we firstly discuss the existence and the stability of the semi-trivial periodic solution. Then, based on the definition of the Poincaré map and its properties, we systematically investigate the bifurcations near the semi-trivial periodic solution with all the key control parameters; consequently, we prove the existence and stability of the positive periodic solutions.

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