本文選題：參數(shù)平面 + Lyapunov指數(shù)��； 參考：《北京交通大學(xué)》2017年碩士論文

【摘要】：神經(jīng)元在中樞神經(jīng)系統(tǒng)處理信息的過程中有著非常重要的地位,神經(jīng)元能夠加工、處理和傳輸信息,而這些過程有豐富的非線性特征。近十年來,Rulkov和Izhikevich分別應(yīng)用離散(map-based)神經(jīng)元數(shù)學(xué)模型,成功地對大腦皮層(cortical layer)和丘腦皮層系統(tǒng)(thermostatically systems)神經(jīng)元的動態(tài)行為進行定性分析和數(shù)值模擬,并與基于常微分方程組(ODEs)的Hodgkin-Huxley模型進行比較,結(jié)果表明:離散神經(jīng)元網(wǎng)絡(luò)能夠模擬生物神經(jīng)元的真實行為,離散神經(jīng)元模型在計算時間、計算算法的透明性、計算資源和數(shù)據(jù)存儲等方面具有明顯的優(yōu)越性。它們不僅可以在大型的數(shù)值計算方面具有明顯的優(yōu)越性,而且可以調(diào)節(jié)混沌動力系統(tǒng),產(chǎn)生豐富的聚合行為。目前,離散神經(jīng)元數(shù)學(xué)模型作為研究大腦神經(jīng)元系統(tǒng)的一種簡化數(shù)學(xué)形式,已經(jīng)被廣泛地用到大腦數(shù)值模擬中。在第一章中,我們介紹了與本文相關(guān)的背景知識。在第二章中,討論了基于Rulkov映射模型中周期擾動的影響。通過固定外部調(diào)節(jié)振幅σ和參數(shù)η,改變外部周期擾動角頻率ω,對參數(shù)平面的不動點、周期解、擬周期解和混沌進行了詳細的論述。在二維參數(shù)平面中,我們發(fā)現(xiàn)一個蝦型周期區(qū)域淹沒在一個混沌區(qū)域當(dāng)中。此外,在參數(shù)平面中我們還可以觀察到倍周期分岔結(jié)構(gòu)�；煦缈梢员豢醋魇且种浦芷谛源翱谇度朐诨煦鐓^(qū)域窗口的結(jié)果。在第三章中,我們利用統(tǒng)計學(xué)相關(guān)知識對第二章的數(shù)值計算結(jié)果進行了進一步的分析,主要討論第二章出現(xiàn)的結(jié)果是否正確。根據(jù)數(shù)據(jù)的最小值、最大值、均值以及方差等分析數(shù)據(jù)、檢驗結(jié)果。最后,對本文的研究內(nèi)容進行了總結(jié)。
[Abstract]:Neurons play an important role in the processing of information in the central nervous system. Neurons can process, process and transmit information, and these processes have rich nonlinear characteristics. In the last ten years, Rulkov and Izhikevich have successfully carried out qualitative analysis and numerical simulation of the dynamic behavior of cortical layersand thalamic cortical system by using discrete map-based neuron mathematical model, respectively. Compared with the Hodgkin-Huxley model based on ordinary differential equations, the results show that the discrete neuron network can simulate the real behavior of the biological neurons, and the computation time of the discrete neuron model and the transparency of the algorithm are obtained. Computing resources and data storage have obvious advantages. They not only have obvious advantages in large-scale numerical calculation, but also can adjust chaotic dynamical system and produce rich aggregation behavior. At present, as a simplified mathematical form to study the brain neuron system, discrete neuron mathematical model has been widely used in brain numerical simulation. In the first chapter, we introduce the background of this paper. In chapter 2, we discuss the effect of periodic perturbation based on Rulkov mapping model. The fixed amplitude 蟽 and parameter 畏 are adjusted to change the external periodic disturbance angular frequency 蠅. The fixed point, periodic solution, quasi periodic solution and chaos of the parameter plane are discussed in detail. In the two-dimensional parametric plane, we find that a shrimp periodic region is submerged in a chaotic region. In addition, the periodic bifurcation structure can be observed in the parameter plane. Chaos can be seen as the result of suppressing periodic windows embedded in chaotic regions. In the third chapter, we use the relevant knowledge of statistics to further analyze the numerical results of the second chapter, and mainly discuss whether the results in the second chapter are correct or not. According to the data minimum, maximum, mean and variance analysis data, test results. Finally, the research content of this paper is summarized.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175
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本文編號：1798237