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新型忆阻耦合异质神经元的放电模式和预定义时间混沌同步

贾美美 曹佳伟 白明明

贾美美, 曹佳伟, 白明明. 新型忆阻耦合异质神经元的放电模式和预定义时间混沌同步. 物理学报, 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872
引用本文: 贾美美, 曹佳伟, 白明明. 新型忆阻耦合异质神经元的放电模式和预定义时间混沌同步. 物理学报, 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872
Jia Mei-Mei, Cao Jia-Wei, Bai Ming-Ming. Firing modes and predefined-time chaos synchronization of novel memristor-coupled heterogeneous neuron. Acta Phys. Sin., 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872
Citation: Jia Mei-Mei, Cao Jia-Wei, Bai Ming-Ming. Firing modes and predefined-time chaos synchronization of novel memristor-coupled heterogeneous neuron. Acta Phys. Sin., 2024, 73(17): 170502. doi: 10.7498/aps.73.20240872

新型忆阻耦合异质神经元的放电模式和预定义时间混沌同步

贾美美, 曹佳伟, 白明明

Firing modes and predefined-time chaos synchronization of novel memristor-coupled heterogeneous neuron

Jia Mei-Mei, Cao Jia-Wei, Bai Ming-Ming
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  • 首先提出一种新型局部有源忆阻器, 并分析该忆阻器的频率特性、局部有源性及非易失性. 然后将新型局部有源忆阻器引入二维Hindmarsh-Rose神经元和二维FitzHugh-Nagumo神经元, 构建新型忆阻耦合异质神经元模型. 在数值仿真中, 通过改变耦合强度, 发现该模型具有周期尖峰放电模式、混沌尖峰放电模式、周期簇发放电模式及随机簇发放电模式. 最后基于Lyapunov稳定性理论和预定义时间稳定性理论, 提出一种新型预定义时间同步策略, 并将该策略应用于新型忆阻耦合异质神经元的混沌同步中. 结果表明, 与有限时间同步策略、固定时间同步策略和传统预定义时间同步策略相比, 新型预定义时间同步策略的实际收敛时间最小. 研究新型忆阻耦合异质神经元的放电模式和混沌同步有助于探索大脑的神经功能, 并在神经信号处理及保密通信领域中具有重要意义.
    The processing and transmission of biological neural information are realized via firing activities of neurons in different regions of brain. Memristors are regarded as ideal devices for emulating biological synapses because of their nanoscale size, non-volatility and synapse-like plasticity. Hence, investigating firing modes of memristor-coupled heterogeneous neurons is significant. This work focuses on modelling, firing modes and chaos synchronization of a memristor-coupled heterogeneous neuron. First, a novel locally active memristor is proposed, and its frequency characteristics, local activity, and non-volatility are analyzed. Then, the novel locally active memristor is introduced into the two-dimensional HR neuron and the two-dimensional FHN neuron to construct a novel memristor-coupled heterogeneous neuron model. In numerical simulations, by changing the coupling strength, it is found that the model exhibits the periodic spike firing mode, the chaotic spike firing mode, the periodic burst firing mode, and the random burst firing mode. Besides, the dynamic behavior of the novel memristor-coupled heterogeneous neuron can switch between periodic behavior and chaotic behavior by changing the initial state. Finally, based on the Lyapunov stability theory and the predefined-time stability theory, a novel predefined-time synchronization strategy is proposed and used to realize the chaos synchronization of the novel memristor-coupled heterogeneous neuron. The results show that compared with a finite-time synchronization strategy, a fixed-time synchronization strategy and a traditional predefined-time synchronization strategy, the novel predefined-time synchronization strategy has a short actual convergence time. Studying the firing modes and chaotic synchronization of the novel memristor-coupled heterogeneous neuron can help explore the neural functions of the brain and is also important in processing the neural signal and secure communication fields.
      PACS:
      05.45.-a(Nonlinear dynamics and chaos)
      05.45.Gg(Control of chaos, applications of chaos)
      87.19.lm(Synchronization in the nervous system)
      05.45.Pq(Numerical simulations of chaotic systems)
      通信作者: 贾美美, meimeijia14@163.com
    • 基金项目: 内蒙古自治区直属高校基本科研业务费(批准号: JY20220181)和内蒙古自治区自然科学基金(批准号: 2024MS06006)资助的课题.
      Corresponding author: Jia Mei-Mei, meimeijia14@163.com
    • Funds: Project supported by the Basic Scientific Research Expenses Program of Universities of Inner Mongolia Autonomous Region, China (Grant No. JY20220181) and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2024MS06006).

    自1963年Lorenz发现第一个混沌吸引子, 混沌在生物工程、机械工程、控制工程、保密通信、电子工程等领域都取得了发展. 混沌系统在控制工程中受到人们的广泛关注. 混沌系统的控制与同步是其应用的前提条件. 自1990年Pecora和Carroll[1]对混沌系统的同步进行研究以来, 混沌系统的同步成为了热点课题, 尤其是在保密通信领域. 混沌同步一般是指两个具有不同初始状态的驱动系统和响应系统在同步策略的作用下达到同步. 这就引出了稳定时间的概念. 人们通常利用稳定时间这一指标来评估同步策略的优劣. 稳定时间越短, 系统达到同步的时间越短.

    根据稳定时间的特点, 同步策略分为: 渐近时间同步策略[24]、有限时间同步策略[57]、固定时间同步策略[810]和预定义时间同步策略[1113]. 首先, 渐近时间同步策略在21世纪被广泛应用于各领域. 但是该策略不适用于对时间精度要求较高的场合. 其次, 有限时间同步策略能够使驱动系统和响应系统在有限时间内达到同步. 但是该策略与系统的初始状态有关. 在实际工业生产中, 由于系统的很多参数是未知的, 其初始状态不易获得, 这限制了有限时间同步策略的应用. 然后, 固定时间同步策略被人们应用于实际工业生产中. 这是由于该策略中的固定时间参数是一个与系统的初始状态无关的常数. 但是该策略仍然存在如下不足: 对收敛时间的估计较保守; 收敛时间与控制器的可调参数之间没有明确关系. 最后, 为了克服固定时间同步策略存在的问题, 人们研究了预定义时间同步策略. 预定义时间同步策略中预定义时间是收敛时间的上界, 这便于控制器的设计, 从而满足实际需求.

    目前, 人们提出了许多策略用于混沌系统的同步, 例如主动同步策略[14]、自适应同步策略[15]、基于事件触发的脉冲同步策略[16,17]和滑模同步策略[18,19]. 作为一种常用的同步策略, 滑模同步策略已经取得了许多成果. 该策略对系统扰动及参数摄动具有良好的鲁棒性.

    放电是生物神经元的主要电活动之一. 放电在神经信息的传递和编码过程中具有重要作用. 为了研究神经元的放电, 人们构建了各种神经元模型, 如Rulkov神经元, Morris-Lecar (ML)神经元, FitzHugh-Nagumo (FHN)神经元, Hindmarsh-Rose (HR)神经元, Hopfield神经元和禁忌学习神经元. 在各类神经元模型中, 人们通常利用HR神经元和FHN神经元来研究神经元放电的复杂动力学行为. 异质神经元位于大脑的不同区域, 并具有不同的功能和结构. 事实上, 生物神经功能是由大脑的不同神经元共同实现的. 因此, 研究异质神经元的放电和同步具有重要意义.

    忆阻器是一种天然的非线性纳米电子器件, 被广泛应用于各领域, 例如, 人工智能、模拟电路及神经形态计算等[2025]. 特别地, 由于忆阻器具有纳米级尺寸、可塑性和非易失性等特性, 忆阻器被认为是一种模拟生物神经突触的理想器件[26]. 忆阻器作为整个应用系统的关键器件, 其速度与功耗的提升对整个应用系统的计算能效的提高起到重 要作用. 为了提高计算效率, 研究具有高频特性的忆阻器具有重要意义. 局部有源性是复杂性的起 源[27]. 2014年, Chua[28]首次提出了局部有源忆阻器的概念. 研究表明, 局部有源忆阻器具有强非线性特性和复杂动力学行为[29]. 局部有源忆阻器在局部有源区内具有负阻特性, 能够放大微弱信号并保持复杂振荡, 也能够产生神经形态行为[24]. 本文提出一种具有高频特性的局部有源忆阻器.

    文献[30]基于多稳态局部有源忆阻器、二维HR神经元和一维Hopfield神经元, 构建了一种简单的异质神经网络. 研究表明, 通过调整耦合强度, 该网络能够产生共存的多种放电模式. 此外, 复杂放电仅发生在忆阻器的局部有源区域. 文献[31]通过将蔡氏结型局部有源忆阻器、HR神经元和tabu神经元进行耦合, 构建了一种新型异质神经网络. 研究表明, 通过选择合适的外部激励和耦合强度, 该网络能够产生尖峰放电模式和簇发放电模式. 文献[32]通过将局部有源忆阻器、二维HR神经元和二维Hopfield神经网络进行耦合, 构建了一种新型忆阻耦合异质神经网络. 研究表明, 该网络具有复杂的动力学行为, 例如, 周期簇发放电模式、周期尖峰放电模式和混沌尖峰放电模式等. 文献[3032]在构建局部有源忆阻耦合异质神经网络时, 没有考虑忆阻器是否具有高频特性.

    文献[33]研究了周期磁控忆阻耦合Rulkov神经网络的各种放电模式, 并采用耦合同步策略实现了系统的同步. 文献[34]采用自适应同步策略实现了分数阶忆阻耦合神经网络的同步. 文献[35]通过调整延迟时间实现了具有两个时延的忆阻耦合异质神经元的同步. 文献[3335]在研究神经元及神经网络同步时, 没有考虑内部不确定性及外部扰动.

    本文的主要工作及创新点如下:

    1) 首先基于双三次插值函数和正弦函数, 提出一种新型局部有源忆阻器. 然后分析该忆阻器的频率特性、局部有源性及非易失性. 研究表明该忆阻器具有高频特性.

    2) 首先将新型局部有源忆阻器、二维HR神经元和二维FHN神经元进行耦合, 构建新型忆阻耦合异质神经元模型. 然后基于该模型, 分析其动力学行为并对其进行电路实现, 从而为研究新型忆阻耦合异质神经元的混沌同步奠定基础.

    3) 首先提出一种新型预定义时间同步策略. 该策略的设计主要包括新型预定义时间Lyapunov函数的充分条件, 新型预定义时间滑模面及滑模控制器. 然后分别利用有限时间同步策略、固定时间同步策略、传统预定义时间同步策略与新型预定义时间同步策略来实现新型忆阻耦合异质神经元的混沌同步. 最后验证新型预定义时间同步策略的鲁棒性. 需要说明的是, 本文讨论的混沌同步是指在一定的耦合强度条件下, 两个处于混沌状态的新型忆阻耦合异质神经元在预定义时间内达到同步.

    双三次插值函数表示为

    $$ \psi \left(x\right)=\begin{cases}\left(a+2\right)\cdot {\left|x\right|}^{3}-\left(a+3\right)\cdot {\left|x\right|}^{2}+1,& \left|x\right|\leqslant 1,\\ a\cdot {\left|x\right|}^{3}-5\cdot a\cdot {\left|x\right|}^{2}+8\cdot a\cdot \left|x\right|-4\cdot a,& 1 < \left|x\right| < 2,\\ 0,& 其他,\end{cases}$$ (1)

    其中, $a$为参数.

    受文献[36]的启发, 本文基于双三次插值函数和正弦函数, 设计一种新型局部有源忆阻器:

    $$ {i = G\left( \varphi \right) \cdot v,} ~~~~ {{{{\text{d}}\varphi }}/{{{\text{d}}t}} = v,} $$ (2)

    其中, $\varphi $表示状态变量; $v$, $i$分别表示输入电压和输出电流; $ G\left( \varphi \right) $表示忆导函数, 与电导具有相同的量纲, 物理单位为西门子(${\text{S}}$), 其数学表达式为

    $$ G( \varphi ) = - ( {a + 2} ) \cdot {| \varphi |^3} + ( {a {+} 3} ) \cdot {| \varphi |^2} + b {\cdot} \sin ( {c {\cdot} \varphi } ), $$ (3)

    其中, $a > 0$, $b > 0$和$c > 0$为可调参数.

    在新型局部有源忆阻器(2)两端, 施加正弦激励$v = {V_{\text{m}}} \cdot \sin \left( {\omega \cdot t} \right)$, 并假设$ \varphi {\left( t \right)_{t = {t_0}}} = \varphi \left( 0 \right) = 0 $, 可得

    $$ \begin{split} \varphi \left( t \right) = \int_0^t {v\left( \tau \right){\text{d}} \tau } = \int_0^t {{V_{\text{m}}} \cdot \sin \left( {\omega \cdot \tau } \right){\text{d}} \tau } = \frac{{{V_{\text{m}}}}}{\omega } \cdot \left[ {1 - \cos \left( {\omega \cdot t} \right)} \right]. \end{split} $$ (4)

    令$ {b^*} = b \cdot \sin \left( {c \cdot \varphi } \right) $, 将(4)式代入(2)式, 可得

    $$ \begin{split} i(t) = {b^*} \cdot v (t) - (a+2) \cdot {\left| {\frac{{{V_{\text{m}}}}}{\omega } \cdot \left[ {1 - \cos (\omega \cdot t)} \right]} \right|^3} \cdot v(t) + (a + 3) \cdot {\left| {\frac{{{V_{\text{m}}}}}{\omega } \cdot \left[ {1 - \cos (\omega \cdot t)} \right]} \right|^2} \cdot v (t). \end{split} $$ (5)

    由于$\sin \left( {\omega \cdot t} \right) = {v}/{{{V_{\text{m}}}}}$, 可得$\cos \left( {\omega \cdot t} \right)$:

    $$ \begin{split} \cos \left( {\omega \cdot t} \right) = \begin{cases} { {{\sqrt {V_{\text{m}}^2 - {v^2}} }}/{{{V_{\text{m}}}}},}&{t \in \left( {0,\dfrac{{\text{π }}}{{2\omega }}} \right) \cup \left( {\dfrac{{3{\text{π }}}}{{2\omega }},\dfrac{{{2{\text{π }}}}}{\omega }} \right),} \\ { - {{\sqrt {V_{\text{m}}^2 - {v^2}} }}/{{{V_{\text{m}}}}},}&{t \in \left( {\dfrac{{\text{π }}}{{2\omega }},\dfrac{{3{\text{π }}}}{{2\omega }}} \right). } \end{cases} \end{split} $$ (6)

    当$ t \in \left( {0, \dfrac{{\text{π }}}{{2\omega }}} \right) \cup \left( {\dfrac{{3{\text{π }}}}{{2\omega }}, \dfrac{{{\text{2}\pi }}}{\omega }} \right) $时, 将(6)式中$ \cos \left( {\omega \cdot t} \right) = {{\sqrt {V_{\text{m}}^2 - {v^2}} }}\big/{{{V_{\text{m}}}}} $代入(5)式, 可得

    $$ \begin{split} i\left( t \right) = & \left[ {{b^*} - 4 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} + 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot v + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^2}}}} \right] \cdot \sqrt {V_{\text{m}}^2 - {v^2}} \cdot v \\ & + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^3}}} - \left( {a + 3} \right) \cdot \frac{1}{{{\omega ^2}}}} \right] \cdot {v^3} + \left( {a + 2} \right) \cdot \frac{1}{{{\omega ^3}}} \cdot {\left( {\sqrt {V_{\text{m}}^2 - {v^2}} } \right)^3} \cdot v.\\[-1pt] \end{split} $$ (7)

    当$ t \in \left( {\dfrac{{\text{π }}}{{2\omega }}, \dfrac{{3{\text{π }}}}{{2\omega }}} \right) $时, 将(6)式中$ \cos \left( {\omega \cdot t} \right) = - {{\sqrt {V_{\text{m}}^2 - {v^2}} }}\big/{{{V_{\text{m}}}}} $代入(5)式, 可得

    $$ \begin{split} i\left( t \right) = & \left[ {{b^*} - 4 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} + 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot v - \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^2}}}} \right] \cdot \sqrt {V_{\text{m}}^2 - {v^2}} \cdot v \\ & + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^3}}} - \left( {a + 3} \right) \cdot \frac{1}{{{\omega ^2}}}} \right] \cdot {v^3} - \left( {a + 2} \right) \cdot \frac{1}{{{\omega ^3}}} \cdot {\left( {\sqrt {V_{\text{m}}^2 - {v^2}} } \right)^3} \cdot v. \end{split} $$ (8)

    根据(7)式和(8)式, 输出电流$i\left( t \right)$由线性部分${i_1}\left( t \right)$和非线性部分${i_2}\left( t \right)$组成:

    $$ {i_1}\left( t \right) = \left[ {{b^*} - 4 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} + 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot v, $$ (9)
    $$ \begin{split} {i_2}\left( t \right) = & \pm \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^2}}}} \right] \cdot \sqrt {V_{\text{m}}^2 - {v^2}} \cdot v \\ & \pm \left( {a + 2} \right) \cdot \frac{1}{{{\omega ^3}}} \cdot {\left( {\sqrt {V_{\text{m}}^2 - {v^2}} } \right)^3} \cdot v + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^3}}} - \left( {a + 3} \right) \cdot \frac{1}{{{\omega ^2}}}} \right] \cdot {v^3}. \end{split} $$ (10)

    线性部分${i_1}\left( t \right)$和非线性部分${i_2}\left( t \right)$之间的关系[37]可表示为:

    $$ \begin{split} {\rho _1} = & \frac{{\left| {{i_2}\left( t \right)} \right|}}{{\left| {{i_1}\left( t \right)} \right|}} = \bigg| \bigg\{ \pm \left[ {3 \cdot ( {a + 2} ) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{{V_{\text{m}}}}}{{{\omega ^2}}}} \right] \cdot \sqrt{V_{\text{m}}^2 - {v^2}} \pm (a + 2) \cdot \frac{1}{{{\omega ^3}}} \cdot \left( {\sqrt {V_{\text{m}}^2 - {v^2}} } \right)^3 \\ & + \left[ {3 \cdot (a + 2) \cdot \dfrac{{{V_{\text{m}}}}}{{{\omega ^3}}} - (a + 3) \cdot \frac{1}{{{\omega ^2}}}} \right] \cdot {v^2} \bigg\} \cdot v \bigg| \cdot {{\left| {\left[ {{b^*} - 4 \cdot \left( {a + 2} \right) \cdot \dfrac{{V_{\text{m}}^3}}{{{\omega ^3}}} + 2 \cdot \left( {a + 3} \right) \cdot \dfrac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot v} \right|}}^{-1} \\ & < \frac{{7 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 3 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}{{\left| {{b^*} \cdot {\omega ^3} - 4 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 2 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega } \right|}}. \end{split} $$ (11)

    由(11)式可知, 比值${\rho _1}$随频率的增大而减小.

    假设当比值${\rho _1}$小于阈值${\rho _2}$($0 < {\rho _2} < 1$), 即${\rho _1} < {\rho _2}$时, 新型局部有源忆阻器简化为线性电阻. 由于$a > 0$, $b > 0$, $c > 0$, 不等式成立:

    $$ \begin{split} {\rho _2} > \frac{{7 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 3 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}{{\left| {{b^*} \cdot {\omega ^3} - 4 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 2 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega } \right|}} > \frac{{7 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 3 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}{{\left| {{b^*}} \right| \cdot {\omega ^3} + 4 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 2 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}.\end{split} $$ (12)

    由于$0 < {\rho _2} < 1$, $ \left( {2 \cdot {\rho _2} - 3} \right) \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega < 0 $, 不等式(12)可缩放为:

    $$ {\rho _2} > \frac{{7 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 3 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega + \left( {2 \cdot {\rho _2} - 3} \right) \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}{{\left| {{b^*}} \right| \cdot {\omega ^3} + 4 \cdot \left( {a + 2} \right) \cdot V_{\text{m}}^3 + 2 \cdot \left( {a + 3} \right) \cdot V_{\text{m}}^2 \cdot \omega }}. $$ (13)

    假设${b^*} \ne 0$, 通过计算不等式(13), 可得

    $$ {f_{\max }} > \left[ {\frac{{\sqrt[{{ {^3}}}]{{\left( {a + 2} \right) \cdot \left( {7 - 4{\rho _2}} \right)}}}}{{2{\text{π }} \cdot \sqrt[{{ {^3}}}]{{{\rho _2} \cdot \left| {{b^*}} \right|}}}} \cdot {V_{\text{m}}}} \right]\,{\text{Hz}}. $$ (14)

    不等式(14)中, 由于${b^*} = b \cdot \sin \left( {c \cdot \varphi } \right)$, 所以$ - b \leqslant {b^*} \leqslant b$. 根据不等式(14), 当${b^*}$相当小, 且${b^*} \ne 0$时, 忆阻频率带宽[36,37]趋于无穷大. 因此, 新型局部有源忆阻器在高频条件下仍然能够保持紧磁滞回线.

    设置正弦激励$v = {V_{\text{m}}} \cdot \sin \left( {\omega \cdot t} \right) = {V_{\text{m}}} \cdot \sin \left(2{\text{π }} \cdot f \cdot t \right)$的频率为$f = 20\;{\text{GHz}}$, 幅值分别为${V_{\text{m}}} = 1\;{\text{V}}$, ${V_{\text{m}}} = 2\;{\text{V}}$, ${V_{\text{m}}} = 3\;{\text{V}}$, 新型局部有源忆阻器参数为a=104, b=105, c=105. 新型局部有源忆阻器在不同幅值下的紧磁滞回线, 如图1(a)所示.

    图 1 新型局部有源忆阻器的紧磁滞回线 (a) f = 20 GHz, 不同幅值; (b) Vm= 1 V, 不同频率\r\nFig. 1. Pinched hysteresis loop of the novel locally active memristor: (a) Different amplitudes for f = 20 GHz; (b) different frequencies for Vm= 1 V.
    图 1  新型局部有源忆阻器的紧磁滞回线 (a) f = 20 GHz, 不同幅值; (b) Vm= 1 V, 不同频率
    Fig. 1.  Pinched hysteresis loop of the novel locally active memristor: (a) Different amplitudes for f = 20 GHz; (b) different frequencies for Vm= 1 V.

    设置正弦激励$v = {V_{\text{m}}} \cdot \sin \left( {\omega \cdot t} \right) = {V_{\text{m}}} \cdot \sin \left( 2{\text{π }} \cdot f \cdot t \right)$的幅值为Vm= 1 V, 频率分别为$f = 3\;{\text{GHz}}$, $f = 5\;{\text{GHz}}$, $f = 9\;{\text{GHz}}$, $f = 20\;{\text{GHz}}$和$f = 80\;{\text{GHz}}$, 新型局部有源忆阻器参数为a=104, b=105, c=105. 新型局部有源忆阻器在不同频率下的紧磁滞回线, 如图1(b)所示.

    图1可知, 新型局部有源忆阻器的紧磁滞回线均过原点. 由图1(a)可知, 紧磁滞回线的旁瓣面积随幅值增大而增大; 由图1(b)可知, 紧磁滞回线的旁瓣面积随频率增大而减小. 图1(b)中, 当频率达到80 GHz时, 紧磁滞回线将收缩为一条直线, 也就是新型局部有源忆阻器的最高频率略大于80 GHz, 此时该忆阻器简化为线性电阻.

    现存忆阻器的最高频率一般为几赫兹、几千 赫兹、几兆赫兹等[36,38]. 因此, 相较于现存忆阻器的最高频率, 新型局部有源忆阻器的最高频率更大.

    忆阻器的局部有源性可由其瞬时功率及忆导函数图来验证[36,38]. 将(4)式代入(3)式, 新型局部有源忆阻器的忆导函数可写为

    $$ \begin{split} G\left( t \right) = &\; {b^*} - \left( {a + 2} \right) \cdot {\left| {\frac{{{V_{\text{m}}}}}{\omega } \cdot \left[ {1 - \cos \left( {\omega \cdot t} \right)} \right]} \right|^3} + \left( {a + 3} \right) \cdot {\left| {\frac{{{V_{\text{m}}}}}{\omega } \cdot \left[ {1 - \cos \left( {\omega \cdot t} \right)} \right]} \right|^2} \\ = &\; {b^*} - \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} + \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}} + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot \cos \left( {\omega \cdot t} \right) \\ & - \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} - \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^2}}{{{\omega ^2}}}} \right] \cdot {\cos ^2}\left( {\omega \cdot t} \right) + \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^3}}{{{\omega ^3}}} \cdot {\cos ^3}\left( {\omega \cdot t} \right). \end{split} $$ (15)

    根据$P\left( t \right) = v\left( t \right) \cdot i\left( t \right)$和$i\left( t \right) = G\left( t \right) \cdot v\left( t \right)$, 可得新型局部有源忆阻器的瞬时功率:

    $$ \begin{split} & P\left( t \right) = G\left( t \right) \cdot {v^2}\left( t \right) \\ = &\;{b^*} \cdot V_{\text{m}}^2 - \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}} + \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^4}}{{{\omega ^2}}} + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^4}}{{{\omega ^2}}}} \right] \cdot \cos \left( {\omega \cdot t} \right) \\ & - \left[ {{b^*} \cdot V_{\text{m}}^2 + 2 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}}} \right] \cdot {\cos ^2}\left( {\omega \cdot t} \right) - \left[ {2 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}} - 2 \cdot \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^4}}{{{\omega ^2}}}} \right] \cdot {\cos ^3}\left( {\omega \cdot t} \right) \\ & + \left[ {3 \cdot \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}} - \left( {a + 3} \right) \cdot \frac{{V_{\text{m}}^4}}{{{\omega ^2}}}} \right] \cdot {\cos ^4}\left( {\omega \cdot t} \right) - \left( {a + 2} \right) \cdot \frac{{V_{\text{m}}^5}}{{{\omega ^3}}} \cdot {\cos ^5}\left( {\omega \cdot t} \right).\end{split} $$ (16)

    由(16)式可知, 在一定参数条件下, 新型局部有源忆阻器的瞬时功率是周期函数.

    (3)式中, 取参数$a = 5$, $b = 1$, $c = 3$, 可得忆导函数图, 如图2所示.

    图 2 忆导函数图\r\nFig. 2. Diagram of the memductance function $ G\left( \varphi \right) $.
    图 2  忆导函数图
    Fig. 2.  Diagram of the memductance function $ G\left( \varphi \right) $.

    根据图2及(16)式, 可得如下结论: 在一个周期范围内, 存在使瞬时功率$ P\left( t \right) < 0 $的区域, 即局部有源区, 例如图2中$\varphi \in \left( { - 0.447, 0} \right)$的区域. 因此, 新型局部有源忆阻器(2)具有局部有源性.

    非易失性定理[39]: 所有非易失性忆阻器的断电图是水平轴的子集.

    本文利用断电图(power-off plot, POP)和动态路线图(dynamic route diagram, DRM)来分析新型局部有源忆阻器的非易失性.

    图3表示新型局部有源忆阻器的断电图和动态路线图. 由断电图可知, 当新型局部有源忆阻器断电时, 即$ v = 0\;{\text{V}} $, 断电图与水平轴重合. 当不同幅值的正弦激励(Vm= 1 V, Vm= 2 V, Vm= 3 V, Vm= –1 V, Vm= –2 V, Vm= –3 V)作用于新型局部有源忆阻器时, 其动态路线图由一些平行水平线组成. 在上半平面, 轨迹从左向右移动; 在下半平面, 轨迹从右向左移动. 这里, 箭头方向是其渐近稳定的方向. 因此, 新型局部有源忆阻器具有非易失性.

    图 3 新型局部有源忆阻器的断电图与动态路线图\r\nFig. 3. POP and DRM of the novel locally active memristor.
    图 3  新型局部有源忆阻器的断电图与动态路线图
    Fig. 3.  POP and DRM of the novel locally active memristor.

    在新型局部有源忆阻器两端施加幅值为${V_{\text{m}}}$的正电压脉冲, 其脉宽为${W_{12}}$. 该脉宽用于实现从状态$ {\varphi _1} $到状态$ {\varphi _2} $的切换, 其中$ {\varphi _1} $和$ {\varphi _2} $分别对应$G\left( {{\varphi _1}} \right)$和$G\left( {{\varphi _2}} \right)$.

    通过对状态方程$ \dfrac{{\text{d}}\varphi }{\text{d} t} = v(V_{\text{m}}) $进行积分, 可得

    $$ {W_{12}} = \int_{{\varphi _1}}^{{\varphi _2}} {\frac{1}{{v\left( {{V_{\text{m}}}} \right)}}{\text{d}}\varphi } . $$ (17)

    当脉宽${W_{12}}$满足(17)式时, 初始状态$ {\varphi _1} $(假设$ {\varphi _1} > 0 $)可以切换到目标状态$ {\varphi _2} $(假设$ {\varphi _2} > {\varphi _1} $), 其中初始状态$ {\varphi _1} $对应低电平$G\left( {{\varphi _1}} \right)$, 目标状态$ {\varphi _2} $对应高电平$G\left( {{\varphi _2}} \right)$.

    图4(a)所示, 设置Vm= 1 V, 且脉宽${W_{12}}$在${t_0} < t < {t_1}$(${t_1} = {t_0} + {W_{12}}$)时作用于忆阻器, 初始状态$ {\varphi _1} $在${t_0}$时刻跳跃到Vm= 1 V的路径上, 然后向右移动, 最后在${t_1}$时刻跌落到断电图的$\left( {{\varphi _2}, 0} \right)$点上. 如图4(b)所示, 通过施加正电压脉冲, 低电平忆导$G\left( {{\varphi _1}} \right)$切换到高电平忆导$G\left( {{\varphi _2}} \right)$. 在这种情况下, 局部有源忆阻器有两种状态, 截止和导通, 这意味着忆导处于低电平(对应截止状态)或高电平(对应导通状态).

    图 4 (a)状态$ {\varphi _1} $切换到状态$ {\varphi _2} $(正电压脉冲的幅值Vm= 1 V); (b)低电平忆导$G\left( {{\varphi _1}} \right)$切换到高电平忆导$G\left( {{\varphi _2}} \right)$(正电压脉冲的幅值Vm= 1 V)\r\nFig. 4. (a) Switching from the state $ {\varphi _1} $ to the state $ {\varphi _2} $ (a positive voltage pulse with amplitude Vm= 1 V); (b) switching from the low-level memductance $G\left( {{\varphi _1}} \right)$ to the high-level memductance $G\left( {{\varphi _2}} \right)$(a positive voltage pulse with amplitude Vm= 1 V).
    图 4  (a)状态$ {\varphi _1} $切换到状态$ {\varphi _2} $(正电压脉冲的幅值Vm= 1 V); (b)低电平忆导$G\left( {{\varphi _1}} \right)$切换到高电平忆导$G\left( {{\varphi _2}} \right)$(正电压脉冲的幅值Vm= 1 V)
    Fig. 4.  (a) Switching from the state $ {\varphi _1} $ to the state $ {\varphi _2} $ (a positive voltage pulse with amplitude Vm= 1 V); (b) switching from the low-level memductance $G\left( {{\varphi _1}} \right)$ to the high-level memductance $G\left( {{\varphi _2}} \right)$(a positive voltage pulse with amplitude Vm= 1 V).

    图5(a)所示, 设置Vm= –2 V, 且脉宽${W_{21}} = {t_2} - {t_0}$作用于忆阻器, 初始状态$ {\varphi _2} $在${t_0}$时刻跌落到Vm= –2 V的路径上, 然后向左移动, 最后在${t_2}$时刻跳跃到断电图的$\left( {{\varphi _1}, 0} \right)$点上. 如图5(b)所示, 通过施加负电压脉冲, 高电平忆导$G\left( {{\varphi _2}} \right)$切换到低电平忆导$G\left( {{\varphi _1}} \right)$.

    图 5 (a) 状态$ {\varphi _2} $切换到状态$ {\varphi _1} $ (负电压脉冲的幅值Vm= –2 V); (b) 高电平忆导$G\left( {{\varphi _2}} \right)$切换到低电平忆导$G\left( {{\varphi _1}} \right)$ (负电压脉冲的幅值Vm= –2 V)\r\nFig. 5. (a) Switching from the state $ {\varphi _2} $ to the state $ {\varphi _1} $ (a negative voltage pulse with amplitude Vm= –2 V); (b) switching from the high-level memductance $G\left( {{\varphi _2}} \right)$ to the low-level memductance $G\left( {{\varphi _1}} \right)$(a negative voltage pulse with amplitude Vm= –2 V).
    图 5  (a) 状态$ {\varphi _2} $切换到状态$ {\varphi _1} $ (负电压脉冲的幅值Vm= –2 V); (b) 高电平忆导$G\left( {{\varphi _2}} \right)$切换到低电平忆导$G\left( {{\varphi _1}} \right)$ (负电压脉冲的幅值Vm= –2 V)
    Fig. 5.  (a) Switching from the state $ {\varphi _2} $ to the state $ {\varphi _1} $ (a negative voltage pulse with amplitude Vm= –2 V); (b) switching from the high-level memductance $G\left( {{\varphi _2}} \right)$ to the low-level memductance $G\left( {{\varphi _1}} \right)$(a negative voltage pulse with amplitude Vm= –2 V).

    综上所述, 通过选择合适的方波电压脉冲, 新型局部有源忆阻器能够由一种状态切换到另一种状态(例如, 由截止切换到导通或由导通切换到截止). 该方波电压脉冲的幅值和脉宽可以调整. 因此, 如文献[36,39,40]所示, 新型局部有源忆阻器能够表示两个二进制状态0和1, 该特性将使其在存储器和数字逻辑电路中具有应用价值.

    孤立神经元中生物电信号的活动模式相对简单, 而不同的神经元之间具有异质性, 神经元活动具有很大的差异, 因此耦合异质神经元的动力学行为更加丰富. 本文通过新型局部有源忆阻器将二维HR神经元模型[41]和二维FHN神经元模型[41]进行耦合, 构成新型忆阻耦合异质神经元模型:

    $$ \left\{\begin{aligned} {{\dot x}_1} = &\;{x_2} - {\beta _1} \cdot x_1^3 + {\beta _2} \cdot x_1^2 \\ & + k \cdot G\left( \varphi \right) \cdot \left( {{x_1} - {x_3}} \right), \\ {{\dot x}_2} =&\; {\beta _3} - {\beta _4} \cdot x_1^2 - {x_2}, \\ {{\dot x}_3} =& \;\dfrac{1}{{{\beta _5}}} \cdot \left( {{x_3} - \dfrac{1}{3} \cdot x_3^3 - {x_4}} \right) \\ & - k \cdot G\left( \varphi \right) \cdot \left( {{x_1} - {x_3}} \right), \\ {{\dot x}_4} =&\; {\beta _5} \cdot {x_3} - {\beta _6} \cdot {x_4} + {\beta _7}, \\ \dot \varphi =&\; {x_1} - {x_3}, \end{aligned}\right. $$ (18)

    其中, ${x_1}$, ${x_2}$, ${x_3}$, ${x_4}$, $\varphi $为状态变量; $G\left( \varphi \right)$为忆导函数(如(3)式所示); $k$为耦合强度. 不失一般性, 设置系统参数$ {\beta _1} = 1 $, ${\beta _2} = 3$, ${\beta _3} = 1$, ${\beta _4} = 5$, ${\beta _5} = 5$, ${\beta _6} = 1$, ${\beta _7} = 1$; 忆阻器参数$a = 3$, $b = 2$, $c = 1$.

    令(18)式中的${\dot x_1} = {\dot x_2} = {\dot x_3} = {\dot x_4} = \dot \varphi = 0$, 可得

    $$ \begin{cases} 0 = {x_2} - x_1^3 + 3x_1^2, ~~~~ 0 = 1 - 5x_1^2 - {x_2}, \\ 0 = 3{x_3} - x_3^3 - 3{x_4}, ~~~~ 0 = 5{x_3} - {x_4} + 1,\\ 0 = {x_1} - {x_3}. \end{cases} $$ (19)

    由(19)式可知, 其不存在任何数学解, 即新型忆阻耦合异质神经元无平衡点. 根据隐藏吸引子的定义[42,43], 无平衡点的新型忆阻耦合异质神经元所产生的动力学行为均是隐藏的, 这在工程应用中具有重要价值.

    (19)式的雅可比矩阵为

    $$ \left[ {\begin{array}{*{20}{c}} { - 3x_1^2 + 6{x_1} + k \cdot G\left( \varphi \right)}&1&{ - k \cdot G\left( \varphi \right)}&0&{k \cdot \dot G\left( \varphi \right) \cdot \left( {{x_1} - {x_3}} \right)} \\ { - 10{x_1}}&{ - 1}&0&0&0 \\ { - k \cdot G\left( \varphi \right)}&0&{\dfrac{1}{5} - \dfrac{1}{5}x_3^2 + k \cdot G\left( \varphi \right)}&{ - \dfrac{1}{5}}&{ - k \cdot \dot G\left( \varphi \right) \cdot \left( {{x_1} - {x_3}} \right)} \\ 0&0&5&{ - 1}&0 \\ 1&0&{ - 1}&0&0 \end{array}} \right], $$ (20)

    其中, $ G\left( \varphi \right) = - 5{\left| \varphi \right|^3} + 6{\left| \varphi \right|^2} + 2\sin \left( \varphi \right) $, $ \dot G\left( \varphi \right) = - 15\varphi \cdot \left| \varphi \right| + 12\varphi {\text{ }} + {\text{ }}2\cos \left( \varphi \right) $. 基于(20)式, 可计算得到新型忆阻耦合异质神经元的李雅普诺夫指数.

    设置初始状态$\left[ {x_1}\left( 0 \right), {x_2}\left( 0 \right), {x_3}\left( 0 \right), {x_4}\left( 0 \right), \varphi \left( 0 \right) \right] = \left( {0, 0, 0, 0, 0} \right)$. 图6(a)表示耦合强度$k$变化时分岔图. 由图6(a)可知, 随耦合强度$k$增大, 新型忆阻耦合异质神经元出现倍周期分岔、混沌及大小不同并且周期数不等的周期窗口等现象. 图6(b)表示耦合强度$k$变化时前3个李雅普诺夫指数. 由图6(b)可知, 李雅普诺夫指数具有和分岔图相同的变化趋势, 对应分岔图中的周期行为与混沌行为.

    图 6 耦合强度$k$变化时分岔图和李雅普诺夫指数 (a)耦合强度$k$变化时分岔图; (b)耦合强度$k$变化时李雅普诺夫指数\r\nFig. 6. Bifurcation diagram and Lyapunov exponents with the coupling strength $k$ changing: (a) Bifurcation diagram with the coupling strength $k$ changing; (b) Lyapunov exponents with the coupling strength $k$ changing.
    图 6  耦合强度$k$变化时分岔图和李雅普诺夫指数 (a)耦合强度$k$变化时分岔图; (b)耦合强度$k$变化时李雅普诺夫指数
    Fig. 6.  Bifurcation diagram and Lyapunov exponents with the coupling strength $k$ changing: (a) Bifurcation diagram with the coupling strength $k$ changing; (b) Lyapunov exponents with the coupling strength $k$ changing.

    表1表示放电模式. 由相图及时域波形图(见图7图8)可知, 取不同耦合强度$k$时新型忆阻耦合异质神经元出现周期1, 2, 4, 5, 6, 8尖峰放电模式、混沌尖峰放电模式、周期4和周期8簇发放电模式及随机簇发放电模式.

    表 1  放电模式
    Table 1.  Firing modes.
    耦合强度$k$放电模式相图编号时域波形图编号
    0.0070周期1尖峰放电图7(a)图7(b)
    0.0400周期2尖峰放电图7(c)图7(d)
    0.1200周期4尖峰放电图7(e)图7(f)
    0.1691周期5尖峰放电图7(g)图7(h)
    0.1428周期6尖峰放电图7(i)图7(j)
    0.1290周期8尖峰放电图7(k)图7(l)
    0.1800混沌尖峰放电图7(m)图7(n)
    0.4150周期4簇发放电图8(a)图8(b)
    0.3600周期8簇发放电图8(c)图8(d)
    0.4800随机簇发放电图8(e)图8(f)
    下载: 导出CSV 
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    图 7 不同耦合强度$k$, 尖峰放电模式的相图及时域波形图\r\nFig. 7. Phase diagrams and time domain waveform diagrams of spiking firing modes, with different coupling strengths $k$.
    图 7  不同耦合强度$k$, 尖峰放电模式的相图及时域波形图
    Fig. 7.  Phase diagrams and time domain waveform diagrams of spiking firing modes, with different coupling strengths $k$.
    图 8 不同耦合强度$k$, 簇发放电模式的相图及时域波形图\r\nFig. 8. Phase diagrams and domain waveform diagrams of bursting firing modes, with different coupling strengths $k$.
    图 8  不同耦合强度$k$, 簇发放电模式的相图及时域波形图
    Fig. 8.  Phase diagrams and domain waveform diagrams of bursting firing modes, with different coupling strengths $k$.

    下文以初始状态$\varphi \left( 0 \right)$为例, 研究初始状态对新型忆阻耦合异质神经元动力学行为的影响. 图9表示初始状态$\varphi \left( 0 \right)$变化时分岔图, $\varphi \left( 0 \right)$的取值范围为$\left[ { - 0.4, 0.4} \right]$. 由图9可知, 随初始状态$\varphi \left( 0 \right)$增大, 新型忆阻耦合异质神经元的动力学行为在周期行为与混沌行为之间切换.

    图 9 初始状态$\varphi \left( 0 \right)$变化时分岔图, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$\r\nFig. 9. Bifurcation diagram with the initial state $\varphi \left( 0 \right)$ changing, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$.
    图 9  初始状态$\varphi \left( 0 \right)$变化时分岔图, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$
    Fig. 9.  Bifurcation diagram with the initial state $\varphi \left( 0 \right)$ changing, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$.

    当耦合强度$k = 0.18$时, 李雅普诺夫指数, 如图10所示. 图10中李雅普诺夫指数分别为$ {\text{L}}{{\text{E}}_1} = {0}{.04916} $, $ {\text{L}}{{\text{E}}_2} = 0.000137 $, $ {\text{L}}{{\text{E}}_3} = - 0.{68487} $, $ {\text{L}}{{\text{E}}_4} = - 1.03458 $和$ {\text{L}}{{\text{E}}_5} = - 6.50428 $. 由图10可知, 新型忆阻耦合异质神经元具有一个正的李雅普诺夫指数, 说明该系统处于混沌状态.

    图 10 李雅普诺夫指数\r\nFig. 10. Lyapunov exponents.
    图 10  李雅普诺夫指数
    Fig. 10.  Lyapunov exponents.

    本文利用0-1测试方法[44]来判别新型忆阻耦合异质神经元是否具有混沌行为. 0-1测试方法能够为系统在$\left( {p, s} \right)$平面内轨迹的检验提供可视化测试, 即在$\left( {p, s} \right)$平面内的有界轨迹表明系统是一个规则动态系统, 而类似于布朗运动的无界轨迹表明系统是一个混沌的动态系统.

    0-1测试如图11所示. 由图11可知, 新型忆阻耦合异质神经元的运动轨迹呈现出随时间变化而无界增长的现象, 具有布朗运动的特征, 对应混沌行为.

    图 11 0-1测试\r\nFig. 11. 0-1 test.
    图 11  0-1测试
    Fig. 11.  0-1 test.

    令${V_{{x_1}}} = {x_1}$, $ {V_{{x_2}}} = {x_2} $, ${V_{{x_3}}} = {x_3}$, $ {V_{{x_4}}} = {x_4} $, ${V_\varphi } = \varphi $, 根据基尔霍夫定律将新型忆阻耦合异质神经元的状态方程(18)转化为等效电路状态方程:

    $$ \begin{cases} {C_0} \cdot \dfrac{{{\text{d}}{V_{{x_1}}}}}{{{\text{d}}t}} = - \dfrac{1}{{{R_6}}} \cdot \left( { - {V_{{x_2}}}} \right) - \dfrac{1}{{{R_5}}} \cdot V_{{x_1}}^3 - \dfrac{1}{{{R_7}}} \cdot \left( { - V_{{x_1}}^2} \right) - \dfrac{1}{{{R_k}}} \cdot \left[ { - G\left( {{V_\varphi }} \right) \cdot \left( {{V_{{x_1}}} - {V_{{x_3}}}} \right)} \right], \\ {C_0} \cdot \dfrac{{{\text{d}}{V_{{x_2}}}}}{{{\text{d}}t}} = - \dfrac{1}{{{R_8}}} \cdot \left( { - {V_{{\text{HR}}}}} \right) - \dfrac{1}{{{R_9}}} \cdot V_{{x_1}}^2 - \dfrac{1}{{{R_{10}}}} \cdot {V_{{x_2}}}, \\ {C_0} \cdot \dfrac{{{\text{d}}{V_{{x_3}}}}}{{{\text{d}}t}} = - \dfrac{1}{{{R_{12}}}} \cdot \left( { - {V_{{x_3}}}} \right) - \dfrac{1}{{{R_{11}}}} \cdot V_{{x_3}}^3 - \dfrac{1}{{{R_{13}}}} \cdot {V_{{x_4}}} - \dfrac{1}{{{R_k}}} \cdot G\left( {{V_\varphi }} \right) \cdot \left( {{V_{{x_1}}} - {V_{{x_3}}}} \right), \\ {C_0} \cdot \dfrac{{{\text{d}}{V_{{x_4}}}}}{{{\text{d}}t}} = - \dfrac{1}{{{R_{14}}}} \cdot \left( { - {V_{{\text{FHN}}}}} \right) - \dfrac{1}{{{R_{16}}}} \cdot {V_{{x_4}}} - \dfrac{1}{{{R_{15}}}} \cdot \left( { - {V_{{x_3}}}} \right), \\ {C_0} \cdot \dfrac{{{\text{d}}{V_\varphi }}}{{{\text{d}}t}} = - \dfrac{1}{R} \cdot \left[ { - \left( {{V_{{x_1}}} - {V_{{x_3}}}} \right)} \right], \end{cases} $$ (21)

    其中, $ G\left( {{V_\varphi }} \right) = - \dfrac{R}{{{R_3}}} \cdot {\left| {{V_\varphi }} \right|^3} - \dfrac{R}{{{R_2}}} \cdot \left( { - {{\left| {{V_\varphi }} \right|}^2}} \right) - \dfrac{R}{{{R_4}}} \cdot \left[ { - \sin \left( {{V_\varphi }} \right)} \right] $.

    对新型忆阻耦合异质神经元进行电路实现时, 主要采用如下元件器: 线性电阻、电容、LF353D运算放大器、乘法器和非线性相关电源(nonlinear dependent source)等. 表2表示新型忆阻耦合 异质神经元的电路参数. 新型忆阻耦合异质神经元的电路实现包括3部分: 新型局部有源忆阻器的电路实现, 如图12(a)所示; HR神经元的电路实现, 如图12(b)所示; FHN神经元的电路实现, 如图12(c)所示. 需要说明的是, 通过调整HR神经元电路及FHN神经元电路中的电阻${R_k}$来实现耦合强度$k$的变化. 电阻${R_k}$与耦合强度$k$的关系为${R_k} ={R}/{k}$.

    表 2  新型忆阻耦合异质神经元的电路参数
    Table 2.  Circuit parameters of the novel memristor-coupled heterogeneous neuron.
    电路参数 类型
    $R$, ${R_5}$, ${R_6}$, ${R_8}$, ${R_{10}}$, ${R_{14}}$, ${R_{16}}$ 电阻/kΩ $200$
    ${R_1}$ 电阻/kΩ $ 2700 $
    ${R_2}$ 电阻/kΩ $ 33.333 $
    ${R_3}$, ${R_9}$, ${R_{15}}$ 电阻/kΩ $ 40 $
    ${R_4}$ 电阻/kΩ $ 100 $
    ${R_7}$ 电阻/kΩ $66.667$
    ${R_{11}}$ 电阻/kΩ $ 3000 $
    ${R_{12}}$, ${R_{13}}$ 电阻/kΩ $ 1000 $
    ${R_k}$ 电阻/kΩ $ {{200} \mathord{\left/ {\vphantom {{200} k}} \right. } k} $
    ${C_0}$ 电容/nF $40$
    ${V_{{\text{HR}}}}$, ${V_{{\text{FHN}}}}$ 直流电压源/V $ 1 $
    下载: 导出CSV 
    | 显示表格
    图 12 电路实现 (a) 新型局部有源忆阻器的电路实现; (b) HR神经元的电路实现; (c) FHN神经元的电路实现\r\nFig. 12. Circuit implementations: (a) Circuit implementation of the novel locally active memristor; (b) circuit implementation of the HR neuron; (c) circuit implementation of the FHN neuron.
    图 12  电路实现 (a) 新型局部有源忆阻器的电路实现; (b) HR神经元的电路实现; (c) FHN神经元的电路实现
    Fig. 12.  Circuit implementations: (a) Circuit implementation of the novel locally active memristor; (b) circuit implementation of the HR neuron; (c) circuit implementation of the FHN neuron.

    不失一般性, 取耦合强度$k = 0.04$, Rk = 5000 kΩ, 电路实现的相图及时域波形图如图13所示. 由图13可知, 新型忆阻耦合异质神经元产生周期2尖峰放电模式. 取耦合强度$k = 0.18$, Rk = 1111.111 kΩ, 电路实现的相图及时域波形图如图14所示. 由图14可知, 新型忆阻耦合异质神经元产生混沌尖峰放电模式. 由上所述, 电路实现结果与数值仿真结果(见图7(c), (d), (m), (n))基本一致.

    图 13 电路实现的相图及时域波形图(周期2尖峰放电模式) (a) 相图; (b) 时域波形图\r\nFig. 13. Phase diagram and time domain waveform diagram of circuit implementation (period-2 spiking firing mode): (a) Phase diagram; (b) time domain waveform.
    图 13  电路实现的相图及时域波形图(周期2尖峰放电模式) (a) 相图; (b) 时域波形图
    Fig. 13.  Phase diagram and time domain waveform diagram of circuit implementation (period-2 spiking firing mode): (a) Phase diagram; (b) time domain waveform.
    图 14 电路实现的相图及时域波形图(混沌尖峰放电模式) (a)相图; (b)时域波形图\r\nFig. 14. Phase diagram and time domain waveform diagram of circuit implementation (chaotic spiking firing mode): (a) Phase diagram; (b) time domain waveform.
    图 14  电路实现的相图及时域波形图(混沌尖峰放电模式) (a)相图; (b)时域波形图
    Fig. 14.  Phase diagram and time domain waveform diagram of circuit implementation (chaotic spiking firing mode): (a) Phase diagram; (b) time domain waveform.

    驱动系统(22)和响应系统(23)如下所示:

    $$ \begin{cases} {{\dot x}_1} = {f_1}\left( {{\boldsymbol{x}},t} \right) + \Delta {f_1}\left( {{\boldsymbol{x}},t} \right) + d_1^{\text{m}}\left( t \right), \\ {{\dot x}_2} = {f_2}\left( {{\boldsymbol{x}},t} \right) + \Delta {f_2}\left( {{\boldsymbol{x}},t} \right) + d_2^{\text{m}}\left( t \right), \\ ~~~~~~~ \vdots \\ {{\dot x}_n} = {f_n}\left( {{\boldsymbol{x}},t} \right) + \Delta {f_n}\left( {{\boldsymbol{x}},t} \right) + d_n^{\text{m}}\left( t \right); \end{cases} $$ (22)
    $$\begin{cases} {{\dot y}_1} = {g_1}\left( {{\boldsymbol{y}},t} \right) + \Delta {g_1}\left( {{\boldsymbol{y}},t} \right) + d_1^{\text{s}}\left( t \right) + {u_1}\left( t \right), \\ {{\dot y}_2} = {g_2}\left( {{\boldsymbol{y}},t} \right) + \Delta {g_2}\left( {{\boldsymbol{y}},t} \right) + d_{2}^{\text{s}}\left( t \right) + {u_2}\left( t \right), \\ ~~~~~~~~ \vdots \\ {{\dot y}_n} = {g_n}\left( {{\boldsymbol{y}},t} \right) + \Delta {g_n}\left( {{\boldsymbol{y}},t} \right) + d_n^{\text{s}}\left( t \right) + {u_n}\left( t \right),\end{cases} $$ (23)

    其中, ${\boldsymbol{x}} = {\left( {{x_1}, {x_2}, \ldots , {x_n}} \right)^{\text{T}}}$和${\boldsymbol{y}} = {\left( {{y_1}, {y_2}, \ldots , {y_n}} \right)^{\text{T}}}$为状态变量; $ \Delta {f_i}\left( {{\boldsymbol{x}}, t} \right) $和$ \Delta {g_i}\left( {{\boldsymbol{y}}, t} \right) $为内部不确定性; $ d_i^{\text{m}}\left( t \right) $和$ d_i^{\text{s}}\left( t \right) $为外部扰动; ${u_i}\left( t \right)$为控制器, $i = 1, 2, \cdots , n$.

    假设1 假设系统内部不确定性有界:

    $$\begin{split} & \| {\Delta {f_i} ( {{\boldsymbol{x}},t} )} \| \leqslant {\eta _i} \| {\boldsymbol{x}} \|, ~\| {\Delta {g_i}( {{\boldsymbol{y}},t} )} \| \leqslant {\lambda _i} \| {\boldsymbol{y}} \| , \\ & \qquad \qquad i = 1,2, \cdots ,n, \end{split} $$

    其中, $\| \cdot \|$表示欧几里得范数. $ {\eta _i} $, $ {\lambda _i} $均为正常数.

    假设2 假设系统外部扰动有界:

    $$\begin{split} & \left| {d_i^{\text{m}} (t)} \right| \leqslant D_i^{\text{m}},~~\left| {d_i^{\text{s}} (t)} \right| \leqslant D_i^{\text{s}}, \\ & ~~~~~~~~~i = 1,2, \cdots ,n, \end{split} $$

    其中, $ D_i^{\text{m}} $和$ D_i^{\text{s}} $均为正常数.

    由驱动系统和响应系统可得误差系统(24):

    $$\begin{cases} {{\dot e}_1} = {g_1}\left( {{\boldsymbol{y}},t} \right) - {f_1}\left( {{\boldsymbol{x}},t} \right) + \Delta {g_1}\left( {{\boldsymbol{y}},t} \right) - \Delta {f_1}\left( {{\boldsymbol{x}},t} \right) + d_1^{\text{s}}\left( t \right) - d_1^{\text{m}}\left( t \right) + {u_1}\left( t \right), \\ {{\dot e}_2} = {g_2}\left( {{\boldsymbol{y}},t} \right) - {f_2}\left( {{\boldsymbol{x}},t} \right) + \Delta {g_2}\left( {{\boldsymbol{y}},t} \right) - \Delta {f_2}\left( {{\boldsymbol{x}},t} \right) + d_2^{\text{s}}\left( t \right) - d_2^{\text{m}}\left( t \right) + {u_2}\left( t \right), \\ ~~~~~~~~\vdots \\ {{\dot e}_n} = {g_n}\left( {{\boldsymbol{y}},t} \right) - {f_n}\left( {{\boldsymbol{x}},t} \right) + \Delta {g_n}\left( {{\boldsymbol{y}},t} \right) - \Delta {f_n}\left( {{\boldsymbol{x}},t} \right) + d_n^{\text{s}}\left( t \right) - d_n^{\text{m}}\left( t \right) + {u_n}\left( t \right). \end{cases} $$ (24)

    定义1 有限时间稳定性[45]

    假设误差系统(24)是全局渐近稳定的, 且误差系统(24)的任意解在有限时间内到达平衡点, 则称误差系统(24)是全局有限时间稳定性系统, 即

    $$ \forall t \geqslant T\left( {{{\boldsymbol{e}}_0}} \right),~~{\boldsymbol{e}}\left( {t,{{\boldsymbol{e}}_0}} \right) = \it{\bf{0}}, $$

    其中, $ T\left( {{{\boldsymbol{e}}_0}} \right) $为误差系统(24)的实际收敛时间.

    定义2 固定时间稳定性[46]

    假设误差系统(24)满足如下两个条件, 则称误差系统(24)为全局固定时间稳定性系统. ①能在有限时间内达到稳定状态; ②稳定时间$ T\left( {{{\boldsymbol{e}}_0}} \right) $是全局有界的, 并与误差系统的初始状态无关, 即:

    $$ \exists {T_{\max }} > 0,~~\forall {{\boldsymbol{e}}_0} \in {{{\mathbb{ R}}}^n},~~ T\left( {{{\boldsymbol{e}}_0}} \right) \leqslant {T_{\max }}, $$

    其中, $ {T_{\max }} $为误差系统(24)的收敛时间估计.

    定义3 预定义时间稳定性[47]

    对于预定义参数$ {T_{\text{c}}} > 0 $ ($ {T_{\text{c}}} $是可调参数), 若误差系统(24)是固定时间稳定的, 且稳定时间函数$ T $: $ {{{\mathbb{ R}}}^n} \to {{\mathbb{ R}}} $使得

    $$ \forall {{\boldsymbol{e}}_0} \in {{{\mathbb{ R}}}^n},~~T\left( {{{\boldsymbol{e}}_0}} \right) \leqslant {T_{\text{c}}}. $$

    则称误差系统(24)为全局预定义时间稳定性系统.

    4.2.1   新型预定义时间Lyapunov函数的充分条件

    定理1 针对误差系统(24), 若Lyapunov函数$V({\boldsymbol{e}})$, 满足如下不等式:

    $$ \dot V \leqslant - \frac{{\ln 4}}{{\alpha \cdot {T_{\text{c}}}}} \cdot {V^{1 - \alpha /2}} \cdot \big[ {1 + {\text{exp}}\big( {{V^{\alpha /2}}} \big)} \big],~~\alpha \in (0,1), $$ (25)

    则称误差系统(24)是预定义时间稳定的, 预定义时间$ {T_{\text{c}}} > 0 $.

    需要说明的是, 误差系统(24)的实际收敛时间$ T\left( {{{\boldsymbol{e}}_0}} \right) $依赖于预定义时间${T_{\text{c}}}$ ($T\left( {{{\boldsymbol{e}}_0}} \right) \leqslant {T_{\text{c}}}$).

    证明 根据不等式(25), 可得收敛时间$ T\left( {{{\boldsymbol{e}}_{0}}} \right) $满足如下条件:

    $$ \begin{split} T\left( {{{\boldsymbol{e}}_0}} \right) & \leqslant - \frac{{{T_{\text{c}}}}}{{\ln 2}} \cdot \int_{{V_{{t_0}}}}^{{V_{{t_{\text{f}}}}}} {\frac{{{\text{d}}{V^{\alpha /2}}}}{{\left[ {1 + \exp \left( {{V^{\frac{\alpha }{2}}}} \right)} \right]}}} \\ & = \frac{{{T_{\text{c}}}}}{{\ln 2}} \cdot \int_0^{{V_{{t_0}}}} {\frac{{{\text{d}}{V^{\alpha /2}}}}{{\left[ {1 + \exp \left( {{V^{\alpha /2}}} \right)} \right]}}} \\ &= \left. { - \frac{{{T_{\text{c}}}}}{{\ln 2}} \cdot \ln \left[ {1 + \exp \left( { - {V^{\alpha /2}}} \right)} \right]} \right|_0^{{V_{{t_0}}}} \\ & = \frac{{{T_{\text{c}}}}}{{\ln 2}}\left\{ {\ln 2 - \ln \left[ {1 + \exp \left( { - V_{{t_0}}^{^{\alpha /2}}} \right)} \right]} \right\} \\ & \leqslant {T_{\text{c}}}. \end{split} $$ (26)

    由(26)式可得$T\left( {{{\boldsymbol{e}}_0}} \right) \leqslant {T_{\text{c}}}$, ${T_{\text{c}}} = \sup \, T\left( {{{\boldsymbol{e}}_0}} \right)$.

    4.2.2   新型预定义时间滑模面设计

    设计新型预定义时间滑模面:

    $$ \begin{split} {s_i} = &\; {e_i} + \int_0^t \big\{{l_1} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{1 - \alpha }} \\ & \times \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] + {l_2} \cdot {\text{signr}}\left( {{e_i}} \right)\big\}{\text{d}}\tau ,\end{split} $$ (27)

    其中, $\alpha \in \left( {0, 1} \right)$, ${l_1} = \dfrac{{\sqrt {{2^\alpha }} \cdot \ln 2}}{{\alpha \cdot {T_{{c_1}}}}}$, ${l_2} > 0$. 非线性函数${\text{signr}}\left( \cdot \right)$[48]的表达式如下:

    $$ {\text{signr}}\left( {{e_i}} \right) = {{{e_i}}} \Big/{{\sqrt {e_i^2 + {{0.01}^2}} }}. $$

    定理2 若误差系统(24)使用新型预定义时间滑模面(27), 则误差系统(24)沿新型预定义时间滑模面在预定义时间${T_{{{\text{c}}_{1}}}}$内收敛到零.

    证明 当误差系统(24)到达新型预定义时间滑模面后, 即${s_i} = 0$, ${\dot s_i} = 0$, 可得

    $$\begin{split} {\dot e_i} = - {l_1} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {\left| {{e_i}} \right|^{1 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] - {l_2} \cdot {\text{signr}}\left( {{e_i}} \right). \end{split} $$ (28)

    选取Lyapunov函数${V_1} = \dfrac{1}{2} \cdot e_i^2$, 对其求导可得

    $$ \begin{split} {{\dot V}_1} & = {e_i} \cdot {{\dot e}_i} = {e_i} \cdot \left\{ - {l_1} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{1 - \alpha }} \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] - {l_2} \cdot {\text{signr}}\left( {{e_i}} \right) \right\} \\ & = - {l_1} \cdot {\left| {{e_i}} \right|^{2 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] - {l_2} \cdot \frac{{e_i^2}}{{\sqrt {e_i^2 + {{0.01}^2}} }} \\ & \leqslant - {l_1} \cdot {\left| {{e_i}} \right|^{2 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] \leqslant - {l_1} \cdot {\left| {{e_i}} \right|^{2 - \alpha }} \cdot \left\{ {1 + {\text{exp}}\left[ {{{\left( {\frac{1}{2} \cdot e_i^2} \right)}^{\alpha /2}}} \right]} \right\}. \end{split} $$ (29)

    将${l_1} = \dfrac{{\sqrt {{2^\alpha }} \cdot \ln 2}}{{\alpha \cdot {T_{{c_1}}}}}$代入不等式(29)可得

    $$ \begin{split} {\dot V_1} & \leqslant - {l_1} \cdot {\left| {{e_i}} \right|^{2 - \alpha }} \cdot \left\{ {1 + {\text{exp}}\left[ {{{\left( {\frac{1}{2} \cdot e_i^2} \right)}^{\alpha /2}}} \right]} \right\} = - \frac{{\ln 4}}{{\alpha \cdot {T_{{{\text{c}}_{1}}}}}} \cdot V_1^{1 - \alpha /2} \cdot \left[ {1 + {\text{exp}}\left( {V_1^{\alpha /2}} \right)} \right]. \end{split} $$ (30)

    (30)式满足定理1. 因此, 误差系统(24)沿新型预定义时间滑模面在预定义时间${T_{{{\text{c}}_{1}}}}$内收敛到零.

    4.2.3   新型预定义时间滑模控制器设计

    设计新型预定义时间滑模控制器:

    $$ \begin{split} {u_i} = & - \left[ {{g_i}\left( {\boldsymbol{y}} \right) - {f_i}\left( {\boldsymbol{x}} \right)} \right] - \left\{ {{l_1} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{1 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] + {l_2} \cdot {\text{signr}}\left( {{e_i}} \right)} \right\} \\ & - \left\{ {{l_3} \cdot {\text{sign}}\left( {{s_i}} \right) \cdot {{\left| {{s_i}} \right|}^{1 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {s_i^\alpha } \right)} \right] + {l_4} \cdot {\text{kas}}\left( {{s_i},\gamma ,\delta } \right)} \right\} \\ & - \left( {{\eta _i}\left\| {\boldsymbol{x}} \right\| + {\lambda _i}\left\| {\boldsymbol{y}} \right\| + D_i^{\text{m}} + D_i^{\text{s}}} \right) \cdot {\text{sign}}\left( {{s_i}} \right), \end{split} $$ (31)

    其中, ${l_1} = \dfrac{{\sqrt {{2^\alpha }} \cdot \ln 2}}{{\alpha \cdot {T_{{{\text{c}}_{1}}}}}}$, ${l_2} > 0$, ${l_3} = \dfrac{{\sqrt {{2^\alpha }} \cdot \ln 2}}{{\alpha \cdot {T_{{{\text{c}}_{2}}}}}}$, ${l_4} > 0$, $\alpha \in \left( {0, 1} \right)$, 非线性函数$ {\text{kas}}\left( \cdot \right) $[49]的表达式如下:

    $$ {\text{kas}}\left( {{s_i},\gamma ,\delta } \right) = \begin{cases} \begin{aligned} \left[ {\gamma \cdot {\delta ^{\gamma - 1}} - \frac{{\left( {1 - \gamma } \right) \cdot {\delta ^\gamma } \cdot \cos \left( \delta \right)}}{{\sin \left( \delta \right) - \delta \cdot \cos \left( \delta \right)}}} \right] \cdot {s_i} + \frac{{\left( {1 - \gamma } \right) \cdot {\delta ^\gamma }}}{{\sin \left( \delta \right) - \delta \cdot \cos \left( \delta \right)}} \cdot \sin \left( {{s_i}} \right), \\ \end{aligned} &{\left| {{s_i}} \right| \leqslant \delta ,} \\ {{{\left| {{s_i}} \right|}^\gamma } \cdot {\text{sign}}\left( {{s_i}} \right),}&{\left| {{s_i}} \right| > \delta ,} \end{cases} $$

    其中, $ \gamma = 0.25 $, $ \delta = 0.1 $.

    需要说明的是, 设计新型预定义时间滑模控制器$ {u_i} $时, 不妨选取趋近律${\dot s_i} = - \big\{ {l_3} \cdot {\text{sign}} ({{s_i}} ) \cdot {{\left| {{s_i}} \right|}^{1 - \alpha }} \times \left[ {1 + {\text{exp}}\left( {s_i^\alpha } \right)} \right] + {l_4} \cdot {\text{kas}} ( {{s_i}, \gamma , \delta }) \big\}$.

    定理3 若误差系统(24)利用新型预定义时间滑模控制器(31), 则误差系统(24)在预定义时间${T_{{{\text{c}}_{2}}}}$内到达新型预定义时间滑模面.

    证明 选取Lyapunov函数${V_2} = \dfrac{1}{2} \cdot s_i^2$, 对其求导可得

    $$ \begin{split} {{\dot V}_2} & = {s_i} \cdot {{\dot s}_i} = {s_i} \cdot \big\{ {{\dot e}_i} + {l_1} \cdot {\text{sign}} ( {{e_i}} ) \cdot {{| {{e_i}} |}^{1 - \alpha }} \cdot [ {1 + {\text{exp}}( {e_i^\alpha } )} ] + {l_2} \cdot {\text{signr}} ( {{e_i}} ) \big\} \\ & = {s_i} \cdot \big\{ {g_i}( {\boldsymbol{y}} ) - {f_i}( {\boldsymbol{x}} ) + \Delta {g_i}( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}} + {u_i} \\ & \quad + {l_1} \cdot {\text{sign}} ( {{e_i}} ) \cdot {| {{e_i}} |^{1 - \alpha }} \cdot [{1 + {\text{exp}}( {e_i^\alpha } )} ] + {l_2} \cdot {\text{signr}}( {{e_i}} ) \big\} \\ & = {s_i} \cdot \big\{ \Delta {g_i} ( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}} - ( {{\eta _i} \| {\boldsymbol{x}} \| + {\lambda _i} \| {\boldsymbol{y}} \| + D_i^{\text{m}} + D_i^s} ) \cdot {\text{sign}}({{s_i}} ) \\ & \quad - {l_3} \cdot {\text{sign}}( {{s_i}}) \cdot {| {{s_i}} |^{1 - \alpha }} \cdot \left[ {1 + {\text{exp}}( {s_i^\alpha } )} \right] - {l_4} \cdot {\text{kas}} ({{s_i},\gamma ,\delta } ) \big\} \\ &= \big\{ {s_i} \cdot \left[ {\Delta {g_i}( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}}} \right] - ( {{\eta _i} \| {\boldsymbol{x}} \| + {\lambda _i} \| {\boldsymbol{y}} \| + D_i^{\text{m}} + D_i^{\text{s}}} ) \cdot | {{s_i}} | \\ & \quad - {l_3} \cdot {| {{s_i}} |^{2 - \alpha }} \cdot \left[ {1 + {\text{exp}}( {s_i^\alpha } )} \right] - {l_4} \cdot {s_i} \cdot {\text{kas}}( {{s_i},\gamma ,\delta } ) \big\} \\ & \leqslant - {l_3} \cdot {\left| {{s_i}} \right|^{2 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {s_i^\alpha } \right)} \right] \leqslant - \frac{{\ln 4}}{{\alpha \cdot {T_{{{\text{c}}_{2}}}}}} \cdot V_2^{^{1 - \alpha /2}} \cdot \left[ {1 + {\text{exp}}\left( {V_2^{^{\alpha /2}}} \right)} \right]. \end{split} $$ (32)

    (32)式满足定理1. 因此, 误差系统(24)在预定义时间${T_{{{\text{c}}_{2}}}}$内到达新型预定义时间滑模面(27).

    综上可知, 通过利用新型预定义时间滑模面(27)和新型预定义时间滑模控制器(31), 误差系统(24)在预定义时间内收敛到零, 也就是两个混沌系统(22)和(23)同步.

    1) 不同稳定时间同步策略的对比. 本文分别利用有限时间同步策略、固定时间同步策略和新型预定义时间同步策略来实现新型忆阻耦合异质神经元的混沌同步.

    2) 与传统预定义时间同步策略的对比. 本文利用传统预定义时间同步策略来实现新型忆阻耦合异质神经元的混沌同步.

    3) 鲁棒性的验证. 本文分别在理想条件下(无内部不确定性和无外部干扰)与非理想条件下(有内部不确定性和有均匀分布噪声), 利用新型预定义时间同步策略来实现新型忆阻耦合异质神经元的混沌同步, 从而验证该新型预定义时间同步策略的鲁棒性.

    基于新型忆阻耦合异质神经元模型(18), 并考虑内部不确定性$ \Delta {f_i}\left( {\boldsymbol{x}} \right) $及外部扰动$ d_i^{\text{m}} $, 可得驱动系统(33). 需要说明的是, 为方便起见, 令(18)式中的$\varphi = {x_5}$.

    $$\left\{\begin{aligned} {{\dot x}_1} = &\; {x_2} - x_1^3 + 3x_1^2 + k \cdot G\left( {{x_5}} \right)\cdot \left( {{x_1} - {x_3}} \right) + \Delta {f_1}\left( {\boldsymbol{x}} \right) + d_1^{\text{m}}, \qquad {{\dot x}_2} = 1 - 5 \cdot x_1^2 - {x_2} + \Delta {f_2}\left( {\boldsymbol{x}} \right) + d_2^{\text{m}}, \\ {{\dot x}_3} =&\; \dfrac{1}{5} \cdot \left( {{x_3} - \dfrac{1}{3} \cdot x_3^3 - {x_4}} \right) - k \cdot G\left( {{x_5}} \right)\cdot\left( {{x_1} - {x_3}} \right) + \Delta {f_3}\left( {\boldsymbol{x}} \right) + d_3^{\text{m}}, \\ {{\dot x}_4} =&\; 5{x_3} - {x_4} + 1 + \Delta {f_4}\left( {\boldsymbol{x}} \right) + d_4^{\text{m}}, \qquad {{\dot x}_5} = {x_1} - {x_3} + \Delta {f_5}\left( {\boldsymbol{x}} \right) + d_5^{\text{m}}. \end{aligned} \right.$$ (33)

    响应系统为:

    $$\left\{ \begin{aligned} {{\dot y}_1} =&\; {y_2} - y_1^3 + 3y_1^2 + k \cdot G\left( {{y_5}} \right) \cdot\left( {{y_1} - {y_3}} \right) + \Delta {g_1}\left( {\boldsymbol{y}} \right) + d_1^{\text{s}} + {u_1}, \\ {{\dot y}_2} =&\; 1 - 5 \cdot y_1^2 - {y_2} + \Delta {g_2}\left( {\boldsymbol{y}} \right) + d_2^{\text{s}} + {u_2}, \\ {{\dot y}_3} = &\;\dfrac{1}{5} \cdot \left( {{y_3} - \dfrac{1}{3} \cdot y_3^3 - {y_4}} \right) - k \cdot G\left( {{y_5}} \right) \cdot \left( {{y_1} - {y_3}} \right) + \Delta {g_3}\left( {\boldsymbol{y}} \right) + d_3^{\text{s}} + {u_3}, \\ {{\dot y}_4} =&\; 5{y_3} - {y_4} + 1 + \Delta {g_4}\left( {\boldsymbol{y}} \right) + d_4^{\text{s}} + {u_4}, \qquad {{\dot y}_5} = {y_1} - {y_3} + \Delta {g_5}\left( {\boldsymbol{y}} \right) + d_5^{\text{s}} + {u_5}. \end{aligned}\right. $$ (34)

    不失一般性, 内部不确定性取为$ \Delta {f_1}\left( {\boldsymbol{x}} \right) = 3 \cdot \sin \left( t \right) \cdot {x_1} $, $ \Delta {f_2}\left( {\boldsymbol{x}} \right) = \cos \left( t \right) \cdot {x_2} $, $ \Delta {f_3}\left( {\boldsymbol{x}} \right) = 2 \cdot \cos \left( t \right) \cdot {x_3} $, $ \Delta {f_4}\left( {\boldsymbol{x}} \right) = 1.5 \cdot \sin \left( t \right) \cdot {x_4} $, $ \Delta {f_5}\left( {\boldsymbol{x}} \right) = 3 \cdot \cos \left( t \right) \cdot {x_5} $, $ \Delta {g_1}\left( {\boldsymbol{y}} \right) = 2 \cdot \cos \left( t \right) \cdot {y_1} $, $ \Delta {g_2}\left( {\boldsymbol{y}} \right) = 1.5 \cdot \cos \left( t \right) \cdot {y_2} $, $ \Delta {g_3}\left( {\boldsymbol{y}} \right) = 2 \cdot \sin \left( t \right) \cdot {y_3} $, $ \Delta {g_4}\left( {\boldsymbol{y}} \right) = 3 \cdot \cos \left( t \right) \cdot {y_4} $, $ \Delta {g_5}\left( {\boldsymbol{y}} \right) = \sin \left( t \right) \cdot {y_5} $; 幅值为2的均匀分布噪声作为外部扰动$ d_i^{\text{m}} $, 幅值为2.5的均匀分布噪声作为外部扰动$ d_i^{\text{s}} $; 耦合强度取为$k = 0.18$; 驱动系统(33)的初始状态取为${\boldsymbol{x}}\left( {0} \right) = [ - 2.5, 0, - 1, 0, - 1 ]^{\text{T}}$, 响应系统(34)的初始状态取为${\boldsymbol{y}}\left( 0 \right) = [ 2.5, 3, 1, 4, 1.5]^{\text{T}}$.

    4.3.1   三种稳定时间同步策略的对比

    本文分别利用3种稳定时间同步策略(有限时间同步策略、固定时间同步策略和新型预定义时间同步策略)使驱动系统(33)和响应系统(34)达到同步. 需要说明的是, 为了更加有效地对比3种稳定时间同步策略的实际收敛时间大小, 本文设置3种稳定时间同步策略中相似项的参数相同.

    仿真中, 设置3种稳定时间同步策略的参数$\alpha = {\varsigma _1} = 0.5$, ${\varsigma _2} = 1.5$, ${l_1} = {l_3} = {l_9} = {l_{11}} = {2^{2.25}} \cdot \ln 32$, ${l_2} = {l_4} = {l_5} = {l_6} = {l_7} = {l_8} = 15$, ${l_{10}} = {l_{12}} = {2^{3.75}} \cdot \ln 32$. 设置新型预定义时间同步策略(27)和(31)中滑动段的预定义时间${T_{{{\text{c}}_{1}}}} = 0.1$, 到达段的预定义时间$ {T_{{{\text{c}}_{2}}}} = 0.1 $.

    有限时间同步策略[50]如下:

    滑模面:

    $$ {s_i} = {e_i} + \int_0^t {\left[ {{l_5} \cdot {e_i} + {l_6} \cdot {\text{sign}}\left( {{e_i}} \right)} \right]{\text{d}}\tau } ; $$ (35)

    滑模控制器:

    $$ \begin{split} {u_i} = & - \left[ {{g_i}\left( {\boldsymbol{y}} \right) - {f_i}\left( {\boldsymbol{x}} \right)} \right] - \left[ {{l_5} \cdot {e_i} + {l_6} \cdot {\text{sign}}\left( {{e_i}} \right)} \right] \\ &- \left[ {{l_7} \cdot {s_i} + {l_8} \cdot {\text{sign}}\left( {{s_i}} \right)} \right] - \big( {\eta _i} \| {\boldsymbol{x}} \| \\ & + {\lambda _i} \| {\boldsymbol{y}} \| + D_i^{\text{m}} + D_i^{\text{s}} \big) \cdot {\text{sign}}( {{s_i}} ), \end{split} $$ (36)

    其中, ${l_5}$, ${l_6}$, ${l_7}$, ${l_8}$均为正常数.

    非理想条件下, 基于有限时间同步策略新型忆阻耦合异质神经元的混沌同步效果, 如图15所示.

    图 15 有限时间同步策略作用下滑模面与同步误差的响应曲线 (a)有限时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$\r\nFig. 15. Response curves of sliding mode surfaces and synchronization errors when the finite-time synchronization strategy acts: (a) Finite-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.
    图 15  有限时间同步策略作用下滑模面与同步误差的响应曲线 (a)有限时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$
    Fig. 15.  Response curves of sliding mode surfaces and synchronization errors when the finite-time synchronization strategy acts: (a) Finite-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.

    固定时间同步策略[51]如下:

    滑模面:

    $$ \begin{split} {s_i} =&\; {e_i} + \int_0^t\left[ {l_9} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{{\varsigma _1}}}\right.\\ & \left.+ {l_{10}} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{{\varsigma _2}}} \right]{\text{d}}\tau ; \end{split} $$ (37)

    滑模控制器:

    $$ \begin{split} & {u_i} = - \left[ {{g_i}\left( {\boldsymbol{y}} \right) - {f_i}\left( {\boldsymbol{x}} \right)} \right] \\ & ~~- \left[ {{l_9} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{{\varsigma _1}}} + {l_{10}} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{{\varsigma _2}}}} \right] \\ & ~~- \left[ {{l_{11}} \cdot {\text{sign}}\left( {{s_i}} \right) \cdot {{\left| {{s_i}} \right|}^{{\varsigma _1}}} + {l_{12}} \cdot {\text{sign}}\left( {{s_i}} \right) \cdot {{\left| {{s_i}} \right|}^{{\varsigma _2}}}} \right] \\ & ~~- \left( {{\eta _i}\left\| {\boldsymbol{x}} \right\| + {\lambda _i}\left\| {\boldsymbol{y}} \right\| + D_i^{\text{m}} + D_i^{\text{s}}} \right) \cdot {\text{sign}}\left( {{s_i}} \right), \end{split}$$ (38)

    其中, ${l_9}$, ${l_{10}}$, ${l_{11}}$, ${l_{12}}$均为正常数, $0 < {\varsigma _1} < 1 < {\varsigma _2}$.

    非理想条件下, 基于固定时间同步策略新型忆阻耦合异质神经元的混沌同步效果, 如图16所示.

    图 16 固定时间同步策略作用下滑模面与同步误差的响应曲线 (a) 固定时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) 同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$\r\nFig. 16. Response curves of sliding mode surfaces and synchronization errors when the fixed-time synchronization strategy acts: (a) Fixed-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.
    图 16  固定时间同步策略作用下滑模面与同步误差的响应曲线 (a) 固定时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) 同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$
    Fig. 16.  Response curves of sliding mode surfaces and synchronization errors when the fixed-time synchronization strategy acts: (a) Fixed-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.

    非理想条件下, 基于新型预定义时间同步策略新型忆阻耦合异质神经元的混沌同步效果, 如图17图18所示.

    图 17  新型预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)新型预定义时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$\r\nFig. 17. Response curves of sliding mode surfaces and synchronization errors when the novel predefined-time synchronization strategy acts: (a) Novel predefined-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$
    图 17   新型预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)新型预定义时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$
    Fig. 17.  Response curves of sliding mode surfaces and synchronization errors when the novel predefined-time synchronization strategy acts: (a) Novel predefined-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$
    图 18 新型预定义时间同步策略作用下相图 (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $\r\nFig. 18. Phase diagrams when the novel predefined-time synchronization strategy acts: (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $.
    图 18  新型预定义时间同步策略作用下相图 (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $
    Fig. 18.  Phase diagrams when the novel predefined-time synchronization strategy acts: (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $.

    图15图17可知, 非理想条件下, 滑模面及同步误差可收敛到零. 图15(a)中, 有限时间滑模面的实际收敛时间$ T_0=0.07557\, \text{s} $; 图16(a)中, 固定时间滑模面的实际收敛时间$ T_2=0.04244\, \text{s} $; 图17(a)中, 新型预定义时间滑模面的实际收敛时间$ T_4= 0.03533\, \text{s} $. 图15(b)中, 有限时间同步策略作用下同步误差的实际收敛时间$ T_1=0.12014\, \text{s} $; 图16(b)中, 固定时间同步策略作用下同步误差的实际收敛时间$ T_3=0.10592\, \text{s} $; 图17(b)中, 新型预定义时间同步策略作用下同步误差的实际收敛时间$ T_5=0.05423\, \text{s} $.

    由上所述, 新型预定义时间滑模面的实际收敛时间${T_4}$最小, 即${T_4} < {T_2} < {T_0}$; 新型预定义时间同步策略作用下同步误差的实际收敛时间${T_5}$最小, 即${T_5} < {T_3} < {T_1}$. 此外, 对于新型预定义时间同步策略, ${T_4}$小于到达段的预定义时间${T_{{{\text{c}}_{2}}}} = 0.1$; ${T_5}$小于总预定义时间${T_{{{\text{c}}_{1}}}} + {T_{{{\text{c}}_{2}}}} = 0.2$. 需要说明的是, 根据实际需求, 可通过设置不同的总预定义时间来调整同步误差的实际收敛时间.

    图18表示新型预定义时间同步策略作用下的相图. 图18中, 虚线表示过渡过程的相图; 实线表示驱动系统和响应系统达到同步后的相图. 由图18可知, 非理想条件下, 驱动系统和响应系统经过过渡过程可达到同步.

    4.3.2   与传统预定义时间同步策略的对比

    下文利用传统预定义时间同步策略使驱动系统(33)和响应系统(34)达到同步.

    引理1[47,52] 针对误差系统(24), 若Lyapunov函数$V\left( {\boldsymbol{e}} \right)$, 满足如下不等式:

    $$ \dot V \leqslant - \frac{1}{{\sigma \cdot {T_{\text{s}}}}} \cdot {V^{1 - \sigma }} \cdot {\text{exp}}\left( {{V^\sigma }} \right),~~\sigma \in \left( {0,1} \right). $$ (39)

    则称误差系统(24)是预定义时间稳定的, 预定义时间${T_{\text{s}}} > 0$.

    定理4 针对误差系统(24), 若传统预定义时间同步策略设计为

    滑模面:

    $$ {s_i} = {e_i} + \int_0^t \Big[{l_{13}} \cdot {\text{sign}} (e_i) \cdot |e_i|^{1-\mu} \cdot \exp(e_i^\mu) \Big]{\text{d}}\tau ; $$ (40)

    滑模控制器:

    $$ \begin{split} & {u_i} = - \left[ {{g_i}\left( {\boldsymbol{y}} \right) - {f_i}\left( {\boldsymbol{x}} \right)} \right] - {l_{13}} {\text{sign}}\left( {{e_i}} \right) \cdot {\left| {{e_i}} \right|^{1 - \mu }} \\ & ~~\times {\text{exp}}\left( {e_i^\mu } \right) -{l_{14}} \cdot {\text{sign}}\left( {{s_i}} \right) \cdot {\left| {{s_i}} \right|^{1 - \mu }} \cdot {\text{exp}}\left( {s_i^\mu } \right) \\ &~~ - \left( {{\eta _i}\left\| {\boldsymbol{x}} \right\| + {\lambda _i}\left\| {\boldsymbol{y}} \right\| + D_i^{\text{m}} + D_i^{\text{s}}} \right) \cdot {\text{sign}}\left( {{s_i}} \right),\\[-1pt] \end{split} $$ (41)

    其中, ${l_{13}} = \dfrac{{\sqrt {{2^\mu }} }}{{\mu \cdot {T_{{{\text{s}}_{1}}}}}}$, $ {l_{14}} = \dfrac{{\sqrt {{2^\mu }} }}{{\mu \cdot {T_{{{\text{s}}_{2}}}}}} $, $\mu \in \left( {0, 1} \right)$, 则误差系统(24)满足引理1, 也就是误差系统在预定义时间${T_{{{\text{s}}_{2}}}}$内到达滑模面(40), 然后沿滑模面在预定义时间$ {T_{{{\text{s}}_{1}}}} $内收敛到零.

    证明 证明过程分为到达段和滑动段.

    1) 到达段: 选取Lyapunov函数${V_3} = \dfrac{1}{2} \cdot s_i^2$, 对其求导可得

    $$ \begin{split} \; {{\dot V}_3} =\; &{s_i} \cdot {{\dot s}_i} = {s_i} \cdot \big[ {{{\dot e}_i} + {l_{13}} \cdot {\text{sign}}( {{e_i}} ) \cdot {{| {{e_i}} |}^{1 - \mu }} \cdot {\text{exp}}( {e_i^\mu } )} \big] \\ =\;& {s_i} \cdot \big[ {g_i}( {\boldsymbol{y}} ) - {f_i}( {\boldsymbol{x}} ) + \Delta {g_i}( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}} + {u_i} + {l_{13}} \cdot {\text{sign}}( {{e_i}} ) \cdot {| {{e_i}} |^{1 - \mu }} \cdot {\text{exp}}( {e_i^\mu } ) \big] \\ =\;& {s_i} \cdot \big[ \Delta {g_i}( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}} - ( {{\eta _i}\| {\boldsymbol{x}} \| + {\lambda _i}\| {\boldsymbol{y}} \| + D_i^{\text{m}} + D_i^{\text{s}}} ) \cdot {\text{sign}}( {{s_i}} ) - {l_{14}} \cdot {\text{sign}}( {{s_i}} ) \cdot {| {{s_i}} |^{1 - \mu }} \cdot {\text{exp}}( {s_i^\mu } ) \big] \\ = \;& \big\{ {s_i} \cdot [ {\Delta {g_i}( {\boldsymbol{y}} ) - \Delta {f_i}( {\boldsymbol{x}} ) + d_i^{\text{s}} - d_i^{\text{m}}} ] - ( {{\eta _i}\| {\boldsymbol{x}} \| + {\lambda _i}\| {\boldsymbol{y}} \| + D_i^{\text{m}} + D_i^{\text{s}}} ) \cdot | {{s_i}} | - {l_{14}} \cdot {| {{s_i}} |^{2 - \mu }} \cdot {\text{exp}}( {s_i^\mu } ) \big\} \\ \leqslant &\; - {l_{14}} \cdot {| {{s_i}} |^{2 - \mu }} \cdot {\text{exp}}( {s_i^\mu } ) \leqslant - \frac{2}{{\mu \cdot {T_{{{\text{s}}_{2}}}}}} \cdot V_3^{1 - \mu /2} \cdot {\text{exp}}\big( {V_3^{\mu /2}} \big).\\[-1pt] \end{split} $$ (42)

    (42)式满足引理1. 因此, 误差系统(24)在预定义时间${T_{{{\text{s}}_{2}}}}$内到达滑模面(40).

    2)滑动段: 当误差系统(24)到达滑模面(40)后, 即${s_i} = 0$, ${\dot s_i} = 0$, 可得

    $$ {\dot e_i} = - {l_{13}} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {\left| {{e_i}} \right|^{1 - \mu }} \cdot {\text{exp}}\left( {e_i^\mu } \right). $$ (43)

    选取Lyapunov函数${V_4} = \dfrac{1}{2} \cdot e_i^2$, 对其求导可得

    $$ \begin{split} {{\dot V}_4}& = {e_i} \cdot {{\dot e}_i} \\ &= {e_i} \cdot \big[ { - {l_{13}} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{1 - \mu }} \cdot {\text{exp}}\left( {e_i^\mu } \right)} \big] \\ & = - {l_{13}} \cdot {\left| {{e_i}} \right|^{2 - \mu }} \cdot {\text{exp}}\left( {e_i^\mu } \right) \\ & \leqslant - \frac{2}{{\mu \cdot {T_{{{\text{s}}_{1}}}}}} \cdot V_4^{1 - \mu /2} \cdot {\text{exp}}\big( {V_4^{\mu /2}}\big).\\[-1pt] \end{split} $$ (44)

    (44)式满足引理1. 因此, 误差系统(24)沿滑模面(40)在预定义时间${T_{{{\text{s}}_{1}}}}$内收敛到零.

    仿真中, 设置传统预定义时间同步策略的参数$\mu = 0.5$, $ {l_{13}} = {l_{14}} = 5 \cdot {2^{2.25}} $. 设置传统预定义时间同步策略(40)式和(41)式中滑动段的预定义时间${T_{{{\text{s}}_{1}}}} = 0.1$, 到达段的预定义时间${T_{{{\text{s}}_{2}}}} = 0.1$.

    非理想条件下, 基于传统预定义时间同步策略新型忆阻耦合异质神经元的混沌同步效果, 如图19所示. 由图19可知, 非理想条件下, 滑模面及同步误差可收敛到零. 图19(a)中, 传统预定义时间滑模面的实际收敛时间$ {T_6} $=0.03999 s; 图19(b)中, 传统预定义时间同步策略作用下同步误差的实际收敛时间${T_7}$=0.08516 s.

    图 19 传统预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)传统预定义时间滑模面${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b)同步误差${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$\r\nFig. 19. Response curves of sliding mode surfaces and synchronization errors when the traditional predefined-time synchronization strategy acts: (a) Traditional predefined-time sliding mode surfaces ${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b) synchronization errors ${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$.
    图 19  传统预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)传统预定义时间滑模面${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b)同步误差${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$
    Fig. 19.  Response curves of sliding mode surfaces and synchronization errors when the traditional predefined-time synchronization strategy acts: (a) Traditional predefined-time sliding mode surfaces ${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b) synchronization errors ${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$.

    当$ {T_{{{\text{c}}_{2}}}} = {T_{{{\text{s}}_{2}}}} = 0.1 $时, 新型预定义时间滑模面的实际收敛时间${T_4}$(见图17(a))小于传统预定义时间滑模面的实际收敛时间$ {T_6} $(见图19(a)). 需要说明的是, $ {T_{{{\text{c}}_{2}}}} $和$ {T_{{{\text{s}}_{2}}}} $分别表示新型预定义时间同步策略和传统预定义时间同步策略中到达段的预定义时间.

    当${T_{{{\text{c}}_{1}}}} + {T_{{{\text{c}}_{2}}}} = {T_{{{\text{s}}_{1}}}} + {T_{{{\text{s}}_{2}}}} = 0.2$时, 新型预定义时间同步策略作用下同步误差的实际收敛时间${T_5}$(见图17(b))小于传统预定义时间同步策略作用下同步误差的实际收敛时间${T_7}$(见图19(b)). 需要说明的是, ${T_{{{\text{c}}_{1}}}} + {T_{{{\text{c}}_{2}}}}$和${T_{{{\text{s}}_{1}}}} + {T_{{{\text{s}}_{2}}}}$分别表示新型预定义时间同步策略和传统预定义时间同步策略中的总预定义时间(总预定义时间是指滑动段的预定义时间与到达段的预定义时间之和).

    4.3.3   鲁棒性的验证

    下文利用新型预定义时间同步策略分别在理想条件下(无内部不确定性和无外部干扰)和非理想条件下(有内部不确定性和有均匀分布噪声)使驱动系统和响应系统达到同步.

    理想条件下, 设计新型预定义时间滑模控制器:

    $$ \begin{split} {u_i} = & - \left[ {{g_i} ({\boldsymbol{y}}) - {f_i} ( {\boldsymbol{x}} )} \right] - \big\{ {l_1} \cdot {\text{sign}}\left( {{e_i}} \right) \cdot {{\left| {{e_i}} \right|}^{1 - \alpha }} \\ & \times \left[ {1 + {\text{exp}}\left( {e_i^\alpha } \right)} \right] + {l_2} \cdot {\text{signr}}\left( {{e_i}} \right) \big\} \\ &- \big\{{l_3} \cdot {\text{sign}}\left( {{s_i}} \right) \cdot {{\left| {{s_i}} \right|}^{1 - \alpha }} \cdot \left[ {1 + {\text{exp}}\left( {s_i^\alpha } \right)} \right] \\ & + {l_4} \cdot {\text{kas}}\left( {{s_i},\gamma ,\delta } \right) \big\}.\\[-1pt] \end{split} $$ (45)

    理想条件下和非理想条件下, 基于新型预定义时 间同步策略新型忆阻耦合异质神经元的混沌同步效果, 如图20所示. 图20中, 虚线表示理想条件下的同步误差, 实线表示非理想条件下的同步误差. 由图20可知, 理想条件下和非理想条件下, 同步误差均可收敛到零, 且同步误差曲线几乎重合, 也就是新型预定义时间同步策略具有良好的鲁棒性.

    图 20 鲁棒性的验证\r\nFig. 20. Verification of robustness.
    图 20  鲁棒性的验证
    Fig. 20.  Verification of robustness.

    首先提出一种新型局部有源忆阻器. 该忆阻器在几十GHz的频率范围内能够表现出良好的紧磁滞回线. 其最高频率可达80 GHz, 这高于大多数现存忆阻器的最高频率. 然后基于新型局部有源忆阻器, 构建新型忆阻耦合异质神经元系统模型. 随后研究系统的平衡点、耦合强度对系统动力学行为的影响、李雅普诺夫指数和0-1测试. 研究表明, 系统无平衡点, 也就是系统的动力学行为均是隐藏的. 随耦合强度的改变, 系统具有不同的放电模式, 例如, 周期1, 2, 4, 5, 6, 8尖峰放电模式、混沌尖峰放电模式、周期4和周期8簇发放电模式及随机簇发放电模式. 此外, 通过李雅普诺夫指数和0-1测试可知, 在一定的耦合强度条件下, 系统处于混沌状态. 最后提出一种新型预定义时间同步策略, 并将该策略应用于新型忆阻耦合异质神经元的混沌同步中. 研究表明, 该策略能够实现快速同步, 并对内部不确定性和外部干扰具有良好的鲁棒性. 此外, 该策略的滑模面和滑模控制器的参数可以预先选取, 这是预定义时间同步策略的一个优点. 本文工作的局限性以及未来的研究方向主要包括: 对新型预定义时间同步策略的参数进行优化; 研究其他非理想条件下, 新型预定义时间同步策略的混沌同步效果; 将新型预定义时间同步策略应用于其他神经元及神经网络的同步中; 将高频忆阻器应用于高速通信领域中.

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  • 图 1  新型局部有源忆阻器的紧磁滞回线 (a) f = 20 GHz, 不同幅值; (b) Vm= 1 V, 不同频率

    Fig. 1.  Pinched hysteresis loop of the novel locally active memristor: (a) Different amplitudes for f = 20 GHz; (b) different frequencies for Vm= 1 V.

    图 2  忆导函数图

    Fig. 2.  Diagram of the memductance function $ G\left( \varphi \right) $.

    图 3  新型局部有源忆阻器的断电图与动态路线图

    Fig. 3.  POP and DRM of the novel locally active memristor.

    图 4  (a)状态$ {\varphi _1} $切换到状态$ {\varphi _2} $(正电压脉冲的幅值Vm= 1 V); (b)低电平忆导$G\left( {{\varphi _1}} \right)$切换到高电平忆导$G\left( {{\varphi _2}} \right)$(正电压脉冲的幅值Vm= 1 V)

    Fig. 4.  (a) Switching from the state $ {\varphi _1} $ to the state $ {\varphi _2} $ (a positive voltage pulse with amplitude Vm= 1 V); (b) switching from the low-level memductance $G\left( {{\varphi _1}} \right)$ to the high-level memductance $G\left( {{\varphi _2}} \right)$(a positive voltage pulse with amplitude Vm= 1 V).

    图 5  (a) 状态$ {\varphi _2} $切换到状态$ {\varphi _1} $ (负电压脉冲的幅值Vm= –2 V); (b) 高电平忆导$G\left( {{\varphi _2}} \right)$切换到低电平忆导$G\left( {{\varphi _1}} \right)$ (负电压脉冲的幅值Vm= –2 V)

    Fig. 5.  (a) Switching from the state $ {\varphi _2} $ to the state $ {\varphi _1} $ (a negative voltage pulse with amplitude Vm= –2 V); (b) switching from the high-level memductance $G\left( {{\varphi _2}} \right)$ to the low-level memductance $G\left( {{\varphi _1}} \right)$(a negative voltage pulse with amplitude Vm= –2 V).

    图 6  耦合强度$k$变化时分岔图和李雅普诺夫指数 (a)耦合强度$k$变化时分岔图; (b)耦合强度$k$变化时李雅普诺夫指数

    Fig. 6.  Bifurcation diagram and Lyapunov exponents with the coupling strength $k$ changing: (a) Bifurcation diagram with the coupling strength $k$ changing; (b) Lyapunov exponents with the coupling strength $k$ changing.

    图 7  不同耦合强度$k$, 尖峰放电模式的相图及时域波形图

    Fig. 7.  Phase diagrams and time domain waveform diagrams of spiking firing modes, with different coupling strengths $k$.

    图 8  不同耦合强度$k$, 簇发放电模式的相图及时域波形图

    Fig. 8.  Phase diagrams and domain waveform diagrams of bursting firing modes, with different coupling strengths $k$.

    图 9  初始状态$\varphi \left( 0 \right)$变化时分岔图, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$

    Fig. 9.  Bifurcation diagram with the initial state $\varphi \left( 0 \right)$ changing, $\varphi \left( 0 \right) \in \left[ { - 0.4, 0.4} \right]$.

    图 10  李雅普诺夫指数

    Fig. 10.  Lyapunov exponents.

    图 11  0-1测试

    Fig. 11.  0-1 test.

    图 12  电路实现 (a) 新型局部有源忆阻器的电路实现; (b) HR神经元的电路实现; (c) FHN神经元的电路实现

    Fig. 12.  Circuit implementations: (a) Circuit implementation of the novel locally active memristor; (b) circuit implementation of the HR neuron; (c) circuit implementation of the FHN neuron.

    图 13  电路实现的相图及时域波形图(周期2尖峰放电模式) (a) 相图; (b) 时域波形图

    Fig. 13.  Phase diagram and time domain waveform diagram of circuit implementation (period-2 spiking firing mode): (a) Phase diagram; (b) time domain waveform.

    图 14  电路实现的相图及时域波形图(混沌尖峰放电模式) (a)相图; (b)时域波形图

    Fig. 14.  Phase diagram and time domain waveform diagram of circuit implementation (chaotic spiking firing mode): (a) Phase diagram; (b) time domain waveform.

    图 15  有限时间同步策略作用下滑模面与同步误差的响应曲线 (a)有限时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$

    Fig. 15.  Response curves of sliding mode surfaces and synchronization errors when the finite-time synchronization strategy acts: (a) Finite-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.

    图 16  固定时间同步策略作用下滑模面与同步误差的响应曲线 (a) 固定时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) 同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$

    Fig. 16.  Response curves of sliding mode surfaces and synchronization errors when the fixed-time synchronization strategy acts: (a) Fixed-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$.

    图 17   新型预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)新型预定义时间滑模面${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b)同步误差${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$

    Fig. 17.  Response curves of sliding mode surfaces and synchronization errors when the novel predefined-time synchronization strategy acts: (a) Novel predefined-time sliding mode surfaces ${s_1}$, ${s_2}$, ${s_3}$, ${s_4}$, ${s_5}$; (b) synchronization errors ${e_1}$, ${e_2}$, ${e_3}$, ${e_4}$, ${e_5}$

    图 18  新型预定义时间同步策略作用下相图 (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $

    Fig. 18.  Phase diagrams when the novel predefined-time synchronization strategy acts: (a) $ \left( {{x_1}, {y_1}} \right) $; (b) $ \left( {{x_2}, {y_2}} \right) $; (c) $ \left( {{x_3}, {y_3}} \right) $; (d) $ \left( {{x_4}, {y_4}} \right) $; (e) $ \left( {{x_5}, {y_5}} \right) $.

    图 19  传统预定义时间同步策略作用下滑模面与同步误差的响应曲线 (a)传统预定义时间滑模面${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b)同步误差${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$

    Fig. 19.  Response curves of sliding mode surfaces and synchronization errors when the traditional predefined-time synchronization strategy acts: (a) Traditional predefined-time sliding mode surfaces ${s_1}, {s_2}, {s_3}, {s_4}, {s_5}$; (b) synchronization errors ${e_1}, {e_2}, {e_3}, {e_4}, {e_5}$.

    图 20  鲁棒性的验证

    Fig. 20.  Verification of robustness.

    表 1  放电模式

    Table 1.  Firing modes.

    耦合强度$k$放电模式相图编号时域波形图编号
    0.0070周期1尖峰放电图7(a)图7(b)
    0.0400周期2尖峰放电图7(c)图7(d)
    0.1200周期4尖峰放电图7(e)图7(f)
    0.1691周期5尖峰放电图7(g)图7(h)
    0.1428周期6尖峰放电图7(i)图7(j)
    0.1290周期8尖峰放电图7(k)图7(l)
    0.1800混沌尖峰放电图7(m)图7(n)
    0.4150周期4簇发放电图8(a)图8(b)
    0.3600周期8簇发放电图8(c)图8(d)
    0.4800随机簇发放电图8(e)图8(f)
    下载: 导出CSV

    表 2  新型忆阻耦合异质神经元的电路参数

    Table 2.  Circuit parameters of the novel memristor-coupled heterogeneous neuron.

    电路参数 类型
    $R$, ${R_5}$, ${R_6}$, ${R_8}$, ${R_{10}}$, ${R_{14}}$, ${R_{16}}$ 电阻/kΩ $200$
    ${R_1}$ 电阻/kΩ $ 2700 $
    ${R_2}$ 电阻/kΩ $ 33.333 $
    ${R_3}$, ${R_9}$, ${R_{15}}$ 电阻/kΩ $ 40 $
    ${R_4}$ 电阻/kΩ $ 100 $
    ${R_7}$ 电阻/kΩ $66.667$
    ${R_{11}}$ 电阻/kΩ $ 3000 $
    ${R_{12}}$, ${R_{13}}$ 电阻/kΩ $ 1000 $
    ${R_k}$ 电阻/kΩ $ {{200} \mathord{\left/ {\vphantom {{200} k}} \right. } k} $
    ${C_0}$ 电容/nF $40$
    ${V_{{\text{HR}}}}$, ${V_{{\text{FHN}}}}$ 直流电压源/V $ 1 $
    下载: 导出CSV
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