Effects of Time Delay on Bifurcation and Synchronization of Flux-Coupled and Chemically Coupled Neurons
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摘要:
以化学突触耦合神经元模型为基础，讨论了抑制性及兴奋性条件下达到同步的区别及同步的类型。并根据磁通耦合对神经元放电的影响，讨论了具有时滞、磁通耦合和化学耦合Morris-Lecar (ML)神经元模型的放电状态、分岔类型及其同步情况。发现具有磁通耦合和化学耦合ML神经元系统在不同参数下会产生丰富的逆倍周期分岔或加周期分岔行为。而时滞的引入，虽然可以增加系统的周期性，但同时也会破环系统同步。相反，适当的耦合强度能够增加同步。
Abstract:Based on the chemical synaptic coupled neuron model, the differences and types of synchronization under inhibitory and excitatory conditions were discussed. According to the effect of the magnetic flow coupling on neuron discharge, the discharge states, bifurcation types and synchronization of the Morris-Lecar (ML) neuron models with time delay, magnetic flux coupling and chemical coupling, were analyzed. The results show that, the ML neuronal systems with magnetic flow coupling and chemical coupling can produce rich inverse periodic bifurcation or incremental periodic bifurcation behaviors under different parameters. The introduction of time delay, although can increase the periodicity of the system, will also break the system synchronization. Conversely, an appropriate coupling strength can enhance synchronization.
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Key words:
- time delay /
- flux-coupled /
- bifurcation /
- synchronization
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图 1 系统(1)在抑制性与兴奋性条件下的相似函数图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。
Figure 1. Similarity function diagrams of system (1) under inhibitory and excitatory conditions
图 2 系统(1)在抑制性条件下的时间历程图与相图：(a)
$D = 1.98$ 时的时间历程图；(b)$D = 1.98$ 时的相图；(c)$D = 2.04$ 时的时间历程图；(d)$D = 2.04$ 时的相图Figure 2. Time history diagrams and phase diagrams of system (1) under inhibition: (a) the time history diagram at time
$D = 1.98$ ; (b) the phase diagram at time$D = 1.98$ ; (c) the time history diagram at time$D = 2.04$ ; (d) the phase diagram at time$D = 2.04$ 
图 3 系统(1)在兴奋性条件下的时间历程图与相图：(a)
$D = 0.081$ 时的时间历程图；(b)$D = 0.081$ 时的相图；(c)$D = 0.175$ 时的时间历程图；(d)$D = 0.175$ 时的相图Figure 3. Time history diagrams and phase diagrams of system (1) under excitatory condition: (a) the time history diagram at time
$D = 0.081$ ; (b) the phase diagram at time$D = 0.081$ ; (c) the time history diagram at time$D = 0.175$ ; (d) the phase diagram at time$D = 0.175$ 
图 4 系统(2)反馈增益
$ {k_1} $ 的峰-峰间期分岔图Figure 4. The peak-to-peak bifurcation diagram of the feedback gain of system (2)
图 5 系统(2)在不同反馈增益下的时间历程图与相图：(a)
$ {k_1} = 0.2 $ 的时间历程图；(b)$ {k_1} = - 0.5 $ 的时间历程图；(c)$ {k_1} = - 1.08 $ 的时间历程图；(d)$ {k_1} = 0.2 $ 的相图；(e)$ {k_1} = - 0.5 $ 的相图；(f)$ {k_1} = - 1.08 $ 的相图Figure 5. Time history diagrams and phase diagrams of system (2) under different feedback gains: (a) the time history diagram at time
$ {k_1} = 0.2 $ ; (b) the time history diagram at time$ {k_1} = - 0.5 $ ; (c) the time history diagram at time$ {k_1} = - 1.08 $ ; (d) the phase diagram at time$ {k_1} = 0.2 $ ; (e) the phase diagram at time$ {k_1} = - 0.5 $ ; (f) the phase diagram at time$ {k_1} = - 1.08 $ 
图 6 系统(2)在不同时滞下反馈系数
$ {k_1} $ 的峰-峰间期分岔图Figure 6. The peak-to-peak bifurcation diagrams of feedback coefficients for system (2) with different delays
图 7 系统(2)在不同时滞下参数
$ \alpha $ 的峰-峰间期分岔图Figure 7. The peak-to-peak bifurcation diagrams of system (2) with different time delays
图 8 反馈增益
$ {k_1} $ ，$ {k_2} $ 与不同参数的双参数分岔图：(a)$ {k_1} $ 与$V_{\rm{K}}$ 双参数分岔图；(b)$ {k_1} $ 与$ V_{\rm Ca} $ 双参数分岔图；(c)$ {k_2} $ 与$ g_{\rm Ca} $ 双参数分岔图；(d)$ {k_2} $ 与$ g_{\rm{K}} $ 双参数分岔图Figure 8. Two-parameter bifurcation diagrams with feedback gains and different parameters: (a) the
$ {k_1} $ and$ V_{\rm{K}} $ two-parameter bifurcation diagram; (b) the$ {k_1} $ and$ V_{\rm Ca} $ two-parameter bifurcation diagram; (c) the$ {k_2} $ and$ g_{\rm Ca} $ two-parameter bifurcation diagram; (d) the$ {k_2} $ and$ g_{\rm{K}} $ two-parameter bifurcation diagram
图 9 系统(2)不同时滞下的相似函数图
Figure 9. Similar function diagrams of system (2) with different time delays
图 10 系统(2)在相同耦合强度不同时滞下的相图与时间历程图
Figure 10. Phase diagrams and time history diagrams of system (2) under the same coupling strength and different delays
图 11 系统(2)在无时滞状态下的相似函数图：(a) 反馈增益
${k_1}$ 与参数$\alpha $ 相似函数图；(b) 反馈增益${k_1}$ 与参数${V_{{\rm{syn}}}}$ 相似函数图；(c) 反馈增益${k_1}$ 与耦合强度D相似函数图Figure 11. Similar function diagrams of system (2) without delay: (a) the feedback gain
${k_1}$ and parameter$\alpha $ similarity function diagram; (b) the feedback gain${k_1}$ and parameter${V_{{\rm{syn}}}}$ similarity function diagram; (c) the feedback gain${k_1}$ and coupling strength similarity function diagram -
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