非线性突触耦合光敏神经元的同步分析

doi:10.19596/j.cnki.1001-246x.8807

摘要/Abstract

摘要：

在神经元电路中加入光电管构造一个典型的光敏神经元电路, 该神经元电路在不同频率的周期光电流刺激下呈现生物神经元典型的放电模态, 经过标度变换得到无量纲的光敏神经元模型和能量函数。研究发现: 神经元的放电模态和系统能量相关。用滤波后的混沌光电流信号刺激时, 该神经元呈现混沌放电模态且随着滤波信号强度的增大, 神经元放电模态和能量发生显著变化。考虑到神经突触的多样性及可塑性, 用一个非线性电阻耦合两个光敏神经元电路, 研究周期信号及经滤波后的混沌光电流信号刺激下两个光敏神经元电路在不同初始放电模态下的同步情况及能量在系统内部的流动与转移, 同时研究噪声对同步状态及系统能量的影响。结果表明: 周期光电流刺激下, 耦合系统在任何初始模态下都不能实现完全同步, 耦合强度始终在波动, 两系统能量不能平衡; 一定强度噪声有利于周期光电流刺激下初始态为簇放电模态的两个耦合全同光敏神经元电路的同步但不能使初始态为混沌态的神经元达到同步。适当强度的滤波混沌光电流信号能调节耦合系统间歇性相位锁定甚至于实现近似的相位同步。

关键词: 光敏神经元, 能量分布, 非线性突触, 同步

Abstract:

A phototube is connected to a simple neural circuit for developing a light-sensitive neural circuit, which can be regulated to trigger suitable firing modes by taming the frequency or amplitude of external stimulus. Furthermore, the photoelectric neuron and its Hamilton energy are obtained through scale transformation. It is discerned that the firing modes of neuron is dependent on the energy level of the neuron. Chaotic signals are filtered and encoded to excite the neuron, then chaotic firing modes are obtained and the modes transition and energy value are changed significantly by increasing the intensity of the filtered signals from chaotic system. Considering the complexity and plasticity of the synapse, a nonlinear resistor is used to couple two photosensitive neural circuits. The synchronization stability and energy exchange between two coupled neurons presenting with different initial firing modes are investigated by applying periodic and filtered signals on the neurons. Furthermore, additive noise is applied to investigate the synchronization stability and energy propagation between the two neurons. When photocurrent is selected in periodic type, two coupled neurons are blocked to reach complete synchronization, the coupling intensity is fluctuated with time and energy balance is broken between two neurons. By taming the noise intensity, two bursting neurons can reach complete synchronization while two chaotic neurons seldom reach synchronization when the neurons are driven by peridic currents. When filtered signals are used to excite the two coupled neurons, intermittent phase lock and phase synchronization can be realized between two neurons.

Key words: photosensitive neuron, energy distribution, nonlinear synapse, zynchronization

李艳妮, 王春妮. 非线性突触耦合光敏神经元的同步分析[J]. 计算物理, 2025, 42(1): 106-117.

Yanni LI, Chunni WANG. Synchronous Stability of Two Photosensitive Neurons Coupled by Nonlinear Synapse[J]. Chinese Journal of Computational Physics, 2025, 42(1): 106-117.

图/表 10

图1 感光神经元电路

Fig.1 Schematic diagram of photosensitive neural circuit

图2 非线性电阻耦合两光敏神经元电路

Fig.2 Photoelectric neural circuits coupled by nonlinear resistor

图3 周期光电流信号驱动时单个神经元电路的振动分岔及能量分布(a) 膜电位x的峰峰间隔ISI关于周期信号频率ω分岔图(小图是局部放大图。); (b) 系统总哈密顿能量H随ω的演化图; (c) 对应模态下电场能量占比PC及磁场能量占比PL随刺激频率ω的演化曲线(a=0.7, b=0.8, c=0.1, ζ=0.25, A=0.6，初始值选取[0.2, 0.1]。)

Fig.3 Bifurcation, Hamilton energy and energy proportion of single neural circuit driven by periodic signals (a) bifurcation diagram of ISI vs. the frequency of the periodic stimulus signal (The small image in the figure is a partial enlargement.); (b) dependence of Hamilton energy on frequency of the Periodic stimulus signal; (c) evolution of the electric energy proportion PC and the magnetic energy proportion PL vs. frequency of the periodic stimulus signal (a=0.7, b=0.8, c=0.1, ζ=0.25, A=0.6, the initial value is [0.2, 0.1].)

图4 不同频率的周期光电流刺激下, 单个神经元电路的时序图、相位图、能量演化曲线及电场能与磁场能的比值P的演化曲线(a) ω=0.01; (b) ω=0.1; (c) ω=0.81 (A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25，初始值为[0.2, 0.1]。)

Fig.4 Sampled time series of membrane potential, attractor, the Hamilton energy and evolution of the ratio P between electric field energy and magnetic field energy of single neuron circuit stimulated by periodic photocurrents of different frequencies ω (a) ω=0.01; (b) ω=0.1; (c) ω=0.81 (a=0.7, b=0.8, c=0.1, ζ=0.25, A=0.6. The initial value is [0.2, 0.1].)

图5 单个神经元电路由滤波后的光电流信号驱动时的放电模态及能量分布(a)ISI关于振幅E的分岔图; (b)系统关联度CV关于振幅E的演化；(c)系统总哈密顿能量H随振幅E的演化图；(d)~(f)行分别为滤波光电流信号幅值E不同时神经元电路的时域演化图(小图为对应的相位图。)、哈密顿能量H及电场能与磁场能的比值P随时间的演化曲线；(d)E=0.69; (e) E=1.43; (f) E=8.4 (a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75。初始值选择为[0.1, 0.1, 1, 0.2, 0.1]。)

Fig.5 Mode transition, CV distribution and energy evolution of neuron driven by filtered photocurrent signal (a) bifurcation diagram of ISI vs. E. (b) CV vs. E.(c) H vs. E. (d)~(f) are sampled time series of membrane potential (the small image in the figure is the corresponding phase diagram.), Hamiltonian energy H and ratio P of electric field energy to magnetic field energy of neuronal circuit with different amplitude E of filtered photocurrent signal; (d) E=0.69; (e) E=1.43; (f) E=8.4 (a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75. The initial value is [0.1, 0.1, 1, 0.2, 0.1].)

图6 在不同频率的周期光电流刺激下，不同初始态的两耦合神经元系统的能量差、同步误差、相位差、耦合强度和耦合通道能量消耗演化(a)簇放电ω=0.01；(b)尖峰放电ω=0.1；(c)混沌放电ω=0.81 (A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25，初始值选择为[0.2, 0.1, 0.02, 0.01]。)

Fig.6 Evolution of energy difference, synchronization error function, phase error, coupling intensity and coupling channel energy consumption of two coupled neurons with different initial states driven by period photocurrent with different frequency (a) bursting pattern, ω=0.01;(b) spiking pattern, ω=0.1;(c) chaos pattern, ω=0.81 (A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25. The initial value is [0.2, 0.1, 0.02, 0.01].)

图7 噪声对初始态为簇放电的两个耦合神经元同步状态的影响(a)D=0.05；(b)D=0.2；(c)D=2.0. (ω=0.01, A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25。初始值为[0.2, 0.1, 0.02, 0.01]。)

Fig.7 Effect of noise on synchronization state of two coupled bursting neurons (a) D=0.05;(b) D=0.2;(c) D=2.0 (ω=0.01, A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25.The initial value is [0.2, 0.1, 0.02, 0.01].)

图8 噪声存在时，周期光电流刺激下初始态均为混沌放电的两耦合神经元电路的能量误差、误差函数、相位差、耦合强度和耦合通道热消耗的演化(a)D=0.05；(b)D=0.3；(c)D=1.8 (ω=0.81, A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25. 初始值选择为[0.2, 0.1, 0.02, 0.01]。)

Fig.8 Evolution of energy diversity, error function, phase error, coupling intensity and coupling channel energy consumption in two coupled neuronal circuit with chaotic discharge driven by noise and periodic stimulus (a) D=0.05;(b) D=0.3; (c) D=1.8 (ω=0.81, A=0.6, a=0.7, b=0.8, c=0.1, ζ=0.25. The initial value is [0.2, 0.1, 0.02, 0.01].)

图9 经滤波后的混沌光电流刺激时，耦合神经元电路的同步情况(a)E=0.69; (b)E=1.43; (c)E=8.4 (a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75. 初始值选择为[0.1, 0.1, 1, 0.2, 0.1, 0.02, 0.01]。)

Fig.9 Evolution of energy diversity, error function, phase error, coupling intensity and coupling channel energy consumption in the two coupled neurons by the filtered chaotic photocurrent (a)E=0.69; (b)E=1.43; (c)E=8.4 (a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75. The initial value is[0.1, 0.1, 1, 0.2, 0.1, 0.02, 0.01].)

图10 噪声存在时，经滤波后的混沌光电流刺激时，耦合神经元电路的同步情况(a)D=0.04; (b)D=1.0; (c)D=4.0 (E=0.69, a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75. 初始值选择为[0.1, 0.1, 1, 0.2, 0.1, 0.02, 0.01]。)

Fig.10 Evolution of energy diversity, error function, phase error, coupling intensity and energy consumption of two coupled neurons driven by filtered chaotic photocurrent with noise (a)D=0.04; (b)D=1.0; (c)D=4.0 (E=0.69, a=0.7, b=0.8, c=0.1, ζ=0.25, α=9.78, β=14.97, γ=0.0, m0=-1.31, m1=-0.75. The initial value is [0.1, 0.1, 1, 0.2, 0.1, 0.02, 0.01].)

参考文献 34

1	马军. 功能神经元建模及动力学若干问题[J]. 广西师范大学学报(自然科学版), 2022, 40 (5): 307- 323.
2	MA Jun , ZHANG Ge , HAYAT T , et al. Model electrical activity of neuron under electric field[J]. Nonlinear Dynamics, 2019, 95 (2): 1585- 1598. DOI
3	XU Ying , GUO Yeye , REN Guodong , et al. Dynamics and stochastic resonance in a thermosensitive neuron[J]. Applied Mathematics and Computation, 2020, 385, 125427. DOI
4	LIU Yong , XU Wanjiang , MA Jun , et al. A new photosensitive neuron model and its dynamics[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21 (9): 1387- 1396.
5	ZHOU Ping , YAO Zhao , MA Jun , et al. A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus[J]. Chaos Solitons & Fractals, 2021, 145, 110751.
6	ZHANG Yin , ZHOU Ping , TANG Jun , et al. Mode selection in a neuron driven by Josephson junction current in presence of magnetic field[J]. Chinese Journal of Physics, 2021, 71, 72- 84. DOI
7	刘丽君, 韦笃取. 忆阻Rulkov神经网络同步研究[J]. 计算物理, 2023, 40 (3): 389- 400.
8	唐利红, 贺宗梅, 姚延立. 忆阻Hopfield神经网络动力学分析及其电路实现[J]. 计算物理, 2022, 39 (2): 244- 252.
9	WANG Hao , WANG Jiahui , THOW X Y , et al. Unveiling stimulation secrets of electrical excitation of neural tissue using a circuit probability theory[J]. Frontiers in Computational Neuroscience, 2020, 14, 50. DOI
10	WANG Jiahui , WANG Hao , THAKOR N V , et al. Self-Powered direct muscle stimulation using a triboelectric nanogenerator (TENG) integrated with a flexible Multiple-Channel intramuscular electrode[J]. ACS Nano, 2019, 13 (3): 3589- 3599. DOI
11	TSODYKS M , PAWELZIK K , MARKRAM H . Neural networks with dynamic synapses[J]. Neural Computation, 1998, 10 (4): 821- 835. DOI
12	薛晓丹, 王美丽, 邵雨竹, 等. 基于抑制性突触可塑性的神经元放电率自稳态机制[J]. 物理学报, 2019, 68 (7): 078701.
13	UZUNTARLA M , OZER M , ILERI U , et al. Effects of dynamic synapses on noise-delayed response latency of a single neuron[J]. Physical Review E, 2015, 92 (6): 062710.
14	FUHRMANN G , SEGEV I , MARKRAM H , et al. Coding of temporal information by activity-dependent synapses[J]. Journal of Neurophysiology, 2002, 87 (1): 140- 148. DOI
15	MARKRAM H , WANG Y , TSODYKS M . Differential signaling via the same axon of neocortical pyramidal neurons[J]. Proceedings of the National Academy of Sciences of the United States of America, 1998, 95 (9): 5323- 5328.
16	WANG Yun , MARKRAM H , GOODMAN P H , et al. Heterogeneity in the pyramidal network of the medial prefrontal cortex[J]. Nature Neuroscience, 2006, 9 (4): 534- 542. DOI
17	THOMSON A M , DEUCHARS J . Temporal and spatial properties of local circuits in neocortex[J]. Trends in Neurosciences, 1994, 17 (3): 119- 126. DOI
18	LENDVAI B , STERN E A , CHEN B , et al. Experience-dependent plasticity of dendritic spines in the developing rat barrel cortex in vivo[J]. Nature, 2000, 404 (6780): 876- 881. DOI
19	ZUO Yi , YANG Guang , KWON E , et al. Long-term sensory deprivation prevents dendritic spine loss in primary somatosensory cortex[J]. Nature, 2005, 436 (7048): 261- 265. DOI
20	于文婷, 张娟, 唐军. 动态突触、神经耦合与时间延迟对神经元发放的影响[J]. 物理学报, 2017, 66 (20): 200201. DOI
21	李国芳, 孙晓娟. 小世界神经元网络随机共振现象: 混合突触和部分时滞的影响[J]. 物理学报, 2017, 66 (24): 240501. DOI
22	VARRÓ P , SZEMERSZKY R , BÁRDOS G , et al. Changes in synaptic efficacy and seizure susceptibility in rat brain slices following extremely low-frequency electromagnetic field exposure[J]. Bioelectromagnetics, 2009, 30 (8): 631- 640. DOI
23	WANG Chunni , TANG Jun , MA Jun . Minireview on signal exchange between nonlinear circuits and neurons via field coupling[J]. The European Physical Journal Special Topics, 2019, 228, 1907- 1924. DOI
24	谭安杰, 韦笃取, 覃英华. 电磁场耦合忆阻神经网络的放电模式及同步行为研究[J]. 广西师范大学学报(自然科学版), 2020, 38 (1): 107- 113.
25	MVOGO A , TAKEMBO C N , EKOBENA FOUDA H P , et al. Pattern formation in diffusive excitable systems under magnetic flow effects[J]. Physics Letters A, 2017, 381 (28): 2264- 2271. DOI
26	LIU Zhilong , WANG Chunni , ZHANG Ge , et al. Synchronization between neural circuits connected by hybrid synapse[J]. International Journal of Modern Physics B, 2019, 33 (16): 1950170. DOI
27	AULD D S , ROBITAILLE R . Glial cells and neurotransmission: An inclusive view of synaptic function[J]. Neuron, 2003, 40 (2): 389- 400. DOI
28	VOLTERRA A , MELDOLESI J . Astrocytes, from brain glue to communication elements: The revolution continues[J]. Nature Reviews. Neuroscience, 2005, 6 (8): 626- 640. DOI
29	GIAUME C , KOULAKOFF A , ROUX L , et al. Astroglial networks: A step further in neuroglial and gliovascular interactions[J]. Nature Reviews Neuroscience, 2010, 11 (2): 87- 99. DOI
30	XU Ying , JIA Ya , MA Jun , et al. Synchronization between neurons coupled by memristor[J]. Chaos, Solitons, and Fractals, 2017, 104, 435- 442. DOI
31	BOURNE J N , HARRIS K M . Balancing structure and function at hippocampal dendritic spines[J]. Annual Review of Neuroscience, 2008, 31, 47- 67. DOI
32	HIGLEY M J , SABATINI B L . Calcium signaling in dendrites and spines: Practical and functional considerations[J]. Neuron, 2008, 59 (6): 902- 913. DOI
33	ZHOU Ping , MA Jun , XU Ying . Phase synchronization between neurons under nonlinear coupling via hybrid synapse[J]. Chaos, Solitons, and Fractals, 2023, 169, 113238. DOI
34	周倩, 韦笃取. 场耦合忆阻神经网络的电活动[J]. 计算物理, 2020, 37 (6): 750- 756.

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