Transition to chaos in random neuronal networks

Jonathan Kadmon, H. Sompolinsky
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引用次数: 142

Abstract

Firing patterns in the central nervous system often exhibit strong temporal irregularity and heterogeneity in their time averaged response properties. Previous studies suggested that these properties are outcome of an intrinsic chaotic dynamics. Indeed, simplified rate-based large neuronal networks with random synaptic connections are known to exhibit sharp transition from fixed point to chaotic dynamics when the synaptic gain is increased. However, the existence of a similar transition in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work we investigate rate based dynamics of neuronal circuits composed of several subpopulations and random connectivity. Nonzero connections are either positive-for excitatory neurons, or negative for inhibitory ones, while single neuron output is strictly positive; in line with known constraints in many biological systems. Using Dynamic Mean Field Theory, we find the phase diagram depicting the regimes of stable fixed point, unstable dynamic and chaotic rate fluctuations. We characterize the properties of systems near the chaotic transition and show that dilute excitatory-inhibitory architectures exhibit the same onset to chaos as a network with Gaussian connectivity. Interestingly, the critical properties near transition depend on the shape of the single- neuron input-output transfer function near firing threshold. Finally, we investigate network models with spiking dynamics. When synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a sharp transition from fast spiking fluctuations and static firing rates to a state with slow chaotic rate fluctuations. When the synaptic time constants are finite, the transition becomes smooth and obeys scaling properties, similar to crossover phenomena in statistical mechanics
随机神经网络向混沌的过渡
中枢神经系统的放电模式在时间平均反应特性上往往表现出强烈的时间不规则性和异质性。以往的研究表明,这些特性是内在混沌动力学的结果。事实上,当突触增益增加时,具有随机突触连接的基于简化速率的大型神经网络表现出从固定点到混沌动力学的急剧转变。然而,在具有更真实的结构和放电动力学的神经元电路模型中存在类似的过渡尚未建立。在这项工作中,我们研究了由几个亚群和随机连接组成的神经元电路的基于速率的动力学。非零连接要么是兴奋性神经元的正连接,要么是抑制性神经元的负连接,而单个神经元的输出是严格正的;符合许多生物系统的已知限制。利用动态平均场理论,我们得到了描述稳定不动点、不稳定动态和混沌速率波动状态的相图。我们描述了系统在混沌过渡附近的性质,并表明稀释兴奋-抑制结构表现出与具有高斯连接的网络相同的混沌开始。有趣的是,过渡附近的关键性质取决于放电阈值附近的单个神经元输入-输出传递函数的形状。最后,我们研究了具有尖峰动力学的网络模型。当突触时间常数相对于平均反向放电速率缓慢时,神经网络经历了从快速尖峰波动和静态放电速率到缓慢混沌速率波动状态的急剧转变。当突触时间常数有限时,跃迁变得平滑并服从标度性质,类似于统计力学中的交叉现象
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