具有不应期的脉冲神经元群体密度动力学的谱理论
Spectral theory for population density dynamics of spiking neurons with refractoriness
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中文总结 AI 辅助
研究具有不应期的脉冲神经元群体密度动力学问题，通过建立算子理论框架，推导精确传递函数，揭示不应期对振荡的影响，为计算神经科学谱分解方法提供严格基础。
中文摘要 AI 辅助
尽管有证据表明不应期会强烈影响非线性传递函数和网络稳定性，但将绝对不应期纳入脉冲神经元的群体密度方法仍是一个未解决的问题。我们通过扩充状态空间以包含不应期历史，并将问题表述为福克 - 普朗克算子的非自伴边界特征值问题，为具有有限不应期的神经元群体动力学建立了一个严格的算子理论框架。这给出了生成器的完整谱特征，证明了耗散性和收缩半群的存在，并将有缺陷的特征值识别为振荡模式从合并的弛豫模式中出现的例外点。在线性响应理论框架内，我们还推导了一个考虑外部输入调制边界条件的精确传递函数，纠正了先前的启发式推导并揭示了额外的阈值噪声贡献。在平均场近似下使用此传递函数，我们进一步表明相互作用神经元群体中的不应期可促进极限环的出现，即放电率的稳定振荡。这些结果为计算神经科学中的谱分解方法提供了严格基础，为进一步的严格数学分析开辟了道路。
英文摘要
Incorporating an absolute refractory period into the population density approach for spiking neurons remains an open problem, despite evidence that refractoriness can strongly affect nonlinear transfer functions and network stability. We develop a rigorous operator-theoretic framework for neuronal population dynamics with a finite refractory time by augmenting the state space to include refractory history and formulating the problem as a non-self-adjoint boundary eigenvalue problem for the Fokker-Planck operator. This yields a complete spectral characterization of the generator, proves dissipativity and the existence of a contraction semigroup, and identifies defective eigenvalues as exceptional points where oscillatory modes emerge from coalescing relaxational modes. Within the framework of linear response theory, we also derive an exact transfer function that accounts for boundary conditions modulated by external input, correcting previous heuristic derivations and revealing additional threshold-noise contributions. Using this transfer function under a mean-field approximation, we further show that refractoriness in populations of interacting neurons can facilitate the onset of limit cycles, that is, stable oscillations in the firing rate. These results provide a rigorous foundation for spectral decomposition methods in computational neuroscience, opening the way to their further rigorous mathematical analysis.
