The circuits comprising superconducting optoelectronic synapses, dendrites, and neurons are described by numerically cumbersome and formally opaque coupled differential equations. Reference 1 showed that a phenomenological model of superconducting loop neurons eliminates the need to solve the Josephson circuit equations that describe synapses and dendrites. The initial goal of the model was to decrease the time required for simulations, yet an additional benefit of the model was increased transparency of the underlying neural circuit operations and conceptual clarity regarding the connection of loop neurons to other physical systems. Whereas the original model simplified the treatment of the Josephson-junction dynamics, essentially by only considering low-pass versions of the dendritic outputs, the model resorted to an awkward treatment of spikes generated by semiconductor transmitter circuits that required explicitly checking for threshold crossings and distinct treatment of time steps wherein somatic threshold is reached. Here we extend that model to simplify the treatment of spikes coming from somas, again making use of the fact that in neural systems the downstream recipients of spike events almost always perform low-pass filtering. We provide comparisons between the first and second phenomenological models, quantifying the accuracy of the additional approximations. We identify regions of circuit parameter space in which the extended model works well and regions where it works poorly. For some circuit parameters it is possible to represent the downstream dendritic response to a single spike as well as coincidences or sequences of spikes, indicating the model is not simply a reduction to rate coding. The governing equations are shown to be nearly identical to those ubiquitous in the neuroscience literature for modeling leaky-integrator dendrites and neurons.

翻译：构成超导光电突触、树突和神经元的电路通常由数值计算繁琐且形式晦涩的耦合微分方程描述。参考文献1表明，超导环路神经元的现象学模型消除了求解描述突触和树突的约瑟夫森电路方程的需求。该模型的初始目标是缩短模拟所需时间，然而模型的额外益处在于提高了底层神经电路操作的透明度，并增强了环路神经元与其他物理系统连接的概念清晰度。原始模型通过仅考虑树突输出的低通版本，简化了对约瑟夫结动力学的处理，但在处理半导体发射电路产生的尖峰信号时，采用了较为笨拙的方法，需要显式检查阈值穿越并对达到胞体阈值的时间步进行特殊处理。本文扩展该模型以简化来自胞体的尖峰信号处理，再次利用神经系统中的下游尖峰事件接收器几乎总是执行低通滤波这一事实。我们提供了第一代与第二代现象学模型的比较，量化了额外近似处理的精度。我们识别了扩展模型表现良好的电路参数区域以及表现较差的区域。对于某些电路参数，该模型能够表征下游树突对单个尖峰以及尖峰重合或序列的响应，表明该模型并非简单地退化为速率编码。研究证明其控制方程与神经科学文献中广泛用于模拟漏积分树突和神经元的方程几乎完全相同。