Analyzing Neural Network Dynamics as Forward and Inverse Problems in the Connection Weights

经过几十年的神经科学和动力系统理论的研究，我们对孤立的脑细胞(神经元)的动力学的理解已经有了很大的发展。事实上，我们现在可以预测神经元电活动的动力学，就像我们可以预测单摆的动力学一样。这一壮举是通过将神经元建模为常微分方程组(ODE)来实现的。然而，当我们将这些ODE作为网络中交互的节点耦合在一起时，我们就失去了这种可预测性。这是不幸的，因为我们现在可以从数以千计的真实神经元中进行记录，并创建复杂的、基于实验的网络模型。如果不了解这些连接权重如何与神经动力学相互作用来创建网络动力学，实验神经科学家就对这些大脑电路使用的潜在功能和解决问题的算法视而不见。有趣的是，机器学习界在训练人工神经网络(ANN)后也面临着同样的问题。由于不知道经过训练的连接权重如何在人工神经网络中创建网络动态，机器学习的用户仍然对这些黑盒如何解决问题视而不见。在这里，我们的无知是危险的，因为ANN可能会在无人驾驶汽车和自动金融交易算法等关键应用程序中犯下灾难性错误，导致生命或资金损失。因此，为了解决我们对功能网络动力学如何出现在生物电路和人工训练网络中的无知，我的研究计划有两个长期目标：1.第一个长期目标是通过研究通常称为前向问题的问题来阐明网络动力学、节点动力学和节点连通性之间的关系：任意的连通性矩阵如何确定模型神经元网络(ODE)的大规模行为？正问题的解采用平均场系统的形式，这是网络动力学的低维和解析描述符。2.我的研究计划的第二个长期目标是同时研究逆问题：给定模型神经元网络的大规模动力学、功能或行为的知识，产生这些动力学的全套连接权重矩阵是什么？逆问题的解决方案是一组潜在的连通性配置文件，可在网络中产生预期的动态。通过双向处理正问题和逆问题，我们最终可以解决如何在相互作用的神经元网络中出现功能和复杂的网络动力学。这种双向方法通过提供对人工和生物网络如何解决问题的机械理解，将给机器学习和计算神经科学带来革命性的变化。就人工神经网络而言，这项研究计划将有助于解决更广泛的公众对在无人驾驶汽车或机器人手术等关键应用中信任这些算法的犹豫。

Through decades of research in neuroscience and dynamical systems theory, our understanding of the dynamics of isolated brain cells (neurons) has evolved considerably. In fact, we can now predict the dynamics of a neuron's electrical activity as well as we can predict the dynamics of a simple pendulum. This feat is performed by modelling neurons as Ordinary Differential Equations (ODE's). However, we lose this predictability when we couple these ODE's together as nodes interacting in a network. This is unfortunate because we can now record from thousands of real neurons and create elaborate, experimentally based network models.  Without an understanding of how these connection weights interact with neural dynamics to create network dynamics, experimental neuroscientists are blind to the potential functions and problem-solving algorithms employed by these brain circuits. Interestingly, the same problem is faced by the machine learning community after training Artificial Neural Networks (ANNs). Without knowing how trained connection weights create network dynamics in ANN's, users of machine learning remain blind to how these black-boxes solve problems. Here, our ignorance is dangerous as ANNs can make catastrophic errors in critical applications like driverless cars, and automated financial trading algorithms, leading to loss of life or capital. Thus, to address our ignorance in how functional network dynamics emerge in both biological circuits and artificial trained networks, my research program has two long-term objectives: 1. The first long-term objective is to elucidate the relationship between network dynamics, node dynamics, and node connectivity by investigating what is commonly called the forward problem: How does an arbitrary connectivity matrix determine the large-scale behaviors of networks of model neurons (ODE's)? The solution to the forward problem takes the form of a mean-field system, a low dimensional and analytical descriptor of the network dynamics. 2. The second long-term objective of my research program is to simultaneously investigate the inverse problem: Given knowledge of the large-scale dynamics, function, or behaviour of a network of model neurons, what is the full set of connection weight matrices that yield these dynamics? The solution to the inverse problem is a set of potential connectivity profiles that yield the intended dynamics in a network. With a bi-directional approach to the forward and inverse problem, we can finally address how functional and complex network dynamics emerge in networks of interacting neurons. This bi-directional approach would revolutionize both machine learning and computational neuroscience by providing a mechanistic understanding of how artificial and biological networks solve problems. In the case of ANNs, this research program will help resolve the broader public's hesitance in trusting these algorithms in critical applications like driverless cars or robotic surgeries.