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）动力学的理解有了很大的发展。事实上，我们现在可以预测神经元电活动的动态，就像我们可以预测一个简单钟摆的动态一样。这项壮举是通过将神经元建模为常微分方程（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，因为我们现在可以记录数千个真实的神经元，并创建精细的、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以创建网络动力学，实验神经科学家就无法了解这些脑回路所使用的潜在功能和解决问题的算法。有趣的是，在训练人工神经网络（ann）之后，机器学习社区也面临着同样的问题。在不知道训练过的连接权重如何在人工神经网络中创建网络动态的情况下，机器学习的用户仍然对这些黑盒如何解决问题一无所知。在这方面，我们的无知是危险的，因为人工神经网络可能会在无人驾驶汽车和自动金融交易算法等关键应用中犯下灾难性错误，导致生命或资本损失。

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.