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

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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.