Automatic Implementation of Neural Networks through Reaction Networks—Part II: Error Analysis

Neural network related machine learning algorithms, inspired by biological neuron interaction mechanisms, are advancing rapidly in the field of computing. This development may be leveraged in reverse to advance synthetic biology progress. A challenging exploration is to implement neural network functionalities through biochemical reaction networks (BCRNs), a language that is inherently compatible with in vivo, with difficulties specifically in constructing an appropriate BCRN that respects computation and information processing steps involved in neural networks. By addressing these difficulties, this paired article manages to develop a biochemical fully connected neural network (BFCNN) that has the potential to behave like a neural network independently in vivo. In Part I, BFCNN is designed according to five different modules with concrete theoretical support on the stability of every module. In this article, we establish a systematic framework for error analysis of the designed BFCNN. We introduce current error evaluating error generation within an individual computational module by measuring the deviation between finite-time concentration and steady-state concentration, and the accumulative error characterizing error propagation across different modules connected via oscillatory signals and within the repeated modules due to multiple iterations. We further derive the formula for the total error upper bound about the iteration number and illustrate its exponential convergence order about non-zero duration time of an oscillatory signal concentration. Ultimately, the numerical experiments on two classification examples are made to exhibit the change tendency of total error bounds about the non-zero duration time and iteration number.