A Framework for Analyzing, Designing, and Visualizing Spiking Neural Networks Part I: Linear Response Surfaces

In this first of two closely related papers, we set the foundation for a new framework, called Response Surfaces (RSs), to address fundamental problems of analyzing, designing, and visualizing spiking neurons and networks. An RS is a plot of the direct transfer function between input and output firing times of a spiking neuron and shows all the patterns of input spike times that fire a neuron at a given time. Thus, an RS is a graphical tool for visualizing, analyzing, and designing spiking neural networks. In this paper, we develop a linear RS framework based on triangular, post-synaptic-potential waveforms and apply it to the following problems: graphing the transfer function of a linear spiking neuron, designing an efficient spiking-XOR gate, analyzing phase tracking, and calculating spike times of arbitrarily large recurrent spiking networks. As another fundamental result of the linear RS framework, we show that the output firing time of a spiking neuron is equivalent to the center-of-mass of input spike times (acting as positions) and weights (acting as masses) plus a fixed delay. In the second paper, we explore the possibilities of more biologically realistic post-synaptic-potential waveforms and create a nonlinear RS framework as an extension of the linear RS framework presented here. Although our application examples in both papers focus on design and analysis, we touch on why the RS framework is helpful for understanding the effects of weight changes in learning algorithms.