Abstract
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.
Highlights
Spiking Neural Networks (SNNs) have always held a fascination for researchers due to the brain’s astonishing ability to compute with high accuracy and low power-consumption
In this paper we focus on excitatory Linear PSPs (LPSPs), the mathematics and the neuron model may be extended to inhibitory synapses
The primary difference between our use of Response Surfaces (RSs) and other work on phase-locking, limit cycles, or bifurcations is that we present a spike-by-spike analysis that directly reveals stability or instability of phase-tracking. (Because phase-locking may fail to exist, we use the more general term ‘‘phase-tracking’’ to describe our results.) We show that three basic modes of phase-tracking may occur in a basic case study: stable, convergent phase-locking; unstable, finite spike trains that die out after a fixed number of spikes; and chaotic, persistent phase-tracking
Summary
Spiking Neural Networks (SNNs) have always held a fascination for researchers due to the brain’s astonishing ability to compute with high accuracy and low power-consumption. Each axis represents a synaptic input’s firing time, and each point on the RSs corresponds to a 3-input spike pattern that fire the neuron at time zero on a rising slope of total activity. 1-dimensional solutions happen when a single weighted-LPSP is large enough to fire the neuron in the absence of the other two synapses By extending the 2-D RSs in 3-D space, we get the solutions for the case when the combination of two synaptic responses is large enough to cause the neuron to fire and the third synapse is firing remotely in time (instead of being inactive). The possible number of χ -regions might seem to grow rapidly with the number of synapses, the number of relevant χ-regions in a modest design problem is usually quite small, as in the XOR design process we present
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