Abstract
In this second part of a two-part study, we extend to nonlinear synaptic responses a new framework, called Response Surfaces (RSs), for analyzing, designing, and visualizing spiking neurons and networks. An RS is the 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. Here in Part II, we present four mathematically tractable functions to model more realistic post-synaptic-potential waveforms, and we build on the linear RS framework of Part I to create a nonlinear RS framework. We then discuss the qualitative differences between the linear and nonlinear RSs frameworks and revisit problems of Part I using the nonlinear RSs framework: graphing the transfer function of a nonlinear spiking neuron, designing an efficient spiking-XOR gate, analyzing phase-tracking in recurrent SNNs, and developing transfer functions for calculating the output firing times of a spiking neuron given a set of input firing times. For the nonlinear RS framework, we show that the output firing time of a nonlinear 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 plus a second-order correction. The second-order centroid correction is one of the main differences between the linear and nonlinear RS frameworks. For recurrent networks, the nonlinear RS framework also shows a more stable behavior compared to linear SNNs, owing to the smooth shapes of nonlinear post-synaptic-potential waveforms.
Highlights
Spiking Neural Networks (SNNs), as exemplified by the brain, are of great interest owing to their potentials for performing complex tasks such as pattern- and speechrecognition [1,2,3,4,5,6] with a low power-consumption.The challenge in realizing the potential of SNNs lies in their complexity
To understand how more bio-realistic spiking neurons use the detailed timing of spikes, in this paper we extend the Linear SRM" (LSRM) neuron to the Nonlinear Spike Response Model (SRM) (NSRM) using more realistic Nonlinear PSPs (NPSPs)
Since the ideas and definitions we present for the LSRM are relevant to NSRM models, this review forms a foundation for the derivation of the NSRMs
Summary
Spiking Neural Networks (SNNs), as exemplified by the brain, are of great interest owing to their potentials for performing complex tasks such as pattern- and speechrecognition [1,2,3,4,5,6] with a low power-consumption. LINEAR SPIKE RESPONSE MODEL REVIEW A challenging problem in the LSRM, and in general in spiking neuron models, is to derive a transfer function that maps input-spike times to exact output spike times rather than averaging or blurring timing information. Each c-region envelopes possible patterns of input spike times firing an output spike at t = 0, and each c-region corresponds to a different combination of synaptic slopes or c-values, each of which has three possible states: rising, falling, or idle. See [19] for a full discussion of LRSs and their properties
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