Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929237382922240 |
|---|---|
| author | Moser, Bernhard A. Lunglmayr, Michael |
| author_facet | Moser, Bernhard A. Lunglmayr, Michael |
| contents | In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05772 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds Moser, Bernhard A. Lunglmayr, Michael Neural and Evolutionary Computing Discrete Mathematics Signal Processing Metric Geometry 82C32, 92B99, 41A65 C.1.3 In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon. |
| title | Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds |
| topic | Neural and Evolutionary Computing Discrete Mathematics Signal Processing Metric Geometry 82C32, 92B99, 41A65 C.1.3 |
| url | https://arxiv.org/abs/2305.05772 |