Paper
Numerical analysis for leaky-integrate-fire networks under Euler--Maruyama
Authors
Xu'an Dou, Frank Chen, Kevin K Lin, Zhuo-Cheng Xiao
Abstract
Leaky integrate-and-fire (LIF) networks are canonical models in computational neuroscience and a standard substrate for neuromorphic AI. We study Euler--Maruyama simulation of current-based LIF networks with exponential synapses and instantaneous resets. Since diffusion enters only through the synaptic current, each spike is a threshold hit for a deterministically advected voltage with random crossing speed, so numerical error is concentrated at event times. For layered feedforward networks we prove finite-horizon mean-square strong bounds and weak order-one bounds for smooth bounded observables. The strong analysis uses a pruning-and-balance strategy: path space is split into a good set, where exact and numerical spike histories match and each matched spike satisfies \(A=I-\Ith\ge α\), and a bad set containing near-tangential crossings and terminal spike-count mismatch. On the good set, spike-time error is the local Euler--Maruyama error times \(A^{-1}\). A threshold-flux estimate gives \(\E[A^{-2}\1_{\{A\geα\}}]\lesssim α_0^{-2}+Tρ_{\max}\log(α_0/α)\), while near-tangential events have probability \(O(Tα^2)\). Balancing these terms yields mean-square error \(h\) times polylogarithmic factors, with explicit dependence on time, depth, and weights; away from terminal mismatch, this matches the classical Euler--Maruyama \(1/2\) rate up to logarithmic losses. For weak error, a semigroup/backward-Kolmogorov argument adapted to resets splits the one-step defect into an interior Taylor term and a boundary-strip term for missed and extra spikes, yielding order \(O(Th)\). We also derive a Lyapunov exponent formula coupling stationary threshold flux to the reset saltation factor, and outline recurrent extensions, including loop-truncated strong/weak bounds controlled by synaptic cycles and rate/density estimates.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.10854v1</id>\n <title>Numerical analysis for leaky-integrate-fire networks under Euler--Maruyama</title>\n <updated>2026-03-11T15:04:28Z</updated>\n <link href='https://arxiv.org/abs/2603.10854v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.10854v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>Leaky integrate-and-fire (LIF) networks are canonical models in computational neuroscience and a standard substrate for neuromorphic AI. We study Euler--Maruyama simulation of current-based LIF networks with exponential synapses and instantaneous resets. Since diffusion enters only through the synaptic current, each spike is a threshold hit for a deterministically advected voltage with random crossing speed, so numerical error is concentrated at event times.\n For layered feedforward networks we prove finite-horizon mean-square strong bounds and weak order-one bounds for smooth bounded observables. The strong analysis uses a pruning-and-balance strategy: path space is split into a good set, where exact and numerical spike histories match and each matched spike satisfies \\(A=I-\\Ith\\ge α\\), and a bad set containing near-tangential crossings and terminal spike-count mismatch. On the good set, spike-time error is the local Euler--Maruyama error times \\(A^{-1}\\). A threshold-flux estimate gives \\(\\E[A^{-2}\\1_{\\{A\\geα\\}}]\\lesssim α_0^{-2}+Tρ_{\\max}\\log(α_0/α)\\), while near-tangential events have probability \\(O(Tα^2)\\). Balancing these terms yields mean-square error \\(h\\) times polylogarithmic factors, with explicit dependence on time, depth, and weights; away from terminal mismatch, this matches the classical Euler--Maruyama \\(1/2\\) rate up to logarithmic losses.\n For weak error, a semigroup/backward-Kolmogorov argument adapted to resets splits the one-step defect into an interior Taylor term and a boundary-strip term for missed and extra spikes, yielding order \\(O(Th)\\). We also derive a Lyapunov exponent formula coupling stationary threshold flux to the reset saltation factor, and outline recurrent extensions, including loop-truncated strong/weak bounds controlled by synaptic cycles and rate/density estimates.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.PR'/>\n <published>2026-03-11T15:04:28Z</published>\n <arxiv:primary_category term='math.NA'/>\n <author>\n <name>Xu'an Dou</name>\n </author>\n <author>\n <name>Frank Chen</name>\n </author>\n <author>\n <name>Kevin K Lin</name>\n </author>\n <author>\n <name>Zhuo-Cheng Xiao</name>\n </author>\n </entry>"
}