On Leaky-Integrate-and Fire as Spike-Train-Quantization Operator on Dirac-Superimposed Continuous-Time Signals
CoRR(2024)
摘要
Leaky-integrate-and-fire (LIF) is studied as a non-linear operator that maps
an integrable signal f to a sequence η_f of discrete events, the spikes.
In the case without any Dirac pulses in the input, it makes no difference
whether to set the neuron's potential to zero or to subtract the threshold
ϑ immediately after a spike triggering event. However, in the case of
superimpose Dirac pulses the situation is different which raises the question
of a mathematical justification of each of the proposed reset variants. In the
limit case of zero refractory time the standard reset scheme based on threshold
subtraction results in a modulo-based reset scheme which allows to characterize
LIF as a quantization operator based on a weighted Alexiewicz norm ._A,
α with leaky parameter α. We prove the quantization formula
η_f - f_A, α < ϑ under the general condition of local
integrability, almost everywhere boundedness and locally finitely many
superimposed weighted Dirac pulses which provides a much larger signal space
and more flexible sparse signal representation than manageable by classical
signal processing.
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