---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
_id: '19068'
abstract:
- lang: eng
  text: Whether or not the neuron emits a spike in response to stimulation by an excitatory
    current pulse is determined by a strength-duration curve (SDC) for the pulse parameters.
    The SDC is a dependence of the minimal pulse amplitude required to elicit the
    spiking response on either the pulse duration or its decay time. Excitatory neurons
    affect the others through pulses of excitatory postsynaptic current. A simple
    yet plausible approximation for the time course of such a pulse is the alpha function,
    with linear rise at the start and exponential decay at the end. However, an exact
    analytical SDC for this case is hitherto not known, even for the leaky integrate-and-fire
    (LIF) neuron, the simplest spiking neuron model used in practice. We have obtained
    general SDC equations for the LIF neuron. Using the Lambert W function — a widely-implemented
    special function, we have found the exact analytical SDC for the spiking response
    of the LIF neuron stimulated by an excitatory current pulse in the form of the
    alpha function. To compare results in a unified way, we have also derived the
    analytical SDCs for (i) rectangular pulse, (ii) ascending ramp pulse, and (iii)
    instantly rising and exponentially decaying pulse. In the limit of no leakage,
    we show that the SDC is reduced to the classical hyperbola for all considered
    cases.
acknowledgement: "The author thanks T.S. Zemskova and N.D. Efimova for verifying some
  of the results. This work was supported by a European Research Council Consolidator
  Grant (SYNAPSEEK, 819603, to Tim P. Vogels).\r\nThe Supplementary Material for this
  article contains (i) the data for graphs in Figure 1 and (ii) ready-to-use MATLAB
  codes for reproducing the data. It is available online at https://doi.org/10.6084/m9.figshare.24081849."
article_number: '100548'
article_processing_charge: Yes
article_type: original
author:
- first_name: Alexander
  full_name: Paraskevov, Alexander
  id: d05e3c56-9262-11ed-9231-be692464e5ac
  last_name: Paraskevov
citation:
  ama: Paraskevov A. Analytical strength-duration curve for the spiking response of
    the LIF neuron to an alpha-function-shaped excitatory current pulse. <i>Results
    in Applied Mathematics</i>. 2025;25. doi:<a href="https://doi.org/10.1016/j.rinam.2025.100548">10.1016/j.rinam.2025.100548</a>
  apa: Paraskevov, A. (2025). Analytical strength-duration curve for the spiking response
    of the LIF neuron to an alpha-function-shaped excitatory current pulse. <i>Results
    in Applied Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.rinam.2025.100548">https://doi.org/10.1016/j.rinam.2025.100548</a>
  chicago: Paraskevov, Alexander. “Analytical Strength-Duration Curve for the Spiking
    Response of the LIF Neuron to an Alpha-Function-Shaped Excitatory Current Pulse.”
    <i>Results in Applied Mathematics</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.rinam.2025.100548">https://doi.org/10.1016/j.rinam.2025.100548</a>.
  ieee: A. Paraskevov, “Analytical strength-duration curve for the spiking response
    of the LIF neuron to an alpha-function-shaped excitatory current pulse,” <i>Results
    in Applied Mathematics</i>, vol. 25. Elsevier, 2025.
  ista: Paraskevov A. 2025. Analytical strength-duration curve for the spiking response
    of the LIF neuron to an alpha-function-shaped excitatory current pulse. Results
    in Applied Mathematics. 25, 100548.
  mla: Paraskevov, Alexander. “Analytical Strength-Duration Curve for the Spiking
    Response of the LIF Neuron to an Alpha-Function-Shaped Excitatory Current Pulse.”
    <i>Results in Applied Mathematics</i>, vol. 25, 100548, Elsevier, 2025, doi:<a
    href="https://doi.org/10.1016/j.rinam.2025.100548">10.1016/j.rinam.2025.100548</a>.
  short: A. Paraskevov, Results in Applied Mathematics 25 (2025).
corr_author: '1'
date_created: 2025-02-23T23:01:55Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-04-14T07:54:31Z
day: '01'
ddc:
- '570'
- '510'
department:
- _id: TiVo
doi: 10.1016/j.rinam.2025.100548
ec_funded: 1
file:
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  creator: dernst
  date_created: 2025-02-24T13:18:47Z
  date_updated: 2025-02-24T13:18:47Z
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file_date_updated: 2025-02-24T13:18:47Z
has_accepted_license: '1'
intvolume: '        25'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: 0aacfa84-070f-11eb-9043-d7eb2c709234
  call_identifier: H2020
  grant_number: '819603'
  name: Learning the shape of synaptic plasticity rules for neuronal architectures
    and function through machine learning.
publication: Results in Applied Mathematics
publication_identifier:
  eissn:
  - 2590-0374
publication_status: published
publisher: Elsevier
related_material:
  link:
  - relation: software
    url: https://doi.org/10.6084/m9.figshare.24081849
scopus_import: '1'
status: public
title: Analytical strength-duration curve for the spiking response of the LIF neuron
  to an alpha-function-shaped excitatory current pulse
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2025'
...