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Related Concept Videos

Integration of Synaptic Events01:28

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Synaptic integration mainly includes the summation of graded potentials. Graded potentials, regardless of their type, cause subtle alterations in membrane voltage, resulting in either depolarization or hyperpolarization. These incremental changes, when combined or summed, can propel the neuron toward its threshold. Consider, for example, a membrane experiencing a +15 mV shift, causing it to depolarize from -70 mV to -55 mV. In this scenario, graded potentials govern the membrane's ability to...
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When the neuron of a motor unit fires an action potential, it triggers a series of events, leading to a twitch contraction in the muscle fibers. The process of excitation-contraction coupling is crucial in relaying the action potential to the muscle fibers.
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A postsynaptic neuron usually receives numerous impulses from several other presynaptic neurons. The axon hillock of the postsynaptic neuron integrates all these signals and determines the likelihood of firing an action potential.
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Neural Circuits01:25

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Neural circuits and neuronal pools are two of the main structures found in the nervous system. Neural circuits are networks of neurons that work together to carry out a specific task or process. They consist of interconnected neurons and glial cells, which provide structural and metabolic support.
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The propagation of an action potential refers to the process by which a nerve impulse, or "action potential," travels along a neuron.
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The contraction strength of muscles is regulated by motor neurons, which modulate the frequency of action potentials dispatched to the motor units based on the body's requirements. This process of varying the muscle stimulation frequency allows muscles to contract with a force that is precisely tailored to the needs of the moment, whether lifting a feather or a heavy box.
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Locally Contractive Dynamics in Generalized Integrate-and-Fire Neurons.

Nicolas D Jimenez1, Stefan Mihalas1, Richard Brown2

  • 1Allen Institute for Brain Science, Seattle, WA 98103.

SIAM Journal on Applied Dynamical Systems
|February 4, 2014
PubMed
Summary

This study introduces a generalized integrate-and-fire neuron model exhibiting both tonic spiking and neuronal bursting. It reveals that bursting arises from discontinuities in piecewise contractive maps, linking dynamics to map properties.

Keywords:
Mihalas–Niebur neuronburstingcontraction analysishybrid dynamical systemsintegrate-and-firepiecewise contractions

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Integrate-and-fire models are fundamental to simulating biological neurons.
  • Standard models with simple voltage resets primarily exhibit tonic spiking.
  • Neuronal bursting is a complex yet common firing pattern requiring more sophisticated models.

Purpose of the Study:

  • To investigate a generalized integrate-and-fire model incorporating spike-induced currents.
  • To analyze the conditions leading to tonic spiking and neuronal bursting.
  • To establish a mathematical framework connecting bursting dynamics to map properties.

Main Methods:

  • Analytical reduction of the model to a one-dimensional return map.
  • Investigation of local and global contractivity properties of the map.
  • Detailed analysis of piecewise contractive maps, termed 'bursting maps'.

Main Results:

  • The generalized model reproduces both tonic spiking and neuronal bursting.
  • A sufficient condition for global map contractivity, leading to tonic spiking, was derived.
  • Discontinuities in the return map were identified as the cause of bursting behavior.
  • Bursting maps were shown to robustly generate stable bursting.

Conclusions:

  • The study establishes a novel link between neuronal bursting and piecewise contractive maps.
  • The model provides a tractable framework for studying complex neuronal dynamics.
  • Bifurcations in the return map explain transitions between different spiking patterns.