Mathematical foundations of neuroscience
G. Bard Ermentrout,David Terman +1 more
- 01 Jan 2010
- Vol. 35
TL;DR: The Hodgkin-Huxley Equations are applied to the model of Neuronal Networks to describe the “spatially distributed” networks.
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Abstract: The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak Coupling.- Neuronal Networks: Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks.
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Citations
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Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models
Albert Granados,Gemma Huguet +1 more
TL;DR: In this article, the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold was studied, and it was shown that the model possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations.
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A framework for macroscopic phase-resetting curves for generalised spiking neural networks
TL;DR: In this article , the authors present a theoretical framework and methodology to compute the phase-resetting curve (PRC) of generic spiking networks with emergent collective oscillations, where neurons are described by the time since their last action potential.
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•Dissertation
Multiscale Modeling of Coupled Oscillators with Applications to the Mammalian Circadian Clock.
Daniel DeWoskin
- 01 Jan 2015
TL;DR: Multiscale Modeling of Coupled Oscillators with Applications to the Mammalian Circadian Clock and their applications to the mammaliancircadian clock are studied.
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Investigating the role of gap junctions in seizure wave propagation
TL;DR: A voltage-based neural field model is used to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination and it is found that in this model gap junction-like connectivity determines the propagation of one-bumping or two-bump traveling wave solutions with features consistent with the clinical data.
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Spatially localized cluster solutions in inhibitory neural networks.
TL;DR: The results demonstrate that an inhibitory network with non-monotonic, distance-dependent connectivity can exhibit cluster solutions where adjacent cells fire together, and predicts the long-term behavior for networks of neurons.
References
Semigroups and Ordered Categories. I. The Reduced Case
TL;DR: In this article, the authors investigated the relationship between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U -semiabundant semigroups, first studied by El-Qallali and by de Barros.
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Expansions, free inverse semigroups, and Schützenberger product
Stuart W. Margolis,J.E Pin +1 more
TL;DR: In this article, the authors present a new construction of the free inverse monoid on a set X. Contrary to previous constructions of [9, 11], their construction is symmetric and originates from classical ideas of language theory.
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