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Autapse Modulated Bursting [chapter]

Guang-Hong Wang, Ping Jiang
2006 Lecture Notes in Computer Science  
The model autapse is applied to a neuron with conductance-based minimal model to construct a fast-slow buster.  ...  As one of the major neuro-computational properties, bursting occurs due to the interplay of fast currents responsible for spiking activity, and slow currents that modulate the activity.  ...  H-H model is not only the starting point for detailed neuron models which account for numerous ion channels, different types of synapse, and the specific spatial geometry of an individual neuron, but also  ... 
doi:10.1007/11759966_52 fatcat:oh6wmecp5vfwpojqkb2bqljsqe

Synchrony of Fast-Spiking Interneurons Interconnected by GABAergic and Electrical Synapses

Masaki Nomura, Tomoki Fukai, Toshio Aoyagi
2003 Neural Computation  
Applying the phase reduction method to such a network, we find that if the two neurons are coupled by only electrical synapses, then they exhibit stable synchrony, whereas, if they are also coupled by  ...  We investigate the combined effects of electrical synapses (gap junctions) and GABAergic synapses on the synchronization of neuronal firing in a network of fast-spiking (FS) interneurons.  ...  We have thus found that the results obtained with the phase reduction method are qualitatively in accordance with those found numerically for the two-neuron network.  ... 
doi:10.1162/089976603322297340 pmid:12959671 fatcat:hifd7qbkxrdrxlplz47fgcxwt4

Simple Cortical and Thalamic Neuron Models for Digital Arithmetic Circuit Implementation

Takuya Nanami, Takashi Kohno
2016 Frontiers in Neuroscience  
This model is intended to provide another meeting point for above trade-off that satisfies the demand for large-scale neuronal network simulation with closer-to-biology models.  ...  We expanded the DSSN model and found appropriate parameter sets with which it reproduces the dynamical behaviors of the ionic-conductance models of four classes of cortical and thalamic neurons.  ...  The DSSN model qualitatively reproduces the behaviors of the four neuron classes by numerical integration with the Euler's method (dt = 0.0001 s).  ... 
doi:10.3389/fnins.2016.00181 pmid:27242397 pmcid:PMC4865656 fatcat:pldvni6wqjauzi75no244axsba

Modeling large populations of spiking neurons with a universal population density solver

Marc de Kamps
2013 BMC Neuroscience  
Apfaltrer F, Ly C, Tranchina D: Population density methods for stochastic neurons with realistic synaptic kinetics: Firing rate dynamics and fast computational methods.  ...  In [2] a numerical method, based on the method of characteristics, was presented that is manifestly stable and very efficient; in particular it is not restricted to the diffusion limit, and therefore can  ...  In [2] a numerical method, based on the method of characteristics, was presented that is manifestly stable and very efficient; in particular it is not restricted to the diffusion limit, and therefore  ... 
doi:10.1186/1471-2202-14-s1-p90 fatcat:uonihgi7azcste35aqytfk7ls4

Decision-making and interacting neuron populations [article]

S Mancini
2014 arXiv   pre-print
The slow-fast characterization of the solutions finally leads us to a complexity reduction of the model by the definition of a one-dimensional stochastic differential equation and its associated one-dimensional  ...  In this article we present the modeling of bi-stability view problems described by the activity or firing rates of two interacting population of neurons.  ...  ψ 1 , ψ 2 are the dynamical part of the equations and model the neuronal activity.  ... 
arXiv:1411.7165v1 fatcat:2ubl4w5skbdi7ebdjtrnpvahhu


John Guckeheimer, Joseph H. Tien, Allan R. Willms
2005 Bursting  
Acknowledgments This work was partially supported by the National Science Foundation and the Department of Energy.  ...  We regard this as a minimal model for action potentials, representative of the fast dynamics of more complex model neurons containing additional conductances.  ...  First, the methods for computing these curves in multiple time scale systems discussed in the previous section were specific to planar vector fields, and robust algorithms that address these numerical  ... 
doi:10.1142/9789812703231_0004 fatcat:phs3t4p7qbcp7pmimpwuw5amri

Synthesizing attractors of Hindmarsh–Rose neuronal systems

Marius-F. Danca, Qingyun Wang
2010 Nonlinear dynamics  
Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh-Rose neuronal system and matches  ...  In this paper a periodic parameter switching scheme is applied to the Hindmarsh-Rose neuronal system to synthesize certain attractors.  ...  Therefore, it controls the difference between the slow and the fast dynamics of HR neuron model corresponding to the difference between fast fluxes of ions across the membrane and slow ones.  ... 
doi:10.1007/s11071-010-9730-6 fatcat:fkentzc4kve6dhiepcfmpw6qf4

Mechanism of bistability: Tonic spiking and bursting in a neuron model

Andrey Shilnikov, Ronald L. Calabrese, Gennady Cymbalyuk
2005 Physical Review E  
The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between  ...  This model can exhibit two coexisting types of oscillations: tonic spiking and bursting with a large amplitude. Which regime occurs depends on the initial state of the neuron model.  ...  Turaev for his helpful comments. The numeric analysis of system (1) utilized the software packages CONTENT and MATCONT [9] . A.S. acknowledge the RFBR grants No. 02-01-00273 and No. 01-01-00975.  ... 
doi:10.1103/physreve.71.056214 pmid:16089641 fatcat:lz4moil3vre7zi7qtbko5w5cau

Time-delay effect on the bursting of the synchronized state of coupled Hindmarsh-Rose neurons

Y. G. Zheng, Z. H. Wang
2012 Chaos  
Thus, Eq. (4) undergoes bursting oscillations, which is confirmed by numerical results, as shown in Fig. 3 , where the Gear method for stiff problems is used in numerical integration with tolerance 0.001  ...  Ding, 19 where it is found that a stable synchronized state exists at low coupling strengths for significant time delays, and by formulating a master stability equation for the delay-coupled HR neurons  ... 
doi:10.1063/1.4768664 pmid:23278062 fatcat:z76wjzryevhzdnzilsbi4spwgu

Global Isochrons and Phase Sensitivity of Bursting Neurons

Alexandre Mauroy, Blane Rhoads, Jeff Moehlis, Igor Mezić
2014 SIAM Journal on Applied Dynamical Systems  
To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I.  ...  Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models.  ...  Section 3 presents the numerical methods, i.e., the Fourier average method and the adaptive (quadtree and octree) grids.  ... 
doi:10.1137/130931151 fatcat:6vhqbjq2s5c2dkrp4emj2cf52e

A unified physiological framework of transitions between seizures, status epilepticus and depolarization block at the single neuron level [article]

Damien Depannemaecker, Anton Ivanov, Davide Lillo, Len Spek, Christophe Bernard, Viktor Jirsa
2020 bioRxiv   pre-print
The two equations for the slow subsystem describe ion concentration variations and the two equations of the fast subsystem delineate the electrophysiological activities of the neuron.  ...  Here we investigated SLEs at the single cell level using a biophysically relevant neuron model including a slow/fast system of four equations.  ...  We used a numerical methods with SymPy [63] and SciPy [64] libraries, to find the roots and the eigenvalues of the Jacobians of the 2D fast subsystem, and thus the stability considering the existence  ... 
doi:10.1101/2020.10.23.352021 fatcat:acemtp43qnfjndejib65amifx4

Suppression of spontaneous oscillations in high-frequency stimulated neuron models

Kęstutis Pyragas, Peter A. Tass
2017 Lithuanian Journal of Physics  
Using a multiple scale method we separate the solutions of the neuron equations into slow and fast components and derive averaged equations for the slow components.  ...  We demonstrate the universality of this effect for typical neuron models such as FitzHugh–Nagumo, Morris–Lecar, and Hodgkin–Huxley neurons as well as for the normal form of the supercritical Hopf bifurcation  ...  An accurate numerical analysis of such systems requires special numerical methods and large computation times.  ... 
doi:10.3952/physics.v56i4.3419 fatcat:i2nkyluf7nc4bihyrzt2rjwbay

Stochastic Phenomena in One-Dimensional Rulkov Model of Neuronal Dynamics

Irina Bashkirtseva
2015 Discrete Dynamics in Nature and Society  
We study the nonlinear Rulkov map-based neuron model forced by random disturbances. For this model, an overview of the variety of stochastic regimes is given.  ...  It is shown how such random transitions can generate a new neuronal regime of the stochastic bursting and transfer the system from order to chaos.  ...  So, for the investigation of stochastic dynamics, a method of direct numerical simulation is widely used. This method is a time-consuming for the parametric analysis.  ... 
doi:10.1155/2015/495417 fatcat:yb6mgmvrcvduvcxydg6fi3tboy

Special issue on nonlinear phenomena in physics: new techniques and applications

Anastasios Bountis, Eusebius J. Doedel, Elbert E. N. Macau, Panayotis Panayotaros, Carlos L. Pando Lambruschini
2018 The European Physical Journal Special Topics  
of coupled oscillators, and numerical analysis of nonlinear oscillations.  ...  These topics are presented in individual articles by physicists, computational scientists, and applied mathematicians, with expertise in computational methods and experimental techniques, who have contributed  ...  Special thanks are due to the Managing Editor of EPJ, Jürgen Kurths, for his support and for bringing the opportunity of a special issue to our attention.  ... 
doi:10.1140/epjst/e2018-00103-0 fatcat:ajfojklovnccthph7jr2rkzvza

Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models

Eva Kaslik
2017 Fractional Calculus and Applied Analysis  
AbstractA theoretical analysis of two- and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect  ...  With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system  ...  Indeed, based on the method of dissection of neuronal bursting [13] , setting ε = 0 in (2) and studying the fast subsystem by treating z as a bifurcation parameter, typically, the fast subsystem exhibits  ... 
doi:10.1515/fca-2017-0033 fatcat:jyr2mar6ufgdnf33inlwgsxemy
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