BURSTING TYPES AND STABLE DOMAINS OF RULKOV NEURON NETWORK WITH MEAN FIELD COUPLING

2013 ◽  
Vol 23 (12) ◽  
pp. 1330041 ◽  
Author(s):  
HONGJUN CAO ◽  
YANGUO WU
Keyword(s):  
Parameter Space ◽  
Single Neuron ◽  
Complex Dynamics ◽  
Mean Field ◽  
Flip Bifurcation ◽  
Neuron Network ◽  
Square Wave ◽  
Field Coupling ◽  
The Mean ◽  
Stable Domains

Based on the detailed bifurcation analysis and the master stability function, bursting types and stable domains of the parameter space of the Rulkov map-based neuron network coupled by the mean field are taken into account. One of our main findings is that besides the square-wave bursting, there at least exist two kinds of triangle burstings after the mean field coupling, which can be determined by the crisis bifurcation, the flip bifurcation, and the saddle-node bifurcation. Under certain coupling conditions, there exists two kinds of striking transitions from the square-wave bursting (the spiking) to the triangle bursting (the square-wave bursting). Stable domains of fixed points, periodic solutions, quasiperiodic solutions and their corresponding firing regimes in the parameter space are presented in a rigorous mathematical way. In particular, as a function of the intrinsic control parameters of each single neuron and the external coupling strength, a stable coefficient of the Neimark–Sacker bifurcation is derived in a parameter plane. These results show that there exist complex dynamics and rich firing regimes in such a simple but thought-provoking neuron network.

scholarly journals Single-Neuron Discharge Properties and Network Activity in Dissociated Cultures of Neocortex

2004 ◽  
Vol 92 (2) ◽  
pp. 977-996 ◽  
Author(s):  
M. Giugliano ◽  
P. Darbon ◽  
M. Arsiero ◽  
H.-R. Lüscher ◽  
J. Streit
Keyword(s):  
Single Cell ◽  
Single Neuron ◽  
Neuronal Response ◽  
Mean Field ◽  
Network Activity ◽  
Model Parameters ◽  
Cell Discharge ◽  
Collective Activity ◽  
The Mean

Cultures of neurons from rat neocortex exhibit spontaneous, temporally patterned, network activity. Such a distributed activity in vitro constitutes a possible framework for combining theoretical and experimental approaches, linking the single-neuron discharge properties to network phenomena. In this work, we addressed the issue of closing the loop, from the identification of the single-cell discharge properties to the prediction of collective network phenomena. Thus, we compared these predictions with the spontaneously emerging network activity in vitro, detected by substrate arrays of microelectrodes. Therefore, we characterized the single-cell discharge properties to Gauss-distributed noisy currents, under pharmacological blockade of the synaptic transmission. Such stochastic currents emulate a realistic input from the network. The mean ( m) and variance ( s2) of the injected current were varied independently, reminiscent of the extended mean-field description of a variety of possible presynaptic network organizations and mean activity levels, and the neuronal response was evaluated in terms of the steady-state mean firing rate ( f). Experimental current-to-spike–rate responses f( m, s2) were similar to those of neurons in brain slices, and could be quantitatively described by leaky integrate-and-fire (IF) point neurons. The identified model parameters were then used in numerical simulations of a network of IF neurons. Such a network reproduced a collective activity, matching the spontaneous irregular population bursting, observed in cultured networks. We finally interpret such a collective activity and its link with model details by the mean-field theory. We conclude that the IF model is an adequate minimal description of synaptic integration and neuronal excitability, when collective network activities are considered in vitro.

Dynamics of globally delay-coupled neurons displaying subthreshold oscillations

2009 ◽  
Vol 367 (1901) ◽  
pp. 3255-3266 ◽  
Author(s):  
Cristina Masoller ◽  
M. C. Torrent ◽  
Jordi García-Ojalvo
Keyword(s):  
Coupled Oscillators ◽  
Single Neuron ◽  
Abrupt Change ◽  
Mean Field ◽  
Flip Bifurcation ◽  
Natural Period ◽  
Subthreshold Oscillations ◽  
The Common ◽  
Global Coupling ◽  
The Individual

We study an ensemble of neurons that are coupled through their time-delayed collective mean field. The individual neuron is modelled using a Hodgkin–Huxley-type conductance model with parameters chosen such that the uncoupled neuron displays autonomous subthreshold oscillations of the membrane potential. We find that the ensemble generates a rich variety of oscillatory activities that are mainly controlled by two time scales: the natural period of oscillation at the single neuron level and the delay time of the global coupling. When the neuronal oscillations are synchronized, they can be either in-phase or out-of-phase. The phase-shifted activity is interpreted as the result of a phase-flip bifurcation, also occurring in a set of globally delay-coupled limit cycle oscillators. At the bifurcation point, there is a transition from in-phase to out-of-phase (or vice versa) synchronized oscillations, which is accompanied by an abrupt change in the common oscillation frequency. This phase-flip bifurcation was recently investigated in two mutually delay-coupled oscillators and can play a role in the mechanisms by which the neurons switch among different firing patterns.

The synchronization of calcium oscillations in coupled hepatocytes: The mean field coupling

2010 ◽  
Vol 495 (4-6) ◽  
pp. 270-274
Author(s):  
Jian Cheng Shi ◽  
Tao Dong ◽  
Chu Sheng Huang
Keyword(s):  
Mean Field ◽  
Calcium Oscillations ◽  
Field Coupling ◽  
The Mean

Classical Kinetic Theory

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Kinetic Equation ◽  
Equation Of State ◽  
Mean Field ◽  
Ideal Gas ◽  
Hydrodynamic Equations ◽  
Free Particles ◽  
Internal Mechanism ◽  
The Mean ◽  
Energy Gains ◽  
Non Locality

The classical non-ideal gas shows that the two original concepts of the pressure based of the motion and the forces have eventually developed into drift and dissipation contributions. Collisions of realistic particles are nonlocal and non-instant. A collision delay characterizes the effective duration of collisions, and three displacements, describe its effective non-locality. Consequently, the scattering integral of kinetic equation is nonlocal and non-instant. The non-instant and nonlocal corrections to the scattering integral directly result in the virial corrections to the equation of state. The interaction of particles via long-range potential tails is approximated by a mean field which acts as an external field. The effect of the mean field on free particles is covered by the momentum drift. The effect of the mean field on the colliding pairs causes the momentum and the energy gains which enter the scattering integral and lead to an internal mechanism of energy conversion. The entropy production is shown and the nonequilibrium hydrodynamic equations are derived. Two concepts of quasiparticle, the spectral and the variational one, are explored with the help of the virial of forces.

scholarly journals Evidence of exactness of the mean-field theory in the nonextensive regime of long-range classical spin models

2000 ◽  
Vol 61 (17) ◽  
pp. 11521-11528 ◽  
Author(s):  
Sergio A. Cannas ◽  
A. C. N. de Magalhães ◽  
Francisco A. Tamarit
Keyword(s):  
Field Theory ◽  
Long Range ◽  
Mean Field Theory ◽  
Mean Field ◽  
Spin Models ◽  
Classical Spin ◽  
The Mean ◽  
Classical Spin Models

scholarly journals The Mean-field Behavior of Processor Sharing Systems with General Job Lengths Under the SQ(d) Policy

2019 ◽  
Vol 46 (3) ◽  
pp. 54-55
Author(s):  
Thirupathaiah Vasantam ◽  
Arpan Mukhopadhyay ◽  
Ravi R. Mazumdar
Keyword(s):  
Mean Field ◽  
Processor Sharing ◽  
Field Behavior ◽  
The Mean

scholarly journals The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Hui Huang ◽  
Jinniao Qiu
Keyword(s):  
Existence Of Solutions ◽  
Particle System ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Unique Existence ◽  
Interacting Particle ◽  
Microscopic Derivation ◽  
The Mean ◽  
Limit Result

AbstractIn this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

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