scholarly journals Stochastic Nonlinear Equations Describing the Mesoscopic Voltage-Gated Ion Channels

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Mauricio Tejo
Keyword(s):  
Membrane Potential ◽  
Nonlinear System ◽  
Particle System ◽  
Approximation Technique ◽  
Voltage Sensors ◽  
Interacting Particle ◽  
Stochastic Nonlinear System ◽  
Voltage Gated Ion Channels ◽  
Gating Process ◽  
Jump Markov Processes

We propose a stochastic nonlinear system to model the gating activity coupled with the membrane potential for a typical neuron. It distinguishes two different levels: a macroscopic one, for the membrane potential, and a mesoscopic one, for the gating process through the movement of its voltage sensors. Such a nonlinear system can be handled to form a Hodgkin-Huxley-like model, which links those two levels unlike the original deterministic Hodgkin-Huxley model which is positioned at a macroscopic scale only. Also, we show that an interacting particle system can be used to approximate our model, which is an approximation technique similar to the jump Markov processes, used to approximate the original Hodgkin-Huxley model.

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.

scholarly journals A Local Barycentric Version of the Bak–Sneppen Model

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Philip Kennerberg ◽  
Stanislav Volkov
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

AbstractWe study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

scholarly journals On the macroscopic limit of Brownian particles with local interaction

2020 ◽  
Vol 20 (06) ◽  
pp. 2040007
Author(s):  
Franco Flandoli ◽  
Marta Leocata ◽  
Cristiano Ricci
Keyword(s):  
Explicit Form ◽  
Diffusion Processes ◽  
Particle System ◽  
Local Interaction ◽  
Nonlinear Pde ◽  
Rigorous Results ◽  
Precise Form ◽  
Interacting Particle ◽  
Brownian Particles ◽  
Main Ideas

An interacting particle system made of diffusion processes with local interaction is considered and the macroscopic limit to a nonlinear PDE is investigated. Few rigorous results exists on this problem and in particular the explicit form of the nonlinearity is not known. This paper reviews this subject, some of the main ideas to get the limit nonlinear PDE and provides both heuristic and numerical informations on the precise form of the nonlinearity which are new with respect to the literature and coherent with the few known informations.

scholarly journals Hydrodynamic limit for a nongradient interacting particle system with stochastic reservoirs

2000 ◽  
Vol 45 (4) ◽  
pp. 694-717 ◽  
Author(s):  
Claudio Landim ◽  
Claudio Landim ◽  
Claudio Landim ◽  
Claudio Landim ◽  
Mustapha Mourragui ◽  
...  
Keyword(s):  
Hydrodynamic Limit ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

Segregation in an interacting particle system

2000 ◽  
Vol 271 (1-2) ◽  
pp. 92-99 ◽  
Author(s):  
Kei-ichi Tainaka ◽  
Nariyuki Nakagiri
Keyword(s):  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

scholarly journals Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 518 ◽  
Author(s):  
Pierre Hodara ◽  
Ioannis Papageorgiou
Keyword(s):  
Membrane Potential ◽  
Point Process ◽  
Markov Processes ◽  
Relevant Information ◽  
Neural System ◽  
Brain Neurons ◽  
Actual Position ◽  
The Past ◽  
Poincaré Type Inequalities ◽  
Jump Markov Processes

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.

scholarly journals A multitype contact process with frozen sites: a spatial model of allelopathy

2005 ◽  
Vol 42 (04) ◽  
pp. 1109-1119
Author(s):  
Nicolas Lanchier
Keyword(s):  
Phase Transitions ◽  
Spatial Model ◽  
Biological Process ◽  
Particle System ◽  
Contact Process ◽  
Interacting Particle System ◽  
Interacting Particle

In this paper, we introduce a generalization of the two-color multitype contact process intended to mimic a biological process called allelopathy. To be precise, we have two types of particle. Particles of each type give birth to particles of the same type, and die at rate 1. When a particle of type 1 dies, it gives way to a frozen site that blocks particles of type 2 for an exponentially distributed amount of time. Specifically, we investigate in detail the phase transitions and the duality properties of the interacting particle system.

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