scholarly journals Synaptic propagation in neuronal networks with finite-support space dependent coupling

2020 ◽  
Author(s):  
Ricardo Erazo Toscano ◽  
Remus Osan

1AbstractTraveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves’ evolution are synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model to investigate the propagation characteristics of traveling wave activity. We formulated a set of evolution equations based on the network connectivity parameters. We numerically investigated the stability of the traveling wave propagation with a series of perturbations with biological relevance.

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.


2019 ◽  
Author(s):  
Sayak Bhattacharya ◽  
Matthieu B. Le Cauchois ◽  
Pablo A. Iglesias ◽  
Zhe S. Chen

AbstractPropagation of neural activity in spatially structured neuronal networks has been observed in awake, anesthetized and sleeping brains. However, it remains unclear how traveling waves are coordinated temporally across recurrently connected brain structures, and how network connectivity affects spatiotemporal neural activity. Here we develop a computational model of a two-dimensional thalamocortical network that enables us to investigate traveling wave characteristics in space-time. We show that thalamocortical and intracortical network connectivity, excitation/inhibition balance, thalamocortical/corticothalamic delay can independently or jointly change the spatiotemporal patterns (radial, planar and rotating waves) and characteristics (speed, direction and frequency) of cortical and thalamic traveling waves. Simulations of our model further predict that increased thalamic inhibition induces slower cortical wave frequency, and enhanced cortical excitation increases cortical wave speed and oscillation frequencies. Overall, the model study provides not only theoretical insight into the basis for spatiotemporal wave patterns, but also experimental predictions that potentially control these dynamics.Author SummaryCognition or sensorimotor control requires the coordination of neural activity across widespread brain circuits. Propagating waves of oscillatory neural activities have been observed at both macroscopic and mesoscopic levels, with various frequencies, spatial coverage, and modalities. However, a complete understanding how thalamocortical traveling waves are originated and temporally coordinated in the thalamus and cortex are still unclear. Furthermore, it remains unknown how the network connectivity, excitation/inhibition balance, thalamocortical or corticothalamic delay determine the spatiotemporal wave patterns and characteristics of cortical and thalamic traveling waves. Here we develop a computational model of a two-dimensional thalamocortical network to investigate the thalamic and neocortical traveling wave characteristics in space-time, which allows us to quantitatively assess the impact of thalamocortical network properties on the formation and maintenance of complex traveling wave patterns. Our computational model provides strong theoretical insight into the basis of spatiotemporal wave propagation, as well as experimental predictions that control these wave dynamics.


2022 ◽  
Vol 4 (1) ◽  
pp. 77-85
Author(s):  
Mohammad Ghani

We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.


2000 ◽  
Vol 10 (09) ◽  
pp. 2061-2073 ◽  
Author(s):  
WEN-XIN QIN

In this paper we study the existence of traveling waves with any rational velocity in coupled circle maps. We introduce an induced map, then a traveling wave with velocity p/q corresponds to a fixed point of the induced map. Moreover, the stability of a traveling wave is equivalent to that of the corresponding fixed point. Space translational system is chaotic on the set of traveling waves with rational velocities. In addition, time evolution operator exhibits sensitivity-like behavior with respect to initial conditions. By investigating the spatial periodicity of traveling waves, we obtain infinitely many space-time periodic structures. We also consider spatial asymptoticity of these traveling waves, which leads to the existence of fronts, defect solutions and soliton-like solutions. The abundance of traveling waves may be regarded as a signature of the spatial-temporal complexity in extended systems.


2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Cong Sun ◽  
Bo Jiang

We study the existence and orbital stability of smooth periodic traveling waves solutions of the(n +1)-dimensional coupled nonlinear Klein-Gordon equations. Such a system occurs in quantum mechanics, fluid mechanics, and optical fiber communication. Inspired by Angulo Pava’s results (2007), and by applying the stability theory established by Grillakis et al. (1987), we prove the existence of periodic traveling waves solutions and obtain the orbital stability of the solutions to this system.


Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Ahmad Alhasanat ◽  
Chunhua Ou

In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.


Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 48
Author(s):  
Miguel Rodriguez ◽  
Jing Li ◽  
Zhijun Qiao

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.


2010 ◽  
Vol 20 (11) ◽  
pp. 1967-1998 ◽  
Author(s):  
TONG LI ◽  
ZHI-AN WANG

Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler1,2 and was modeled by Keller and Segel.15 For a quasilinear hyperbolic–parabolic system that arises as a non-diffusive limit of the Keller–Segel model with nonlinear kinetics, we establish the existence and nonlinear stability of traveling wave solutions with large amplitudes. The numerical simulations are performed to show the stability of the traveling waves under various perturbations.


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