
Previous Article
On a spike train probability model with interacting neural units
 MBE Home
 This Issue

Next Article
Gaussdiffusion processes for modeling the dynamics of a couple of interacting neurons
FitzHughNagumo equations with generalized diffusive coupling
1.  Department of Mathematical Sciences, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
This approach enables us to address three fundamental issues. Firstly, each neuron is described using the wellknown FitzHughNagumo model which might allow to differentiate their individual behaviour. Furthermore, exploiting the Laplacian matrix, a well defined connection structure is formalized. Finally, random networks and an ensemble of excitatory and inhibitory synapses are considered.
Several simulations are performed to graphically present how dynamics within a network evolve. Thanks to an appropriate initial stimulus a wave is created: it propagates in a selfsustained way through the whole set of neurons. A novel graphical representation of the dynamics is shown.
References:
[1] 
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, Linear Algebra Appl., 436 (2012), 99111. doi: 10.1016/j.laa.2011.06.035. 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling, Phys. Rev. E (3), 436 (2012), 99111. doi: 10.1103/PhysRevE.67.066222. 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445466. doi: 10.1016/S00063495(61)869026. 
[5] 
A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve, J. Physiol., 117 (1952), 500544. 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction third edition, SpringerVerlag, New York, 2002. 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks, Progress of Theoretical Physics Supplements, 161 (2006), 389392. doi: 10.1143/PTPS.161.389. 
[8] 
A. C.Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487533. 
show all references
References:
[1] 
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, Linear Algebra Appl., 436 (2012), 99111. doi: 10.1016/j.laa.2011.06.035. 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling, Phys. Rev. E (3), 436 (2012), 99111. doi: 10.1103/PhysRevE.67.066222. 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445466. doi: 10.1016/S00063495(61)869026. 
[5] 
A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve, J. Physiol., 117 (1952), 500544. 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction third edition, SpringerVerlag, New York, 2002. 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks, Progress of Theoretical Physics Supplements, 161 (2006), 389392. doi: 10.1143/PTPS.161.389. 
[8] 
A. C.Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487533. 
[1] 
Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a sixdimensional FitzHughNagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems  B, 2011, 16 (2) : 457474. doi: 10.3934/dcdsb.2011.16.457 
[2] 
Joachim Crevat. Meanfield limit of a spatiallyextended FitzHughNagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 13291358. doi: 10.3934/krm.2019052 
[3] 
Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHughNagumo model with recovery variable. Evolution Equations and Control Theory, 2018, 7 (4) : 571585. doi: 10.3934/eect.2018027 
[4] 
Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete and Continuous Dynamical Systems  S, 2020, 13 (3) : 781795. doi: 10.3934/dcdss.2020044 
[5] 
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the FitzhughNagumo system. Discrete and Continuous Dynamical Systems  B, 2021, 26 (2) : 775794. doi: 10.3934/dcdsb.2020134 
[6] 
Arnold Dikansky. FitzhughNagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216224. doi: 10.3934/proc.2005.2005.216 
[7] 
Willem M. SchoutenStraatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHughNagumo equation with infiniterange interactions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 50175083. doi: 10.3934/dcds.2019205 
[8] 
Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Crossdiffusion effects on stationary pattern formation in the FitzHughNagumo model. Discrete and Continuous Dynamical Systems  B, 2022 doi: 10.3934/dcdsb.2022063 
[9] 
Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367376. doi: 10.3934/proc.2009.2009.367 
[10] 
Yiqiu Mao. Dynamic transitions of the FitzhughNagumo equations on a finite domain. Discrete and Continuous Dynamical Systems  B, 2018, 23 (9) : 39353947. doi: 10.3934/dcdsb.2018118 
[11] 
Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHughNagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559586. doi: 10.3934/eect.2017028 
[12] 
John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHughNagumo equation: The singularlimit. Discrete and Continuous Dynamical Systems  S, 2009, 2 (4) : 851872. doi: 10.3934/dcdss.2009.2.851 
[13] 
Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHughNagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems  B, 2021, 26 (3) : 15491563. doi: 10.3934/dcdsb.2020172 
[14] 
Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhughnagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems  B, 2018, 23 (4) : 16891720. doi: 10.3934/dcdsb.2018072 
[15] 
Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHughNagumo system on the timevarying domains. Discrete and Continuous Dynamical Systems  B, 2017, 22 (10) : 36913706. doi: 10.3934/dcdsb.2017150 
[16] 
Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete and Continuous Dynamical Systems  B, 2016, 21 (6) : 17291755. doi: 10.3934/dcdsb.2016020 
[17] 
Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHughNagumo neuron model through a modified Van der Pol equation with fractionalorder term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems  S, 2021, 14 (7) : 22292243. doi: 10.3934/dcdss.2020397 
[18] 
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHughNagumo type ReactionDiffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 21332156. doi: 10.3934/cpaa.2017106 
[19] 
Takashi Kajiwara. The subsupersolution method for the FitzHughNagumo type reactiondiffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 24412465. doi: 10.3934/dcds.2018101 
[20] 
B. Ambrosio, M. A. AzizAlaoui, V. L. E. Phan. Global attractor of complex networks of reactiondiffusion systems of FitzhughNagumo type. Discrete and Continuous Dynamical Systems  B, 2018, 23 (9) : 37873797. doi: 10.3934/dcdsb.2018077 
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]