
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), 99. doi: 10.1016/j.laa.2011.06.035. Google Scholar 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222. Google Scholar 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). Google Scholar 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. Google Scholar 
[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), 500. Google Scholar 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). Google Scholar 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389. Google Scholar 
[8] 
A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. Google Scholar 
show all references
References:
[1] 
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs,, Linear Algebra Appl., 436 (2012), 99. doi: 10.1016/j.laa.2011.06.035. Google Scholar 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222. Google Scholar 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). Google Scholar 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. Google Scholar 
[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), 500. Google Scholar 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). Google Scholar 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389. Google Scholar 
[8] 
A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. Google Scholar 
[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 & 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 & 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 & 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 & Continuous Dynamical Systems  S, 2020, 13 (3) : 781795. doi: 10.3934/dcdss.2020044 
[5] 
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 
[6] 
Arnold Dikansky. FitzhughNagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216224. doi: 10.3934/proc.2005.2005.216 
[7] 
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the FitzhughNagumo system. Discrete & Continuous Dynamical Systems  B, 2020 doi: 10.3934/dcdsb.2020134 
[8] 
Willem M. SchoutenStraatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHughNagumo equation with infiniterange interactions. Discrete & Continuous Dynamical Systems  A, 2019, 39 (9) : 50175083. doi: 10.3934/dcds.2019205 
[9] 
Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHughNagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559586. doi: 10.3934/eect.2017028 
[10] 
John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHughNagumo equation: The singularlimit. Discrete & Continuous Dynamical Systems  S, 2009, 2 (4) : 851872. doi: 10.3934/dcdss.2009.2.851 
[11] 
Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhughnagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems  B, 2018, 23 (4) : 16891720. doi: 10.3934/dcdsb.2018072 
[12] 
Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHughNagumo system on the timevarying domains. Discrete & Continuous Dynamical Systems  B, 2017, 22 (10) : 36913706. doi: 10.3934/dcdsb.2017150 
[13] 
Yiqiu Mao. Dynamic transitions of the FitzhughNagumo equations on a finite domain. Discrete & Continuous Dynamical Systems  B, 2018, 23 (9) : 39353947. doi: 10.3934/dcdsb.2018118 
[14] 
Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHughNagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems  B, 2020 doi: 10.3934/dcdsb.2020172 
[15] 
Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete & Continuous Dynamical Systems  B, 2016, 21 (6) : 17291755. doi: 10.3934/dcdsb.2016020 
[16] 
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 & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020397 
[17] 
Takashi Kajiwara. The subsupersolution method for the FitzHughNagumo type reactiondiffusion system with heterogeneity. Discrete & Continuous Dynamical Systems  A, 2018, 38 (5) : 24412465. doi: 10.3934/dcds.2018101 
[18] 
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHughNagumo type ReactionDiffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 21332156. doi: 10.3934/cpaa.2017106 
[19] 
Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHughNagumo equations. Discrete & Continuous Dynamical Systems  B, 2016, 21 (4) : 12031223. doi: 10.3934/dcdsb.2016.21.1203 
[20] 
Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHughNagumo systems driven by deterministic nonautonomous forcing. Discrete & Continuous Dynamical Systems  B, 2013, 18 (3) : 643666. doi: 10.3934/dcdsb.2013.18.643 
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]