American Institute of Mathematical Sciences

Advanced
2014, 11(2): 203-215. doi: 10.3934/mbe.2014.11.203

FitzHugh-Nagumo equations with generalized diffusive coupling

Anna Cattani 1,

1. 

Department of Mathematical Sciences, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  September 2012 Revised  March 2013 Published  October 2013

The aim of this work is to investigate the dynamics of a neural network, in which neurons, individually described by the FitzHugh-Nagumo model, are coupled by a generalized diffusive term. The formulation we are going to exploit is based on the general framework of graph theory. With the aim of defining the connection structure among the excitable elements, the discrete Laplacian matrix plays a fundamental role. In fact, it allows us to model the instantaneous propagation of signals between neurons, which need not be physically close to each other.
    This approach enables us to address three fundamental issues. Firstly, each neuron is described using the well-known FitzHugh-Nagumo 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 self-sustained way through the whole set of neurons. A novel graphical representation of the dynamics is shown.
Keywords: stability analysis, FitzHugh-Nagumo model, Biological neural network, travelling pulses., diffusive coupling.
Mathematics Subject Classification: Primary: 92C42; Secondary: 68R1.
Citation: Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203
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 FitzHugh-Nagumo 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/S0006-3495(61)86902-6.  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 FitzHugh-Nagumo 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 FitzHugh-Nagumo 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/S0006-3495(61)86902-6.  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 FitzHugh-Nagumo 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 six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[2]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[3]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. 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) : 781-795. 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) : 367-376. doi: 10.3934/proc.2009.2009.367
[6]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
[7]

Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
[8]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028
[9]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851
[10]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072
[11]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[12]

Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
[13]

Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1729-1755. doi: 10.3934/dcdsb.2016020
[14]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[15]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[16]

Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203
[17]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
[18]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[19]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[20]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences