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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Bifurcations and Dynamics in Modified Two Population Neuronal Network Models

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 237--257 | DOI:10.5890/DNC.2021.06.006

S. Roy Choudhury , Gizem S. Oztepe

Department of Mathematics, University of Central Florida, Orlando, FL32816, USA Department of Mathematics, Ankara University, Ankara, 06100, Turkey

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A canonical modified two population neuronal network model of Laplace convolution type is considered via the 'linear chain trick'. Linear stability analysis of this system and conditions for Hopf bifurcation initiating spatiotemporal oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. For more steeply negative firing-rate functions, the Hopf bifurcations occur at larger values of both the delay and the inhibitory time constant. Other bifurcations such as double Hopf or generalized Hopf modes occurring from the homogeneous background state are also shown to be impossible for our model. In this first model, the Hopf-generated limit cycles turn out to be remarkably stable under very large variations of all four system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. Numerical simulations reveal strong distortion of the limit cycle shapes in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as the increase of the oscillation amplitudes of the physical variables on the limit cycle attractor, as well as decrease of their time periods, as both the delay and the inhibitory time constant are increased. The final section considers alternative Fourier convolution models with general functional forms for the synaptic connectivity functions. In particular, we develop an approach to derive the large variable or asymptotic behaviors in both space and time for arbitrary functional forms of the connectivity functions.


  1. [1]  Dayan, P. and Abbott, L.F. (2001), Theoretical Neuroscience, MIT Press: Cambridge.
  2. [2]  Ermentrout, B. (1998), Neural networks as spatio-temporal patternforming systems, Rep. Prog. Phys., 61, 353-430.
  3. [3]  Amari, S. (1977), Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., 27, 77-87.
  4. [4]  Pinto, D.J. and Ermentrout, G.B. (2001), Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses, SIAM J. Appl. Math., 62, 226-243.
  5. [5]  Laing, C.R., Troy, W.C., Gutkin, B., and Ermentrout, G.B. (2002), Multiple bumps in a neuronal network model of working memory, SIAM J. Appl. Math., 63, 62-97.
  6. [6]  Laing, C.R. and Troy, W.C. (2003), Two-bump solutions af Amari-type models of neuronal pattern formation, Physica D, 178, 90-218.
  7. [7]  Coombes, S., Lord, G.J., and Owen, M.R. (2003), Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178, 219-241.
  8. [8]  Rubin, J.E. and Troy, W.C. (2004), Sustained spatial patterns of activity in neuronal populations without recurrent excitation, SIAM J. Appl. Math., 64, 1609-1635.
  9. [9]  Folias, S.E. and Bressloff, P.C. (2004), Breathing pulses in an excitatory network, SIAM J. Appl. Dyn. Syst., 3, 378-407.
  10. [10]  Coombes, S. and Owen, M.R. (2005), Bumps, breathers, and waves in a neural network with spike frequency adaptation, Phys. Rev. Lett., 148102.
  11. [11]  Guo, Y. and Chow, C.C. (2005), Existence and stability of standing pulses in neural networks: I. Existence, SIAM J. Appl. Dyn. Syst., 4, 217-248.
  12. [12]  Guo, Y. and Chow, C.C. (2005), Existence and stability of standing pulses in neural networks: II. Stability, SIAM J. Appl. Dyn. Syst., 4, 249-281.
  13. [13]  Blomquist, P., Wyller, J., and Einevoll, G.T. (2005), Localized activity patterns in two-population neuronal networks, Physica D, 206, 180-212.
  14. [14]  Ermentrout, G.B. and Cowan, J.D. (1980), Large scale spatially organized activity in neural nets, SIAM J. Appl. Math., 38, 1-21.
  15. [15]  Hutt, A., Bestehorn, M., and Wennekers, T. (2003), Pattern formation in intracortical neuronal fields, Netw.: Comput. Neural Syst., 14, 351-368.
  16. [16]  Hutt, A. and Atay, F.M. (2005), Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203, 30-54.
  17. [17]  Atay, F.M. and Hutt, A. (2005), Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM J. Appl. Math., 65, 644-666.
  18. [18]  Ermentrout, G.B. and Cowan, J.D. (1979), Temporal oscillations in neuronal nets, J.Math. Biol., 7, 265-280.
  19. [19]  Ermentrout, G.B. and Cowan, J.D. (1980), Secondary bifurcations in neuronal nets, SIAM J. Appl. Math., 39, 323-340.
  20. [20]  Curtu, R. and Ermentrout, B. (2004), Pattern formation in a network of excitatory and inhibitory cells with adaptation, SIAM J. Appl. Dyn. Syst., 3, 191-231.
  21. [21]  Laing, C. and Coombes, S. (2006), The importance of different timings of excitatory and inhibitory pathways in neural field models, Netw.: Comput. Neural Syst., 17, 151-172.
  22. [22]  Idiart, M.A.P. and Abbott, L.F. (1993), Propagation of excitation in neural networks, Networks, 4, 285-294.
  23. [23]  Pinto, D.J. and Ermentrout, G.B. (2001), Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses, SIAM J. Appl. Math., 62, 206-225.
  24. [24]  Bressloff, P.C. and Folias, S.E. (2004), Front bifurcations in an excitatory neural network, SIAM J. Appl. Math., 65, 131-151.
  25. [25]  Coombes, S. and Owen, M.R. (2004), Evans functions for integral neural field equations with Heaviside firing rate functions, SIAM J. Appl. Dyn. Syst., 3, 574-600.
  26. [26]  Pinto, D.J., Jackson, R.K., and Wayne, C.E. (2005), Existence and stability of traveling pulses in a continuous neuronal network, SIAM J. Appl. Dyn. Syst., 4, 954-984.
  27. [27]  Wilson, H.R. and Cowan, J.D. (1973), A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybern, 13, 55-80.
  28. [28]  Wyller, J., Blomquist, P., and Einevoll, G.T. (2007), Turing instability and pattern formation in a two-population neuronal network model, Phys. D, 225, 75-93.
  29. [29]  Nayfeh, A.H. and Balachandran, B. (1995), Applied Nonlinear Dynamics, John Wiley: New York.
  30. [30]  Glendinning, P. (1994), Stability, Instability and Chaos: An Introduction to The Theory of Nonlinear Differential Equations. Vol. 11, Cambridge University Press: Cambridge.
  31. [31]  MacDonald, N. (1978), Time Lags in Biological Models. In: Lecture Notes In Biomathematics, Vol. 27, Springer-Verlag: Berlin.
  32. [32]  Davis, H.T. (1962), Introduction to Nonlinear Differential and Integral Equations, Dover: New York.
  33. [33]  Cushing, J.M. (1977), Integro Differential Equations and Delay Models in Population Dynamics. In: Lecture Notes in Biomathematics, Vol.20, Springer-Verlag: Berlin. \enlargethispage{\baselineskip}
  34. [34]  Smitalova, K. and Sujan, S. (1991), A Mathematical Treatment of Dynamical Models in Biological Science, Ellis Horwood: New York, and references therein.
  35. [35]  Hale, J. (1977), Theory of Functional Differential Equations, Springer-Verlag: Berlin.
  36. [36]  MacDonald, N. (1976), Time delay in prey-predator models, Mathematical Biosciences 28, 3-4, 321-330.
  37. [37]  MacDonald, N. (1977), Time delay in prey-predator models-II, Bifurcation theory, Mathematical Biosciences 33, 3-4, 227-234.
  38. [38]  Vogel, T. (1965), Systemes evolutifs, Gautier-Villars: Paris.
  39. [39]  Farkas, M. (1984), Stable oscillations in a predator-prey model with time lag, J. Math. Anal. Appl., 102, 175-188.
  40. [40]  El-Owaidy, H. and Ammar, A.A. (1988), Stable oscillations in a predator-prey model with time lag, J. Math. Anal. Appl., 130, 191-199.
  41. [41]  Krise, S. and Choudhury, S.R. (2003), Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons and Fractals, 16, 59-77.
  42. [42]  Choudhury, S.R. (1992), On bifurcations and chaos in predator-prey models with delay, Chaos, Solitons and Fractals, 2.4, 393-409.
  43. [43]  Sivasamya, R., Sathiyanathana, K., and Balachandran, K. (2012), Dynamics of a Modified Leslie-Gower model with Crowley-Martin functional response and prey harvesting. J. of Appl. Non. Dynmcs, 1, 1-6.
  44. [44]  Bender, C. and Orszag, S.A. (1991), Advanced Mathematical Methods for Scientists and Engineers, Springer: New York.
  45. [45]  Davies, B. (1978), Integral Transforms and their Applications, Springer: New York.
  46. [46]  Born, M. and Wolf, E. (2002), Principles of Optics, Cambridge Univ. Press: Cambridge.
  47. [47]  Jeffreys, H. and Jeffreys, B. (1999), Methods of Mathematical Physics, Cambridge Univ. Press: Cambridge.