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April  2021, 26(4): 1867-1887. doi: 10.3934/dcdsb.2020363

Delay-induced spiking dynamics in integrate-and-fire neurons

Chang-Yuan Cheng 1, , Shyan-Shiou Chen 2,, and  Rui-Hua Chen 3,

1. 

Department of Applied Mathematics, National Pingtung University Pingtung, Taiwan, R.O.C., No.4-18, Minsheng Rd., Pingtung City, Pingtung County 90003, Taiwan, R.O.C
2. 

Department of Mathematics, National Taiwan Normal University, No. 88, Sec. 4, Ting-chou Rd., Taipei 116, Taiwan
3. 

Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan, R.O.C., Fang Liao High School, Pingtung, Taiwan, R.O.C., No.3, Yimin Rd., Fangliao Township, Pingtung County 940, Taiwan, R.O.C

* Corresponding author: Shyan-Shiou Chen

(Dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas)

Received  August 2020 Revised  October 2020 Published  December 2020

Fund Project: S. S. Chen and C. Y. Cheng were partially supported by the Ministry of Science and Technology of Taiwan, R.O.C. (Grant Nos. MOST 108-2115-M-003 -011 and MOST 108-2115-M-153-003)

Experiments showed that a neuron can fire when its membrane potential (an intrinsic quality related to its membrane electrical charge) reaches a specific threshold. On theoretical studies, there are two crucial issues in exploring cortical neuronal dynamics: (i) what model describes spiking dynamics of each neuron, and (ii) how the neurons are connected [E. M. Izhikevich, IEEE Trans. Neural Networks, 15 (2004)]. To study the first issue, we propose the time delay effect on the well-known integrate-and-fire (IF) model which is classically introduced to study the spiking behaviors in neural systems by using the spike-and-reset procedure. Under the consideration of delayed adaptation on the membrane potential, the parameter range for the IF model with spiking dynamics becomes wider due to undergoing subcritical Hopf bifurcation and the existence of an unstable orbit. To study the second issue, we consider the system with two coupled identical IF units where time delay takes place in the coupling structure. We also demonstrate spiking behaviors in the coupled system when the delay time is large enough, and it contributes an original viewpoint of the connection between neurons. In contrast with the emergence of delay-induced spiking in a single-neuron system, a coupled two-neuron system involve both emergence and death of spiking according to different values of delay times. We also discuss the ranges of different parameters in which it allows occurrence of spiking behaviors.

Keywords: Spiking dynamics, Integrate-and-fire, Delayed coupling, Hopf Bifurcation.
Mathematics Subject Classification: Primary:34K18, 34K20, 34K25.
Citation: Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1867-1887. doi: 10.3934/dcdsb.2020363
References:
[1]

S. A. Campbell, Time delays in neural systems., In Handbook of Brain Connectivity (eds A. R. McIntosh & V. K. Jirsa), 65–90, Berlin, Germany: Springer, 2007. doi: 10.1007/978-3-540-71512-2_2.  Google Scholar

[2]

S. S. Chen, C. Y. Cheng and Y. R. Lin, Application of a two-dimensional Hinmarsh-Rose type model for bifurcation analysis, Int. J. Bifurcation and Chaos, 23 (2013), 1350055, 21pp. doi: 10.1142/S0218127413500557.  Google Scholar

[3]

S. S. Chen and C. Y. Cheng, Delay-induced mixed-mode oscillations in a 2d HindMarsh-Rose type model with recurrent neural feedback, Discrete Conti. Dyn. Sys.-B, 21 (2016), 37-53.  doi: 10.3934/dcdsb.2016.21.37.  Google Scholar

[4]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[5]

S. Coombes and C. Laing, Delays in activity-based neural networks, Phil. Trans. R. Soc. A, 367 (2009), 1117-1129.  doi: 10.1098/rsta.2008.0256.  Google Scholar

[6]

S. Ditlevsen and P. Greenwood, The Morris-Lecar neuron model embeds a leaky integrate-and-fire model, J. Math. Biol., 67 (2013), 239-259.  doi: 10.1007/s00285-012-0552-7.  Google Scholar

[7]

G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8.  Google Scholar

[8]

N. Fourcaud-Trocme, D. Hansel, C. van Vreeswijk and N. Brunel, How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci, 23 (2003), 11628-11640.  doi: 10.1523/JNEUROSCI.23-37-11628.2003.  Google Scholar

[9]

E. Foxall, R. Edwards, S. Ibrahim and P. van den Driessche, A contraction argument for two-dimensional spiking neuron models, SIAM J. Appl. Dyn. Sys., 11 (2012), 540-566.  doi: 10.1137/10081811X.  Google Scholar

[10] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.   Google Scholar
[11]

E. M. Izhikevich, Which model to use for cortical spiking neurons?, IEEE Trans. Neural Networks, 15 (2004), 1063-1070.  doi: 10.1109/TNN.2004.832719.  Google Scholar

[12]

W. Nicola and S. A. Campbell, Bifurcations of large networks of two-dimensional integrate and fire neurons, J. Comp. Neurosci., 35 (2013), 87-108.  doi: 10.1007/s10827-013-0442-z.  Google Scholar

[13]

L. Prignano, O. Sagarra and A. Díaz-Guilera, Tuning synchronization of integrate-and-fire oscillators through mobility, Phys. Rev. Lett., 110 (2013), 114101. doi: 10.1103/PhysRevLett.110.114101.  Google Scholar

[14]

I. Ratas and K. Pyragas, Macroscopic oscillations of a quadratic integrate-and-fire neuron network with global distributed-delay coupling, Phys. Rev. E, 98 (2018), 052224, 11pp. doi: 10.1103/physreve.98.052224.  Google Scholar

[15]

S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.  doi: 10.1090/qam/1811101.  Google Scholar

[16]

M. A. Schwemmer and T. J. Lewis, Bistability in a leaky integrate-and-fire neuron with a passive dendrite, SIAM J. Appl. Dyn. Sys., 11 (2012), 507-539.  doi: 10.1137/110847354.  Google Scholar

[17]

E. Shlizerman and P. Holmes, Neural dynamics, bifurcations and firing rates in a quadratic integrate-and-fire model with a recovery variable. I: Deterministic behavior, Neural Comput., 24 (2012), 2078-2118.  doi: 10.1162/NECO_a_00308.  Google Scholar

[18]

J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM J. Appl. Math., 68 (2008), 1045-1079.  doi: 10.1137/070687268.  Google Scholar

[19]

J. Touboul and R. Brette, Dynamics and bifurcations of the adaptive exponential integrate-and-fire model, Biolog. Cybernet., 99 (2008), 319-334.  doi: 10.1007/s00422-008-0267-4.  Google Scholar

[20]

J. Touboul, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Comput., 21 (2009), 2114-2122.  doi: 10.1162/neco.2009.09-08-853.  Google Scholar

[21]

J. Touboul and R. Brette, Spiking dynamics of bidimensional integrate-and-fire neurons, SIAM J. Appl. Dyn. Sys., 8 (2009), 1462-1506.  doi: 10.1137/080742762.  Google Scholar

[22]

G. Zheng and A. Tonnelier, Chaotic solutions in the quadratic integrate-and-fire neuron with adaptation, Cogn. Neurodyn., 3 (2009), 197-204.  doi: 10.1007/s11571-008-9069-6.  Google Scholar

show all references

References:
[1]

S. A. Campbell, Time delays in neural systems., In Handbook of Brain Connectivity (eds A. R. McIntosh & V. K. Jirsa), 65–90, Berlin, Germany: Springer, 2007. doi: 10.1007/978-3-540-71512-2_2.  Google Scholar

[2]

S. S. Chen, C. Y. Cheng and Y. R. Lin, Application of a two-dimensional Hinmarsh-Rose type model for bifurcation analysis, Int. J. Bifurcation and Chaos, 23 (2013), 1350055, 21pp. doi: 10.1142/S0218127413500557.  Google Scholar

[3]

S. S. Chen and C. Y. Cheng, Delay-induced mixed-mode oscillations in a 2d HindMarsh-Rose type model with recurrent neural feedback, Discrete Conti. Dyn. Sys.-B, 21 (2016), 37-53.  doi: 10.3934/dcdsb.2016.21.37.  Google Scholar

[4]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[5]

S. Coombes and C. Laing, Delays in activity-based neural networks, Phil. Trans. R. Soc. A, 367 (2009), 1117-1129.  doi: 10.1098/rsta.2008.0256.  Google Scholar

[6]

S. Ditlevsen and P. Greenwood, The Morris-Lecar neuron model embeds a leaky integrate-and-fire model, J. Math. Biol., 67 (2013), 239-259.  doi: 10.1007/s00285-012-0552-7.  Google Scholar

[7]

G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8.  Google Scholar

[8]

N. Fourcaud-Trocme, D. Hansel, C. van Vreeswijk and N. Brunel, How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci, 23 (2003), 11628-11640.  doi: 10.1523/JNEUROSCI.23-37-11628.2003.  Google Scholar

[9]

E. Foxall, R. Edwards, S. Ibrahim and P. van den Driessche, A contraction argument for two-dimensional spiking neuron models, SIAM J. Appl. Dyn. Sys., 11 (2012), 540-566.  doi: 10.1137/10081811X.  Google Scholar

[10] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.   Google Scholar
[11]

E. M. Izhikevich, Which model to use for cortical spiking neurons?, IEEE Trans. Neural Networks, 15 (2004), 1063-1070.  doi: 10.1109/TNN.2004.832719.  Google Scholar

[12]

W. Nicola and S. A. Campbell, Bifurcations of large networks of two-dimensional integrate and fire neurons, J. Comp. Neurosci., 35 (2013), 87-108.  doi: 10.1007/s10827-013-0442-z.  Google Scholar

[13]

L. Prignano, O. Sagarra and A. Díaz-Guilera, Tuning synchronization of integrate-and-fire oscillators through mobility, Phys. Rev. Lett., 110 (2013), 114101. doi: 10.1103/PhysRevLett.110.114101.  Google Scholar

[14]

I. Ratas and K. Pyragas, Macroscopic oscillations of a quadratic integrate-and-fire neuron network with global distributed-delay coupling, Phys. Rev. E, 98 (2018), 052224, 11pp. doi: 10.1103/physreve.98.052224.  Google Scholar

[15]

S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.  doi: 10.1090/qam/1811101.  Google Scholar

[16]

M. A. Schwemmer and T. J. Lewis, Bistability in a leaky integrate-and-fire neuron with a passive dendrite, SIAM J. Appl. Dyn. Sys., 11 (2012), 507-539.  doi: 10.1137/110847354.  Google Scholar

[17]

E. Shlizerman and P. Holmes, Neural dynamics, bifurcations and firing rates in a quadratic integrate-and-fire model with a recovery variable. I: Deterministic behavior, Neural Comput., 24 (2012), 2078-2118.  doi: 10.1162/NECO_a_00308.  Google Scholar

[18]

J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM J. Appl. Math., 68 (2008), 1045-1079.  doi: 10.1137/070687268.  Google Scholar

[19]

J. Touboul and R. Brette, Dynamics and bifurcations of the adaptive exponential integrate-and-fire model, Biolog. Cybernet., 99 (2008), 319-334.  doi: 10.1007/s00422-008-0267-4.  Google Scholar

[20]

J. Touboul, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Comput., 21 (2009), 2114-2122.  doi: 10.1162/neco.2009.09-08-853.  Google Scholar

[21]

J. Touboul and R. Brette, Spiking dynamics of bidimensional integrate-and-fire neurons, SIAM J. Appl. Dyn. Sys., 8 (2009), 1462-1506.  doi: 10.1137/080742762.  Google Scholar

[22]

G. Zheng and A. Tonnelier, Chaotic solutions in the quadratic integrate-and-fire neuron with adaptation, Cogn. Neurodyn., 3 (2009), 197-204.  doi: 10.1007/s11571-008-9069-6.  Google Scholar

Figure 1.  Left: The bifurcation curves of the equation in $ I-b $ parameter space with other parameter values $ a = 1 $, $ d = 1 $. The curve $ \Gamma_1:b^2 = 4dI $ describes the saddle-node bifurcation, and the line $ \Gamma_2:I = \frac{ab}{2d}-\frac{a^2}{4d} $ denotes where the stability of equilibrium changes. $ D_1 $: no fixed equilibrium, $ \Gamma_1 $: s saddle equilibrium, $ D_2 $: a saddle and a repulse equilibria, $ D_3 $: a saddle and an attractive equilibrium. Right: Illustration of $ I-b $ parameter space for the system (3) to undergoes Hopf bifurcation near $ \underline{\xi} $, where the curve $ \Gamma_3:B^2 = C^2 $ separates the parameter spaces whether Hopf bifurcation occurs or not. $ D_{31} $: Hopf bifurcation occurs near the equilibrium $ \underline{\xi} $, $ D_{32} $: Hopf bifurcation never occur near any equilibrium
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Figure 2.  Left: Graphs of $ s_0(b) $ in different values of $ d $. The solid curve: $ d = 0.85 $, the dash curve: $ d = 1 $, the dot curve: $ d = 1.15 $. Right: Zoomed out graphs as in the left panel showing that $ s_0(b)\rightarrow 0 $ as $ b\rightarrow +\infty $
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Figure 3.  Two trajectories near and not near the equilibrium $ \underline{\xi} $ of the system (3) without the reset process and under fixed parameters $ a = 1 $, $ b = 1.2 $, $ d = 1 $, $ I = 0.2 $. Left: Choosing delay time $ s = 0.5 $, the equilibrium $ \underline{\xi} $ is locally stable. Right: Choosing delay time $ s = 0.52 $, the equilibrium $ \underline{\xi} $ becomes unstable, and two solutions blow up in finite time
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Figure 4.  Delay-induced spiking in the equation (3). (a) The membrane potential tends to silence when $ s = 5 $; (b) a spiking emerges when $ s = 5.2 $ (due to the reset process)
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Figure 5.  Graphs of $ v_1 = F(v_2) $ and $ v_2 = F(v_1) $. Herein, parameters $ b = 1 $ and $ d = 0.3 $ are fixed. (a) $ c = 2 $ and $ I = 1.2 $, (b) $ c = 1 $ and $ I = 0.3 $, (a) $ c = 0.3 $ and $ I = 0.3 $
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Figure 6.  Regions of local stability of the equilibrium $ \underline{\xi} $ in Eq. (38) with $ a = 0.22 $, $ b = 1 $, $ d = 0.3 $, $ I = 0.3 $ and other parameter values as indicated. Herein, we fix all parameters except the coupling strength $ c $ to observe how the bifurcation values $ \tau^{\pm}_j $ and $ \tilde{\tau}^{\pm}_j $ depend on $ c $. Colored curves are $ \tau^{\pm}_j(c) $ and $ \tilde{\tau}^{\pm}_j(c) $ respectively and $ \Omega $ is the curve $ \tau^+_0 $. The equilibrium $ \underline{\xi} $ is stable (unstable) with $ (c,\tau) $ locating in the s (u) regions. When $ c<c^*\approx 0.1874 $ the equilibrium $ \underline{\xi} $ is stable fore all $ \tau\geq 0 $; when $ c>c^* $ stability of the equilibrium $ \underline{\xi} $ switches as the delay time increases
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Figure 7.  Emergence and death of spiking behaviors in the system (38). (a) The membrane potential tends to silence when $ \tau = 6.5 $; (b) a spiking emerges when $ \tau = 7.1 $; (c) death of spiking occurs when $ \tau = 13 $; (d) a spiking emerges again when $ \tau = 18 $
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Table 1.  Hopf bifurcation values $ \tau^{\pm}_j $, $ \tilde{\tau}^{\pm}_j $ and stability switches of the equilibrium $ \underline{\xi} $. It applies the real function $ f(v) = dv^2 $, $ d = 0.3 $, and sets parameters $ b = 1 $, $ c = 0.3 $ and $ I = 0.3 $. The symbol O (T) indicates that the bifurcation value is the type of $ \tau^{\pm}_j $ ($ \tilde{\tau}^{\pm}_j $) and $ \mathcal{S} $ denotes the sign of the transversality condition $ \frac{d}{d\tau}{\rm Re}\lambda(\cdot) $. The system (38) undergoes Hopf bifurcation at $ \tau = \tau^{\pm}_j $, $ \tilde{\tau}^{\pm}_j $, and the transversality condition holds that $ \frac{d}{d\tau}{\rm Re}\lambda(\tau)>0(<0) $ for $ \tau = \tau^+_j,\tilde{\tau}^+_j(\tau^-_j,\tilde{\tau}^-_j) $ and $ j = 0,1,2,\cdots $. When $ a = 0.18 $, the equilibrium $ \underline{\xi} $ is unstable at $ \tau = 0 $, two times switch from instability to stability and back instability occurs as the delay time increases, and $ \underline{\xi} $ is unstable for $ \tau $ is large enough. When $ a = 0.22 $, $ \underline{\xi} $ is locally asymptotically stable at $ \tau = 0 $, five times switch from stability to instability and back stability occurs as the delay time increases, and $ \underline{\xi} $ is unstable for $ \tau $ is large enough
$ a=0.18 $ $ a=0.22 $
(unstable for $ \tau=0 $) (stable for $ \tau=0 $)
$ \tau^u_j $/$ \tilde{\tau}^u_j $ O/T $ j $ $ u $ $ \mathcal{S} $ $ \tau^u_j $/$ \tilde{\tau}^u_j $ O/T $ j $ $ u $ $ \mathcal{S} $
0.3312 T 0 - -1 4.8967 O 0 + 1
4.4199 O 0 + 1 6.6875 O 0 - -1
8.9989 O 0 - -1 10.9079 T 0 + 1
10.3973 T 0 + 1 13.7393 T 0 - -1
16.3747 O 1 + 1 16.9192 O 1 + 1
17.6667 T 1 - -1 20.7912 O 1 - -1
22.3521 T 1 + 1 22.9304 T 1 + 1
26.3344 O 1 - -1 27.8431 T 1 - -1
28.3296 O 2 + 1 28.9416 O 2 + 1
34.3070 T 2 + 1 34.8949 O 2 - -1
35.0021 T 2 - -1 34.9529 T 2 + 1
43.6699 O 2 - -1 40.9641 O 3 + 1
40.2844 O 3 + 1 41.9469 O 2 - -1
46.2618 T 3 + 1 46.9754 T 3 + 1
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
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$ a=0.18 $ $ a=0.22 $
(unstable for $ \tau=0 $) (stable for $ \tau=0 $)
$ \tau^u_j $/$ \tilde{\tau}^u_j $ O/T $ j $ $ u $ $ \mathcal{S} $ $ \tau^u_j $/$ \tilde{\tau}^u_j $ O/T $ j $ $ u $ $ \mathcal{S} $
0.3312 T 0 - -1 4.8967 O 0 + 1
4.4199 O 0 + 1 6.6875 O 0 - -1
8.9989 O 0 - -1 10.9079 T 0 + 1
10.3973 T 0 + 1 13.7393 T 0 - -1
16.3747 O 1 + 1 16.9192 O 1 + 1
17.6667 T 1 - -1 20.7912 O 1 - -1
22.3521 T 1 + 1 22.9304 T 1 + 1
26.3344 O 1 - -1 27.8431 T 1 - -1
28.3296 O 2 + 1 28.9416 O 2 + 1
34.3070 T 2 + 1 34.8949 O 2 - -1
35.0021 T 2 - -1 34.9529 T 2 + 1
43.6699 O 2 - -1 40.9641 O 3 + 1
40.2844 O 3 + 1 41.9469 O 2 - -1
46.2618 T 3 + 1 46.9754 T 3 + 1
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
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