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Delay-induced spiking dynamics in integrate-and-fire neurons
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |







(unstable for |
(stable for |
||||||||
O/T | O/T | ||||||||
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 |
(unstable for |
(stable for |
||||||||
O/T | O/T | ||||||||
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 |
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