• Previous Article
    Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups
  • ERA Home
  • This Issue
  • Next Article
    Averaging principle on infinite intervals for stochastic ordinary differential equations
doi: 10.3934/era.2021034

Firing patterns and bifurcation analysis of neurons under electromagnetic induction

1. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

2. 

School of Science and Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China

* Corresponding author: Shenquan Liu

Received  November 2020 Revised  March 2021 Published  April 2021

Fund Project: The first author is supported by NSF of China under Grant Nos. 11872183 and 11572127

Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter $ k_0 $, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.

Citation: Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, doi: 10.3934/era.2021034
References:
[1]

J. F. BarryM. J. TurnerJ. M. SchlossD. R. GlennY. SongM. D. LukinH. Park and R. L. Walsworth, Optical magnetic detection of single-neuron action potentials using quantum defects in diamond, Proc. Natl. Acad. Sci. U.S.A., 113 (2016), 14133-14138.  doi: 10.1073/pnas.1601513113.  Google Scholar

[2]

R. Bertram and J. E. Rubin, Multi-timescale systems and fast-slow analysis, Math. Biol., 287 (2017), 105-121.  doi: 10.1016/j.mbs.2016.07.003.  Google Scholar

[3]

F. A. CarrilloF. Verduzco and J. Delgado, Analysis of the Takens-Bogdanov bifurcation on m-parameterized vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 995-1005.  doi: 10.1142/S0218127410026277.  Google Scholar

[4]

L. DuanQ. CaoZ. Wang and J. Su, Dynamics of neurons in the pre-B$ \ddot{o} $tzinger complex under magnetic flow effect, Nonlinear Dynam, 94 (2018), 1961-1971.  doi: 10.1007/s11071-018-4468-7.  Google Scholar

[5]

L. DuanQ. Lu and Q. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomputing, 72 (2008), 341-351.  doi: 10.1016/j.neucom.2008.01.019.  Google Scholar

[6]

H. Gu, Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker, Chaos, 23 (2013), 023126. doi: 10.1063/1.4810932.  Google Scholar

[7]

F. Han, Z. Wang, Y. Du, X. Sun and B. Zhang, Robust synchronization of bursting Hodgkin-Huxley neuronal systems coupled by delayed chemical synapses, Int. J. Non. Linear. Mech, 70 (2015), 105-111. doi: 10.1016/j. ijnonlinmec. 2014.10.010.  Google Scholar

[8]

E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266.  doi: 10.1142/S0218127400000840.  Google Scholar

[9]

M. S. Kafraj, F. Parastesh and S. Jafari, Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise, Chaos, Solitons Fractals, 137 (2020), 109782, 11 pp. doi: 10.1016/j. chaos. 2020.109782.  Google Scholar

[10]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^nd$ edition, Springer-Verlag, New York, 2004. Google Scholar

[11]

J. Li, Y. Wu, M. Du and W. Liu, Dynamic behavior in firing rhythm transitions of neurons under electromagnetic radiation, Acta Phys. Sin, 64 (2015), 030503. doi: 10.7498/aps. 64.030503.  Google Scholar

[12]

Y. LoewensteinS. MahonP. ChaddertonK. KitamuraH. SompolinskyY. Yarom and M. Häusser, Bistability of cerebellar Purkinje cells modulated by sensory stimulation, Nature Neurosci., 8 (2005), 202-211.  doi: 10.1038/nn1393.  Google Scholar

[13]

B. Lu, S. Liu, X. Liu, X. Jiang and X. Jang, Bifurcation and spike adding transition in Chay-Keizer model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650090, 13 pp. doi: 10.1142/S0218127416500905.  Google Scholar

[14]

M. Lv and J. Ma, Multiple modes of electrical activities in a new neuron model under electromagnetic radiation, Neurocomputing, 205 (2016), 375-381.  doi: 10.1016/j.neucom.2016.05.004.  Google Scholar

[15]

A. MondalR. K. Upadhyay and J. Ma et al., Bifurcation analysis and diverse firing activities of a modified excitable neuron model, Cognitive Neurodynamics, 13 (2019), 393-407.  doi: 10.1007/s11571-019-09526-z.  Google Scholar

[16]

A. MvogoC. N. TakemboH. P. Ekobena Fouda and T. C. Kofan$\acute{e}$, Pattern formation in diffusive excitable systems under magnetic flow effects, Phys. Lett. A, 381 (2017), 2264-2271.  doi: 10.1016/j.physleta.2017.05.020.  Google Scholar

[17]

C. S. NunemakerR. BertramA. ShermanK. Tsaneva-AtanasovaC. R. Daniel and L. S. Satin, Glucose modulates ${[Ca^{2+}]}_i$ oscillations in pancreatic islets via ionic and glycolytic mechanisms, Biophys. J, 91 (2006), 2082-2096.  doi: 10.1529/biophysj.106.087296.  Google Scholar

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model, Ordinary Partial Differ. Equations, Springer Berlin Heidelberg, Berlin, Heidelberg 1151 (1985), 304-316. doi: 10.1007/BFb0074739.  Google Scholar

[19]

J. E. RubinJ. Signerska-RynkowskaJ. D. Touboul and A. Vidal, Wild oscillations in a nonlinear neuron model with resets: (I) bursting, spike-adding and chaos, Discrete. Contin. Dyn. Syst. Ser. B, 22 (2017), 3967-4002.  doi: 10.3934/dcdsb.2017204.  Google Scholar

[20]

A. SaitoT. TeraiK. MakinoM. Takahashi and al. et, Real-time detection of stimulus response in cultured neurons by high-intensity intermediate-frequency magnetic field exposure, Integr. Biol, 10 (2018), 442-449.  doi: 10.1039/C8IB00097B.  Google Scholar

[21]

A. Saito, K. Wada, Y. Suzuki and S. Nakasono, The response of the neuronal activity in the somatosensory cortex after high-intensity intermediate-frequency magnetic field exposure to the spinal cord in rats under anesthesia and waking states, Brain Res., 1747 (2020), 147063. doi: 10.1016/j. brainres. 2020.147063.  Google Scholar

[22]

W. TekaK. Tsaneva-AtanasovaR. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition, Bull. Math. Biol., 73 (2011), 1292-1311.  doi: 10.1007/s11538-010-9559-7.  Google Scholar

[23]

K. Tsaneva-AtanasovaH. M. OsingaT. Rie$ \beta $ and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theor. Biol., 264 (2010), 1133-1146.  doi: 10.1016/j.jtbi.2010.03.030.  Google Scholar

[24]

X. J. Wang, Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle, Phys. D, 62 (1993), 263-274.  doi: 10.1016/0167-2789(93)90286-A.  Google Scholar

[25]

F. ZhanS. LiuX. ZhangJ. Wang and B. Lu, Mixed-mode oscillations and bifurcation analysis in a pituitary model, Nonlinear Dynam., 94 (2018), 807-826.  doi: 10.1007/s11071-018-4395-7.  Google Scholar

show all references

References:
[1]

J. F. BarryM. J. TurnerJ. M. SchlossD. R. GlennY. SongM. D. LukinH. Park and R. L. Walsworth, Optical magnetic detection of single-neuron action potentials using quantum defects in diamond, Proc. Natl. Acad. Sci. U.S.A., 113 (2016), 14133-14138.  doi: 10.1073/pnas.1601513113.  Google Scholar

[2]

R. Bertram and J. E. Rubin, Multi-timescale systems and fast-slow analysis, Math. Biol., 287 (2017), 105-121.  doi: 10.1016/j.mbs.2016.07.003.  Google Scholar

[3]

F. A. CarrilloF. Verduzco and J. Delgado, Analysis of the Takens-Bogdanov bifurcation on m-parameterized vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 995-1005.  doi: 10.1142/S0218127410026277.  Google Scholar

[4]

L. DuanQ. CaoZ. Wang and J. Su, Dynamics of neurons in the pre-B$ \ddot{o} $tzinger complex under magnetic flow effect, Nonlinear Dynam, 94 (2018), 1961-1971.  doi: 10.1007/s11071-018-4468-7.  Google Scholar

[5]

L. DuanQ. Lu and Q. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomputing, 72 (2008), 341-351.  doi: 10.1016/j.neucom.2008.01.019.  Google Scholar

[6]

H. Gu, Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker, Chaos, 23 (2013), 023126. doi: 10.1063/1.4810932.  Google Scholar

[7]

F. Han, Z. Wang, Y. Du, X. Sun and B. Zhang, Robust synchronization of bursting Hodgkin-Huxley neuronal systems coupled by delayed chemical synapses, Int. J. Non. Linear. Mech, 70 (2015), 105-111. doi: 10.1016/j. ijnonlinmec. 2014.10.010.  Google Scholar

[8]

E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266.  doi: 10.1142/S0218127400000840.  Google Scholar

[9]

M. S. Kafraj, F. Parastesh and S. Jafari, Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise, Chaos, Solitons Fractals, 137 (2020), 109782, 11 pp. doi: 10.1016/j. chaos. 2020.109782.  Google Scholar

[10]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^nd$ edition, Springer-Verlag, New York, 2004. Google Scholar

[11]

J. Li, Y. Wu, M. Du and W. Liu, Dynamic behavior in firing rhythm transitions of neurons under electromagnetic radiation, Acta Phys. Sin, 64 (2015), 030503. doi: 10.7498/aps. 64.030503.  Google Scholar

[12]

Y. LoewensteinS. MahonP. ChaddertonK. KitamuraH. SompolinskyY. Yarom and M. Häusser, Bistability of cerebellar Purkinje cells modulated by sensory stimulation, Nature Neurosci., 8 (2005), 202-211.  doi: 10.1038/nn1393.  Google Scholar

[13]

B. Lu, S. Liu, X. Liu, X. Jiang and X. Jang, Bifurcation and spike adding transition in Chay-Keizer model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650090, 13 pp. doi: 10.1142/S0218127416500905.  Google Scholar

[14]

M. Lv and J. Ma, Multiple modes of electrical activities in a new neuron model under electromagnetic radiation, Neurocomputing, 205 (2016), 375-381.  doi: 10.1016/j.neucom.2016.05.004.  Google Scholar

[15]

A. MondalR. K. Upadhyay and J. Ma et al., Bifurcation analysis and diverse firing activities of a modified excitable neuron model, Cognitive Neurodynamics, 13 (2019), 393-407.  doi: 10.1007/s11571-019-09526-z.  Google Scholar

[16]

A. MvogoC. N. TakemboH. P. Ekobena Fouda and T. C. Kofan$\acute{e}$, Pattern formation in diffusive excitable systems under magnetic flow effects, Phys. Lett. A, 381 (2017), 2264-2271.  doi: 10.1016/j.physleta.2017.05.020.  Google Scholar

[17]

C. S. NunemakerR. BertramA. ShermanK. Tsaneva-AtanasovaC. R. Daniel and L. S. Satin, Glucose modulates ${[Ca^{2+}]}_i$ oscillations in pancreatic islets via ionic and glycolytic mechanisms, Biophys. J, 91 (2006), 2082-2096.  doi: 10.1529/biophysj.106.087296.  Google Scholar

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model, Ordinary Partial Differ. Equations, Springer Berlin Heidelberg, Berlin, Heidelberg 1151 (1985), 304-316. doi: 10.1007/BFb0074739.  Google Scholar

[19]

J. E. RubinJ. Signerska-RynkowskaJ. D. Touboul and A. Vidal, Wild oscillations in a nonlinear neuron model with resets: (I) bursting, spike-adding and chaos, Discrete. Contin. Dyn. Syst. Ser. B, 22 (2017), 3967-4002.  doi: 10.3934/dcdsb.2017204.  Google Scholar

[20]

A. SaitoT. TeraiK. MakinoM. Takahashi and al. et, Real-time detection of stimulus response in cultured neurons by high-intensity intermediate-frequency magnetic field exposure, Integr. Biol, 10 (2018), 442-449.  doi: 10.1039/C8IB00097B.  Google Scholar

[21]

A. Saito, K. Wada, Y. Suzuki and S. Nakasono, The response of the neuronal activity in the somatosensory cortex after high-intensity intermediate-frequency magnetic field exposure to the spinal cord in rats under anesthesia and waking states, Brain Res., 1747 (2020), 147063. doi: 10.1016/j. brainres. 2020.147063.  Google Scholar

[22]

W. TekaK. Tsaneva-AtanasovaR. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition, Bull. Math. Biol., 73 (2011), 1292-1311.  doi: 10.1007/s11538-010-9559-7.  Google Scholar

[23]

K. Tsaneva-AtanasovaH. M. OsingaT. Rie$ \beta $ and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theor. Biol., 264 (2010), 1133-1146.  doi: 10.1016/j.jtbi.2010.03.030.  Google Scholar

[24]

X. J. Wang, Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle, Phys. D, 62 (1993), 263-274.  doi: 10.1016/0167-2789(93)90286-A.  Google Scholar

[25]

F. ZhanS. LiuX. ZhangJ. Wang and B. Lu, Mixed-mode oscillations and bifurcation analysis in a pituitary model, Nonlinear Dynam., 94 (2018), 807-826.  doi: 10.1007/s11071-018-4395-7.  Google Scholar

Figure 1.  (a) The one-parameter bifurcation versus $ I_{ext} $ in improved endocrine model. H is Hopf bifurcation point, $ {\rm LP}_i\ \ \left(i=1,\ 2\right) $ are fold bifurcation points. (b) The bifurcation diagram of $ k_0 $. LP is fold bifurcation point, $ {\rm H}_1 $ is Hopf bifurcation point, $ {\rm H}_2 $ is neutral saddle
Figure 2.  (a) The bifurcation diagram of ISIs with respect to $ I_{ext} $. (b) The bifurcation diagram of ISIs with respect to $ k_0 $
Figure 3.  (a) The inverse period-double bifurcation of ISIs over the range $ \left[0.234,0.265\right] $ of $ I_{ext} $. (b) The period-double bifurcation of ISIs over the range $ \left[0.49,0.53\right] $ of $ I_{ext} $. (c) The period-double bifurcation of ISIs over the range [0.006, 0.007] of $ k_0 $. (d), (e), and (f) The first (red) and second (blue) Lyapunov exponents $ \lambda_{1,2} $ corresponding to (a), (b), and (c) respectively
Figure 4.  The diagrams on the left are the firing patterns generated by $ I_{ext} $, and the diagrams on the right are the firing patterns only produced by induced current. (a) $ I_{ext}=-0.1 $, (c) $ I_{ext}=-0.03 $, (e) $ I_{ext}=0.21 $, (g) $ I_{ext}=0.2406 $, (b) $ k_0=0.0116 $, (d) $ k_0=0.0104 $, (f) $ k_0=0.0079 $, (g) $ k_0=0.00666. $
Figure 5.  The fast-slow analysis of fast subsystem under different external forcing currents. The dotted green curve is slow-nullcline for $ \dot{c}=0 $, the black and blue lines are stable and unstable equilibria. The trajectory of the system (2) (the blue curve) is superimposed. (a) $ I_{ext}=-0.1 $, (b) $ I_{ext}=-0.03 $, (c) $ I_{ext}=0.21 $, (d)$ I_{ext}=0.2406 $
$ {\rm H}_1 $ represent subcritical Hopf bifurcation. $ {\rm H}_i $ are neutral saddle, $ {\rm LP}_i $ represent fold bifurcation. HC represents the saddle homoclinic bifurcation. LPC is the limit point of cycle">Figure 6.  The diagram of the membrane potential sequence for $ I_{ext}=-0.5 $, $ V_{ml}=-27.5 $. (b) Fast-slow dynamics of"subHopf/homoclinic" bursting via the "fold/homoclinic" hysteresis loop. The point sub$ {\rm H}_1 $ represent subcritical Hopf bifurcation. $ {\rm H}_i $ are neutral saddle, $ {\rm LP}_i $ represent fold bifurcation. HC represents the saddle homoclinic bifurcation. LPC is the limit point of cycle
Figure 7.  Firing patterns of neuron model with variations of $ k_0 $. (a) $ k_0=0.0001 $, (b) $ k_0=0.007 $, (c) $ k_0=0.007408 $ (d) $ I_{ext}=0.00742 $(e) $ k_0=0.00747 $, (f) $ k_0=0.0085 $, (g) $ k_0=0.00866 $, (h) $ k_0=0.009 $
Figure 8.  ISIs bifurcation diagram with respect to $ k_0 $, $ I_{ext} $=0
Figure 9.  Firing patterns of neuron model with variations of $ I_{ext} $, $ k_0=0.01 $.(a) $ I_{ext}=0.075 $, (b) $ I_{ext}=0.75 $, (c) $ I_{ext}=0.85 $, (d) $ I_{ext}=0.895 $, (e) $ I_{ext}=0.9075 $, (f) $ I_{ext}=0.95 $
Figure 10.  ISIs bifurcation diagram with respect to $ I_{ext} $, $ k_0 $=0.01
Figure 11.  Firing patterns of neuron model with variations of $ I_{ext} $, $ k_0=0.02 $.(a) $ I_{ext}=1 $, (b) $ I_{ext}=1.04 $, (c) $ I_{ext}=1.08 $, (d) $ I_{ext}=1.14 $, (e) $ I_{ext}=1.2 $, (f) $ I_{ext}=1.6 $
Figure 12.  ISIs bifurcation diagram with respect to $ I_{ext} $, $ k_0 $=0.02
Figure 13.  Codimension-two bifurcation analysis of the neuron model. (a) Representation of the two-parameter bifurcation diagram in the $ \left(I_{ext},k_0\right) $-plane. (b)-(d) are the partial enlargement of the diagram (a). $ f_1 $ and $ f_2 $ are the fold bifurcation curves; $ h_1 $ and $ h_2 $ are Hopf bifurcation curves
Table 1.  Parameter values used in this paper
Parameter Value Parameter Value Parameter Value
$ f_c $ 0.0001 $ g_{Ca} $ 0.81nS $ k_{PMCA} $ 20$ s^{-1} $
$ d_{cell} $ $ 10\mu m $ $ g_{K\left(Ca\right)} $ 0.2nS $ \tau_n $ 0.03$ s^{-1} $
$ V_{ml} $ -22.5mV $ g_K $ 2.25nS $ \alpha $ 1
$ V_K $ -65mV $ k_0 $ 0.01 $ k_1 $ 1
$ V_{Ca} $ 0mV $ \beta $ 0.0001 $ k_2 $ 3
Parameter Value Parameter Value Parameter Value
$ f_c $ 0.0001 $ g_{Ca} $ 0.81nS $ k_{PMCA} $ 20$ s^{-1} $
$ d_{cell} $ $ 10\mu m $ $ g_{K\left(Ca\right)} $ 0.2nS $ \tau_n $ 0.03$ s^{-1} $
$ V_{ml} $ -22.5mV $ g_K $ 2.25nS $ \alpha $ 1
$ V_K $ -65mV $ k_0 $ 0.01 $ k_1 $ 1
$ V_{Ca} $ 0mV $ \beta $ 0.0001 $ k_2 $ 3
Table 2.  Data related to special points
Poins Parameter values ($ I_{ext}, k_{0} $) Eigenvalues ($ \lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4} $) Normal Form Parameter
$ CP_{1} $ (-64.649, -12.888) $ \lambda_{1}=0,\lambda_{2}=-19.1961, $ $ c=-4.21\times10^{-4} $
$ \lambda_{3}=-3.0007,\lambda_{4}=3447 $
$ CP_{2} $ (-66.356, -1.1266) $ \lambda_{1}=0,\lambda_{2}=-88.818+206.79i $ $ c=-3.22\times{10}^{-4} $
$ \lambda_{3}=-\ 88.818-206.79i,\lambda_{4}=-2.999 $,
$ CP_{3} $ (-78.42135, -4.6267) $ \lambda_{1}=0,\lambda_{2}=-2.9873 $, $ c=1.9\times{10}^{-4} $
$ \lambda_{3}=4.4263,\lambda_{4}=1.003963 $
$ CP_{4} $ (-23.9617, -0.5826) $ \lambda_{1}=0,\lambda_{2}=-29.6947 $, $ c=-2.33\times{10}^{-5} $
$ \lambda_{3}=193.632,\lambda_{4}=-3.0969 $
$ CP_{5} $ (-16.1428, 0.3545) $ \lambda_{1}=0,\lambda_{2}=-14.6374 $, $ c=3.66\times{10}^{-5} $
$ \lambda_{3}=-3.1904,\lambda_{4}=90.4468 $
$ CP_{6} $ (1.599659, 0.025901) $ \lambda_{1}=0,\lambda_{2}=-30.6756 $, $ c=9.29\times{10}^{-5} $
$ \lambda_{3}=-2.71,\lambda_{4}=2.0821 $
$ GH_{1} $ (-5.514, -0.07359) $ \lambda_1=1.0119i,\lambda_2=-1.0119i, $ $ l_1=36,03 $
$ \lambda_3=-3.73+4.19i,\lambda_4=-3.73-4.19i $
$ GH_{2} $ (-13.993, -0.22718) $ \lambda_1=70.688i,\lambda_2=-70.6880i, $ $ l_1=-1.7\times{10}^{-3} $
$ \lambda_3=-2.98418654,\lambda_4=-0.00208283 $
$ GH_{3} $ (1.98293, 0.33597) $ \lambda_1=0.09056i,\lambda_2=-0.09056i, $ $ l_1=-0.117 $
$ \lambda_3=-30.013089,\lambda_4=-2.371837 $
$ GH_{4} $ (-7.88796, -0.10746) $ \lambda_1=20.5112i,\lambda_2=-20.5112i, $ $ l_1=-7.8\times{10}^{-3} $
$ \lambda_3=-2.8259,\lambda_4=-0.016958 $
$ GH_{5} $ (-7.7194, -0.1145) $ \lambda_1=2.35556i,\lambda_2=-2.35556, $ $ l_1=-2.5\times{10}^5 $
$ \lambda_3=0.715+1.936i,\lambda_4=0.715-1.936i $
$ GH_{6} $ (-7.68801, -0.11345) $ \lambda_1=1.577914i,\lambda_2=-1.577914i, $ $ l_1=-4.4\times{10}^6 $
$ \lambda_3=0.582+3.039i,\lambda_4=0.582-3.039i $
$ GH_{7} $ (-67.9198, -1.7171) $ \lambda_1=214.4301i,\lambda_2=-214.4301i, $ $ l_1=-0.03122 $
$ \lambda_3=-2.999953,\lambda_4=-0.000808023 $
$ GH_{8} $ (-67.1444, -1.6382) $ \lambda_1=206.6605i,\lambda_2=-206.6605i, $ $ l_1=0.03811 $
$ \lambda_3=-2.9997,\lambda_4=-0.00073251 $
$ GH_{10} $ (-64.649, -12.888) $ \lambda_1=1.268\times{10}^{-7}i,\lambda_2=-1.268\times{10}^{-7}i, $ $ l_1=8.63\times{10}^{-3} $
$ \lambda_3=2.999109,\lambda_4=1394.19249 $
$ ZH_{1} $ (-73.3066, -6.1271) $ \lambda_1=209.65718i,\lambda_2=-209.65718i, $ $ (s,\theta,E_0)= $
$ \lambda_3=0,\lambda_4=-2.99981307 $ $ (1,-5866.3,-1) $
$ ZH_{2} $ (-67.4984, -1.6958) $ \lambda_1=212.9485i,\lambda_2=-212.9485i, $ $ (s,\theta,E_0)= $
$ \lambda_3=0,\lambda_4=2.999916 $ (-1, 4043.5, -1)
$ BT_{1} $ (-73.124, -6.294215) $ \lambda_1=0,\lambda_2=0, $ $ a=1.19\times{10}^{-3} $
$ \lambda_3=-3,\lambda_4=1420.03 $ $ b=1.097 $
$ BT_{2} $ (0.64939, 0.00913) $ \lambda_1=0,\lambda_2=0, $ $ a=4.54\times{10}^{-4} $
$ \lambda_3=-33.0915,\lambda_4=-2.7655 $ $ b=0.455 $
$ BT_{3} $ (-47.737, 1.87268) $ \lambda_1=0,\lambda_2=0, $ $ a=6.1\times{10}^{-3} $
$ \lambda_3=-2.7509,\lambda_4=416.481 $ $ b=6.147 $
$ HH_{1} $ (-7.4232, -0.10856) $ \lambda_1=3.700106i,\lambda_2=-3.700106i, $ $ (p_{11}p_{22},\vartheta,\delta)= $(1, -2, -2)
$ \lambda_3=1.3485054i,\lambda_4=-1.3485054i $ $ (\Theta,\Delta)= $(-50.2, 285)
$ NS_{1} $ (-72.904, -5.822) $ \lambda_1=0,\lambda_2=1285.27, $ None
$ \lambda_3=-3,\lambda_4=3 $
$ NS_{2} $ (-1.3217, -0.03665) $ \lambda_1=0,\lambda_2=-3.054, $ None
$ \lambda_3=-29.495,\lambda_4=29.495 $
$ NS_{3} $ (1.5887, 0.02568) $ \lambda_1=0,\lambda_2=-30.46293, $ None
$ \lambda_3=-2.74899,\lambda_4=2.74899 $
Poins Parameter values ($ I_{ext}, k_{0} $) Eigenvalues ($ \lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4} $) Normal Form Parameter
$ CP_{1} $ (-64.649, -12.888) $ \lambda_{1}=0,\lambda_{2}=-19.1961, $ $ c=-4.21\times10^{-4} $
$ \lambda_{3}=-3.0007,\lambda_{4}=3447 $
$ CP_{2} $ (-66.356, -1.1266) $ \lambda_{1}=0,\lambda_{2}=-88.818+206.79i $ $ c=-3.22\times{10}^{-4} $
$ \lambda_{3}=-\ 88.818-206.79i,\lambda_{4}=-2.999 $,
$ CP_{3} $ (-78.42135, -4.6267) $ \lambda_{1}=0,\lambda_{2}=-2.9873 $, $ c=1.9\times{10}^{-4} $
$ \lambda_{3}=4.4263,\lambda_{4}=1.003963 $
$ CP_{4} $ (-23.9617, -0.5826) $ \lambda_{1}=0,\lambda_{2}=-29.6947 $, $ c=-2.33\times{10}^{-5} $
$ \lambda_{3}=193.632,\lambda_{4}=-3.0969 $
$ CP_{5} $ (-16.1428, 0.3545) $ \lambda_{1}=0,\lambda_{2}=-14.6374 $, $ c=3.66\times{10}^{-5} $
$ \lambda_{3}=-3.1904,\lambda_{4}=90.4468 $
$ CP_{6} $ (1.599659, 0.025901) $ \lambda_{1}=0,\lambda_{2}=-30.6756 $, $ c=9.29\times{10}^{-5} $
$ \lambda_{3}=-2.71,\lambda_{4}=2.0821 $
$ GH_{1} $ (-5.514, -0.07359) $ \lambda_1=1.0119i,\lambda_2=-1.0119i, $ $ l_1=36,03 $
$ \lambda_3=-3.73+4.19i,\lambda_4=-3.73-4.19i $
$ GH_{2} $ (-13.993, -0.22718) $ \lambda_1=70.688i,\lambda_2=-70.6880i, $ $ l_1=-1.7\times{10}^{-3} $
$ \lambda_3=-2.98418654,\lambda_4=-0.00208283 $
$ GH_{3} $ (1.98293, 0.33597) $ \lambda_1=0.09056i,\lambda_2=-0.09056i, $ $ l_1=-0.117 $
$ \lambda_3=-30.013089,\lambda_4=-2.371837 $
$ GH_{4} $ (-7.88796, -0.10746) $ \lambda_1=20.5112i,\lambda_2=-20.5112i, $ $ l_1=-7.8\times{10}^{-3} $
$ \lambda_3=-2.8259,\lambda_4=-0.016958 $
$ GH_{5} $ (-7.7194, -0.1145) $ \lambda_1=2.35556i,\lambda_2=-2.35556, $ $ l_1=-2.5\times{10}^5 $
$ \lambda_3=0.715+1.936i,\lambda_4=0.715-1.936i $
$ GH_{6} $ (-7.68801, -0.11345) $ \lambda_1=1.577914i,\lambda_2=-1.577914i, $ $ l_1=-4.4\times{10}^6 $
$ \lambda_3=0.582+3.039i,\lambda_4=0.582-3.039i $
$ GH_{7} $ (-67.9198, -1.7171) $ \lambda_1=214.4301i,\lambda_2=-214.4301i, $ $ l_1=-0.03122 $
$ \lambda_3=-2.999953,\lambda_4=-0.000808023 $
$ GH_{8} $ (-67.1444, -1.6382) $ \lambda_1=206.6605i,\lambda_2=-206.6605i, $ $ l_1=0.03811 $
$ \lambda_3=-2.9997,\lambda_4=-0.00073251 $
$ GH_{10} $ (-64.649, -12.888) $ \lambda_1=1.268\times{10}^{-7}i,\lambda_2=-1.268\times{10}^{-7}i, $ $ l_1=8.63\times{10}^{-3} $
$ \lambda_3=2.999109,\lambda_4=1394.19249 $
$ ZH_{1} $ (-73.3066, -6.1271) $ \lambda_1=209.65718i,\lambda_2=-209.65718i, $ $ (s,\theta,E_0)= $
$ \lambda_3=0,\lambda_4=-2.99981307 $ $ (1,-5866.3,-1) $
$ ZH_{2} $ (-67.4984, -1.6958) $ \lambda_1=212.9485i,\lambda_2=-212.9485i, $ $ (s,\theta,E_0)= $
$ \lambda_3=0,\lambda_4=2.999916 $ (-1, 4043.5, -1)
$ BT_{1} $ (-73.124, -6.294215) $ \lambda_1=0,\lambda_2=0, $ $ a=1.19\times{10}^{-3} $
$ \lambda_3=-3,\lambda_4=1420.03 $ $ b=1.097 $
$ BT_{2} $ (0.64939, 0.00913) $ \lambda_1=0,\lambda_2=0, $ $ a=4.54\times{10}^{-4} $
$ \lambda_3=-33.0915,\lambda_4=-2.7655 $ $ b=0.455 $
$ BT_{3} $ (-47.737, 1.87268) $ \lambda_1=0,\lambda_2=0, $ $ a=6.1\times{10}^{-3} $
$ \lambda_3=-2.7509,\lambda_4=416.481 $ $ b=6.147 $
$ HH_{1} $ (-7.4232, -0.10856) $ \lambda_1=3.700106i,\lambda_2=-3.700106i, $ $ (p_{11}p_{22},\vartheta,\delta)= $(1, -2, -2)
$ \lambda_3=1.3485054i,\lambda_4=-1.3485054i $ $ (\Theta,\Delta)= $(-50.2, 285)
$ NS_{1} $ (-72.904, -5.822) $ \lambda_1=0,\lambda_2=1285.27, $ None
$ \lambda_3=-3,\lambda_4=3 $
$ NS_{2} $ (-1.3217, -0.03665) $ \lambda_1=0,\lambda_2=-3.054, $ None
$ \lambda_3=-29.495,\lambda_4=29.495 $
$ NS_{3} $ (1.5887, 0.02568) $ \lambda_1=0,\lambda_2=-30.46293, $ None
$ \lambda_3=-2.74899,\lambda_4=2.74899 $
[1]

Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123

[2]

Zhuoqin Yang, Tingting Guan. Bifurcation analysis of complex bursting induced by two different time-scale slow variables. Conference Publications, 2011, 2011 (Special) : 1440-1447. doi: 10.3934/proc.2011.2011.1440

[3]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112

[4]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219

[5]

Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021097

[6]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603

[7]

Lixia Duan, Zhuoqin Yang, Shenquan Liu, Dunwei Gong. Bursting and two-parameter bifurcation in the Chay neuronal model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 445-456. doi: 10.3934/dcdsb.2011.16.445

[8]

Younghae Do, Juan M. Lopez. Slow passage through multiple bifurcation points. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 95-107. doi: 10.3934/dcdsb.2013.18.95

[9]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547

[10]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233

[11]

Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209

[12]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021

[13]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[14]

Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171

[15]

Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069

[16]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[17]

Feng Zhang, Wei Zhang, Pan Meng, Jianzhong Su. Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 637-651. doi: 10.3934/dcdsb.2011.16.637

[18]

Lixia Duan, Dehong Zhai, Qishao Lu. Bifurcation and bursting in Morris-Lecar model for class I and class II excitability. Conference Publications, 2011, 2011 (Special) : 391-399. doi: 10.3934/proc.2011.2011.391

[19]

Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289

[20]

Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621

 Impact Factor: 0.263

Article outline

Figures and Tables

[Back to Top]