# American Institute of Mathematical Sciences

doi: 10.3934/era.2021023

## Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

 1 School of Mathematics and Science, Henan Institute of Science and Technology, Xinxiang 453003, China 2 School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

* Corresponding author: Huaguang Gu (guhuaguang@tongji.edu.cn)

Received  December 2020 Revised  February 2021 Published  March 2021

Fund Project: This work is supported by National Science Foundation of China (Grant Nos. 11872276 and 11572225) and Research Project of Henan province postdoctoral (No. 19030095)

Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution ($I_h$) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without $I_h$ current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

Citation: Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, doi: 10.3934/era.2021023
##### References:
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##### References:
 [1] G. A. Ascoli, S. Gasparini, V. Medinilla and M. Migliore, Local control of postinhibitory rebound spiking in CA1 pyramidal neuron dendrites, J. Neurosci., 30 (2010), 6434-6442.  doi: 10.1523/JNEUROSCI.4066-09.2010.  Google Scholar [2] A. Basu, C. Petre and P. E. Hasler, Dynamics and bifurcations in a silicon neuron, IEEE Trans. Biomed. Circuits Syst., 4 (2010), 320-328.  doi: 10.1109/TBCAS.2010.2051224.  Google Scholar [3] S. Bertrand and J. Cazalets, Postinhibitory rebound during locomotor-like activity in neonatal rat motoneurons in vitro, J. Neurophysiol., 79 (1998), 342-351.   Google Scholar [4] P. Channell, G. Cymbalyuk and A. Shilnikov, Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Lett., 98 (2007), 134101. doi: 10.1103/PhysRevLett.98.134101.  Google Scholar [5] L. Duan, Q. Cao, Z. Wang and J. Su, Dynamics of neurons in the pre-Bötzinger complex under magnetic flow effect, Nonlinear Dyn., 94 (2018), 1961-1971.  doi: 10.1007/s11071-018-4468-7.  Google Scholar [6] L. Duan, Z. Yang, S. Liu and and D. Gong, Bursting and two-parameter bifurcation in the Chay neuronal model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 445-456.  doi: 10.3934/dcdsb.2011.16.445.  Google Scholar [7] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems. A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.  Google Scholar [8] D. Fan, Y. Zheng, Z. Yang and Q. Wang, Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit, Appl. Math. Mech-Engl., 41 (2020), 1287-1302.   Google Scholar [9] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., 17 (1955), 257-278.  doi: 10.1007/BF02477753.  Google Scholar [10] L. Glass, Synchronization and rhythmic processes in physiology, Nature, 410 (2001), 277-284.  doi: 10.1038/35065745.  Google Scholar [11] J.-M. Goaillard, A. L. Taylor, S. R. Pulver and E. Marder, Slow and persistent postinhibitory rebound acts as an intrinsic short-term memory mechanism, J. Neurosci., 30 (2010), 4687-4692.  doi: 10.1523/JNEUROSCI.2998-09.2010.  Google Scholar [12] L. Guan, B. Jia and H. Gu, A novel threshold across which negative stimulation evokes action potential near a saddle-node bifurcation in a neuronal model with $I_h$ current, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950198, 26 pp. doi: 10.1142/S0218127419501980.  Google Scholar [13] J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851-872.  doi: 10.3934/dcdss.2009.2.851.  Google Scholar [14] F. Han, B. Zhen, Y. Du, Y. Zheng and M. Wiercigroch, Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 457-474.  doi: 10.3934/dcdsb.2011.16.457.  Google Scholar [15] A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, J. Physiol., 107 (1948), 165-181.  doi: 10.1113/jphysiol.1948.sp004260.  Google Scholar [16] 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 [17] E. M. Izhikevich, Which model to use for cortical spiking neurons?, IEEE T. Neural Network, 15 (2004), 1063-1070.   Google Scholar [18] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT press, Cambridge, 2007.   Google Scholar [19] P. Jiang, X. Yang and Z. Sun, Dynamics analysis of the hippocampal neuronal model subjected to cholinergic action related with alzheimer's disease, Cogn. Neurodyn., 14 (2020), 483-500.  doi: 10.1007/s11571-020-09586-6.  Google Scholar [20] T. Malashchenko, A. Shilnikov and G. Cymbalyuk, Bistability of bursting and silence regimes in a model of a leech heart interneuron, Phys. Rev. E, 84 (2011), 041910. doi: 10.1103/PhysRevE.84.041910.  Google Scholar [21] Y. Mao, Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.  Google Scholar [22] J. Mitry, M. McCarthy, N. Kopell and M. Wechselberger, Excitable neurons, firing threshold manifolds and canards, J. Math. Neurosci., 3 (2013), Art. 12, 32 pp. doi: 10.1186/2190-8567-3-12.  Google Scholar [23] J. Rinzel and G. Ermentrout, Analysis of neural excitability and oscillations, in Methods in Neuronal Modeling (eds. C. Koch and I. Segev), MIT Press, Cambridge, (1998), Chap. 7,251–291. Google Scholar [24] M. Rush and J. Rinzel, The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models, Bull. Math. Biol., 57 (1995), 899-929.   Google Scholar [25] Z. Song and J. Xu, Codimension-two bursting analysis in the delayed neural system with external stimulations, Nonlinear Dyn., 67 (2012), 309-328.  doi: 10.1007/s11071-011-9979-4.  Google Scholar [26] V. A. Straub and P. Benjamin, Extrinsic modulation and motor pattern generation in a feeding network: A cellular study, J. Neurosci., 21 (2001), 1767-1778.  doi: 10.1523/JNEUROSCI.21-05-01767.2001.  Google Scholar [27] X. Sun and G. Li, Synchronization transitions induced by partial time delay in an excitatory-inhibitory coupled neuronal network, Nonlinear Dyn., 89 (2017), 2509-2520.  doi: 10.1007/s11071-017-3600-4.  Google Scholar [28] A. Tonnelier, Threshold curve for the excitability of bidimensional spiking neurons, Phys. Rev. E, 90(2014), 022701. doi: 10.1103/PhysRevE.90.022701.  Google Scholar [29] F. Q. Wu and H. Gu, Bifurcations of negative responses to positive feedback current mediated by memristor in neuron model with bursting patterns, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2030009, 22 pp. doi: 10.1142/S0218127420300098.  Google Scholar [30] F. Wu, H. Gu and Y. Li, Inhibitory electromagnetic induction current induced enhancement instead of reduction of neural bursting activities, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104924, 15 pp. doi: 10.1016/j.cnsns.2019.104924.  Google Scholar [31] X. Zhang, H. Gu and L. Guan, Stochastic dynamics of conduction failure of action potential along nerve fiber with hopf bifurcation, Sci. China Technol. Sci., 62 (2019), 1502-1511.  doi: 10.1007/s11431-018-9515-4.  Google Scholar [32] Y. Zhang, Y. Xu, Z. Yao and J. Ma, A feasible neuron for estimating the magnetic field effect, Nonlinear Dyn., 102 (2020), 1849-1867.   Google Scholar [33] F. Zhang, W. Zhang, P. Meng and J. Z. Su, Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 637-651.  doi: 10.3934/dcdsb.2011.16.637.  Google Scholar [34] Z. Zhao and H. Gu, Transitions between classes of neuronal excitability and bifurcations induced by autapse, Sci. Rep., 7 (2017), 6760. doi: 10.1038/s41598-017-07051-9.  Google Scholar [35] Z. Zhao, L. Li and H. Gu, Dynamical mechanism of hyperpolarization-activated non-specific cation current induced resonance and spike-timing precision in a neuronal model, Front. Cell. Neurosci., 12 (2018), 62. doi: 10.3389/fncel.2018.00062.  Google Scholar
The bifurcation diagram, firing period and firing frequency with respect to $u$ at $c$ = $-$0.55. (a)The bifurcation diagram. The $S$-shaped blue line is formed by the equilibria, where the lower branch (thick solid blue line) denotes the stable node, the middle branch (short dashed blue line) represents the saddle, and the upper branch is composed of unstable focus (long dashed blue line) and stable focus (thin solid blue line). The symbol SN (black solid circle, $u_{SN}\approx -1.02$) is for a saddle-node bifurcation and H (black hollow circle) for a Hopf bifurcation. The red solid curves, labeled as Vmax and Vmin, denote the maximal and minimal value of the stable limit cycle, respectively. BHom (black arrow, $u_{BHom}$ $\approx$ $-$1.099400401984) represents a bifurcation of BHom orbit. The inset shows the dynamics at the red arrow, which are the stable node (blue solid line), the extreme values of stable limit cycle (green solid line), a saddle-node bifurcation point (red solid circle), a Hopf bifurcation point (red hollow circle) and a common homoclinic orbit bifurcaction (half-filled circle); (b) Firing period (blue) and firing frequency (red) for $u$ near $u_{BHom}$ (black arrow)
The dynamics related to BHom orbit (black) at $u$ = $u_{BHom}$. (a) The phase trajectory of BHom orbit (black loop with anti-clockwise arrow), the nullclines $dV/dt$ = 0 (red curve) and $dw/dt$ = 0 (blue curve), the stable node (solid red circle), saddle (half-filled circle), and unstable focus (hollow red circle); (b) The enlargement of (a) near the saddle and the stable (solid green) and unstable (dashed green) manifolds of the saddle. The black arrow indicates the direction for both the trajectory and manifolds of saddle
Mono-stable node at $u$ = $-$1.12. (a) The changes of the potential of the steady state with respect to time; (b) The threshold curve in phase plane ($V$, $w$), i.e, the border between the subthreshold area (yellow) and suprathreshold area (white), and the equilibria. The solid, half-filled, hollow circle denotes the stable node, saddle, and stable focus, respectively
The threshold curve, manifolds of the saddle, and the response of the stable node to excitatory stimulation with strength $A$ = 0.29 (red) and $0.3$ (black) at $u$ = $-$ 1.12. (a)The time series of the responses (upper) and the stimulation (lower). The purple dashed line corresponds to the membrane potential of saddle; (b)The responses in phase plane ($V$, $w$) together with the threshold curve, i.e, the border between the subthreshold area (yellow) and suprathreshold area (white). The solid, half-filled, hollow circle denotes the stable node, saddle, and stable focus, respectively; (c)The details of the trajectories between the stable node and saddle in (b); (d) The enlargement of (b) near the saddle
The threshold curve, manifolds of the saddle, and the response of the stable node to inhibitory stimulations at $u$ = $-$1.12. (a) The responses of membrane potentials (upper) to pulse stimulations (lower) with strength $A = -0.5$ (blue), $-0.65$ (red), $-0.66$ (black), and $-0.8$ (green); (b) The phase trajectories of the responses for $A$ = $-$0.65 (red), $-$0.66 (black) and the threshold curve, i.e., the border between the subthreshold area (yellow) and suprathreshold area (white). The stable node is denoted by solid red circle, the saddle by half-filled circle, and the unstable focus by hollow red circle. The hollow square denotes the termination phase of the stimulation; (c)The enlargement of (b) near the squares, i.e., the termination phase of the stimulation; (d)The enlargement of (b) near saddle. The solid and dashed green line represents the stable and unstable manifold of saddle
Coexistence of stable node and stable limit cycle at $u$ = $-$1.08. (a) The changes of the resting potential with respect to time (upper) and spiking behavior (lower); (b) Phase portraits of the equilibrium points and limit cycle and their attraction domains. The red line and blue line are for nullclines $dV/dt$ = 0 and $dw/dt$ = 0, respectively. Their intersection points are stable node (solid red circle), saddle (half-filled circle), and unstable focus (hollow red circle). The black curve denotes the stable limit cycle running along the black arrow. The yellow and white regions represent the attraction domain of stable node and stable limit cycle, respectively; (c) The enlargement of (b) near the saddle point. The solid (dashed) green line represents the stable (unstable) manifold of the saddle
Coexistence of stable node and stable limit cycle at $u$ = $-$1.08. (a) The changes of the resting potential with respect to time (upper) and spiking behavior (lower); (b) Phase portraits of the equilibrium points and limit cycle and their attraction domains. The red line and blue line are for nullclines $dV/dt$ = 0 and $dw/dt$ = 0, respectively. Their intersection points are stable node (solid red circle), saddle (half-filled circle), and unstable focus (hollow red circle). The black curve denotes the stable limit cycle running along the black arrow. The yellow and white regions represent the attraction domains of stable node and stable limit cycle, respectively; (c)The details of the trajectories between the stable node and saddle in (b); (d) The enlargement of (b) near the saddle point. The solid (dashed) green line represents the stable (unstable) manifold of the saddle
The threshold curve, manifolds of the saddle, and the responses of the resting state to inhibitory stimulations with strength $A$ = $-$0.51 (red) and $-$0.52 (black) at $u$ = $-$1.08. (a) The responses of membrane potential (upper) to pulse stimulations (lower); (b) The phase trajectories of the responses with the threshold curve, i.e., the border between the yellow and white area, which represent the attraction domain of the stable node and limit cycle respectively. The stable node is denoted by solid red circle, the saddle by half-filled circle, and the unstable focus by hollow red circle. The hollow square denotes the termination phase of the stimulation. The thick magenta curve is for the stable limit cycle; (c) The enlargement of (b) near the termination phase of the stimulation; (d) The enlargement of (b) near saddle. The solid (dashed) green line represents the stable (unstable) manifold of saddle
The bifurcation diagram, firing period and firing frequency with respect to $u$ at $c$ = $-$0.4. (a)The bifurcation diagram. The equilibria form the $S$-shaped blue line, where the lower branch (LB) of the $S$-shaped line is consisted by stable focus (thick blue line) and unstable focus (dashed line), the middle branch (MB) is saddle (short dashed line), and the upper branch (UB) is formed by unstable focus (long dashed blue line) and stable focus (thin solid blue line). The symbol $\rm{H_1}$, $\rm{H_2}$ (black squares), and $\rm{H_3}$ (black circle) denote Hopf bifurcation points. The symbol SN is for a saddle-node bifurcation. The green solid line represents the extreme of the small stable limit cycle. The extreme of the big stable limit cycle is shown by red curve and the maximal (minimal) value is denoted by Vmax (Vmin). The symbol BHom denotes a BHom bifurcation at $u_{BHom}$ $\approx$ $-$0.99447689769051 (black arrow);(b) The firing period (blue) and firing frequency (red) for $u$ near $u_{BHom}$ (black arrow).
The dynamics related to BHom orbit (black) at $u$ = $u_{BHom}$ for $c$ = $-$0.4. (a) The phase trajectory of BHom orbit (black) and equilibrium points. The red line and blue line are for nullclines $dV/dt$ = 0 and $dw/dt$ = 0, respectively. Their intersection points are stable focus (solid red circle), saddle (half-filled circle), and unstable focus (hollow red circle); (b) The details of (a) near saddle point and manifolds of the saddle. The solid (dashed) green line represents the stable (unstable) manifold of saddle. The black arrow indicates the direction for both the trajectory and manifolds of saddle
The threshold curve, manifolds of the saddle, and the response of the stable focus to excitatory pulse current with strength $A$ = 0.57 (red) and 0.58 (black) at $u$ = $-$1.03. (a)The changes of the pulse current (lower) and the corresponding responses (upper) with respect to time. The purple dashed line corresponds to the membrane potential of saddle; (b) the phase trajectories of responses together with the threshold curve, i.e., the border between subthreshold area (yellow) and suprathreshold area (white). The solid, half-filled, hollow circle denotes the stable focus, saddle, and unstable focus respectively. The hollow square denotes the phase point at the end of the stimulation; (c) The enlargement of the trajectories in (b) between the stable focus and saddle; (d) The details near saddle in (b). The solid (dashed) green line represents the stable (unstable) manifold of saddle
The threshold curve, manifolds of the saddle, and the response of the steady state (focus) to inhibitory stimulations with strength $A$ = $-$0.59 (red) and $A$ = $-$0.6 (black) at $u$ = $-$1.03. (a) Inhibitory stimulation (lower) and their evoked membrane potential (upper). The purple dashed line corresponds to constant membrane potential of saddle; (b) The phase trajectory of the response together with threshold curve, i.e., the border between the subthreshold area (yellow) and suprathreshold area (white). The solid circle is for the stable node, half-filled circle for saddle, and hollow circle for unstable focus. The hollow square denotes the termination of the stimulation; (c) The details around the hollow squares in (b); (d) The details around the saddle in (b). The solid (dashed) grey line represents the stable (unstable) manifold of the saddle
Coexistence of sable focus and stable limit cycle at $u$ = $-$0.96. (a) The membrane potential of steady state (upper) and spiking(lower); (b) Phase portraits of the equilibrium points and limit cycle. The red line and blue line are for nullclines $dV/dt$ = 0 and $dw/dt$ = 0, respectively. Their intersection points are stable focus (solid red circle), saddle (half-filled circle), and unstable focus (hollow red circle). The black curve denotes the stable limit cycle and black arrow indicates its running direction. The yellow region is responsible for the attraction domain of stable focus, and the white part for the attraction domain of stable limit cycle; (c) The enlargement of (b) near saddle point (half-filled circle), where the solid (dashed) green line represents the stable (unstable) manifold of saddle
The threshold curve, manifolds of the saddle, and the response of the stable node to excitatory pulse current with strength $A$ = 0.47 (red) and 0.48(black) at $u$ = $-$0.96. (a) The time series of the pulse current (lower) and the responses (upper). The purple dashed line represents the membrane potential of the saddle. (b) The responses in phase plane ($V$, $w$) together with the threshold curve, i.e., the border between the attraction domain of stable focus (yellow) and stable limit cycle (white). The solid, half-filled hollow circle denotes the stable node, saddle, and stable focus, respectively. The thick magenta curve represents the stable limit cycle; (c) The details of the trajectories in (b) between stable focus and the saddle; (d) The enlargement of (b) near the saddle. The solid (dashed) green line represents the stable (unstable) manifold of saddle
The responses of the resting state to inhibitory stimulations with strength $A$ = $-$0.4(red) and $-$0.41 (black) at $u$ = $-$0.96, and the relationships to the threshold curve and the manifolds of the saddle. (a) Inhibitory stimulation (lower) and their evoked membrane potential (upper). The cyan dashed line corresponds to constant membrane potential of saddle; (b) Phase trajectory of the responses and the threshold curve, i.e., the border between yellow and white area. Any initial value from the yellow area induces subthreshold potential and from the white area induces periodic spiking. The solid circle is for the stable node, half-filled circle for saddle, hollow circle for unstable focus, the magenta thick line for the stable limit cycle. The hollow square denotes the phase point at the termination of stimulation; (c) The details around the hollow squares in (b); (d) The details around the saddle in (b). The solid (dashed) grey line represents the stable (unstable) manifold of saddle
The bifurcations with respect to $u$ at different $c$ values. The equilibria form the $S$-shaped blue line. The red line is responsible for the extreme values of the stable limit cycle. (a)$c$ = $-$0.45. The lower branch (LB) of the $S$-shaped line is consisted by stable node (thick blue line) and stable focus (thin blue line), the middle branch (MB) is saddle (short dashed line), and the upper branch (UB) is formed by unstable focus (long dashed blue line) and stable focus (thin solid blue line). The symbol SN (red circle), $\rm{H_{UB}}$ (black circle) and BHom (black arrow) denote the saddle-node bifurcation point, Hopf bifurcation point, and big homoclinic bifurcation point respectively; (b) $c$ = $-$0.615. LB is consisted by stable node (thick blue line), MB is saddle (short dashed line), and UB is formed by unstable focus (long dashed blue line) and stable focus (thin solid blue line). The symbols $\rm{H_{UB}}$(black circle) and SNIC (black arrow) denote the Hopf bifurcation and saddle-node bifurcation on invariant circle, respectively
The double-parameter bifurcation diagram in ($u$, $c$) plane. The 5 codimension-1 bifurcation curves are for the saddle-node bifurcation on an invariant cycle (SNIC, magenta line), the saddle-node bifurcation (SN, blue), the big homoclinic orbit (BHom, black), the Hopf bifurcation on LB ($\rm {H_{LB}}$, dashed red line), and the Hopf bifurcation on UB ($\rm {H_{UB}}$, solid red). The green curve (NF) represents the transition from stable node to stable focus on LB. The SNHO represents a codimension-2 bifurcation point (cyan star), a saddle-node Homoclinic orbit bifurcation point, which is the intersection point between the SN, SNIC, and BHom curves. The ($u, c$) plane is divided into region Ⅰ (orange), Ⅱ (gray), Ⅲ (green), Ⅳ (yellow), Ⅴ (white), Ⅵ (pink), and Ⅶ (cyan), respectively, which corresponds to the mono-stable focus, mono-stable node, the coexistence of two stable limit cycles, coexistence of stable focus on LB and stable limit cycle, coexistence of stable node and stable limit cycle, stable limit cycle, depolarization block, respectively. The stable equilibrium in the region marked by black star ($*$) can elicit PIR spike, marked by red star ($*$) can generate PIR spiking
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