
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 (Ih) 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 Ih 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[J]. Electronic Research Archive, 2021, 29(5): 2987-3015. doi: 10.3934/era.2021023
[1] | 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, 2021, 29(5): 2987-3015. doi: 10.3934/era.2021023 |
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[6] | Jun Pan, Haijun Wang, Feiyu Hu . Revealing asymmetric homoclinic and heteroclinic orbits. Electronic Research Archive, 2025, 33(3): 1337-1350. doi: 10.3934/era.2025061 |
[7] | Zhihui Wang, Yanying Yang, Lixia Duan . Dynamic mechanism of epileptic seizures induced by excitatory pyramidal neuronal population. Electronic Research Archive, 2023, 31(8): 4427-4442. doi: 10.3934/era.2023226 |
[8] | Haijun Wang, Jun Pan, Guiyao Ke . Multitudinous potential homoclinic and heteroclinic orbits seized. Electronic Research Archive, 2024, 32(2): 1003-1016. doi: 10.3934/era.2024049 |
[9] | Moutian Liu, Lixia Duan . In-phase and anti-phase spikes synchronization within mixed Bursters of the pre-Bözinger complex. Electronic Research Archive, 2022, 30(3): 961-977. doi: 10.3934/era.2022050 |
[10] | Qixiang Wen, Shenquan Liu, Bo Lu . Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, 2021, 29(5): 3205-3226. doi: 10.3934/era.2021034 |
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 (Ih) 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 Ih 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.
Nonlinear dynamics has played very important roles in identifying dynamics of neural electrical activities [10,29,32], which are involved in the information processing, locomotion control, cognitive functions, and brain disease [8,19,31]. These activities mainly include the steady state such as the resting state and the firing behavior composed of action potentials or spikes, for example, the spiking and bursting, which has been widely investigated with the conceptions of bifurcations and chaos [6,16,18,23,33]. For example, by modulation to some physiological parameters, the steady state changes to firing behavior via the subcritical or supercritical Hopf bifurcation point, or the saddle-node bifurcation on an invariant cycle (SNIC), or the saddle-node bifurcation (SN) point, or the firing behavior changes to the steady state via the supercritical Hopf bifurcation point, or the fold bifurcation of limit cycle, or the SNIC, or the bifurcation of homoclinic (Hom) orbit [16,18]. The Hom bifurcation usually contains the bifurcation of big homoclinic orbit (BHom) and small homoclinic orbit(SHom, i.e., the common homoclinic orbit), which are named here according to the size of homoclinic loop. In addition, the SNIC, or SN, or Hom are built relationship to type Ⅰ excitability, which means that the firing generates with nearly zero frequency, and the Hopf bifurcation to type Ⅱ excitability, which corresponds to the firing with a nearly fixed frequency [16,18]. All these show that nonlinear dynamics is effective to identify the dynamics and physiological functions of the neural electronic behaviors.
The dynamics for an action potential or spike evoked by external stimulation from steady state has been an important issue in both neuroscience and nonlinear dynamics [15,16,17], which has been related to the nonlinear concept, threshold [15,16,17]. Well known that, a positive, or called excitatory, or called depolarization stimulation pulse with suprathreshold strength can induce membrane potential increased to reach a voltage threshold to elicit an action potential while with subthreshold strength can not evoke a spike but subthreshold membrane potential. In nonlinear dynamics, the threshold has been well explained with the trajectory or manifold in the phase space combined with the bifurcations as well as the types of excitability. For example, Fig. 14 in Ref. [16] describes how the action potential is evoked by excitatory stimulation. For a stable node near SNIC with type Ⅰ excitability [16,17,23], the threshold curve corresponds to the stable manifold of the saddle, which is right to the stable node and has a positive slope, i.e. the membrane potential of the saddle is larger than the stable node. The suprathreshold stimulation induces the membrane potential or phase trajectory increased to run across the stable manifold of the saddle to form an action potential, as depicted in Fig. 14 (left) in Ref. [16]. For the stable focus near Hopf bifurcation with type Ⅱ excitability [16,17,23], the threshold sets locate on three sides of stable focus, left, bottom, and right sides, as depicted in Fig. 14 (right) in Ref. [16]. Similarly to that of SNIC, it is also the part right to the stable focus that acts the threshold for an action potential induced by the excitatory stimulation. Generally speaking, for the excitatory stimulation, it is the right part of threshold curve or sets, which locates right to the steady state, are responsible for the generation of an action potential. In Ref. [28], the threshold curves for multiple two-dimensional models are acquired, however, the bifurcations for these models are not provided. The threshold curve for other bifurcations such as Hom awaits to be studied.
To the contrary, for the negative (or called inhibitory or hyperpolarization) stimulation, the left part of the threshold sets for the Hopf bifurcation is very important for the generation of an action potential, i.e. the post-inhibitory rebound (PIR) spike, which is depicted in Fig. 14 (right) of Ref. [16]. Therein the inhibitory stimulation induces the trajectory to run across the left part of the threshold sets, and then the trajectory locates outside of the threshold sets and rotates in anti-clockwise. After the trajectory approaches the right part of the threshold sets, the membrane potential increases to form an action potential. Therefore, it is the left part of threshold sets, locating left to the stable focus, are responsible for the generation of PIR spike. The PIR spike presents that inhibitory stimulation can enhance neural firing activity [2], which is different from the traditional viewpoint that the inhibitory modulations always suppress the electronic activity. It has been widely accepted that the PIR spike can be evoked from focus near the Hopf bifurcation point with type Ⅱ excitability instead of node near the SNIC with type Ⅰ excitability [16], due to the absence of the part of threshold curve left to the steady state for type Ⅰ excitability. In neurophysiology, the PIR spike has been build relationship to special ionic currents, mainly the hyperpolarization-active caution (
However, the condition for the PIR spike is extended to the SNIC in system with
In the present paper, the threshold curve and PIR spike for the BHom bifurcation are studied in a planar dynamical system without
The rest of the paper is organized as follows. Section 2 shows the model and methods. The main results are displayed in Section 3. Section 4 gives the conclusions and discussions.
The FitzHugh-Nagumo (FHN) model was created by reducing the Hodgkin-Huxley model to two dimensions. This reduced model successfully imitates the generation of action potential and therefore attracts a lot of attention, and has been studied by analytical, numerical and experimental methods [13,21,14]. In the present paper, we study the action potential induced by pulse current with a modified FHN model, which is Eq. (15) in Ref. [16] and reads as
(1){dVdt=V−V3/3−w,dwdt=ϵ(−u+V−s(w)), |
where the variable
In the present paper, the threshold curve is acquired to study the action potential evoked from the steady state. The
In the present paper, to study how the spike or PIR spike is evoked, we adopt a single pulse stimulation
Fourth-order Runge-Kutta method is utilized to integrate the FHN model with time step
The results are divided into 3 parts. Firstly, a novel threshold curve for big homoclinic (BHom) bifurcation related to the stable node is acquired. The parameter value
The bifurcation diagram of equilibrium point (blue) and stable limit cycle (red) is shown in Fig. 1(a). Here the
As
At the bifurcation point
For
A pulse stimulation with
The generation of a spike (black curve) for
Two important characteristics can be found from Fig. 4(d). One is that the stable manifold of the saddle coincides exactly with the part of the threshold curve right to the stable node, and the other is that the trajectory running across the threshold curve results in an action potential. In addition, the unstable manifold (dashed green) drives the trajectory to move to right to form the action potential. Therefore, the dynamics of saddle, not only the stable manifold, but also the unstable manifold, play important roles to evoke a spike. The stable manifold of the saddle locating right to the stable node acts as the threshold curve to evoke an action potential from the stable node, which resembles that of the SNIC [16]. However, the threshold curve for the Hopf bifurcation has no direct relationship to the dynamics of the saddle. Therefore, the threshold curve for the BHom differs from that of Hopf bifurcation. Considering that the threshold curve is also different from that for the SNIC, the threshold curve for the BHom is novel.
The responses (upper panel) of the steady state to inhibitory pulse stimulations (lower panel) with strength
Different from the excitatory stimulation, the membrane potential decreases during the stimulation. The stronger the stimulation strength, the more negative the membrane potential at the termination time of the stimulation, as shown in Fig. 5(a). For weaker stimulations such as
The trajectories corresponding to Fig. 5(a) (
After the stimulation, the red trajectory locates within the yellow area and very close to the threshold curve, approaches the left unstable manifold (dashed green line) of the saddle (half-filled circle) along the red arrow, then goes away from the saddle along the left unstable manifold, at last returns to the steady state (red solid circle). Different from the red trajectory, the black trajectory after the stimulation (square) locates within the blanket area and very close to the threshold curve first, as shown in Fig. 5(b) and (c), after approaching the saddle, exhibits dynamics resembling that of the excitatory stimulation to form action potential. The stable and unstable manifolds of the saddle still play roles in the formation of the PIR spike, as shown in Fig. 5(d).
As shown in Fig. 1(a), when
As depicted in Fig. 6(b), the attraction domain of the stable node (red solid circle) is marked with yellow, and that of the stable limit cycle (black curve) is marked with blanket. That is to say, any initial value in yellow region induces FHN model (Eq. (1)) to generate subthreshold behavior, which converge to the stable node, and that in blanket region results in spiking behavior, which approaches the stable limit cycle. Therefore, the border between the yellow and white region has the role to separate subthreshold behavior and spiking behavior, which resembles that of mono-stable node shown in Fig. 3(b), where the border separates subthreshold behavior and suprathreshold (spike) behavior. The details of Fig. 6(b) around the saddle (half-filled circle) are enlarged in Fig. 6(c), where the solid (dashed) green line represents the stable (unstable) manifold of saddle (half-filled circle). There one striking characteristic is that the stable manifold (solid green line) of the saddle coincides with the border between the yellow and blanket regions, which resembles that shown in Fig. 4(d). Besides, the stable node shown in Fig. 6(b) is surrounded by the white region from left, below, and right side, which resembles to that for the mono-stable node shown in Fig. 3(b). These geometrical characteristics imply that the results for the coexisted node maybe resemble that of the mono-stable node.
For the stable node coexisting with spiking at
For the stable node coexisting with spiking at
As shown in Fig. 8(c), during the stimulation, the trajectory for suprathreshold stimulation (
In addition, from the geometrical viewpoint, the threshold curve for spiking induced by excitatory stimulation (Fig. 7) and inhibitory stimulation (Fig. 8) can be attributed to that the stable node is surrounded all around by white area (the attraction domain of the stable limit cycle), which provides the chance that the stimulation drives the trajectory to transit from the yellow area (the attraction domain of the stable node) to the white area, i.e., to run across the border to form spiking. From viewpoint of the dynamics, it is the stimulation-induced transition from stable node to stable limit cycle.
In this subsection,
Shown in Fig. 9(a) is the bifurcation diagram with respect to
The BHom orbit (black circle) at
When
The phase trajectories of above two responses together with the threshold curve (i.e., the border between yellow and white region) are shown in Fig. 11(b). Here the yellow (white) region is for subthreshold (suprathreshold) area. The enlargement of dynamical behaviors between the stable focus (red solid circle) and saddle (half-filled circle) is depicted in Fig. 11(c), the details near the saddle (half-filled circle) are further enlarged in Fig. 11(d), where the hollow square is for the phase point at the termination time of stimulation. For
In addition, although the result for the focus depicted in Fig. 11(d) in the present paper and Fig. 14 (right) in Ref. [16] seems similar with each other, they have difference in essence from the viewpoint of dynamics. The similarity is that, for focus near both BHom and Hopf bifurcation, the threshold curve for spike evoked by the excitatory stimulation locates right to the focus. However, the former in case of BHom bifurcation has close relationship with the saddle, i.e., coincides with the stable manifold of saddle, as shown in Fig. 11(d), whereas the latter in case of Hopf bifurcation has nothing to do with the saddle, as depicted in Fig. 14 (right) in Ref. [16]. Therefore, the threshold curve for the stable focus near the BHom bifurcation is a novel case different from that for the Hopf bifurcation.
The response of the resting state (stable focus) to inhibitory stimulations with strength
As depicted in Fig. 9(a), when
In case of stable focus coexisting with the stable limit cycle at
As can be found from Fig. 14(b)-(d), the black trajectory runs across the stable manifold and finally moves along the stable limit cycle (thick magenta line), which corresponds to spiking behavior, whereas the red trajectory does not run across the stable manifold and then forms subthreshold damping oscillation. Therefore, resembling the mono-stable focus shown in Fig. 11, the stable manifold of the saddle acts as the threshold curve for the spiking evoked by excitatory stimulation. Besides, the unstable manifold also plays important role in the generation of spiking. As shown in Fig. 14(d), the right part of the unstable manifold forces the black trajectory to move right and thus results in the increase of the membrane potential, which is a key factor to evoke spiking.
At
In the present subsection, firstly, the bifurcations with respect to
For the FHN model, except for the two cases of bifurcations with respect to
To clearly show the 4 different cases of bifurcations with respect to
The (
The region Ⅰ (orange) locates above the NF curve and left to BHom curve, which corresponds to mono-stable focus on LB.
The region Ⅱ (gray) lies below the NF curve, left to HBom curve and SNIC curve, which corresponds to mono-stable node.
The region Ⅲ (green) is upper to
The region Ⅳ (yellow) locates right to BHom curve, left to the right branch of the curve NF, and lower to curve the
The region Ⅴ (white) is surrounded by the BHom curve, the NF curve, and SN curve, which corresponds to the coexistence of stable node and stable limit cycle.
The region Ⅵ (pink) locates right to the SN and SNIC, and left to
The region Ⅶ (cyan) is right to the
In addition, the (
In the present paper, the mono-stable node and stable focus (close and left to the BHom curve) and the stable node or focus coexisting with stable limit cycle (close and right to the BHom curve) are investigated. The action potential, including PIR spike or spiking, can be evoked from these behaviors. The mono-stable equilibrium in the region marked by black star (
The nonlinear concept of threshold and post-inhibitory rebound spike are the fundamental conceptions in both nonlinear dynamics and neurophysiology, which are helpful to identify the physiological functions and modulation measures to firing pattern related to threshold or PIR spike [12,16,18,35]. In the present paper, we obtain the threshold curve for spike or spiking evoked from stable node and focus near bifurcation of big homoclinic (BHom) orbit in a modified FHN model without
Firstly, an example of the threshold curve near the BHom bifurcation with type Ⅰ excitability is presented, as expected in Fig. 15 in Ref. [16]. For case of mono-stable node or focus close and left to the BHom bifurcation, the threshold curve is the border between the initial values in phase plane successful and unsuccessful to induce an action potential. For case of coexistence close and right to the BHom bifurcation, the threshold curve is the border between the attraction domains of the stable equilibrium and the stable limit cycle. For both the stable node and focus, the threshold curve is around the steady state from down-left, below, and right to the stable steady state, with structure resembling that of the focus near Hopf bifurcation to a certain extent. In future, the threshold curves for more kinds of bifurcations [16,18] should be acquired.
Secondly, the threshold curve for the BHom bifurcation is a novel case different from those of the well-known SNIC and Hopf bifurcation[16]. In the present paper, the part of the threshold curve right to the stable equilibrium coincides with the stable manifold of the saddle, which acts as the well-known threshold for spike or spiking evoked by excitatory stimulation. Such a result resembles that of the SNIC but is different from that of the Hopf bifurcations, wherein the threshold sets right to the stable focus is unrelated to saddle. For the BHom bifurcation, the part of the threshold curve down-left to the stable equilibrium, which has a negative slope, acts as the threshold for the spike or spiking induced by inhibitory stimulation, i.e. the PIR spike or spiking, which resembles that of the Hopf bifurcations. However, it is different from that of stable node near SNIC, whose threshold curve fails to extend to the region down-left to the stable node [16]. Therefore, the threshold curve for the BHom bifurcation presents a novel case of threshold curve. Although the threshold curves for many theoretical models are acquired [28], these curves have not been built relationship to bifurcations or PIR spike. In future, the characteristics of these threshold curves should be acquired and build relationships to bifurcations.
Thirdly, in the present paper, both parts of threshold curve down-left to and right to the steady state play important roles for the generation of PIR spike or spiking. The trajectory runs across the part of the threshold curve down-left to the steady state during the inhibitory stimulations, and then evolves to the neighborhood of saddle in the spike region. Near the saddle, the stable manifold of saddle, coinciding with the threshold, plays the role of separating the subthreshold behavior and subthreshold behavior. Furthermore, the unstable manifold plays the role to increase the membrane potential of the suprathreshold behavior by forcing the trajectory to move to right, which is a key step to form action potential or spiking. The detailed dynamical process and role for the threshold curve for the PIR spike are acquired.
Fourthly, the condition for PIR spike is extended from mainly the Hopf bifurcation with type Ⅱ excitability or
Last, the dynamical behavior near or related to the BHom bifurcation are acquired by double-parameter bifurcation analysis. In the double-parameter plane, multiple codimension-1 bifurcation curves, including the BHom curve, are acquired. The mono-stable steady state or steady state coexisting with spiking behavior are acquired, which is related to BHom bifurcation curve and PIR spikes. Furthermore, a codimension-2 bifurcation point related to the BHom bifurcation, the saddle-node Homoclinic orbit bifurcation point (SNHO) which is the intersection point between the codimension-1 bifurcation curves of SN, SNIC as well as BHom, is acquired. Therefore, the results of the present paper are for BHom bifurcation near the SNHO bifurcation. The SNHO bifurcation has been investigated in Refs. [4,16,18,20]. In addition, it is well-known that the common homoclinic orbit is related to another codimension-2 bifurcation, BT bifurcation [16,18,25,34]. In future, the relationship between the homoclinic orbit and SNHO or BT bifurcation should be studied.
The results of the present paper provide comprehensive viewpoint to the threshold for BHom, which extends the concept of threshold and the condition for PIR spike to a large extent, which are very important for both nonlinear science and neuroscience. In the present paper, the voltage threshold for a single pulse stimulation to evoke spike is investigated. In future, more stimulation parameters or stimulus patterns or current threshold such as stimulation directions, stimulation timings, stimulation amplitudes, and stimulation durations to evoke spike or spike patterns should be considered. In addition, the modulations of noise and time delay on the BHom orbit and the dynamics of network composed of neurons with BHom bifurcation should be studied.
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