Bursting types and bifurcation analysis of the temperature-sensitive Purkinje neuron

The bursting discharge behaviour of neurons is affected by many factors, among which temperature is one of the more important factors. In this work, we study the bursting discharge behaviour and dynamics process of two different temperature-sensitive ion channels, the temperature-sensitive potassium current and the temperature-sensitive calcium current. In the case of the temperature-sensitive potassium current, the bursting discharge waveforms, codimension-1 bifurcations and trajectory plots at different temperatures indicate that five different types of bursting discharge (Hopf/Flip, Hopf/Homoclinic, Fold/Homoclinic, Fold/Fold Cycle, Circle/Big Homoclinic) appear with increasing temperature. In the case of temperature-sensitive calcium current, two types of bursting discharge (Circle/Big Homoclinic, Fold/Fold Cycle) emerge. According to the bursting discharge waveforms, the rise in temperature can promote the generation of bursting discharge at the beginning, and finally, the bursting discharge phenomenon disappears. This is consistent with the experimental results that blocking potassium and calcium currents can promote the bursting of Purkinje neurons. Then, it can be seen from the codimension-2 bifurcation and the waveform area distribution diagrams that even if the dynamic paths are consistent, the bursting discharge types and the waveforms may be different. In contrast, even if the bursting discharge type is the same, the dynamic paths and the waveform may be different. These results provide insight into the effect of temperature on the neuronal dynamics and bursting behaviour of temperature-sensitive ion channels.


Introduction
The firing rhythm exhibited by neurons plays an important role in the encoding and transmission of information in the nervous system. The firing rhythm of neurons mainly includes bursting and spiking. When neuron activity alternates between a quiescent state and repetitive spiking, the neuron activity is said to be bursting. Spiking is a discharge state corresponding to a stable limit cycle. Bursting is a pervasive phenomenon in the activation patterns of neurons in the central nervous system [1][2][3][4][5] and spinal cord [6,7]. After many years of research development in neuroscience, studying how bursting are involved in other neural phenomena has become very popular. Areas such as neural synchronization [8][9][10][11][12][13], neural coding [14][15][16], neuropathologies such as epilepsy [17,18], and robust central pattern generators [19][20][21] are closely related to bursting.
To better understand the dynamic mechanism of bursting oscillation in the nervous system, we generally adopt the analysis method of fast-slow dynamics [22]. The so-called fast-slow dynamics analysis generally means that the system consists of fast subsystems and slow subsystems. Fast subsystems consist of fast variables, and slow subsystems consist of slow variables. The fast subsystem is mainly responsible for generating spiking and resting states. The slow subsystem is mainly responsible for the transition between these two states. The analysis method of fast-slow dynamics has been widely used in the study of neuronal bursting [23][24][25]. As early as 1985, Rinzel [26] carried out a systematic theoretical study on the mode of bursting and developed a preliminary classification of bursting oscillation for the first time [27]. With the progression of bursting theory research, Izhikivich [28] proposed a more comprehensive bursting topology classification method. In [28], the bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed. Geometric bifurcation theory was used to expand the existing classification of bursting to include many new types.
Purkinje neurons are a class of GABAergic inhibitory neurons located in the cerebellum [29,30]. They were first discovered and named by Jan Evangelista Purkyně in 1837. The cerebellum plays an important role in sensory coordination and motor control. At the centre of these functional circuits are the Purkinje neurons, which form the only output of the cerebellar cortex. In some neuron models, the neurons generally engage in generating of periodic spiking and bursting [22,31,32], and Purkinje neurons are no exception. However, the transition between bursting oscillation and spiking discharge is accompanied by more complex kinetic characteristics. Potassium channels play an important role in regulating neuronal hyperexcitability and bursting. When potassium channels are blocked, Purkinje cell electrical activity and bursting discharge are enhanced [33,34]. Calcium channels are necessary for spontaneous firing of Purkinje cells, and bursting discharge is enhanced when calcium channels are blocked [35].
Temperature plays a vital role in many biological processes. Each neuron has different ion channels, and these ion channels also have different dependencies on temperature. For example, temperature has a distinct effect on the activation of neural activity by modulat-ing excitability and channel conductance. Some neurons are so sensitive to small changes in temperature that they can induce appropriate firing patterns at different temperatures [36]. According to known experimental studies, the spontaneous activity of Purkinje cells is very sensitive to temperature changes. In tonically firing cells, the spontaneous firing rate of each cell decreases with decreasing temperature, when the temperature is less than 35 • C. In cells with a trimodal pattern, a slight decrease in temperature causes the cells to stop firing [35]. The effect of temperature on the Hodgkin-Huxley model of a Purkinje cell was estimated. The detailed firing behaviours of Purkinje cells was studied using a computer compartmental neuronal model. The simulation results indicated that the firing of Purkinje cells varies dynamically depending on different electrophysiological parameters of these neurons, and the respective properties may play significant roles in the formation of the mentioned characteristics of dynamical firings in the coding strategies for information processing and learning [37]. However, detailed studies on the effect of temperature on the bursting of Purkinje neurons are still scarce.
In this paper, we investigate the generation and transition of bursting discharge in temperature-sensitive Purkinje neuron model. We choose two sets of scaling factors to describe the temperature-sensitive potassium and calcium currents and provided codimension-1, codimension-2 bifurcations and time course plots using XPPAUT [38]. It is found that in Purkinje neurons with the temperature-sensitive potassium/calcium current, bursting discharge is promoted with increasing temperature, which is consistent with the conclusion that blocking potassium/calcium channel can promote bursting discharge in [33][34][35]. This suggests that increasing temperature has a similar effect on the bursting discharge of Purkinje neurons by blocking calcium and potassium currents.
The paper is organized as follows. In Sect. 2, we present descriptions of the Purkinje neuron temperaturesensitive model and the simulation configurations. The detailed results and analysis are given by Sect. 3, followed by the conclusion remark in Sect. 4.

Model
In the process of studying Purkinje neurons, many scholars have established many mathematical mod-els according to different research directions [39,40]. Some of these models are too detailed or simplistic to discuss the detailed biological characteristics as a whole. We chose the mathematical model proposed by Kramer et al. [41]. Therefore, a modified temperaturesensitive Purkinje neuron model is described by the following equations [41,42] here V and J (J = -23) represent the membrane potential and an externally applied current, respectively. g K , g Na , g Ca , g L and g M are the conductance for ion channels, such as potassium, sodium, calcium, leak and M current (a muscarinic receptor suppressed potassium current). V K , V Na , V Ca , V L and V M are the reversal potentials of the potassium current, sodium current, calcium current, leak current and M current, respectively.  (Table 1). m 0 [V] is the equilibrium function for the fast sodium gating variables. Different ion channels have a different sense of temperature. The electrical properties of the ion channels are strongly dependent on the temperature [43]. We use the following formalism to describe the temperature effect on the ion channels [44].
Here, g ref s is the corresponding value at the reference temperature (T ref = 22 • C). In this work, the parameters are shown in Table 2. In this paper, we consider temperature-sensitive neurons in two scenarios. One is for the temperature-sensitive potassium current (Q K = 3, Q Na = 1, Q L = 1, Q Ca = 1, Q M = 1) and the other is for the temperature-sensitive calcium current

Results
Ion channel currents can be divided into input currents, output currents and leak currents in neuron models. We know that calcium currents are input currents and potassium currents are output currents. Based on a current study, we know that temperature has an effect on the ion Table 1 The expressions of α i and β i in the Purkinje neuron model. In addition, we use C = 1 nF for the cell's capacitance   3.1 Topological type analysis of bursting of the temperature-sensitive potassium/calcium current By changing the temperature value of the system, the trajectory diagram of the bursting will show different behaviours under different temperatures. When we change the system temperature, the relative position of the bifurcation diagram and the trajectory diagram of the bursting will change to some extent. The fastslow dynamics is an approximate method to analyse the bursting behaviour. The bursting discharge types caused by the temperature-sensitive potassium current include the bursting discharge types caused by the temperature-sensitive calcium current. Therefore, in this part, we describe the dynamic discharge process of bursting caused by temperature-sensitive potassium current in detail (Figs. 1, 2, 3, 4, 5, 6). However, we will still give the bursting discharge and bifurcation trajectory diagrams of the temperature-sensitive calcium current (Fig. 7).

Hopf/Flip bursting via Fold/Flip hysteresis loop with slow passage effect
The trajectory of the bursting for T = 8.1 • C shown in Fig. 1a is placed in Fig. 1c. Figure 1b shows the time courses of variables n, h, c and M. M is a slow variable. When the bifurcation parameter is the slow variable M, the bifurcation curve of the fast subsystem with respect to M is Z-shaped, as shown in Fig. 1c. As the parameter M of the system decreases, the unstable limit cycle collides with the unstable saddle point, and saddle homoclinic bifurcation (HC 1 ) is generated. There are two stable branches from HC 1 to the subcritical Hopf bifurcation (SubH), which we call bistable. When bursting occurs, the lower resting state disappears through the saddle-node bifurcation (LP 1 ) point and then transitions to the vicinity of the stable focus on the upper branch. From HC 1 to subcritical Hopf bifurcation (SupH) is also bistable. With M increasing, the stable focus on the upper branch disappears via the SupH and becomes an unstable focus. Due to the slow passage effect of the SupH, the trajectory is still near the unstable focus. M continues to increase, and the oscillation amplitude of the bursting increases gradually due to the attraction of the stable limit cycle and the repulsion of the unstable focus. The appearance of period-doubling bifurcation (PD 1 ) bifurcation destabilizes the stable limit cycle and turns it into an unstable limit cycle. From SupH to PD 1 , the stable limit cycle appears, accompanied by two unstable equilibrium points and one stable node. The continuous and repeated discharge state ends via the PD 1 bifurcation, and the trajectory of the bursting returns to the lower resting state of the bifur- cation curve. Then, the unstable limit cycle collides with the unstable saddle point, and HC 2 is generated. This bursting pattern thus exhibits the kinetic properties of "Hopf/Flip" bursting via "Fold/Flip" hysteresis loop.

Hopf/Homoclinic bursting via Fold/Homoclinic hysteresis loop
The trajectory of the bursting for T = 10.55 • C shown in Fig. 2a is placed in Fig. 2c. The dynamics around bifurcations HC 1 and SubH are consistent with Fig. 1. When bursting occurs, the lower resting state disappears through the LP 1 point and then transitions to the vicinity of the stable focus on the upper branch. Due to the convergence effect of the stable focus, the oscillation amplitude of the bursting gradually decreases. Next, as the parameter M increases, the stable focus on the upper branch disappears via SupH and becomes an unstable focus. The oscillation amplitude of the bursting gradually increases due to the attraction of the stable limit cycle and the repulsion of the unstable focus. As the parameter M continues to increase, the stable limit cycle is destabilized by PD 1 bifurcation. The dynamical properties of bifurcations SupH to PD 1 are consistent with Fig. 1. Due to the repulsion of the unstable limit cycle, there is a slight attenuation of the oscillation amplitude. And then, bifurcation HC 2 forms. Thus the continuous and repeated bursting state ends through HC 2 . The trajectory of the bursting returns to the lower resting state of the bifurcation curve. This bursting pattern thus exhibits the kinetic properties of "Hopf/Homoclinic" bursting via "Fold/Homoclinic" hysteresis loop. Compared with Fig. 1, the trajectory of bursting discharge is similar when it jumps upward from the lower resting state, but it undergoes different bifurcations when it returns to the lower resting state.

Fold/Homoclinic bursting via Fold/Homoclinic hysteresis loop
The trajectory of the bursting for T = 15 • C shown in Fig. 3a is placed in Fig. 3c. When bursting occurs, the lower resting state disappears through the LP 1 point and then transitions to the discharge of the stable limit cycle around the upper branch. The peak of the oscillation amplitude of the bursting oscillates back and forth between high and low values due to the attraction of the stable limit cycle and the pulsion of the unstable focus.
As M increases, the stable limit cycle becomes unstable at the bifurcation point PD 1 . The dynamic bifurcation characteristics of HC 1 to PD 1 are the same as those of SupH to PD 1 in Fig. 1. After parameter M passes through PD 1 , the amplitude peak difference of the oscillation decreases gradually due to the repulsion of the unstable limit cycle and the unstable focus. The unstable limit cycle collides with the unstable saddle point  and HC 2 is generated. From PD 1 to HC 2 , two unstable equilibrium points and one stable equilibrium point, accompanied by an unstable limit cycle. After experiencing several discharge peaks with different peaks, the continuous and repeated discharge state ends via HC 2 . The trajectory of the bursting discharge returns to the lower resting state of the bifurcation curve. This bursting pattern thus exhibits the kinetic properties of "Fold/Homoclinic" bursting via "Fold/Homoclinic" hysteresis loop. Compared with Fig. 2, the trajectory of bursting discharge is different when it jumps upward from the lower resting state, but it experiences the same bifurcation when it returns to the lower resting state.

Fold/Fold Cycle bursting via Fold/Fold Cycle hysteresis loop
The trajectory of the bursting for T = 18 • C shown in Fig. 4a is placed in Fig. 4c. As the parameter M of the system decreases, HC 1 and SubH appear sequentially. From HC 1 to SubH, the number and type of equilibrium points are consistent with Fig. 3c. However, there are two unstable limit cycles. When bursting occurs, the lower resting state disappears through the LP 1 point, and then transitions to the discharge state of the unstable limit cycle around the upper branch.
The peak of the oscillation amplitude of the bursting oscillates back and forth between high and low values due to the repulsion of the unstable limit cycle and the unstable focus. As M increases, the unstable limit cycle becomes a stable limit cycle via PD 2 . From HC 1 to PD 2 , a stable node, two unstable equilibrium points and an unstable limit cycle coexist. After parameter M passes through PD 2 , the amplitude peak difference of the oscillation decreases gradually due to the attraction of the stable limit cycle and the repulsion of the unstable focus. Then, the unstable limit cycle collides with the unstable saddle point, and HC 2 is generated. From PD 2 to HC 2 , the number and type of equilibrium points remain the same. However, the unstable limit cycle becomes a stable limit cycle. M increases, the stable limit cycle collides with the unstable limit cycle, and LPC is generated. There is a brief gradual increase in the amplitude of the oscillations before reaching the LPC. From HC 2 to LPC, stable and unstable limit cycles coexist. After experiencing several discharge spikes with different peak values, the continuous and repeated discharge state ends via LPC. The trajectory of the bursting returns to the lower resting state of the bifurcation curve. This bursting pattern thus exhibits the kinetic properties of "Fold/Fold Cycle" bursting via a "Fold/Fold Cycle" hysteresis loop. The trajectory of bursting discharge is the same as in Fig. 3 when it jumps upward from the lower resting state, but the experienced bifurcations are different bifurcations when it returns to the lower resting state. Compared to T = 18 • C, the waveform of the bursting changes slightly, and PD 2 disappears at T = 21 • C. The trajectory diagram is formed by the combined action of the attraction of the stable limit and the repulsion of the unstable focus, as shown in Fig. 5c. The amplitude of the trajectory first decreases and then increases. This bursting pattern is consistent with T = 18 • C (Fig. 4), called "Fold/Fold Cycle" bursting. As seen from Figs. 4 and 5, even if the bursting types are the same, the waveforms may be slightly different.

Circle/Big Homoclinic bursting via Circle/Circle hysteresis loop
The trajectory of the bursting for T = 30 • C shown in Fig. 6a is placed in Fig. 6c. As the system parameter M decreases, HC 1 and SubH still exist. The position diagram of the stable limit cycle at M = 0.01556 and the LP 1 are shown in Fig. 6d. We know that the saddle node LP 1 is located on the stable limit cycle from Fig. 6d. Therefore, this stable limit cycle is a saddlenode homoclinic orbit. When bursting occurs, the lower resting state disappears via the saddle node bifurcation on invariant circle (SNIC) and then transitions to the discharge state of the stable limit cycle. Then, the stable limit cycle of the system hits the saddle point of the middle branch. Therefore, the bursting trajectory returns to the resting state of the lower branch via the Big-homoclinic (BHom) orbit. After experiencing the single peak and large amplitude discharge mode over a short period, the bursting is completed. This bursting pattern thus exhibits the kinetic properties of "Circle/Big Homoclinic" bursting via a "Circle/Circle" hysteresis loop.
In summary, for the two temperature-sensitive Purkinje neuron, we know that when the temperature rises, bursting discharge will be promoted. These phenomena are consistent with the experimental results, indicating that increasing temperature plays the same role as blocking potassium and calcium channels in promoting bursting discharge [33][34][35]. However, for the temperature-sensitive potassium current, the bursting becomes spiking, when the temperature increases to a certain value (T = 27 • C). Finally, the discharge state will disappear. For the temperature-sensitive calcium current, the discharge state will disappear when T > 29 • C. We can also say that when Purkinje neuron generate bursting, temperature-sensitive potassium channel is more likely to stimulate the generation and variety of bursting.

Codimension-2 bifurcations in the M-temperature plane of the temperature-sensitive potassium current
To understand the detailed response dynamics of Purkinje neuron with the temperature-sensitive potassium current, a two-parameter bifurcation diagram (coordinate parameters M and Temperature, respectively) is plotted in Fig. 8. The two-parameter bifurcation plot consists of twenty-three regions at the boundaries of the codimension-1 curve (LP 1 , LP 2 , HC 1 , HC 2 , BHom,  Fig. 8a-f. The bifurcation characters of these twenty-three regions are listed as follows: an unstable and two stable equilibrium points (Region 1 ); a stable and two unstable equilibrium points, a stable cycle (Regions 2 , 5 and 23 ); a stable and two unstable equilib-rium points, an unstable cycle (Region 3 ); a stable and two unstable equilibrium points, a stable and an unstable cycle (Regions 6 , 18 and 22 ); a stable and two unstable equilibrium points, two unstable cycles (Regions 7 and 20 ); a stable and two unstable equilibrium points (Regions 8 and 15 ); a stable equilibrium point (Regions 9 and 10 ); an unstable equilib- ; an unstable equilibrium point and an unstable cycle (Region 12 ); an unstable and two stable equilibrium points, an unstable cycle (Region 16 ); three unstable equilibrium points and a stable cycle (Regions 17 and 21 ); three unstable equilibrium points and an unstable cycle (Region 19 ). There will be two unstable limit cycles, two unstable equilibrium points and a stable equilibrium point in part of Region 4 . After the unstable limit cycle disappears, only two unstable equilibrium points and a stable equilibrium point coexist in Region 4 . Since the discussion in our paper is not closely related to Region 4 , we do not distinguish the specific regions of different kinetic features in Region 4 .
We label these different waveforms as I, II, III, IV, V and VI (Spiking) according to the bursting waveforms in Figs. 1a-6a, as shown in Fig. 9. There are seven pink straight lines in Fig. 9a. The plane is divided into eight regions, which represent different bursting waveform regions, and enlarged views of each region are shown in Fig. 9b-i. The routes of response dynamics at different temperatures can be seen from Fig. 8a, b, f. When T = 8.1 and 10.55 • C, the routes of the response dynamics are all 10 → 15 → 16 → 1 → 2 → 3 → 4 (see Table 3). The same routes of response dynamics mean that the types of bifurcation experienced are consistent  (Fig. 8b). This indicates that the route of response dynamics may be the same even if the bursting discharge types and waveforms are different. The routes of response dynamics are different at T = 15, 18, 21 and 30 • C. However, at these six different temperatures, all undergo LP 1 , SubH and HC 1 bifurcations in turn.
From Table 3, we can clearly see that when T = 18 and 21 • C, the topology types of bursting are the same, but the routes of the response dynamics and the waveforms are different. This suggests that the bursting types are likely to be the same even if the routes of the response dynamics and the waveforms are different. Table 3 summarizes this part.

Codimension-2 bifurcations in the M-temperature plane of the temperature-sensitive calcium current
To understand the detailed response dynamics of Purkinje neuron with the temperature-sensitive calcium current, a two-parameter bifurcation diagram (coordinate parameters M and temperature) is plotted in Fig. 10. The whole two-parameter bifurcation plot consists of twenty-four regions at the boundaries of the codimension-1 curve (LP, HC 1 , HC 2 , BHom, LPC 1 , LPC 2 , LPC 3 , SubH 1 , SubH 2 , SubH 3 , SupH, PD 1 , and PD 2 ). The numbers of the regions are shown in Fig. 10a-f. The bifurcation characteristics of these twenty-four regions are listed as follows: a stable and two unstable equilibrium points, an unstable cycle (Region 7 ); an unstable equilibrium point and an unstable cycle (Region 8 ); a stable and two unstable equilibrium points, a stable cycle (Regions 9 and 15 ); an unstable equilibrium point and a stable cycle (Regions 10 and 11 ); a stable and two unstable equilibrium points (Region 12 ); three unstable equilibrium points and a stable cycle (Region 13 ); a stable and two unstable equilibrium points, a stable and an unstable cycle (Regions 14 and 16 ); three unstable equilibrium points and an unstable cycle (Region 23 ) and a stable and two unstable equilibrium points, two unstable cycles (Region 24 ). The other regions have little to do with the content discussed in the text, so we do not specifically discuss the dynamic characteristics of the other regions here.
We label these different waveforms as I (Spiking), II, III according to the bursting waveforms in Fig. 7a1-a3, as shown in Fig. 11. There are three pink straight lines in Fig. 11a. The plane is divided into four regions, which represent different bursting waveform regions, and the enlarged views of each region are shown in Fig. 11b-e. The routes of the response dynamics at different temperatures can be seen in Fig. 10a-c. The routes of the response dynamics are 11 → 13 → 14 → 15 → 12 , 10 → 13 → 14 → 9 → 16 → 12 , 8 → 23 → 24 → 7 → 9 → 16 → 12 , when T = 5, 25 and 27 • C respectively (see Table 4). However, at these three different temperatures, all of them undergo LP 1 , SubH 1 and HC 1 bifurcations in turn. The topology type of bursting is "Fold/Fold Cycle", when T = 25 and 27 • C. This indicates that even if the bursting topology types are the same, the routes of the response dynamics and the waveforms may be different (Fig. 11c, d). Table 4 summarizes this part. ing behaviour of neurons. In this paper, based on the Purkinje neuron model, we selected two different temperature-sensitive ion channels to study the bursting behaviour of neurons. Here, we refer to the temperature-sensitive potassium current (Q K = 3, Q Na = 1, Q L = 1, Q Ca = 1, Q M = 1) and the temperature-sensitive calcium current (Q K = 1, Q Na = 1, Q L = 1, Q Ca = 3, Q M = 1). In these two temperature-sensitive cases, we can observe the generation of multiple bursting discharge types and the variation between the discharge waveforms. These findings suggest that the effect of temperature on ion channels causes the phenomenon of bursting discharge consistent with the phenomenon that block-  ing ion channels can promote bursting discharge. This means that when discussing the bursting of Purkinje neurons, temperature plays a similar role as blocking ion channels. For the temperature-sensitive potassium current, by changing the temperature, the types of bursting discharge at different temperatures were observed. We can conduct a comprehensive analysis of the detailed dynamic characteristics through the codimension-1 and codimension-2 bifurcation diagrams. When T = 8.1 and 10.55 • C, there are two different types of bursting discharges. Even though the types of bursting discharge are different, their kinetic paths are consistent. When T = 18 and 21 • C, it corresponds to the same bursting discharge type, but the dynamic paths and the waveforms are not consistent. This indicates that there is no absolute relationship between the type of bursting discharge and the kinetic path but instead a relative change. With increasing temperature, the bursting discharge will become spiking at T = 27 • C, and finally the discharge will disappear (see Table 3). Figure 9 shows different discharge waveform regions for the temperaturesensitive potassium current of Purkinje neuron. Each waveform corresponds to a bursting discharge type (see Table 3). We classify bursting discharges according to the bifurcations corresponding to the start and disap-pearance of discharge. For the temperature-sensitive calcium current, the types of bursting discharge are reduced compared to the temperature-sensitive potassium current. When T = 5 • C, it is spiking. When T = 25 and 27 • C, the bursting discharge types are the same, and both are "Fold/Fold Cycle", which is consistent with the T =18 and 21 • C of the temperature-sensitive potassium current, but the kinetic paths are different (see Tables 3, 4). Figure 11 shows different discharge waveform regions of the temperature-sensitive calcium current Purkinje neuron. Consistent with the above scenario of temperature-sensitive potassium current, the same bursting type does not mean the same waveforms and the route of response dynamics.
In previous studies on Purkinje neuron bursting discharges, the effect of ion channels on bursting was mainly investigated [33][34][35][45][46][47]. For example, 4aminopyridine (4-AP) was applied to brain slices to block the Kv potassium channel, thus enhancing electrical activity and bursting discharge in mature Purkinje cells. The role of SK in the bursting discharge induced by 4-AP was studied by injecting enough hyperpolarizing current to suppress spontaneous activity and the bursting discharge was induced by depolarizing pulse. In the neurons treated with 4-AP, apamin reduced the duration of bursting. The results also show that Ca 2+ activated K + channels (SK) play an important role in bursting [33]. Blocking the large conductance calcium-activated potassium (BK) channel with iberiotoxin increases the firing rate and results in irregular firing in tonically Purkinje neurons. During BK channel blocking, the intracellular calcium concentration during spontaneous firing increases [34]. P/Q-Type voltage-gated calcium channels are required for spontaneous firing of Purkinje neurons. Three different antagonists acting on the Purkinje neurons of tonically firing all lead to bursting discharge and then return to silent [35].
In this work, for the two temperature-sensitive Purkinje neurons, the bursting discharge was promoted by increasing the temperature. The results of this paper and the above experimental phenomena all have various bursting discharge types [33][34][35]. However, our waveforms are more regular than the explosive discharge in the experiment. This indicates that increasing temperature plays the same role as blocking potassium and calcium channels in promoting bursting discharge. In the real world, magnetic field [48], synapse, etc., all affect the neuronal dynamic behaviours, such as burst-ing, and synchronization [49,50]. The formation and development of autaptic connections are also closely related to neuronal firing patterns. [51,52]. Therefore, in future studies, it is valuable to study the behaviour of neurons in response to a variety of factors.
It is well known that temperature has a very large effect on neuronal activity, such as repolarization of action potentials and switching of excitability types [53,54]. On the basis of previous studies, this paper applied the temperature term to ion channels and then studied the effect of temperature on the bursting discharge of Purkinje neurons. Through the studies in this paper, the effect of temperature on neuron firing can be further clarified, and the bursting behaviour of neurons can be further regulated.