The influence of hyperpolarization-activated cation current on conduction delay and failure of action potentials along axon related to abnormal functions

Abstract

Conduction delay and failure behaviors of action potentials with a high frequency along nerve fiber are related to the abnormal functions. For instance, upregulation of a hyperpolarization-activated cation current (I_h) is identified to reduce the conduction delay to recover the temporal encoding, and downregulation of the I_h current to enhance the conduction failure rate to ease the pain sensation, with the dynamic mechanisms remaining unclear. In the present paper, the dynamic mechanism is obtained in a chain network model with coupling strength (g_c) and action potentials induced by periodic stimulations with a period (T_s). At first, as the action potentials exhibit a high frequency corresponding to a short T_s and the network has a small g_c, i.e., a short and unrecovered afterpotential and a small coupling current, the conduction delay is reproduced. The conduction failure is reproduced for T_s shorter and g_c smaller than those of the conduction delay, presenting a direct relationship between the two behaviors. Then, the conduction delay and failure are explained with the response time and current threshold of an action potential evoked from the unrecovered afterpotential. The prolonged response time for short T_s and small g_c presents the cause for the conduction delay, and the enhanced threshold for shorter T_s and smaller g_c presents the cause for the conduction failure. Furthermore, reduction of the delay and enhancement of the failure rate respectively induced by upregulation and downregulation of the I_h current are reproduced and explained. The positive I_h current induces Hopf bifurcation advanced and resting membrane potential elevated. Then, upregulation and downregulation of the I_h current induce the afterpotential elevated to shorten the response time and reduced to enhance the threshold, respectively. The results present nonlinear dynamics for the non-faithful conduction behaviors and dynamical mechanism for the modulation effect of the I_h current on the conduction delay and failure related to encoding and pain.

Introduction

It is well-known that the generation of an action potential of a neuron is related to the threshold, which is associated with the complex nonlinear dynamics such as bifurcations (Duan et al. 2020; Guo et al. 2021; Xing et al. 2023; Lyu et al. 2023). The conduction of action potentials along an axon or nerve fiber exhibits complex spatiotemporal dynamics (Zhang et al. 2020; Ding and Jia 2021; Zang et al. 2022), playing important roles in information transmission (Freeman 2000; Shimba et al. 2021; Yao et al. 2022). In general, action potentials faithfully propagate along an axon, which corresponds to normal functions of the nervous system (Bucher and Goaillard 2011). However, complex conduction behaviors of action potentials, such as conduction delay or conduction failure behaviors, may correspond to abnormal functions or diseases. For example, the conduction delay behavior means that the conduction time of action potentials increases with respect to time, i.e., delayed action potentials appear, which are associated with the changes in temporal encoding and dysfunction in learning, thinking, and acting (Ballo et al. 2012; Arancibia-Cárcamo et al. 2017; Zhang et al. 2017). Some action potentials fail to propagate along an axon, forming the conduction failure behavior, which is associated with pathological pain, demyelinating disease, and so on (Lesnick et al. 2007; Wang et al. 2016a; Waxman 2006; Wimmer et al. 2010). Then, identification of mechanism for the spatiotemporal dynamics of different conduction behaviors is helpful to modulate the related abnormal functions.

Conduction delay means that the conduction velocity of action potentials decreases with respect to time (Bucher and Goaillard 2011), which is related to the frequency of action potentials. In some neurological disorders such as the demyelinating disease, conduction velocity of action potentials significantly reduces (Waxman 1977; Kiziltan et al. 2007; Dutta et al. 2018), resulting in altered temporal encoding of information contained in the action potentials (Panzeri et al. 2010). It is well-known that the type or diameter of the nerve fibers determines the conduction velocity. Conduction delay behavior appears as a burst containing multiple action potentials with high frequency propagates along the pyloric dilator axon of lobsters, which is related to the digestive function (Zhang et al. 2017). To investigate the conduction delay behavior, current pulse stimulation with high frequency is used to evoke action potentials. The conduction time (delay) becomes long with increasing the frequency of action potentials (Ballo et al. 2012). A special ionic current, hyperpolarization-activated cation current (called I_h current for convenience)) mediated by the hyperpolarization-activated cyclic-nucleotide-gated (HCN) channels (Lezmy et al. 2021; Khurana et al. 2011; Byczkowicz et al. 2019), plays important roles in influencing the conduction time. Application of dopamine to enhance the I_h current can reduce the conduction time, resulting in enhanced temporal fidelity of action potentials, which may be beneficial for the temporal encoding (Ballo and Bucher 2009; Ballo et al. 2012; Zhang et al. 2017). In many studies on the unmyelinated axon and cerebellar mossy fiber, upregulation of the I_h current can increase the conduction velocity of action potentials, i.e., reduces the conduction delay, and blockage of the I_h current significantly enhances the conduction time of action potentials (Byczkowicz et al. 2019; Roth and Hu 2020), i.e., enhances the conduction delay. These experimental observations show that upregulation of the I_h current is an important modulation target to reduce the conduction delay to recover the normal function. Unfortunately, it is unclear how the I_h current affects the conduction delay behavior. Then, identification of dynamics of the conduction delay modulated by the frequency of action potentials, conduction velocity of the axon, and the I_h current is one objective of the present paper.

Another non-faithful conduction behavior, conduction failure behavior, has been observed in more and more experiments (Zhu et al. 2009; Dilley et al. 2013; Wang et al. 2016b; Hamada et al. 2017; Mao et al. 2019). There are various factors to influence the conduction failure behavior (Bucher and Goaillard 2011; Ratas and Pyragas 2012; Wang et al. 2016b), such as the type of fibers to determine the conduction velocity, the frequency of action potentials, ionic currents, and so on. The conduction velocity of unmyelinated axons is often slower than that of myelinated axons (Gillespie and Stein 1983). For unmyelinated axons, the larger the diameter is, the faster the conduction velocity is, due to that the conduction velocity is positively correlated with the diameter of axon (Hodgkin 1954). The conduction failure behavior is often observed in the unmyelinated axon such as the C-fiber (Sun et al. 2012; Ballo et al. 2012; Zhang et al. 2017). For example, a representative of the conduction failure behavior is associated with diabetes-induced pain information, which is reduced as the conduction failure is enhanced (Sun et al. 2012). The result suggests that the conduction failure behavior may be an intrinsic self-inhibitory mechanism of the pain sensation. Conduction failure behavior has also been confirmed in the C-fibers of rabbits (Zhu et al. 2009), especially for the action potentials with a high frequency induced by an external stimulus with a short period (Ratas and Pyragas 2012; Cross and Robertson 2016). Ion currents such as the I_h, potassium (K⁺), and pump currents play important roles in regulating the conduction failure behavior (Crotty and Levy 2007; Kueh et al. 2016; Wang et al. 2016a; Wang et al. 2016b). Especially, the I_h current plays important roles in inflammatory and neuropathic pain (Chaplan et al. 2003; Weng et al. 2012; Jansen et al. 2021; Bernard Healey et al. 2021). For instance, blockage of the I_h current significantly increases the conduction failure (Zhu et al. 2009), which has an analgesic effect (suppressing the pain sensation) (Chaplan et al. 2003; Wang et al. 2016a). Then, identifying the dynamics of conduction failure modulated by the frequency of action potentials, the conduction velocity, and the I_h current is the second objective of the present paper. In addition, as the conduction time of action potentials increases to a certain extent (Zhu et al. 2009), the following action potential fails to conduct, forming the conduction failure behavior, which presents a possible relationship between the conduction delay and failure behaviors. Then, identification of the direct relationship between the conduction delay and failure behaviors is another objective of the present paper.

The theoretical models of reaction-diffusion equations or chain network are used to simulate the axon (Ding and Jia 2021; Zhang et al. 2019b). The complex spatiotemporal behaviors related to the conduction behaviors are reproduced in various studies, and the spatiotemporal behaviors change with respect to multiple modulation parameters (Zhang et al. 2019c, 2020; Song et al. 2019; Zang and Marder 2021; Zang et al. 2022). For example, the influences of the nerve fiber type, ionic currents, temperature, and memristor on the conduction behaviors are investigated. For the chain network model containing multiple compartments to simulate the axon, a compartment receives coupling pulse current from the neighborhood compartments at a phase of the afterpotential (membrane potentials following an action potential). Then, three factors, the amplitude of the coupling current pulse, the timing or application phase of the pulse, and the level of the afterpotential at the application phase determine the conduction delay or failure behaviors. The application phase is determined by the frequency of action potentials (Zhang and Gu 2019), the amplitude is determined by the coupling strength, and the level of afterpotential is determined by the factors of the compartment such as the I_h current. For instance, conduction delay behavior appears if a delayed action potential is induced by the coupling current from the afterpotential, and conduction failure occurs if an action potential cannot be caused sometimes. Then, with a single neuron model, the threshold of current to evoke an action potential is used to characterize the conduction failure behavior (Zhang et al. 2020), which has not been used to study the modulation effect of the I_h current on the conduction failure. Unfortunately, the dynamical mechanism for the conduction delay behavior remains unclear. To present an effective measure to characterize the dynamics underlying the conduction delay behavior is one goal of the present paper. Then, identification of dynamical mechanisms for the conduction delay with a novel measure and the conduction failure with the current threshold is the last objective of the present paper. Especially, the modulation effects of the I_h current on the conduction delay and failure behaviors are studied.

In this article, the experimental results of the conduction delay and conduction failure behaviors are reproduced in a chain network model with the I_h current (Hodgkin and Huxley 1952; Khurana et al. 2011), and the dynamical mechanisms are presented with nonlinear responses of the afterpotentials in a neuron model, which is helpful to modulate the temporal encoding and pathological pain. Firstly, conduction delay behavior similar to the experimental observations is reproduced in a chain network with the coupling strength labeled as g_c. External pulse stimulations with a period (labeled as T_s) are used to evoke action potentials at one end of the chain. For a short T_s to evoke action potentials with a high frequency, conduction delay behavior appears for a relatively small g_c, due to the unrecovered afterpotential within the short T_s. Conduction time to reflect the conduction delay becomes short as the I_h current increases. When g_c is large or T_s is long, faithful conduction behavior occurs. Secondly, conduction failure behavior resembling experimental observations (Wang et al. 2016a) is reproduced, appearing for a shorter T_s and a lower g_c, compared with the conduction delay behavior, which presents a direct relationship between conduction delay and failure behaviors. As the I_h current decreases, the conduction failure rate increases in wide ranges of T_s and g_c, which contains four cases. Finally, the mechanisms for the conduction delay and failure behaviors are acquired in a single neuron model. The conduction delay and failure are respectively explained with the response time (a novel indicator used in the present paper) and current threshold for an action potential, which is caused by a current pulse applied to different phases of the unrecovered afterpotentials. A small g_c induces a small coupling current pulse which cannot go beyond the threshold sometimes to evoke action potential, presenting the cause for the conduction failure appearing at small g_c. The response time becomes long and the current threshold becomes large with decreasing the application phase of the stimulation current pulse, which can explain the conduction delay and failure appearing at a short T_s. Especially, the I_h current induces Hopf bifurcation advanced and afterpotential related to the Hopf bifurcation elevated, resulting in a shortened response time and reduced current threshold, which can explain the modulation effects of I_h current. The results present dynamical mechanisms for the conduction delay and conduction failure behaviors modulated by the I_h current, which is helpful to understand the viewpoint that the I_h current may be used as a target to regulate the conduction delay and conduction failure, such as upregulation of the I_h current to reduce conduction delay to improve the temporal encoding and blockage of the I_h current to enhance the conduction failure rate to ease pain.

The rest of this article is organized as follows. Sects. ”Model and method”, “Result”, and “Conclusion and Discussion” present the Model and Method, Result, and Conclusion, respectively.

Model and method

Chain network model to simulate axon

The axon is described by a chain network model containing multiple compartments and the compartment in the network is described by Hodgkin-Huxley model (HH) with the I_h current (Hodgkin and Huxley 1952; Khurana et al. 2011). The network model is described as follows:

$$C\frac{\text{d}{V}_{\text{i}}}{\text{d}t}=-(I_{\text{Na,i}}+I_{\text{K,i}}+I_{\text{L,i}}+I_{\text{h,i}})+I_{\text{app,i}}+I_{\text{c,i}}$$ 1

$$\frac{\text{d}{m}_{\text{i}}}{\text{d}t}={\alpha }_{\text{m}}(1-m_{\text{i}})-{\beta }_{\text{m}}{m}_{\text{i}},$$ 2

$$\frac{\text{d}{h}_{\text{i}}}{\text{d}t}={\alpha }_{\text{h}}(1-h_{\text{i}})-{\beta }_{\text{h}}{h}_{\text{i}},$$ 3

$$\frac{\text{d}{n}_{\text{i}}}{\text{d}t}={\alpha }_{\text{n}}(1-n_{\text{i}})-{\beta }_{\text{n}}{n}_{\text{i}},$$ 4

$$\frac{\text{d}{H}_{\text{i}}}{\text{d}t}=\frac{H_{\infty }-H_{\text{i}}}{{\tau }_{\text{H}}},$$ 5

where 100 compartments are considered in the network, and “i” for a compartment denotes its sequential number in the chain network. Each compartment except the first and 100-th ones is connected to the two neighboring compartments, the first compartment is only coupled to the second one, and the compartment 100 to the compartment 99. Then, the coupling current in Eq. (1) is I_c,i = g_c (V_i-1 + V_i+1 − 2V_i) (1 < i < 100), I_c,1 = g_c (V₂ − V₁), and I_c,100 = g_c (V₉₉ − V₁₀₀), where g_c denotes the coupling strength. If I_c,i in Eq. (1) and the sequential numbers “i” in Eqs. (1–5) are ignored, Eqs. (1–5) become the HH model to describe a single neuron, which is introduced as follows:

V, m, h, and n represent the membrane potential, activation, and inactivation of sodium (Na⁺) channel, and activation of potassium (K⁺) channel, respectively. The parameter C is capacitance of membrane, and I_app is the applied current. The currents I_Na, I_K, I_L, and I_h represent the Na⁺ current, K⁺ current, leak current, and hyperpolarization-activated cation current, respectively, which are given as follows:$I_{\text{Na}}=g_{\text{Na}}{m}^{3}h(V-E_{\text{Na}})$, $I_{\text{K}}=g_{\text{K}}{n}^4(V-E_{\text{K}})$, $I_{\text{L}}=g_{\text{L}}(V-E_{\text{L}})$, and $I_{\text{h}}=g_{\text{h}}H(V-E_{\text{h}})$. The parameters g_Na, g_K, g_L, and g_h are the maximal conductances, and E_Na, E_K, E_L, and E_h denote the equilibrium potentials. The related functions are given by: ${\alpha }_{\text{m}}=\frac{0.1(V+45)}{1-{\text{e}}^{-(V+45)/10}}$, ${\beta }_{\text{m}}=4e^{-\frac{V+70}{18}}$, ${\alpha }_{\text{h}}=0.07e^{-\frac{V+70}{20}}$, ${\beta }_{\text{h}}=\frac{1}{1{\text{+ e}}^{-(V+40)/10}}$, ${\alpha }_{\text{n}}=\frac{0.01(V+60)}{1-{\text{e}}^{-(V+60)/10}}$, ${\beta }_{\text{n}}=\frac{1}{8}{\text{e}}^{-\frac{V+70}{80}}$, $H_{\infty }=\frac{1}{1+{\text{e}}^{(V+85)/7.3}}$, and ${\tau }_H=2000{[\frac{-7.4(V+60)}{e^{-(V+60)/0.8}-1+5e^{-(V+60)/22}}+65e^{-(V+56)/23}]}^{-1}$.

I_h current

Especially, for a single neuron or compartment, the I_h current $I_{\text{h}}=g_{\text{h}}H(V-E_{\text{h}})$ is considered in the present paper, where H, g_h, and E_h represent the open probability of the inactivation gate, the maximum conductance, and the equilibrium potential of the I_h current, respectively.

External voltage pulse stimulus

Axon at resting state is chosen as control in the biological experiments, then, one end of the axon (stimulation region) is stimulated by external periodic voltage stimulation (labeled with Sti) to evoke action potentials conducting to the other end (conduction region) (Zhu et al. 2009), as depicted in Fig. 1. The voltage pulse stimulation (please refer to Figs. 7a, 13) similarly to the experiments (Zhu et al. 2009) is applied in the present paper. Pulse width 2 ms and pulse strength 15 mV is used, and then a pulse can induce an action potential. The period of the voltage pulses is labeled as T_s and is set as a control parameter to determine the frequency of action potentials. Compartments 1 to 5 are the stimulation region in the present paper, and the remaining compartments are the conduction region.

Fig. 1.

Sketch map of action potentials that are evoked at one end (stimulation region) by external voltage pulses and conduct along the axon (conduction region)

Fig. 7.

Conduction behaviors. a Conduction failure at g_c = 0.08 mS/cm²; b enlargement of the first six spikes of the panel (a); c conduction time for conduction failure behavior at g_c = 0.08 (black) and 0.087 (red)  mS/cm²; d conduction block behavior at g_c = 0.05 mS/cm²

Fig. 13.

Small (g_c = 0.06 mS/cm², left column) and large (g_c = 0.1 mS/cm², right column) coupling currents respectively for the conduction failure and conduction delay. a1 and b1: External periodic pulse stimulation; a2 and b2: V₅; a3 and b3: Coupling current I_c,6; a4 and b4: V₆. Other parameter values: a short T_s (= 17 ms) and g_h = 0.3 mS/cm²

Parameter values and method

In the present paper, C = 1 μF/cm², g_Na, g_K, and g_L are 120, 36, and 0.3 mS/cm², respectively, E_Na, E_K, E_L, and E_h are 50, − 77, − 59, and − 30 mV, respectively. I_app, g_h, T_s, and g_c are the control parameters.

Fourth Runge–Kutta method is used to integrate the equations, and the integration time step is 0.001 ms. Software XPPAUT is employed to calculate the bifurcations (Cartwright 2010).

Result

Dynamics of conduction delay behavior

The period of external pulse stimulation to evoke action potentials in the stimulation region is labeled as T_s, the coupling strength between compartments of the chain network is labeled as g_c, and the conductance of the I_h current is labeled as g_h.

A representative of conduction delay behavior

For the conduction delay behavior, the conduction time of action potentials increases with respect to time. A representative of the conduction delay behavior for g_c = 0.12 mS/cm², T_s = 18 ms, and g_h = 0 mS/cm² is depicted in Fig. 2a, b. The circular column (left) represents the axon, where the blue region indicates the stimulation region and the gray area denotes the conduction region, and changes of three action potentials for V₅ and V₉₅ are shown as representatives here. For the j-th action potential (j is the sequential number of the action potentials), the conduction time from the compartment 5 to the compartment 95 is labeled as D_j. For Fig. 2a with g_h = 0 mS/cm², D₁ = 164.14 ms, D₂ = 173.79 ms, and D₃ = 180.89 ms, showing that the conduction delay behavior appears. In the network model, I_app = −4 μA/cm² is used.

Fig. 2.

Conduction delay behaviors. a g_h = 0 mS/cm²; b g_h = 0.3 mS/cm²; c Comparison between panels (a) and (b); d Enlargement of V₉₅ in panels (a) and (b)

The conduction delay at g_h = 0.3 mS/cm² is illustrated in Fig. 2b. D₁ = 153.60 ms, D₂ = 157.92 ms, and D₃ = 161.49 ms, which are shorter than those of g_h = 0 mS/cm². The comparison between panels (a) and (b) is shown in Fig. 2c. For the 5-th compartment, action potential for g_h = 0 mS/cm² appears at the same time as that of 0.3 mS/cm². However, for the 95-th compartment, the action potential for g_h = 0.3 mS/cm² (blue) appears earlier, indicating a faster conduction velocity, compared with g_h = 0 mS/cm² (black). As illustrated in Fig. 2d, the local enlargement of V₉₅ shows that the resting membrane potential (before the first action potential) for g_h = 0.3 mS/cm² is higher, compared with 0 mS/cm². For both g_h values, the afterpotential does not recover to the resting potential, due to that T_s = 18 ms is too short.

The influence of the I_h current on conduction delay behavior

For g_h = 0.3 mS/cm², D_j labeled with blue circles gradually increases with respect to time and tends to be a steady value 176.75 ms (after t ≈ 360 ms), as shown in Fig. 3a (g_c = 0.12 mS/cm² and T_s = 18 ms). The time interval between circles in Fig. 3a is T_s, and “t” of the x-axis for the j-th circle is j × T_s, which corresponds to the application time of the stimulation pulse to evoke the j-th action potential of the compartment 5. For g_h = 0 mS/cm², D_j labeled with black squares gradually increases and tends to be a steady value of 223.01 ms (after t ≈ 594 ms). The conduction time for g_h = 0.3 mS/cm² becomes shorter, compared with 0 mS/cm², indicating that increasing I_h current can reduce the conduction time, which is consistent with the experimental results (Ballo et al. 2012). As can be found from Fig. 3a, the conduction time converges to a steady value, showing that the convergence process of the conduction time corresponds to a transient behavior. The convergence processes and the steady values of the convergence time for different values of g_h are different.

Fig. 3.

Changes of conduction time for g_h = 0 (black) and 0.3 mS/cm² (blue). a D_j, b ΔD_j

For g_h = 0 mS/cm², ΔD_j = D_j+1 − D_j (j = 1, 2, 3, …) is shown by a black curve in Fig. 3b, which reflects the speed of change in the conduction time with respect to the application time of the stimulation. ΔD_j decreases from a positive value to zero gradually, showing that the speed of change in the conduction time becomes small. ΔD_j for g_h = 0.3 mS/cm² (blue) is smaller, compared with 0 mS/cm² (black), as illustrated in Fig. 3b. The results show that increases of the I_h current induces the speed of change in the conduction time decreased.

Conduction delay for a short T_s and faithful conduction for a long T_s

When g_h = 0.3 and g_c = 0.1 mS/cm², conduction delay behavior occurs at a shorter T_s (18 ms), as shown in Fig. 4a. Two spikes are chosen as a representative, and the conduction time is 153.6 ms and 157.9 ms, respectively. However, for a large T_s (50 ms), faithful conduction behavior occurs, as depicted in Fig. 4b. The two conduction times are both 153.6 ms, showing that the conduction time does not change with increasing time. The unchanged conduction time for T_s = 50 ms is depicted by the bottom curve (dark blue) in Fig. 4c. The convergence processes of the conduction time for a short T_s are shown by the top four curves in Fig. 4c (T_s = 18, 19, 20, and 21 ms from top to bottom). For the conduction delay behavior at a small T_s, conduction time increases at first, and then tends to be a steady value. The conduction time is long for a small T_s.

Fig. 4.

Conduction behaviors at different T_s values. a Conduction delay for T_s = 18 ms; b faithful conduction for T_s = 50 ms; c changes of conduction time at different values of T_s; d V₅ for T_s = 18 ms in panel (a) (Black solid curve) and for T_s = 50 ms in panel (b) (Red dotted curve); e Enlargement of panel (d)

The local enlargement of V₅ for T_s = 18 ms shown in panel (a) and for T_s = 50 ms shown in panel (b) are depicted by black solid and red dotted curves in Fig. 4d, respectively. The first spikes for T_s = 18 ms and T_s = 50 ms are duplicated, because the first stimulation times are the same (at 11 ms). The local enlargement of Fig. 4d is shown in Fig. 4e. For T_s = 18 ms, the afterpotential (solid black curve) before the second stimulation (at 29 ms), as denoted by the former arrow (black), does not recover. For T_s = 50 ms, the afterpotential before the latter arrow (red) as well as a red dot (at ~ 53 ms) does not recover, while recovers after 53 ms. The second stimulation is applied at 61 ms, then, the membrane potential at the second stimulation recovers. The different levels of the afterpotentials at the second stimulation times are the cause for the conduction delay behavior or the faithful conduction behavior, which is addressed in SubSect. ”Response time curve to explain the conduction delay behavior” and Fig. 15.

Fig. 15.

The response time of an action potential evoked from the unrecovered afterpotentials to a current pulse stimulus applied at a phase Δt (pulse width of 2 ms). At different g_h values (the black and blue curves respectively represent g_h = 0 and 0.3 mS/cm²): a1 the response for Δt = 15 ms with pulse intensity ΔA of 8.5 μA/cm²; a2 changes of the response time. At different ΔA values (red and blue for ΔA = 8.5 and 9.5 μA/cm², respectively): b1 the response for Δt = 15 ms and g_h = 0 mS/cm²; b2 changes of the response time

Different conduction behaviors for different g_c values

At different values of g_c, changes of the conduction time exhibit different cases, showing different conduction behaviors, as illustrated in Fig. 5 (T_s = 18 ms and g_h = 0.3 mS/cm²).

Fig. 5.

Faithful conduction behavior for a large value of g_c (g_c = 3 mS/cm², magenta), and conduction delay for relatively small g_c values (black, red, green, blue, and cyan respectively represent g_c = 0.1, 0.12, 0.14, 0.2, and 0.5 mS/cm²)

For a large g_c, for instance, g_c = 3 mS/cm², the conduction time does not change with respect to time, as shown by magenta (bottom curve) in Fig. 5, corresponding to the faithful conduction behavior.

For relatively small values of g_c, such as g_c = 0.1, 0.12, 0.14, 0.2, and 0.5 mS/cm², the convergence processes of the conduction time are depicted in Fig. 5. The conduction time increases at first and then tends to be a steady value, i.e., conduction delay behavior appears, as shown by the top 5 curves in Fig. 5. The larger the g_c is, the shorter the conduction time is, and the shorter the duration of conduction delay behavior is.

In addition, the conduction failure behavior and conduction block behavior occur for small values of g_c (such as 0.08 and 0.087 mS/cm²) and a much smaller g_c (0.05 mS/cm²), respectively, which are addressed in detail in the following Subsection.

Large g_h induces large I_h currents to reduce the conduction delay

As shown in Fig. 3a, the conduction time for g_h = 0 mS/cm² is longer than that of g_h = 0.3 mS/cm². The role of the I_h current in modulating the conduction delay is shown in Fig. 6, left column for g_h = 0 mS/cm² and right column for g_h = 0.3 mS/cm² (with T_s = 18 ms and g_c = 0.12 mS/cm²). The voltage stimulation pulses applied to the compartment 5, V₅, V₆, I_h,6 (the I_h current of the compartment 6), I_total,6 (the total current of the compartment 6), and enlargement of the I_h,6 current and the I_total,6 current are shown from top to bottom rows, respectively. In Fig. 6a6, g_h = 0 mS/cm², the I_h,6 current (red) is 0 μA/cm², and the I_total,6 current (blue) for the afterpotential is relatively small (the maximum value is about 0.24 μA/cm²). In Fig. 6b6, g_h = 0.3 mS/cm², the maximum value of the I_h,6 current for the afterpotential is about 1.82 μA/cm², and the I_total,6 current is relatively large (the maximum value is about 0.52 μA/cm²). The result shows that the positive I_h current for the afterpotential plays a role in raising the total current, which can induce earlier appearance of the action potentials. Then, the conduction time of the first spike from the fifth compartment to the sixth compartment for g_h = 0.3 mS/cm² (about 1.96 ms) is shorter than that for g_h = 0 mS/cm² (about 2.43 ms). This result can explain that the conduction time for g_h = 0.3 mS/cm² is shorter than that for g_h = 0 mS/cm² in Fig. 3a.

Fig. 6.

The influence of the I_h currents on conduction delay for g_h = 0 mS/cm² (left column) and 0.3 mS/cm² (right column). a1 and b1: External periodic pulse stimulations; a2 and b2: V₅; a3 and b3: V₆; a4 and b4: I_h,6 current; a5 and b5: I_total,6 current; a6 and b6: enlargement of the I_h,6 current (red) and the I_total,6 current (blue). T_s = 18 ms and g_c = 0.12 mS/cm²

Dynamics of conduction failure behavior

Conduction failure behavior and conduction block behavior

A representative of conduction failure behavior is shown in Fig. 7a (T_s = 18 ms, g_c = 0.08 mS/cm², and g_h = 0.3 mS/cm²). It can be found that each stimulation pulse (black labeled with “Sti”) in stimulation region can evoke an action potential for the compartment 5, V₅. Here, 22 spikes are depicted as representatives due to the limited range of the time axe. For the first spike of the compartment 5, the conduction to the compartment 50 and to the compartment 95 is represented by the left line with an arrow in Fig. 7a. For the tenth spike of the compartment 5, the conduction to the compartment 50 and to the compartment 95 is shown by the middle line with an arrow. For the sixteenth spike of the compartment 5, the conduction to the compartment 50 is shown by the right line with an arrow, the conduction to the compartment 95 is not shown due to the limited range of the time axe. For the seventeenth to twenty-second spikes of the compartment 5, the conductions are not illustrated, due to the limited ranges of the time axe in Fig. 7a. As can be found in Fig. 7a, the 3 k-th (k = 1, 2, 3) action potentials (one out of three) of compartment 5 cannot conduct to the conduction region, resulting in a long interspike interval after the 2 k-th (k = 1, 2, 3) action potential of the compartment 50 and 95. To clearly show the dynamics of the conduction failure, the first six spikes of the compartment 5 are plotted in Fig. 7b as representatives. Correspondingly, 4 spikes conduct to the compartment 50 and the compartment 95 and 2 spikes fail to conduct. From the 5-th compartment to the 95-th compartment, the conduction time for the 1-st, 2-nd, 4-th, and 5-th spikes is 214.12 ms, 218.21 ms, 215.45 ms, and 219.19 ms, respectively. The conduction time for the second spike is larger than the first one, showing that the conduction velocity becomes slow. However, no conduction time can be acquired for the 3-rd spike (6-th spike), as depicted by the red dashed line, due to the conduction failure of the 3-rd spike (6-th spike) of the compartment 5.

The conduction time for the conduction failure behavior at g_c = 0.08 (black) and 0.087 (red) mS/cm² is illustrated in Fig. 7c. For g_c = 0.08 mS/cm² (black), there are gaps after the 2 k-th spikes (k = 1, 2, 3, …), due to the conduction failure shown in Fig. 7a, b. For g_c = 0.087 mS/cm², there are gaps after the 4 k-th spikes (k = 1, 2, 3, …), showing that the 5 k-th (k = 1, 2, 3, …) action potentials of the compartment 5 fail to conduct. Especially, before the conduction failure of a spike, the conduction time of action potentials gradually increases, similar to the experimental observations (Zhu et al. 2009). The smaller the g_c is, the longer the conduction time is, and the higher the frequency of the action potentials failing to conduct is.

To distinguish the conduction failure for g_c = 0.08 (black) and 0.087 (red) mS/cm² in Fig. 7c, conduction failure rate for the compartment i denoted as R_i can be defined. As shown in Fig. 7b (g_c = 0.08 mS/cm²), 4 out of 6 action potentials in the stimulation region conduct to the compartments 50 and 95, i.e., 2 out of 6 spikes fail to conduct. Then, the conduction failure rate is defined as R₅₀ = (6–4)/6 = 1/3, and R₉₅ = (6–4)/6 = 1/3. Similarly, R₅₀ = 1/5 and R₉₅ = 1/5 for g_c = 0.08 mS/cm² (black in Fig. 7c). More generally, if the number of the action potentials induced by the stimulation pulses applied in the stimulation region is labeled as M (M is an integer), and the number of the action potentials conducted to the compartment i is labeled as N (N is an integer between 0 and M), the conduction failure rate of the i-th compartment is defined as R_i = (M − N) / M, with a value between 0 and 1. 0 < R_i < 1 denotes that only some action potentials successfully conduct, i.e., conduction failure occurs. R_i = 0 means that no action potentials fail to propagate, which corresponds to conduction delay or faithful conduction behaviors. R_i = 1 means that all action potentials fail to conduct, called conduction block behavior, as shown in Fig. 7d, with g_c further decreased to 0.05 mS/cm².

The influence of T_s and g_c on conduction behaviors

The compartment 95 is chosen as a representative to show the influence of g_c and T_s on R₉₅. For g_h = 0 mS/cm², the distribution of R₉₅ on the parameter plane (T_s, g_c) is depicted in Fig. 8a. Green, red, and blue represent 0 < R₉₅ < 1 (conduction failure), R₉₅ = 1 (conduction block), and R₉₅ = 0 (conduction delay and faithful conduction for long T_s), respectively. The purple curve represents the border between the conduction failure and conduction delay (faithful conduction), and the magenta curve denotes the border between the conduction failure and block behaviors. Two borders are separated when T_s <  ~ 37.5 ms and are near or coincide when T_s >  ~ 37.5 ms. The conduction failure and delay behaviors appear for relatively small g_c and short T_s, and the conduction block for very small g_c. Especially, compared with the conduction delay behavior, the conduction failure behavior occurs for shorter T_s and smaller g_c, which presents a direct relationship between the conduction delay and failure behaviors. A short T_s corresponds to a high frequency of the action potentials and a small g_c to the C-fiber, showing that the conduction failure occurs for stimulations with a short T_s, resembling the experimental results of the C-fibers (Sun et al. 2012). With increasing g_c, the range of T_s for the conduction failure decreases_, i.e., presenting the changing regularity of the purple curve behavior. It is the faithful conduction behavior that locates up-right to the conduction delay behavior (not shown here).

Fig. 8.

The influence of T_s and g_c on the conduction failure rate. a R₉₅ for g_h = 0 mS/cm²; b R₉₅ for g_h = 0.3 mS/cm². Green, blue, and red represent 0 < R₉₅ < 1, R₉₅ = 0, and R₉₅ = 1, respectively; c difference between panels (a) and (b), labeled as ∆R. ∆R = 0, 0 < ∆R < 1, and ∆R = 1 is shown by blue, light blue, and red, respectively

The influence of the I_h current on the conduction failure rate

For g_h = 0.3 mS/cm², the distribution of R₉₅ is shown in Fig. 8b. Green, blue, and red areas respectively represent 0 < R₉₅ < 1, R₉₅ = 0, and R₉₅ = 1. The black curve represents the border between 0 < R₉₅ < 1 and R₉₅ = 0, and the yellow curve denotes the border between R₉₅ = 1 and 0 < R₉₅ < 1 or R₉₅ = 0. The conduction delay/faithful conduction region becomes larger for g_h = 0.3 mS/cm², compared with 0 mS/cm², which indicates that increase of the I_h current facilitates conduction of the action potentials.

∆R, obtained by subtracting R₉₅ for g_h = 0.3 mS/cm² from R₉₅ for g_h = 0 mS/cm², can reflect the difference in failure rate. The distribution of ∆R on (T_s, g_c) plane is depicted in Fig. 8c, which can reflect the changes of conduction failure rate induced by downregulation of the I_h current. ∆R = 0, 0 < ∆R < 1, and ∆R = 1 are shown by blue, light blue, and red, respectively. Combined with the borders between different conduction behaviors and borders between ∆R = 0, 0 < ∆R < 1, and ∆R = 1, (T_s, g_c) plane is divided into 7 regions. ∆R = 0 appears in regions labeled with 5, 6, and 7 (not the focus of the present paper), 0 < ∆R < 1 in regions labeled with 1, 2, and 4, and ∆R = 1 in region 3. ∆R > 0 in regions 1, 2, 3, and 4 shows that the conduction failure rate enhances as g_h decreases, labeled as case-1, -2, -3, and -4, respectively. The four cases show that the downregulation of g_h enhances the conduction failure rate in wide ranges of (T_s, g_c) plane, resembling the experiment results (Wang et al. 2016a). More details for the four cases are addressed in the following paragraphs.

Case-1: enhanced failure rate for the conduction failure behavior

A representative for case-1 is shown in Fig. 9 (g_c = 0.1 mS/cm² and T_s = 17 ms). The spike trains of the compartment 6 for g_h = 0.3 mS/cm² exhibit R₆ = 1/6, as shown in Fig. 9a1. As g_h decreases to 0 mS/cm², R₆ increases to 1/2, as shown in Fig. 9b1. To present a global view, the spatiotemporal behaviors for g_h = 0.3 and 0 mS/cm² are depicted in Fig. 9a2, b2, respectively. The abscissa and ordinate denote the sequential number of compartments in the network and time, respectively. The left bright horizontal lines represent the peaks of action potentials in the stimulation regions (compartments 1 to 5), and the blue regions between two bright horizontal lines represent the membrane potential (afterpotential) between peaks of the action potentials. The bright oblique lines represent peaks of the action potentials in the conduction region (compartments 6 to 100), and the “oblique” reflects the propagations of the action potentials. Compared with stimulation region, the disappearance of bright lines in conduction region corresponds to conduction failure of action potentials. As depicted in Fig. 9a2, one out of six action potentials fail to conduct. As depicted in Fig. 9b2, one of very two action potentials fails to conduct.

Fig. 9.

Downregulation of g_h causes enhancement of the failure rate. Left for g_h = 0.3 mS/cm²: a1 spike trains with R₆ = 1/6; a2 spatiotemporal behaviors. Right for g_h = 0 mS/cm²: b1 spike trains with R₆ = 1/2; b2 spatiotemporal behaviors

Case-2: from conduction delay to conduction failure

A representative of case-2 for g_c = 0.12 mS/cm² and T_s = 17 ms is depicted in Fig. 10. Conduction delay and conduction failure behaviors occur for g_h = 0.3 and 0 mS/cm², respectively, and the corresponding conduction failure rates are 0 and 0.2, respectively. Figure 10a1, b1 show the stable behavior of membrane potential of the 6-th compartment for g_h = 0.3 and 0 mS/cm², respectively, and Fig. 10a2 and Fig. 10b2 show the spatiotemporal behaviors of action potentials. The results show that the downregulation of g_h can enhance the failure rate. Here, Fig. 10a1 shows the membrane potential of conduction delay behavior, when the conduction time tends to be a constant value.

Fig. 10.

Downregulation of g_h induces conduction delay behavior changed to conduction failure. Left for g_h = 0.3 mS/cm²: a1 spike trains with R₆ = 0; a2 spatiotemporal behaviors. Right for g_h = 0 mS/cm²: b1 spike trains with R₆ = 1/5; b2 spatiotemporal behaviors

Case-3: from conduction delay to conduction block

When g_c = 0.06 mS/cm² and T_s = 30 ms, the conduction delay behavior (left column) changes to the conduction block behavior (right column) as g_h decreases from 0.3 to 0 mS/cm², as illustrated in Fig. 11. Membrane potentials of the compartment 6 are depicted in upper row and spatiotemporal behaviors in the lower row. Similar to Fig. 10a1, Fig. 11a1 shows the membrane potential of the conduction delay behavior, when the conduction time tends to be a constant value.

Fig. 11.

Downregulation of g_h induces conduction delay changed to conduction block. Left for g_h = 0.3 mS/cm²: a1 spike trains with R₆ = 0; a2 spatiotemporal behaviors; Right for g_h = 0 mS/cm²: b1 membrane potential for compartment 6; b2 spatiotemporal behaviors

Case-4: from conduction failure to conduction block

For g_c = 0.06 mS/cm² and T_s = 17 ms, as illustrated in Fig. 12, the conduction failure behavior (left column) changes to the conduction block behavior (right column) as g_h changes from 0.3 to 0 mS/cm². Membrane potentials of the compartment 6 are depicted in the upper row and spatiotemporal behaviors are in the lower row. As shown in Fig. 12a1, R₆ = 1/2.

Fig. 12.

Downregulation of g_h induces conduction failure changed to conduction block. Left for g_h = 0.3 mS/cm²: a1 spike trains with R₆ = 1/2; a2 spatiotemporal behaviors. Right for g_h = 0 mS/cm²: b1 membrane potential for the compartment 6; b2 spatiotemporal behaviors

Small g_c induces small coupling current to cause conduction failure from unrecovered afterpotentials for short T_s

As can be found in Fig. 8, conduction failure behavior occurs for smaller g_c and shorter T_s, compared with the conduction delay behavior. For example, for a short T_s (17 ms), the conduction failure occurs at g_c = 0.06 mS/cm² and conduction delay (conduction time tends to be a constant value) occurs at g_c = 0.1 mS/cm² as illustrated in the left and right columns of Fig. 13, respectively. The voltage stimulation pulses applied to the compartment 5, V₅, coupling current I_c,6, and V₆ are shown from top to bottom rows, respectively. For the conduction failure behavior, the second, fourth, and sixth actional potentials fail to conduct, as shown by the stars in Fig. 13a4. The maximal value of the coupling current pulse is relatively small, about 6.0 μA/cm², as illustrated in Fig. 13a3. However, all action potentials conduct for the conduction delay behavior, as illustrated by blue arrows in Fig. 13b4. The maximal value of the coupling current is relatively large, about 9.7 μA/cm², as shown in Fig. 13b3. Then, it is the strength of the coupling current that determines the appearance of the conduction failure or conduction delay behaviors at a fixed T_s value. A large coupling current for a large g_c can evoke action potentials to form the conduction delay behavior, while a small coupling current for a small g_c sometimes cannot evoke action potentials sometimes to form the conduction failure behavior. Furthermore, it can be found that the afterpotential elevates when receiving the coupling pulse current, showing that the afterpotential has not recovered to the resting membrane potential within the short T_s. Then, responses of the unrecovered afterpotential to a current pulse stimulation with different strengths (corresponding to g_c) and different application phases (corresponding to T_s), and the influence of g_h on the responses, are helpful to explain the conduction failure behavior modulated by g_c, T_s, and g_h, which are addressed in SubSect. ”Mechanisms for modulations to the conduction delay and failure behaviors”.

Mechanisms for modulations to the conduction delay and failure behaviors

As mentioned above, both the conduction delay behavior and conduction failure behavior are dependent on strength of the coupling current pulse, phase of the unrecovered afterpotential when stimulated by the current pulse, and level of the unrecovered afterpotential. In fact, the level of the afterpotential is determined by g_h, the stimulation time/phase of the current pulse is related to T_s, and the strength of the coupling current pulse is modulated by g_c. Therefore, in the present Subsection, conduction delay and conduction failure behaviors in the network are explained with a single neuronal model with the I_h current, which is stimulated by a current pulse to simulate the coupling current. The strength ΔA and application time Δt of the pulse correspond to g_c and T_s, respectively. Considering that the conduction delay behavior corresponds to delayed action potentials, response time of an action potential is proposed to characterize the conduction delay behavior. In addition, the current threshold to evoke an action potential is used to explain the conduction failure behavior. Then, two important curves to describe the responses of action potential and the modulations of ΔA, Δt, and g_h on the two curves, are acquired. One curve to describe the response time of an action potential evoked from the afterpotential at different phases, which is called the response time curve, can well explain the conduction delay behavior. The other curve to describe the current strength threshold to induce an action potential from the afterpotential at different phases, i.e., the current threshold curve, can well explain the conduction failure behavior. Especially, the modulations of the I_h current on the afterpotential related to a Hopf bifurcation and on the two curves are obtained.

I_h current induces Hopf bifurcation advanced and afterpotential elevated

Here, the dynamics of HH model of a single neuron in the absence of external stimuli is studied. As depicted in Fig. 14a, black and blue curves represent the bifurcations near the steady state for g_h = 0 and 0.3 mS/cm², respectively. It is well-known that there is a subcritical Hopf bifurcation (H) and a saddle node bifurcation of limit cycles (SNLC). As g_h increases from 0 to 0.3 mS/cm², point H changes from I_app ≈ 3.75 to 3.29 μA/cm², and SNLC changes from I_app ≈ 2.49 to 1.76 μA/cm². With increasing g_h, the Hopf bifurcation advances, due to that the I_h current is positive.

Fig. 14.

Dynamics of the HH model as g_h changes from 0 (black) to 0.3 (blue)  mS/cm². a Bifurcations with respect to I_app. Bold and dotted solid curves respectively for stable and unstable focus, and thin solid and dashed curves for stable and unstable limit cycles, respectively. b The resting potential for I_app =  − 4 μA/cm²; c An action potential induced by a stimulation and the afterpotential following the action potential

The I_h current is positive, then, the resting membrane potential prior to the Hopf bifurcation elevates with increasing g_h. For instance, for I_app =  − 4 μA/cm² (same as that used in the network), the resting membrane potential increases from − 73.16 mV to − 70.65 mV as g_h increases from 0 (black) to 0.3 (blue) mS/cm², as illustrated in Fig. 14b. The behavior before t = 50 ms of Fig. 14c is same as that of Fig. 14b, with y-axis shown in a large scale. Perturbed by a voltage pulse stimulation (V = 15 mV), an action potential is evoked for g_h = 0 and 0.3 mS/cm². The afterpotential for g_h = 0.3 mS/cm² is higher than that for 0 mS/cm². For both g_h = 0 and 0.3 mS/cm², the afterpotential increases monotonically to recover to the resting membrane potential after a relatively long time (nearly 50 ms).

Response time curve to explain the conduction delay behavior

A current pulse with width of 2 ms and intensity ΔA of 8.5 μA/cm² is applied at t = 0 ms, as shown in Fig. 15a1, resulting in the first action potentials for g_h = 0 (black) and 0.3 (blue) mS/cm². The level of the unrecovered afterpotential for g_h = 0.3 mS/cm² (blue) is higher than that for 0 mS/cm² (black). Then, as the same pulse stimulus is applied to the unrecovered afterpotentials at a phase Δt (15 ms in Fig. 15a1) following the first pulse stimulation, the second action potentials are evoked. The application time/phase Δt corresponds to T_s, and ΔA corresponds to g_c. The second action potential appears earlier for g_h = 0.3 mS/cm² (blue), compared with 0 mS/cm² (black), showing that the response times of action potentials evoked at a same time are different for different g_h values. The response time, which is defined as the interval between the beginning phase of the pulse stimulation and the peak of action potential evoked by the pulse, is 3.08 ms and 5.44 ms for g_h = 0.3 and 0 mS/cm², respectively.

The response time changes with respect to Δt, as illustrated in Fig. 15a2, black and blue for g_h = 0 and 0.3 mS/cm², respectively. The responses of the afterpotentials to a stimulation pulse with ΔA = 8.5 μA/cm² (red) and ΔA = 9.5 μA/cm² (blue) are illustrated in Fig. 15b1, b2, respectively, which can explain the conduction delay in three aspects:

1. For g_h = 0 (black) and 0.3 (blue) mS/cm², with increasing Δt, the response time decreases at first and then tends to be a fixed value, which remains unchanged as Δt increases further. The later the phase of current pulse stimulation applied to the afterpotential is, the faster the action potential is, resembling the changes of conduction time (delay) in the network with respect to T_s which determines the application phase of the coupling current pulse. Then, such a result can well explain that the conduction delay occurs for a shorter T_s than that of the faithful conduction behavior, and conduction time for the conduction delay behavior decreases with increasing T_s (Please refer to Fig. 4).

2. The response time for g_h = 0.3 mS/cm² (blue) is lower than that for 0 mS/cm² (black), which means that the larger the g_h is, the faster the response of an action potential is. With increasing g_h, action potentials induced by the current pulse are more quickly, similar to the results induced by the increase of g_h in the network. Such a result can well explain why the conduction time for g_h = 0.3 mS/cm² is shorter than that for 0 mS/cm² (Please refer to Figs. 2, 3).

3. A smaller conduction time appears for a larger g_c value (Please refer to Fig. 5) can be explained. The response time of an action potential for ΔA = 9.5 μA/cm² (blue) becomes shorter, compared with ΔA = 8.5 μA/cm² (red), as shown in Fig. 15b1. The response time is 3.43 ms for ΔA = 9.5 μA/cm² and 5.44 ms for ΔA = 8.5 μA/cm². The response time changes with respect to Δt, as shown in Fig. 15b2, red and blue for ΔA = 8.5 and ΔA = 9.5 μA/cm², respectively. The response time for ΔA = 9.5 μA/cm² is shorter, compared with ΔA = 8.5 μA/cm², which means that the larger the current stimulation pulse strength (ΔA) is, the faster the action potential is. ΔA of the current stimulation pulse is associated with g_c in the network to determine the coupling current strength.

Current threshold curve to explain the conduction failure behavior

A current pulse with application time Δt and stimulus intensity ΔA is applied to the afterpotential, as shown in Fig. 16a. The current pulse can evoke action potential or not, which can be used to explain the conduction delay/faithful conduction or conduction failure. For the afterpotential (g_h = 0.3 mS/cm²), a suprathreshold current pulse (blue) can cause an action potential at the phase Δt, while a subthreshold current pulse (red) does not, as depicted in Fig. 16a. For Δt = 15 ms, pulse strength ΔA = 6 μA/cm² (blue) is used as a suprathreshold stimulation, and ΔA = 4 μA/cm² (red) as a subthreshold stimulation, as illustrated in Fig. 16a. The minimal ΔA to trigger an action potential at Δt is called the current threshold, denoted as A_T(Δt). For different values of Δt, the values of the current threshold intensity A_T(Δt) are different.

Fig. 16.

Current threshold curve to explain the conduction failure. a Responses of the afterpotentials to a current stimulus pulse with different intensities ΔA (blue and red curves for ΔA = 6 μA/cm² and 4 μA/cm², respectively). Pulse width 2 ms, application phase Δt = 15 ms, and g_h = 0.3 mS/cm²; b Changes of the current intensity threshold A_T with increasing ∆t for g_h = 0 (black) and 0.3 (blue) mS/cm²; c The black and blue curves correspond to the border between conduction failure and conduction delay for g_h = 0 in Fig. 8(a) and g_h = 0.3 mS/cm² in Fig. 8(b), respectively

The threshold A_T(Δt) increases with decreasing Δt, for both g_h = 0.3 (blue) and 0 (black) mS/cm², as illustrated in Fig. 16b. The larger the g_h value is, the smaller the A_T(Δt) is. The current threshold curve decreases monotonically. The threshold curve can explain the conduction failure behavior in three aspects.

1. The increases of A_T(Δt) with decreasing Δt for both g_h = 0 and 0.3 mS/cm², resembling changes of the border between the conduction failure and conduction delay behaviors in plane (T_s, g_c), as shown in Fig. 16c, black and blue for g_h = 0 and 0.3 mS/cm², respectively. At a short Δt, A_T(Δt) is very large. Therefore, the conduction failure mostly occurs at short T_s.

2. The current threshold A_T(Δt) is relatively large in a wide range of T_s, showing that pulse stimulation strength ΔA larger than A_T(Δt) can induce action potential while lower than A_T(Δt) not, which can explain why the conduction failure occurs for smaller values of g_c, compared with the conduction delay.

3. The threshold A_T(Δt) becomes large with decreasing g_h value, as illustrated in Fig. 16b, resembling that the border between the conduction delay and conduction failure (Fig. 8) becomes high as g_h decreases, as depicted in Fig. 16c. The threshold A_T(Δt) for g_h = 0 mS/cm² is higher, compared with 0.3 mS/cm². Therefore, the failure rate for g_h = 0 mS/cm² is larger, compared with 0.3 mS/cm². Then, the modulation effect that downregulation of g_h enhances the conduction failure rate is explained.

Conclusion and discussion

Abnormal firings or spatiotemporal behaviors of the nervous system can lead to abnormal functions or brain diseases (Li et al. 2017; Yang et al. 2021; Song et al. 2022; Ma 2023; Wang et al. 2023). Conduction behavior of action potentials is one of the important spatiotemporal behaviors of the nervous system (Chen et al. 2019; Dai et al. 2022; Li et al. 2023). Different from the traditional view that action potentials faithfully conduct along axon to involve in normal function of the nervous system, non-faithful conduction behaviors such as the conduction failure or delay behaviors have been observed in the experiments associated with abnormal functions or diseases (Sun et al. 2012; Ballo et al. 2012; Zhang et al. 2017). For instance, conduction failure behavior is observed in C-fibers related to the pathological pain (Sun et al. 2012), and the conduction delay behavior is observed in the pyloric dilator axons to investigate the temporal fidelity of action potentials and the temporal encoding (Ballo et al. 2012). Furthermore, the influence of the I_h current in modulating conduction behaviors is investigated. Increase of the I_h current reduces conduction delay to improve the temporal encoding (Wang et al. 2016a; Zhang et al. 2017), and blockage of the I_h current significantly increases the conduction failure to ease pain (Wang et al. 2016a). In this article, the experimental observations of the conduction delay and failure behaviors are reproduced in a chain network model, and the dynamical mechanism for the modulations of the I_h current to the conduction delay and failure behaviors is presented with the nonlinear responses of afterpotentials in a neuronal model. The results exhibit significance in the following aspects:

Firstly, the conduction delay and failure behaviors closely matching the experimental results in 3 aspects are reproduced, and the direct relationships between conduction delay and failure behaviors are obtained. (1) Conduction failure and delay behaviors appear for action potentials with a high frequency, i.e., the period (T_s) of the stimulations to evoke action potentials is short. With increasing T_s, the conduction delay and failure reduce at first, and change to faithful conduction behavior at last. (2) Conduction failure and delay behaviors appear for the fiber with little conduction velocity, i.e., little coupling strength (g_c), whereas faithful conduction behavior occurs for large g_c or long T_s. With decreasing g_c, the conduction delay and failure increase. Compared with the conduction delay behavior, the conduction failure behavior appears for smaller g_c and shorter T_s, which presents direct relationships between the conduction delay and failure behaviors. In the previous experiments, no direct relationship between conduction delay and failure behaviors has been presented, since the two behaviors are studied independently in different types of fibers, as mentioned above (Sun et al. 2012; Ballo et al. 2012). In addition, for the conduction failure behavior, conduction time of action potentials increases before the action potential fails to conduct, as shown in Fig. 7c, which presents simulation results to the experimental observation (Zhu et al. 2009). (3) Upregulation of the I_h current reduces the conduction delay, and downregulation of the I_h current enhances the conduction failure rate. The enhancement of the failure rate with changing g_h contains four cases of changes of spatiotemporal behaviors in wide ranges of the plane (T_s, g_c). The result shows that upregulation of the I_h current can help to improve the temporal encoding and blockage of the I_h current can help to ease pain (less action potentials mean lower pain sensation).

Secondly, the response time and current threshold of an action potential evoked from the unrecovered afterpotentials by a current pulse stimulation are acquired in a neuron model, which can well explain the modulations of the coupling strength (g_c) and frequency of action potentials (T_s) on the conduction delay and failure behaviors. Conduction delay and failure behaviors appear for action potentials with a high frequency for a short T_s, due to that the afterpotentials cannot recover to the resting potential within the short T_s. As a stimulation pulse with strength ΔA is introduced to the unrecovered afterpotentials at a phase Δt, an enhanced current threshold and a prolonged response time of an action potential appear, which present cause for the conduction failure and conduction delay, respectively. Δt and ΔA of the current pulse stimulation respectively correspond to T_s and g_c. With increasing Δt, the current threshold and response time decrease, presenting the cause for the decrease of the conduction delay and failure with increasing T_s. The existence of a large threshold at a small Δt shows that stimulation larger than the threshold can evoke an action potential and lower than the threshold not, which can explain that the conduction failure occurs for a smaller g_c, compared with the conduction delay. In the previous studies, conduction delay behavior is observed in experiments and reproduced in theoretical model (Ballo et al. 2012; Zhang et al. 2017). In this article, the conduction delay behavior is further explained with the response time of action potential. In the previous studies, the conduction failure behavior is reproduced in multiple ranges of short T_s values (Zhang and Gu 2019), however, such a discrete distribution of T_s has not been reported in the experimental studies. In this paper, the conduction failure behavior appears in a continuous range of T_s with short values, resembling the experimental observations (Zhu et al. 2009). The different T_s ranges for the conduction failure are due to that the current threshold curve exhibits damping oscillations with multiple cycles in Ref (Zhang and Gu 2019), whereas a continuous range of T_s in the present paper is due to that the current threshold curve decreases monotonically.

Finally, modulations of the I_h current on the conduction delay and failure behaviors are explained with the response time and current threshold. As the I_h current increases, Hopf bifurcation with respect to current advances, and the resting membrane potential before the Hopf bifurcation elevates. Then, the level of the afterpotentials elevates, resulting in shortened response time and reduced current threshold to cause an action potential. Then, the shortened response time can explain decrease of the conduction delay induced by upregulation of the I_h current observed in the experiments (Ballo et al. 2012; Zhang et al. 2017), which lays a theoretical foundation to improve temporal encoding by upregulation of the I_h current. If the I_h current decreases, the afterpotential becomes low, resulting in a prolonged response time and enhanced current threshold. The enhanced current threshold can explain the enhanced failure rate induced by downregulation of the I_h current observed in the experiments (Wang et al. 2016a; Byczkowicz et al. 2019), laying a theoretical foundation to ease pain through blockage of the I_h current. In previous studies, upregulation of K⁺ channel enhances the conduction failure (Zhang et al. 2020), showing another target to modulate the pain sensation. The results of this paper present a comprehensive viewpoint for the conduction delay and failure behaviors of action potentials with a high frequency and lay a theoretical foundation that the I_h current is an important target to modulate temporal encoding and pathological pain.

Although some progress has been made in the present paper, there are many questions related to the nonfaithful conduction behaviors or the modulation effects of the I_h current to be studied. Firstly, except for the temporal encoding and pathological pain (Wang et al. 2016a; Zhang et al. 2017), non-faithful conduction behaviors of action potentials such as conduction failure behavior are associated with a variety of pathological states or diseases such as epilepsy, Parkinson’s diseases, and multiple sclerosis (Waxman 2006; Lesnick et al. 2007; Wimmer et al. 2010). Then, the roles of the nonfaithful conduction in the brain diseases should be an important issue. Secondly, except for action potential frequency and ionic current (Cross and Robertson 2016; Feng et al. 2017; Cho et al. 2017), conduction behaviors of action potentials are related to many factors such as the fiber type or topology (Bucher and Goaillard 2011; Bukh et al. 2023), heterogeneity of the ion channel distribution (Freeman et al. 2016; Zang et al. 2022; Zang and Marder 2023), astrocyte (Lezmy et al. 2021), temperature (Song et al. 2019), and noise (Zhang et al. 2019b). The influence of other important factors on the nonfaithful conduction behaviors should be studied in future. For example, spontaneous firing induced by the uneven distribution of ion channel density along the axon is related to demyelinating diseases (Freeman et al. 2016), and slow inactivation of Na⁺ channel reduces firing frequency of axon to restore the conduction behavior (Zang et al. 2022). Finally, except for the conduction behaviors, the I_h current is also responsible for regulating the dynamics of the neuronal firing to achieve some important or specific functions (Zhang et al. 2019a; Guan et al. 2019, 2021; Takagi et al. 2020) such as sleep, learning, memory, sensation, and some brain diseases. Then, the various nonlinear dynamics related to the I_h current should be studied in future. More interestingly, in a recent study (Lezmy et al. 2021), the increase of Ca²⁺ concentration in astrocyte causes the release of ATP at first, and then induces the I_h current increased through a series of processes, resulting in the reduced conduction velocity of action potentials along axon with myelin, which shows that the modulation effect of the I_h current on the conduction behavior is in contrast to the results of the previous experiments (Ballo et al. 2012; Zhang et al. 2017) and the present paper. Then, the dynamics of the conduction behaviors of action potentials along axon with myelin, which is modulated by the I_h current and astrocyte, should be studied in future.

Data availability

The data used in this study are available upon request from the corresponding author.

Declarations

Conflict of interest

The authors indicate no conflicts of interest related to the content of this paper.