Ionic mechanisms underlying tonic and phasic firing behaviors in retinal ganglion cells

Abstract

In the retina, the firing behaviors that ganglion cells exhibit when exposed to light stimuli are very important due to the significant roles they play in encoding the visual information. However, the detailed mechanisms, especially the intrinsic properties that generate and modulate these firing behaviors is not completely clear yet. In this study, 2 typical firing behaviors—i.e., tonic and phasic activities, which are widely observed in retinal ganglion cells (RGCs)—are investigated. A modified computational model was developed to explore the possible ionic mechanisms that underlie the generation of these 2 firing patterns. Computational results indicate that the generation of tonic and phasic activities may be attributed to the collective actions of 2 kinds of adaptation currents, i.e., an inactivating sodium current and a delayed-rectifier potassium current. The concentration of magnesium ions has crucial but differential effects in the modulation of tonic and phasic firings, when the model neuron is driven by N-methyl-D-aspartate (NMDA) -type synaptic input instead of constant current injections. The proposed model has robust features that account for the ionic mechanisms underlying the tonic and phasic firing behaviors, and it may also be used as a good candidate for modeling some other firing patterns in RGCs.

Introduction

In the visual pathway, the first processing center of visual information is located in the retina, and spike output of the retinal ganglion cells (RGCs) codes the processed information that is transmitted to the lateral geniculate nucleus (LGN) and then to the visual cortex.¹ Thus, it is of great importance to investigate the characteristics of the intrinsic firings that RGCs exhibit, particularly, how the firing is generated and how it can be modulated under different physiological and pharmacological conditions.

Several previous studies suggested that the discharges of RGCs are largely dependent on the kinds of synaptic inputs,^2,3 while some other studies reported that the firing activities of RGCs are mainly determined by the intrinsic ionic channel dynamics.^4-6 For instance, electrophysiological recordings on mouse RGCs have revealed that a persistent sodium current has a crucial effect on the generation of spontaneous activity.⁴ Another experimental study on rat RGCs has demonstrated that the emergence of rebound excitation is probably triggered by the hyperpolarization-activated mixed-cation current.⁷ It was also found that adaptation in the firing activity of RGCs can be generated and modulated by a slowly inactivating sodium current⁸ and a delayed-rectifier potassium current.⁶ Investigations also found that synaptic currents can regulate the firing activities of RGCs, e.g., N-methyl-D-aspartate (NMDA) synaptic current has been shown to regulate resting activity,⁹ spontaneous burst activity,¹⁰ and the delay of spike onset¹¹ in RGCs.

Tonic and phasic activities are two typical firing behaviors observed in many types of neurons, e.g., LGN relay neurons,¹² midbrain dopaminergic neurons,¹³ pallidal neurons,¹⁴ locus coeruleus neurons,¹⁵ olfactory receptor neurons,¹⁶ and RGCs.^17-20 Generally speaking, tonic firing refers to a sustained response, which activates during the course of the stimulus; while phasic firing refers to a transient response with one or few action potentials at the onset of stimulus followed by accommodation. It was reported that tonic and phasic activities observed in guinea pig RGCs could be significantly regulated by the activation of cGMP-gated currents.²¹ However, the mechanisms underlying the generation of these 2 firing patterns are still unclear. Thus, in this study, our primary objective is to investigate the possible ionic mechanisms that generate tonic and phasic activities using computational approaches.

Over the past 2 decades, several computational models have been successfully constructed to model the diverse firing behaviors of RGCs. Based on data from salamander RGCs obtained with whole cell recordings, Fohlmeister, Coleman, and Miller constructed an excitability model, consisting of 5 voltage-gated ion channels, specifically, sodium, delayed-rectifier potassium, calcium, A-type potassium, and calcium-activated potassium channels (FCM model).²² The model has excellent performance in reproducing the repetitive activities of RGCs. Subsequently, in order to analyze the adapting firing behavior of RGCs, Kim and Rieke revised the conventional sodium current by introducing 2 slow variables, aiming to mimic the inactivation of sodium current. Thus, the revised sodium current, together with a leakage current, constituted a new model (KR model).⁸ Numerical results indicated that the KR model can approximate to the adapting behavior seen in experimental observations.

To investigate the generation of tonic and phasic activities, we combined and modified these 2 models (FCM and KR). Our results demonstrate that the new model can not only maintain the capacity in reproducing the repetitive firing and adapting activities seen in the 2 previous models, but it can also exhibit the tonic and phasic activities. Results from our modified model may help identify the ionic mechanisms for these 2 firing patterns. In the last part of this study, potential modulatory effects of NMDA-mediated synaptic input on the tonic and phasic activities were studied, and some interesting phenomena were observed.

Results

Reproducing the repetitive firing and adapting activity of RGCs

Simulation results shown in Figure 1 illustrate that the modified model can reproduce the general repetitive spiking activity similar to that observed in previous experiments and model results,^5,24-26 when the 2 slow variables s₁ and s₂ are not included. The larger the stimulus intensity, more spikes the neuron would elicit.

Figure 1. Repetitive firing activity of retinal ganglion cell without adaptation under constant stimulus intensity, λ = 0. (A-C) The value of I is 0.5, 1.5, 2.5 μA/cm² respectively.

As illustrated in Figure 2A, under a constant stimulus, the model exhibits the adapting firing behavior which is similar to that observed in previous experiments.^27-29 Diagrams illustrated in Figure 2B–D are the variations of the sodium current and the 2 slow variables, the changes of which are analogous to previous reports.⁸

Figure 2. Adapting firing behavior of retinal ganglion cell when subjected to constant current injection. (A) Simulated adapting behavior using the modified model (I = 1.5 μA/cm², λ = 2.0). (B) Variation of sodium current. (C and D) Variations of the 2 slow variables, s₁ and s₂.

Functional effects of different ion channels on the adapting activity

A recent pharmacological experiment on guinea pig RGCs indicated that in addition to sodium channel inactivation, the generation of adapting behavior may depend on delayed-rectifier potassium channels, but has little relation to calcium channels and calcium-dependent potassium channels.⁶ As our model contains these channels, each of them was removed to explore whether the variation of these ion channels can lead to similar conclusions as reported.⁶

It was shown that depolarized and hyperpolarized prepulse exhibit different suppressive effects on the subsequent adapting activities in RGCs,⁶ i.e., spikes in the adaptation period following depolarized prepulse is fewer than that following hyperpolarized prepulse. Results demonstrated in Figure 3A–B can confirm this phenomenon. Results in Figure 3C–F demonstrate that compared with the data in Figure 3B (all channels are included), removal of the calcium channel (Fig. 3C) and the calcium-dependent potassium channel (Fig. 3D) have little influence on the firing rate of the original adapting activity. The delayed-rectifier potassium channel in our model mainly contributes to the repolarization of the action potentials and removal of it makes the membrane potential stay in a high-depolarized state (data not shown). When decreasing g_K to a low level, the corresponding result presented in Figure 3E shows that it changes the adapting firing behavior observably, consistent with experimental results.⁶

Figure 3. The effects of different ion channels on the adapting firing behavior of RGCs. A prepulse precedes test stimulus. I₀ = 0 is the baseline stimulus, I_D is depolarized prepulse, I_H is hyperpolarized prepulse, I = 2.5 is the test stimulus, and λ = 3.0. (A) All the ion channels are included, with depolarized prepulse I_D = 4.0. (B) All the channels are included, with hyperpolarized prepulse I_H = -0.3. CandtoF) Without I_Ca, I_KCa, lowered I_K, and without I_A, respectively.

Blocking the A-type potassium channel leads to an obvious change in the firing behavior of RGCs (Fig. 3F). However, Wieck and Demb⁶ suggested that adding the 4-aminopyridine (4-AP) to specifically block the A-type potassium channels would not change the adapting firing behavior significantly. Thus, there is some inconsistency between model results and the experimental observations.

Tonic and phasic activity are mainly controlled by the collective actions of I_Na and I_K

A recent experimental study together with computer modeling of CA1 pyramidal cells suggested that transient sodium current and delayed-rectifier potassium current activate collectively in determining the response properties of CA1 pyramidal cells, especially the tonic firing activity.³⁰ Since our model contains a sodium current (the inactivating type) and the delayed-rectifier potassium current, whether they contribute to produce the tonic and phasic activities of RGCs was investigated.

Model results in Figure 4A–C show that the neuron exhibits typical tonic activity in response to small stimulus steps, while larger stimuli make the action potentials disappear in the late period of stimulus, consistent with a phenomenon called depolarization block (and comparable with the firing in CA1 pyramidal cells³⁰). This result induced by increased stimulus intensity, is quite similar to what was observed in previous experiments.^17,21

Figure 4. Simulated tonic and phasic firing behaviors of retinal ganglion cell under different stimulus intensity, λ = 1.5. (A-C) Tonic activity (g_Na = 200, g_K = 12 mS/cm²). (D-F) Phasic activity (g_Na = 118; g_K = 113 mS/cm²).

Results in Figure 4D–F show that the neuron generates phasic activity with only one spike during a small stimulus step (Fig. 4D), and that increased stimulus amplitude makes the neuron elicit more spikes (Fig. 4E–F). This trend of phasic activity induced by increased stimulus amplitude is also quite similar to previous experimental observations.^17,21

A more intuitive result is demonstrated in Figure 5A, which shows how the spike counts of tonic and phasic activities vary with increases of injected current. One can also observe from Figure 5A that the spike counts of phasic activities are significantly lower than those of tonic activities. Results in Figure 5B also show that phasic RGC always fire its first spike in a short latency, while tonic RGC has a long latency under weak stimulus, and a short latency under strong stimulus.

Figure 5. Properties of tonic and phasic activities with the increase of stimulus intensity. (A) Spike counts. (B) First spike latencies.

To further illustrate the roles of g_Na and g_K in shaping the tonic and phasic activity in depth, a dynamic map of the 2 firing patterns under different combinations of g_Na and g_K is shown in Figure 6. It is clear that for a constant stimulus (I = 2.8 μA/cm²), phasic activity is preferred when the values of g_Na and g_K are approximate, indicating that rapid accommodation in phasic activity is probably caused by the balanced activation of sodium current and potassium current. Tonic activity is more likely to emerge when g_Na is much bigger than g_K, suggesting that sustained firing is likely to be induced by the larger amplitude of sodium current (which is well recognized to be responsible for the generation of action potentials). It should be noted that the proportion of g_K relative to g_Na in mimicking the phasic activity is large, which may beyond the physiological range. As spike block needs a large proportion of outward potassium channels, and in our study, we only adjusted the conductance of one potassium channel (Delayed-rectifier type), thus, the g_K is much higher than its true value in balancing the effect of g_Na. This is a limitation of our model.

Figure 6. Dynamic map of tonic and phasic activities under different values of g_Na and g_K. Corresponding unit for the maximal conductance is mS/cm², I = 2.8 μA/cm², λ = 0.

Differential roles of magnesium concentration in the modulation of tonic and phasic activities

In the above sections, we discussed tonic and phasic activities elicited in response to constant current steps, without any simulated synaptic inputs. However, previous studies have suggested that firing behaviors of RGCs could be influenced by synaptic inputs,^2,3 especially the NMDA-type.^9-11 Thus, in this section, the tonic and phasic activities of RGCs are analyzed under simulated NMDA-type input instead of square current steps, to investigate whether these 2 kinds of input lead to distinct firing activities.

Since previous evidence has indicated that NMDA currents can be modulated by extracellular magnesium concentrations,^31-33 variations of synaptic conductance (${\overline{g}}_{NMDA}$) with respect to membrane voltage under 3 different magnesium concentrations are illustrated in Figure 7A. It is clear that when no magnesium is present, the value of ${\overline{g}}_{NMDA}$ is constant (according to the expression demonstrated in Box 1). However, when magnesium is present, the value of ${\overline{g}}_{NMDA}$ changes in a sigmoidal manner with respect to voltage, similar to reported results.³⁴ The variation of NMDA current with respect to voltage is also provided in Figure 7B. The manner in which variation of synaptic strength induced by magnesium influences the tonic and phasic activities of RGCs were further studied.

Figure 7. Variations of ${\overline{g}}_{NMDA}$ and I_NMDA with respect to membrane potential under different concentrations of magnesium ions, the maximal conductance is 0.5 mS/cm². (A) ${\overline{g}}_{NMDA}$ vs voltage. (B) I_NMDA vs voltage.

Box 1. Specific expressions of currents and the corresponding gating variables.

In the case of tonic activity, when the magnesium concentration is zero (Figure 8A), it is apparent that the increase of synaptic conductance (when large enough) induces the disappearance of action potentials in the late period of stimulus, which is similar to the results in Figure 4. Under a moderate level of magnesium (0.5 mM in Figure 8C1–D1), the tonic spike count decreases in response to a small synaptic conductance, but it increases under a large synaptic conductance. And when the level of magnesium is relatively high (1.0 mM in Fig. 8E1–F1), spike count in response to small synaptic conductance stimuli continues to decrease, while the spike count during large simulated synaptic conductances tends to decrease.

Figure 8. Modulation of tonic and phasic activities by magnesium concentration (mM) under different maximal synaptic conductances (mS/cm²), λ = 0. (A1-F1) Tonic activity. (A2-F2) Phasic activity. Variations of I_NMDA are shown in the bottom of each subgraphs, and the zero value of I_NMDA was adjusted to -80.

However, in the case of phasic activity, the presence of magnesium ions suppresses the firing of the neuron (Fig. 8A2–F2), regardless of the amplitude of simulated synaptic conductance. The variations of spike counts with the increase of maximal synaptic conductance under 3 different levels of magnesium concentration for the tonic and phasic activities is shown in Figure 9. From this, one can clearly see that the NMDA synaptic current, which is modulated by magnesium concentration, exhibits differential roles in modulating the dynamics of tonic and phasic activities.

Figure 9. Variations of spike counts with respect to g_NMDA under different magnesium concentrations. (A) Tonic activity. (B) Phasic activity.

Discussion

In the present study, we performed a computational investigation of the generation of tonic and phasic firing behaviors of RGCs using an ionic model, and analyzed the modulation of tonic and phasic activities by Mg²⁺ regulated NMDA synaptic currents. Our results show that the modified model provides insights regarding the possible ionic mechanisms underlying the initiation of these 2 firing activities. Our results also demonstrate that the magnesium dependence embedded in the NMDA-mediated synaptic current differentially influences the spike counts of tonic and phasic activities.

As 2 widely observed firing behaviors in neuronal systems, tonic and phasic firing activities have been found in neurons from different brain regions.^12-20 The functional roles for these 2 behaviors have also been subject to intensive investigations during the past decades. For instance, they possess prominent effects in the encoding of reward and punishment signals,^35,36 modulation of conditioned fear behaviors,³⁷ occupation of dopamine receptors,³⁸ mediation of behavioral conditioning,³⁹ and synaptic plasticity.^40,41 Since tonic and phasic firings encode important neural information, the underlying mechanisms should be clarified. Based on our model results, we infer that in RGCs, 2 currents, i.e., the inactivating sodium current and the delayed-rectifier potassium current, contribute collectively to the occurrence of tonic and phasic activities. Similar conclusions have been drawn in CA1 pyramidal cells, in which it was shown that tonic activity is triggered by the concerted actions of transient sodium currents and delayed-rectifier potassium currents.³⁰ However, an important distinction is that the sodium current we used in the RGC model is responsible for adapting activities of ganglion cells, whereas the sodium current in the CA1 pyramidal cell model does not induce adapting behaviors of pyramidal cells. Nevertheless, Bianchi et al.³⁰ also reported that adaptation currents in the pyramidal cell model exhibit some regulatory roles in the tonic activity, consequently, the inference we draw that two adaptation currents (inactivating sodium current and delayed-rectifier potassium current) contribute collectively to the generation of tonic and phasic activities in the RGCs, is reasonable.

Although our modified model has many good features, there is still a disadvantage that the model fails to reproduce the effect of I_A on influencing the adapting activity of RGCs. Weick and Demb⁶ applied 4-AP to specifically block I_A, however, other studies have shown that 4-AP cannot only block I_A, but can also affect calcium channels,⁴² calcium-dependent potassium channels,⁴³ and sodium channels.⁴,⁵⁴⁴ Thus, the effect of 4-AP may be more general and diffuse. In our model, we only altered the activation of I_A, and this might be the reason that our model result is inconsistent with the experimental observation.

NMDA receptors are a major receptor participating in the synaptic transmission of neural signals, and many neurons in the brain have already been found to express this receptor, e.g., midbrain dopaminergic cells,³⁴ hippocampal pyramidal neurons,⁴⁵ and RGCs.^9-11 The roles of magnesium ions in the activation of NMDA current have also been widely reported.^31-33 Thus, changes of magnesium concentration would lead to corresponding variations of NMDA synaptic current with subsequent influences on the firing behavior of postsynaptic neurons. In our model study, we find that magnesium concentration effects embedded in the NMDA synaptic current are vital in regulating the tonic and phasic activities of RGCs, and that the regulation is rather different for the 2 firing patterns.

As the sole output neurons in the retina, the activities of ganglion cells have attracted much attention for their significant roles in the encoding and transmission of visual signals, which may be manifested in a variety of firing patterns. The modified model we proposed in this study has successfully reproduced several firing behaviors, thus, it can be used as a good basis for simulating some other firing patterns, and to further uncover the possible mechanisms that may mediate those firing patterns.

Models and Methods

The model we proposed in this study was integrated from the FCM model and the KR model. Similar to the FCM model, our model mainly contains 5 voltage-gated ion channels, i.e., inactivating sodium (I_Na), delayed-rectifier potassium (I_K), calcium (I_Ca), A-type potassium (I_A), and calcium-activated potassium (I_KCa) channels. The expressions and parameters for the ion channels were adopted from the FCM model,^{5, 22} except for the inactivating sodium channel which was adopted from the KR model.⁸ In the last part of our study, the effects of synaptic input, particularly the NMDA-type was considered, and the expression of I_NMDA was adopted from reference 23.

Detailed description of the modified model is as following:

$$C_m\frac{dV}{dt}=-I_{Na}-I_K-I_{Ca}-I_A-I_{KCa}-I_{NMDA}-I_L+I+\lambda \xi (t)$$ (1)

where, C_m = 1 μF/cm² is the specific membrane capacitance, I is the stimulus current applied to the neuron. ξ(t) is the Gaussian white noise, with zero mean, and $\langle \xi \left(t\right)\xi \left(t-\tau \right)\rangle =\delta \left(\tau \right)$, δ(τ) is the delta function, λ is the noise intensity.

Specific expressions and gating variables for each currents are illustrated in Box 1, where g_Na, g_K, g_Ca, g_A, g_KCa, g_L are the maximal conductances for the corresponding ion channels, and their values are: 80, 12, 2.2, 36, 0.05, and 0.05 mS/cm² respectively; V_Na, V_K, V_Ca are the equilibrium potentials, and the values for V_Na, V_K are 35, -75 mV respectively, while the value of V_Ca is time-dependent (Eq. 2).

$$V_{Ca}=\frac{RT}{ZF}\ln \left(\frac{{\left[C{a}^{2+}\right]}_e}{{\left[C{a}^{2+}\right]}_i\left(t\right)}\right)$$ (2)

where R = 8.314 J/(M·K) is the gas constant, T = 295 K is the temperature in Kelvin, Z is the ionic valency, F = 96485 C/M is the Faraday constant, [Ca²⁺]_e = 1.8 mM is the concentration of extracellular calcium ions, and the variation of intracellular calcium ion concentration [Ca²⁺]_i obeys the Equation 3.⁵

$$\frac{d{\left[C{a}^{2+}\right]}_i}{dt}=\frac{-5I_{Ca}}{Fr}-\frac{{\left[C{a}^{2+}\right]}_i-{\left[C{a}^{2+}\right]}_{res}}{{\tau }_{Ca}}$$ (3)

where r = 22 μm means the depth of the shell beneath the membrane for the calcium pump, and τ_Ca is the time constant for calcium current, which value is 1.5 ms. The residual level of the free intracellular calcium ions is [Ca²⁺]_res = 0.001 mM, and the calcium dissociation constant is [Ca²⁺]_diss = 0.001 mM/dm³.

In the expression of I_Na, s₁ and s₂ are 2 slow variables which can mimic the inactivation of I_Na. As reported by Kim and Rieke, s₁ is voltage-dependent, while s₂ is spike-dependent.⁸

The voltage-dependent gating variables are described below:

$$\frac{dx}{dt}={\alpha }_x(1-x)-{\beta }_{x}x(x=m,h,n,c,a,b,s_1)$$ (4)

And the spike-dependent slow variable s₂ is described by the following equation:

$$\frac{d{s}_2}{dt}={\alpha }_{s_2}(1-s_2)$$ (5)

Except when the neuron fires a spike, the variable s₂ should be decreased by a factor of 0.77.⁸

In the expression of I_NMDA, ${\overline{g}}_{NMDA}$ is the synaptic conductance, and V_NMDA is the synaptic reversal potential, which is 0 mV. g_NMDA is the maximal conductance, and [Mg²⁺] is the extracellular magnesium concentration.

Simulations of the RGCs’ activities were performed in the MATLAB environment (R2010a), and the fourth-order Runge-Kutta algorithm was employed to calculate the voltage values of RGCs with time step of 0.01 ms.