Modeling Recovery of Rhythmic Activity: Hypothesis for the role of a calcium pump

Introduction

Networks that produce rhythmic patterns of activity are commonly involved in behaviors serving basic biological functions (e.g. breathing, mastication, etc). The existence of mechanisms of recovery of function may therefore be essential for survival. One example of such a system is the pyloric network of crustaceans. The pyloric network generates a rhythmic activity pattern that normally depends obligatorily on the actions of neuromodulatory substances released by axon terminals from adjacent ganglia onto the neurons of the network: after action potential transmission along these axons is blocked or destroyed (decentralization), rhythmic activity ceases [1]. However, activity recovers spontaneously within several hours [2-4] following a very complex temporal dynamical process that involves the alternating turning on and off of the rhythm (‘bouting’) [4]. This bouting activity can last several hours, after which a stable pyloric rhythm emerges that is characterized by a lower frequency than control but otherwise similar properties (Fig.1, [3, 4]). Experimental evidence indicates that STG neurons [5, 6], crayfish leg axons [7] and other neuronal types [8] possess feedback mechanisms that sense neuronal activity and can regulate specific ionic currents and pumps. Intracellular Ca⁺⁺, [Ca]_in, is a likely second messenger that can act as a feedback element for conductance regulation because [Ca]_in changes appear correlated with neuronal activity changes [8] and are involved in many intracellular signaling pathways. The recovery process of rhythmic pyloric activity has previously been accounted for theoretically in a simplified pyloric network model in terms of activity-dependent regulation of ionic currents [3]. However, the transition between the quiescent and the stable recovery states in this model was monotonic and it failed to explain the complex bouting dynamics that precedes full recovery.

Figure 1.

Experimental activity recovery after decentralization. Pyloric network activity during the recovery process after decentralization, achieved by cutting the single central input stomatogastric nerve (stn). The repeated turning on and off of the pyloric rhythm is termed “bouting” (inset). After ∼65hrs the pyloric frequency increases towards a stable level.

Here we reevaluate the mechanism of long-term activity-dependent regulation of conductances by introducing an intracellular molecular network of [Ca]_in regulation that involves pumping of Ca⁺⁺ into intracellular Ca⁺⁺ stores (eg. endoplasmic reticulum, ER) and inositol 1,4,5-trisphosphate (IP₃) receptor-dependent (IP₃RCa) Ca⁺⁺ release from the ER. Our model suggests that slow activity-dependent regulation of an intracellular Ca⁺⁺ pump is key to generating the complex temporal dynamics of recovery observed during the bouting period.

Methods

For simplicity we build a single isolated neuron using XPP [9], based on experiments showing that rhythmic activity recovery takes place in isolated neurons of the STG [4]. The model neuron consists of two compartments with a soma/neurite (S/N) compartment generating slow-wave oscillations and an axonal compartment generating Hodgkin & Huxley (H&H)-type action potentials (e.g. Fig. 2C). Ionic currents are modeled closely following those described by Golowasch et al. [3], with only two modifications: V_1/2 of m_Ca = −60.6 mV, and the Ca⁺⁺ equilibrium potential is updated at every integration cycle according to E_Ca = RT/zF ln ([Ca]_out /[Ca]_in). Temperature=10°C and [Ca]_out=13 mM. In summary, the S/N compartment is composed of a leakage current, a voltage-gated Ca⁺⁺ current (I_Ca), a voltage-gated K current (I_K), an A type K current (I_A), a voltage-dependent peptide-activate current found in STG neurons (I_P, see [10]) and a current flowing symmetrically between the S/N and axonal compartments. Thus, the S/N membrane potential (V_m) is given by:

Figure 2.

Model properties. A. Schematic diagram of intracellular Ca⁺⁺- dependent conductance and Ca⁺⁺ pump regulation. G_K and G_Ca are the conductances of I_Ca and I_K, respectively. S_1g, S_2g and S_p are the Ca⁺⁺ sensors, IP₃RCa is the activated IP₃ receptor/channel, and Ca pump is the sole intracellular Ca⁺⁺ uptake process. ER represents a generic intracellular Ca⁺⁺ store. The shaded arrow labeled Diff represents Ca⁺⁺ diffusion. B. Time course of model's membrane potential with definitions of measured properties. Decentralization: I_P=0. Bottom panels show voltage traces from A at higher magnification. Left: Control trace before decentralization. Middle: During bouting. Right: During stable recovery. C. Magnified trace during control, bouting and stable recovery.

$$C_{m}d{V}_m/dt=-(I_{\text{Ca}}+I_K+I_A+I_P+I_{\text{leak}}+I_{\text{axon}})$$ [1]

[Ca]_in is used as a feedback element whose concentration is related to neuronal activity: as activity increases, [Ca]_in increases due to raised Ca⁺⁺ influx through I_Ca. [Ca]_in depends on four terms: calcium influx via I_Ca, calcium diffusion, and intracellular calcium pump plus IP₃-sensitive calcium receptor-channel (IP₃R), both on an intracellular compartment (ER):

$$d{[\text{Ca}]}_{\text{in}}/dt=-\gamma I_{\text{Ca}}-D{[\text{Ca}]}_{\text{in}}-F_{\text{pump}}({[\text{Ca}]}_{\text{in}})+F_{\text{IP}3R}({[\text{Ca}]}_{\text{in}})$$ [2]

[Ca]_in is calculated in a thin cytoplasmic shell adjacent to the plasma membrane. Diffusion removes Ca⁺⁺ out of this shell (D=0.00017 msec⁻¹) and diffusion within the shell is ignored. An intracellular Ca⁺⁺ storage compartment (ER) is evenly distributed and adjacent to the cytoplasmic shell [11]. The charge-to-concentration conversion factor γ (0.00678 μMnA⁻¹msec⁻¹) = f/(zFV_Shell), where f is the cytoplasmic buffering coefficient (0.05), z is the valence (+2), F is Faraday's constant, and V_shell is the volume of a 2 μm thick shell and 80 μm neuron diameter (3.8216*10⁻¹¹ lt). F_IP3R([Ca]_in) represents the calcium release from the ER into the cytoplasm through IP₃Rs [12]:

$$F_{{\text{IP}}_{3}R}([{\text{Ca}}_{\text{in}}])=r({\text{g}}_0+{\text{g}}_1{[{\text{IP}}_3\text{RCa}]}^4){({[\text{Ca}]}_{\text{ER}}-{[\text{Ca}]}_{\text{in}})}^{+}$$ [3]

IP₃R is a tetramer and activation requires binding of one calcium ion to each subunit. IP₃RCa is the open state. Collectively IP₃R channels will release calcium out of ER store with rate g₁ (0.00357 msec⁻¹) but, even if all channels close, a small calcium leak conductance g₀ (0.000286 msec⁻¹) remains. r (0.6) is the ratio of ER volume to cytoplasmic volume, [Ca]_ER is assumed to be constant and ∼10 times larger than the total intracellular Ca concentration (=3.56 μM). The term ([Ca]_ER − [Ca]_in)⁺ means that if [Ca]_ER > [Ca]_in Ca⁺⁺ will flow out of the ER through activated IP₃RCa channels, otherwise the channels are closed. [IP₃RCa] represents the fraction of active IP₃ receptors, and the details of channel opening rate d[IP₃RCa]/dt is exactly as given in [12].

F_pump([Ca]_in) represents calcium uptake by a Ca⁺⁺ pump into the ER [13]:

$$F_{\text{pump}}({[\text{Ca}]}_{\text{in}})=R_{\text{pump}}\frac{{{\displaystyle {[\text{Ca}]}_{\text{in}}}}^2}{{{\displaystyle {[\text{Ca}]}_{\text{in}}}}^2+{{\displaystyle {\alpha }_{\text{pump}}}}^2}$$ [4]

R_pump is the calcium pump rate (in μMmsec⁻¹) and is regulated by activity (see Fig. 2A and below). α_pump is the Ca⁺⁺ concentration for half maximal pump rate (0.2 μM).

Intracellular Ca⁺⁺ regulates 3 sensors, S_1g, S_2g and S_p, which represent Ca-sensitive signaling pathways. S_1g, S_2g regulate the maximum conductances of I_Ca and I_K, $\overline{{\text{G}}_{\text{Ca}}}$ and $\overline{G_K}$, and S_p regulates the Ca⁺⁺ pump activity (R_pump) (Fig. 2A). These sensors are modified from Liu et al. [14] and described by:

$$\begin{array}{ccc}{S}_{1g}=\overline{S_{1g}}{{\displaystyle M_{1g}}}^4{H}_{1g}\hspace{0.17em},& S_{2g}=\overline{S_{2g}}{{\displaystyle M_{2g}}}^2\hspace{0.17em},& S_p=\overline{S_p}{{\displaystyle M_p}}^2\end{array}$$ [5]

The terms $\overline{S_{ig,\hspace{0.17em}p}}$ are maximum values ( $\overline{S_{1g}}=20.5,\hspace{0.17em}\overline{S_{2g}}=\overline{S_p}=1$), and M and H are sigmoidal functions of [Ca]_in: $M_x=1/(1+e^{M_{\text{thr}-x}-{[\text{Ca}]}_{in}})$, $H_{1g}=1/(1+e^{{[\text{Ca}]}_{in}-H_{\text{thr}}})$. M_thr-1,2g represent [Ca]_in thresholds of sensor activation (M_thr-1g=0.9, M_thr-2g=0.5, M_thr-p=0.01, H_thr=0.45 μM). Because of the term H_1g, the S_1g is sensitive to a [Ca]_in range with a maximum at [Ca]_in=0.685 μM.

$\overline{{\text{G}}_{\text{Ca}}}$ and $\overline{G_K}$ are regulated by sensors S_1g and S_2g, but in opposite directions to allow a homeostatic regulation of I_Ca and I_K, implying that when [Ca]_in increases $\overline{{\text{G}}_{\text{Ca}}}$ will tend to decrease and $\overline{G_K}$ will tend to increase, and vice-versa:

$$\begin{array}{l}{\tau }_{\text{g}}d\overline{G_{\text{Ca}}}/dt=\overline{G_{\text{Ca}}}\{(S_{1\text{g-eq}}-S_{1g})+(S_{2\text{g-eq}}-S_{2g})\}\\ {\tau }_{\text{g}}d\overline{G_K}/dt=-\overline{G_K}\{(S_{1\text{g-eq}}-S_{1g})+(S_{2\text{g-eq}}-S_{2g})\}\end{array}$$ [6]

$\overline{{\text{G}}_{\text{Ca}}}$ and $\overline{G_K}$ vary slowly (τ_g=100000) while both sensors are away from equilibrium with their time averaged steady-state equilibrium values (S_1g-eq = S_2g-eq=0.2). The sensors used here differ from those used in the previous model [3] in that they allow the values of $\overline{{\text{G}}_{\text{Ca}}}$ and $\overline{G_K}$ to vary without bounds. This method is more consistent with observations of the biological STG neurons, which show large variability of conductances while activity remains largely preserved [14].

R_pump is regulated by S_p according to:

$${\tau }_{\text{P}}{dR}_{\text{pump}}/dt=R_{\text{pump}}(S_{\text{p-eq}}-S_p)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{S}_{\text{p-eq}}(R_{\text{pump}})=S+\frac{2(W-S)}{1+e^{(R_{\text{pump-Thr}}-R_{\text{pump}})/z}}$$ [7]

where τ_p is the time constant of this process (the slowest time constant in the system, τ_p=500000). The time courses of change of S_1g, S_2g and S_p as functions of [Ca]_in are much faster than those of either $\overline{{\text{G}}_{\text{Ca}}}$ and $\overline{G_K}$ or R_pump and are assumed to be instantaneous. While S_1g-eq and S_2g-eq are assumed to be constant, we have found that S_p-eq is required to depend on R_pump. When R_pump is much lower than the threshold pump rate value (R_pump-Thr=4.5 nMmsec⁻¹), S_p-eq approaches its maximum value S=0.353. However, when R_pump approaches R_pump-Thr, S_p-eq tends to its lower bound value W=0.303 (z =0.0001 μMmsec⁻¹). Time has arbitrary units, hence is unitless.

Results

The single-cell model, with all ionic currents activated, including the neuromodulator-activated current I_P, generates stable bursting activity (Fig. 2B, bottom left and 2C, top trace).

There are several signature features of the process of pyloric activity recovery after decentralization [4]: 1) The time required to produce the first bout after decentralization on average takes >3hrs; 2) The average period of bouting activity before stable recovery takes >52hrs; 3) The average bout duration is 104 ± 63 sec; 4) The average time between individual bouts (i.e. interbout period) is 23.2 ± 38.1 min. Consequently, we used the following criteria to develop a model that correctly reproduced the process of activity recovery: 1) the period of bouting lasted ≥10 times the duration of the silent period immediately after decentralization; 2) the individual bout duration was shorter than the interbout duration; 3) the average pyloric frequency of the stable recovered rhythm is lower than the control frequency.

Derivation of successful model

Golowasch et al [3] used a single sensor of activity in the form of a sigmoidal function of the difference between Ca⁺⁺ influx and a set cytoplasmic Ca⁺⁺ influx equilibrium level. This proved enough to reproduce the recovery of pyloric activity after decentralization but no bouting activity could be observed before the activity stably recovered. We reasoned that a second slower intracellular process had to take place that interacts intermittently with the regulation of ionic currents. The operation of a Ca⁺⁺ pump and of an IP₃- and Ca⁺⁺-activated Ca⁺⁺ channel in intracellular compartments provides a mechanism that explains the observed slow [Ca]_in oscillations in several systems [12, 15]. Because the original model of activity recovery [3] relied on the regulation of ionic conductances by a slow Ca⁺⁺-dependent mechanism, [Ca]_in oscillations may be sufficient to generate bouting of activity. We incorporated Othmer and Tang's [12] model of [Ca]_in oscillations (described by F_pump([Ca]_in) and F_IP3R([Ca]_in) in Methods, eqs. 2-4, Fig. 2A) to the original model and we did indeed observe bouts, however they had durations >10 times longer than the average interbouts, bouts were ∼2 times longer than the initial silent period before bouting began, and no stable recovery of activity could be produced in spite of extensive parameter search.

In a second approach we replaced the simple sigmoidal function of Ca⁺⁺ flux of the original model [3] with two sensors of [Ca]_in¸ S_1g and S_2g, representing Ca⁺⁺-dependent enzymatic pathways that feed back and regulate exclusively the ionic currents I_Ca and I_K in the manner described in Methods and represented in Fig. 2A but without sensor S_p. This form allows the maximal conductances of these currents to vary without bounds [14]. Thus, we reasoned, overshoots and undershoots of these conductances as [Ca]_in oscillates may generate bouting activity. This model indeed generates bouting activity, with bout durations similar but not shorter than interbout durations, and much shorter than the silent period immediately following decentralization. Nevertheless, no recovery of stable bursting activity followed the bouting period. Stable activity could be obtained by either a) increasing the calcium pump rate, R_pump, b) decreasing the IP₃ receptor/channel conductance, g₁, c) decreasing the rate constants leading to, or d) increasing the rate constants leaving, the conducting state of the IP₃ receptor/channel. However, all these modifications lead to bout/interbout duration ratios >1. Conversely, changes in the opposite sense to the parameters a)-d) just mentioned resulted in a lowering of the bout/interbout duration ratio to values <1 but stable recovery is lost.

In summary, the analysis described above for our second model shows that the calcium pump rate R_pump is a sensitive parameter in regulating recovery of rhythmic activity. Small increases of R_pump result in a monotonic increase of rhythmic activity until stable activity is recovered with no intervening bouting activity, while small decreases of R_pump restores bouting activity that qualitatively matched the temporal properties of experimentally recorded bouts but destroys stable activity recovery.

Successful model

The general features of the final model's activity as described by Eqs. 1-7 are shown in Fig. 2B and incorporate an intracellular Ca⁺⁺ storage/release mechanism in which the uptake is regulated by activity through a Ca-dependent sensor [14]. The regular bursting activity observed before decentralization (Fig. 2B, lower left panel, and Fig. 2C, top trace) rapidly ceases after the maximum conductance of I_P is set to zero (decentralization). After a silent period in which the membrane potential gradually increases (“time to first bout”), bouting activity ensues, which lasts for a period significantly longer than the initial silent period, satisfying Criterion 1 (Fig 2B, top and lower middle panel). During this bouting period, bursting activity transiently resumes (Fig. 2B, bottom middle panel, and Fig. 2C, middle trace) and shows a progression of burst duration. In the beginning stage of the bouting phase, bouts are significantly shorter than the interbouts, satisfying Criterion 2 (bout duration=0.25 ± 0.02; interbout duration 0.49 ± 0.01; bout/interbout ratio=0.51), but later the ratio gradually increases to 1.45 before stable activity resumes. Finally, after stable recovery of rhythmic activity, the mean oscillation period (0.027) is longer than the control period (0.023), satisfying Criterion 3.

We performed a sensitivity analysis of the different parameters in the regulatory paths of the model. Large variations of the conductance parameters (g₀ and g_1; eq. 3) have little effect on recovery. $\overline{S_{ig,\hspace{0.17em}p}}$ values (eq. 5) can only be varied by approximately ±5% without completely disrupting the bouting properties; within this window of ±5%, bout/interbout ratio is slightly affected. The sensor terms M_thr-xg and S_xg-eq (eqs. 5-6) are functionally coupled: modification of one must accompany opposite changes of the other. With changes larger than ±10% from the listed values (Methods) stable recovery is never obtained. The sensitivity of the recovery process to the properties of the Ca⁺⁺ pump mentioned before dictates the complex relationship between [Ca]_in and R_pump (Eqs. 7). The S_p-eq(R_pump) value varies between S, immediately after decentralization (when the pump activity is low), and W shortly before stable recovery (when the pump activity is high). Consequently, the duration of the bouting period (but not of either bout or interbout duration, or the time to first bout) is sensitively related to how fast S_p-eq(R_pump) shifts from S to W.

The other parameters that sensitively regulate the temporal properties of activity recovery after decentralization are τ_g and τ_p (Fig. 3). The time constant of conductance regulation, τ_g, strongly and inversely affects the duration of the bouting period (Fig. 3A), and directly affects the individual bout and interbout durations (Fig. 3B). The time constant of pump regulation, τ_p, directly affects the duration of bouting (Fig. 3C) and the interbout duration (Fig. 3D), but inversely affects the individual bout duration (Fig. 3D). While τ_p has no effect on the time to first bout (Fig. 3C), τ_g has a slight and direct effect (Fig. 3A).

Figure 3.

Effects of sensor time constants on activity recovery. A and C show the effects of τ_g (A) and τ_p (C) on the duration of the silent interval between decentralization and the first bout, and on the duration of the bouting period until stable activity appears (see Fig. 2B for definitions). B and D show the effects of τ_g (B) and τ_p (D) on the duration of individual bouts and on the interval between bouts. τ_g is expressed in units of τ_p (τ_p =300000), and τ_p in units of τ_g (τ_g=100000).

Discussion

We have developed a model to explain the complex temporal dynamics of the pyloric network activity following the removal of central input to the STG. We assume that STG neurons recover rhythmic activity as they sense changes in their activity [3, 6, 8]. [Ca]_in changes regulate three Ca⁺⁺ sensors acting as activity transducers (representing Ca-dependent metabolic pathways) to regulate the maximal conductances of Ca⁺⁺ and K⁺ currents, and the intracellular Ca⁺⁺ pump activity. Parameter sensitivity analysis indicates that the kinetics of regulation of both ionic conductances and Ca⁺⁺ pump activity, not the ionic conductance values and the Ca⁺⁺ pump rate themselves, are the most critical parameters in determining the dynamic temporal properties of rhythmic pyloric activity recovery.

The mechanism of activity recovery can be explained as follows: decentralization removes an inward current (I_P) that helps sustain membrane potential oscillations, which are partly due to activation of I_Ca. With I_Ca now inactive, [Ca]_in drops and rapid activation of all three sensors occurs. As a consequence, $\overline{{\text{G}}_{\text{Ca}}}$ increases (and $\overline{G_K}$ decreases) with time constant τ_g until the new balance of I_Ca and I_K transiently enables the generation of bursting activity and bouting ensues (Fig. 4A, arrows). R_pump also increases with similar rate until a point when this increase decelerates (Fig. 4B, arrow). This point corresponds to the beginning of bouting activity, which is marked by an overshoot of $\overline{{\text{G}}_{\text{Ca}}}$ (Fig. 4A, arrow) and of [Ca]_in (which before decentralization is on average <1 μM) (Fig. 4C, top). The increase in [Ca]_in that results from I_Ca activation leads to enhanced sensor activity (Fig. 4C, middle) and a consequent decrease in I_Ca conductance (Fig. 4C, bottom), which terminates the oscillations after a few cycles (Fig. 2B and 4C). During the interbout, [Ca]_in remain low and both R_pump and $\overline{{\text{G}}_{\text{Ca}}}$ increases (and $\overline{G_K}$ decreases) again until a new bout begins. Thus, the key to the generation of bouting activity is the nature of the activity sensors we use, which are based on biologically more realistic assumptions [14] than used previously [3]. However, the key to a recovery of stable rhythmic activity lies in the gradual increase of R_pump (Fig. 4B), which by the end of the bouting period has increased enough to effectively control the changes in [Ca]_in on a cycle-by-cycle basis, preventing the overshoots that characterize the bouting period (Fig. 4A), allowing stable periodic oscillations of the membrane potential.

Figure 4.

Regulation of model variables after decentralization. A ${\overline{\text{G}}}_{\text{Ca}}$ and ${\overline{G}}_K$ changes during the entire recording period. B. Change of R_pump during the entire recording period. Decentralization happens at time=0. Arrow marks the beginning of the bouting period. C. Changes of [Ca]_in, S_1g and ${\overline{\text{G}}}_{\text{Ca}}$ during bouts and interbouts of the bouting period.

The regularity of our model's bouting activity contrasts with the biological system's variability. We believe this to be due to the extreme simplicity of the model, in which many signaling paths probably involved and intrinsic variability of the components (i.e. noise) has been left out in favor of an understanding of basic principles. Hence, our model predicts that decentralization leads to 1) a significant increase in intracellular Ca⁺⁺ pump activity that plays a crucial role in the recovery of stable activity; 2) a rapid increase in I_Ca and intermittent down- and up-regulation of G_Ca as the Ca⁺⁺ pump activity gradually increases (simultaneously I_K shows an opposite trend to I_Ca), which accounts for the bouting activity. These testable predictions will be evaluated experimentally.

Biography

Yili Zhang is a graduate student in Biological Sciences (Computational Biology) Department at Rutgers University. She obtained a first Master of Science in Pathology from National University of Singapore in 2000 and a Master of Science in Computer Science from Virginia Polytechnic Institute in 2002.

Jorge Golowasch received his PhD in Biophysics from Brandeis University. He is currently an associate professor in Mathematical Sciences at New Jersey Institute of Technology and Biological Sciences at Rutgers University.