Conductance relationships across compartments associated with bursting output in a model of a motor neuron in the crustacean cardiac ganglion

Abstract

The crustacean cardiac ganglion network coordinates rhythmic contractions of the heart muscle to control the circulation of blood. The specific network of the crab (Cancer borealis) consists of 9 cells: 5 large cell motor neurons (LC1–5) and 4 small endogenous pacemaker cells (SCs). We report a new three-compartmental biophysical LC model that includes synaptic inputs from SCs onto gap-junction coupled spike-initiation-zone (SIZ) compartments. To determine physiologically viable LC models in this realistic configuration, we sampled maximal conductances from a biologically constrained 9D-parameter space, followed by a selection protocol that had three levels. Our results provide previously unknown structure-function insights related to the crustacean cardiac ganglion large cell, including predictions about morphology, SIZ, and the differential roles of active conductances in the three compartments. An analysis of conductance relationships in model neurons revealed a lack of notable correlations among active conductances in the model population, despite clear reports of such relationships in biological neurons. When combined with the interpretations from other model systems, we hypothesize that modes of bursting driven by a strong presynaptic influence (i.e., “forced” bursting) may not require such conductance relationships, whereas endogenous bursters may require them. We further suggest that conductance relationships in a forced burster neuron will more likely serve to shape the characteristics of the firing pattern in the burst, once generated, rather than contribute to a generative mechanism for bursting itself.

Graphical Abstract

News & Noteworthy:

Our results support the hypotheses that, despite being prevalent in biological neurons, correlated membrane conductances are not a necessary feature of synaptically-driven bursting in CPG motor neurons and averaging conductances across motor neurons in a network can maintain biologically realistic output. Our results may be most relevant to similar networks, where “forced bursting” is a result of highly interconnected neurons with strong synaptic input organizing the output.

INTRODUCTION

Neurons are endowed with a rich and complex set of intrinsic and synaptic conductances that control their electrical activity [1; 2]. Although the role of such a varied set of conductances is not fully understood, it is natural to expect that neurons of the same cell type would possess similar active and passive properties, especially within the same animal. However, experimental findings suggest that maximal conductance levels of individual currents can vary two- to six-fold among same cell types, even within the same animal [3–7] and that different combinations of conductances preserve activity at the single cell level [8; 9]. Computational modeling continues to shed light on the role of such conductance variations in conserving cellular output [2; 10–20]. For instance, Prinz et al. [2] explored the maximal conductance space of a single-compartment model neuron to quantify the types of spiking and bursting models possible, and showed that similar patterns of activity could be produced by many different parameter sets, both for single neurons [2] and within small networks [21].

Beyond the broad range of conductance combinations that are associated with convergent outputs among neurons of the same type, there are also numerous reports that, among populations of neurons, different sets of ionic conductances [5] and ion channel mRNA levels [7; 22] can be correlated in a cell-type specific manner. This suggests that an on-going, rather than developmentally fixed, regulation of specific sets of conductances may be necessary to provide stable output of neurons and networks over the lifetime of an animal. These correlated mRNA and conductance levels can arise from a relatively simple set of feedback control algorithms in computational models [23]. However, there have been few studies that directly demonstrate that these conductance or mRNA relationships are necessary to generate appropriate, ongoing neuronal activity in biological neurons. Indeed, compelling computational work has demonstrated that – at least theoretically – such relationships are not necessary to generate robust output in a population of model neurons. For example, one such study demonstrated that using a model selection methodology focused on single-cell output in a multi-compartment model results in a population of cells with only weak or no correlations among conductances [14]. However, most studies to date have focused on selecting models based on isolated neuron activity. Therefore, to further extend these analyses, we performed multiple levels of selection that included generating model networks with multiple neurons of a given type and only selecting cells that perform within biological parameters of the full network output for inclusion in our population. We then searched for conductance correlations among the populations of neurons in our simulated model networks.

For this work, we used a conductance-based model of the crustacean cardiac ganglion (CG) network, based on the crab, Cancer borealis. This relatively simple network consists of 4 pacemaker neurons (Small Cells, SCs) and 5 Large Cell (LC) motor neurons that innervate the heart muscle. The present study extends previous computational investigations that focused on a two-compartment cardiac ganglion LC [15; 16; 21] by considering the potential role of conductances in multiple compartments of an LC on the spiking output of the SIZ. Specifically, we developed a new more morphologically complex three-compartment LC model of the CG network in Cancer borealis that incorporates a compartment that both receives the presynaptic inputs of pacemaker neurons and acts as the spike-initiation zone to integrate those inputs into firing output. We then investigated how the distribution as well as potential covariations of intrinsic conductances affected this output (defined henceforth as the spike and burst characteristics of the SIZ). Using a rejection sampling approach with a 10-D parameter space of maximal conductances, we report, as in previous studies [15; 16], an unbiased approach to determine the role of various conductances in shaping cellular and network function.

We extended previous studies by performing model neuron selection in complete CG networks, with constraints drawn from intact network activity as well as from single cell electrophysiology and responses to current injection data. This population of model LCs then provided predictions related to the differential roles of conductances in the soma vs. neurite in shaping neuronal output. This included an explanation for the variable cellular voltage responses seen in LCs with application of the selective potassium channel blocker tetraethyl ammonium (TEA) [24]. Furthermore, this population of LCs pooled across all networks allowed us to explore emergent conductance relationships both within and across these compartments. Finally, we compared these correlations with those generated from an alternative approach where LC conductances were first averaged within a network, new constituent model neurons generated from averaged conductances, and then simulated, re-selected and pooled across averaged networks.

RESULTS

Morphologically realistic LC model and SC stimulus reproduces experimental profiles

Building on our previous two-compartment LC model [15; 16; 21], we added morphological realism to the LC by adding a third compartment, the spike initiation zone (SIZ; Fig. 1A). We chose to the model the SIZ as a multi-function compartment (see Methods), serving as both the spike initiation and the site of presynaptic input from pacemaker neurons, as well as the site of electrical coupling among all 5 LCs of the network (Fig. 1A).

Figure 1.

Biologically realistic model of intact LC reproduces experimental data. (A) Schematic of the biological cardiac ganglion the crab (Cancer borealis) showing the Large Cells (LC) 1–5 as well as the Small Cell (SC) pacemakers. (B) Simultaneous biological membrane potential recordings from the somata of 5 LCs of the same ganglion as well as a time matched extracellular recording of the nerve trunk showing network activity and individual LC and SC spikes as indicated. Vertical scale bar for voltage is 2 mV and horizontal bar for time is 1 s. (C) Biophysical model of an LC with three compartments and conductances. (D) Network model with 5 LCs. Gap junctions (resistor symbols) exist between LC4 and LC5 [35], and between LC1 and LC2 [35], and we inferred that all SIZs share gap junctions with each other. SC input goes to the SIZ of every cell. Right panel shows control recordings from the network model for all five cells. Vertical scale bar is 10 mV.

We first matched passive properties (see methods) and waveform data from intact cells (Fig. 1A,B: expt., and Fig. 1C,D: model); [21]). We note that among the five biological LCs, only LC3–5 are easily accessible, and that LC4 is gap-junction coupled strongly to LC5 at their somata [24]. Our biological data (Fig. 1B) are from LCs 1–5, and our model predicts network performance for all five LCs, assuming LCs1–2 have similar gap junction coupling at their somata as do LC4–5. As shown in Fig. 1A, all five LCs are gap junction coupled at the SIZ.

Developing selection criteria from experimental data.

We briefly describe the key characteristics of a modified version of our previous selection protocol [21] to select model LCs that match experimental data. For additional details see Methods.

SELECTION LEVEL 1.

In the first selection level of the overall rejection sampling protocol, we sampled a 10-D parameter space of maximal conductances to generate a pool of ligated LCs (soma + limited neurite with passive conductances; henceforth termed Level-1 cell) for which the passive properties of resting potential and input resistance were within measured ranges for cells which then went on to have voltage clamp measurements of active conductances made in previously published studies [21; 24; 25].

SELECTION LEVEL 2.

To Level-1 cells (those that passed level-1 selection), we then added active conductances in the neurite (picked randomly – see below) and added a SIZ compartment (with fixed conductances). To this model, we added a synapse to the SIZ compartment and provided synaptic input from the SC with spike frequency that varied within the experimentally observed range of 16–32 Hz and retained only cells that had at least one spike with the SC input (see Table 3 in Methods). This ensured that selection level 2 did not pass cells that had membrane potential depolarizations but no spikes, reducing the load on the computationally intensive selection level 3. To verify the need for level 2, we also ran a separate trial by skipping level 2, which produced no passing networks. This was because even one cell in the 5-cell network that did not produce spikes could cause the entire network to fail. Cells that passed level 2 were termed ‘Level-2’ cells.

Table 3.

Connection and input parameters in levels 2 and 3 of the selection protocol

Parameter	Value
Exp2Syn Tau1	10 ms
Exp2Syn Tau2	120 ms
Exp2Syn reversal potential	−15 mV
NetCon weight	0.09
VecStim spiking frequency	16Hz to 32Hz
Input resistance of soma	1.54 MΩ
Input resistance of SIZ	200 MΩ

SELECTION LEVEL 3.

In selection level 3, we combined five random Level-2 cells into a network (see methods for more details). The network itself is deemed viable if it satisfies the waveform selection criteria of level 2, as well as the two experimental observations related to LC3 and LC5: (i) a synchrony value between waveforms of LC3 and LC5 >0.95 in control (Pearson’s R-squared); and (ii) the same value <0.89 in TEA (see methods). Results from each of these selection levels are discussed next. Cells that passed level 3 were termed ‘Level-3’ cells.

Developing morphology of the single cell for Level-2 screening:

The Level-1 model cells were tested for passive properties based on experimental data [21; 24]. From among 20,000 ligated model cells with random conductances selected from the 10-D parameter space (see methods), 15,161 passed selection level 1 of the protocol. For tests in levels 2 and 3, we designed an SC input to represent experimentally determined SC input characteristics (as described next) and then provided that to the synapse attached to the SIZ for one burst cycle of period 2.1 sec. As seen in Fig. 1B the inter-burst interval (IBI) in experiments is ~2 sec. We confirmed that all relevant currents in the model were deinactivated in that duration. Studying the role of the inactivating currents for shorter IBI values is a topic for future research.

Designing the SC input to LCs:

In the crustacean CG, the five-cell LC network receives input from a cluster of four small cells, and we assumed that all five LCs receive a common synchronized input from this SC cell cluster due to lack of more detailed information about the SC cluster. Figure 1B shows simultaneous experimental recordings for all five LCs as well as SC and LC spiking. We designed the SC input as a spike train to synapses on the SIZ, the properties of which are tuned to mimic experimental voltage responses in the SIZ [26]. Analysis of experimental recordings initially collected for other studies ([21; 24]) showed that the spike frequency of SC input varied across individual animals between 16 and 32 Hz (unpublished data). Also, two components of the burst pattern were noted in the experimental data over a typical duration of 1000 ms: a steady one that continued over the entire duration, and a second one that lasted for 600 ms, starting from 300 ms and ending at 900 ms. As described in methods, we designed the SC input with the two components and tested the cell (in Level-2) or network (in Level-3) with a random spike frequency selected within every 1 Hz range from 16–32 Hz, e.g., 16.1 within 16–17, 17.3 within17–18 Hz, and so on (see methods and Fig. S1).

Matching responses of intact single cells.

To make the foregoing analysis tractable, we considered the case where the intact cells in a network did not receive input from the other four model LCs, i.e., all gap junction coupling between the LCs was disconnected (e.g., schematic in Fig. 1C). We consider the SIZ gap-junction coupled case in a later section.

For Level-1 cells, we initially assumed a passive neurite, i.e., only leak conductances in the neurite. To repeat, for each Level-1 cell we added a SIZ and synapse, and provided the SC spike train input described in the previous section. Interestingly, none of the Level-1 cells were able to pass selection level 2. This was because the cells had a spiking frequency above 8 Hz in control and did not exhibit a TEA response. However, the SIZ responses did match biological reports. Specifically, the model membrane potential responses at the soma had a depolarization of 10 mV for 1000 ms, and with spikes on top of the depolarization that reached 20 mV in height. This response matched the soma membrane potential response characteristics from our lab that had a depolarization bump of 10 mV for 1000 ms, and spike height of 15 mV (Fig. 1B, scale bars are 10 mV). Additionally, the model SIZ spike height (Fig. 3C) attenuated by a factor of 4 (60 to 15 mV) (Fig. 3C vs Fig. 2A), matching the experimental SIZ recording in [26].

Figure 3.

Parameter space of network model output for the 650 cells that passed selection level 3. (A): Parameter space of three electrophysiological features of the output for all cells, with gap junction coupling present: Spikes per Burst, Peak Height (mV) of the action potential, and Area Under the Curve (AUC) of the voltage waveform (mV²). The four colored circles (purple, red, green, blue) indicate different networks which represent extremes of the parameter space. (B): Voltage traces for the network output for cases represented by the networks corresponding to the four colored circles in the top figure, showing the variations in output features with vs without gap junction coupling. Scale bars represent 10 mV. (C): Representative traces from one model of the synaptic current inputs across one burst activity, including the output (V_M) trace of the cell, synaptic current (to SIZ of LC1; I_SYN) and gap junction currents in the soma (from LC2 to LC1; I_GAP-SOMA) and SIZ (from LC2 to LC1; I_GAP-SIZ).

Figure 2.

Model predicts presence and roles of active conductances in neurite. (A) Control and TEA response of model network shown in Fig. 1D. (B) A different representative model neuron voltage response and simultaneous underlying current for all three compartments in control and TEA conditions.

The functional reason for the cells failing in selection level 2 was determined to be the excessive hyperpolarizing leak through the neurite, i.e., although sufficient current entered the neurite from the SIZ, this leakage diminished the amount that reached the soma for raising the response in TEA. To explore this further, we determined the electrotonic drop from the SIZ to the soma and found it to be 0.57 indicating that the soma was not too distant electrotonically. We also explored reducing the diameter, and therefore surface area, to decrease leakage. However, the neurite diameter had to be 5 μm to accomplish this, which was below the biologically estimated minimum diameter value of ~10 μm. So, as the logical next step, we considered the addition of active conductances in the neurite.

In summary, a three-compartment biophysical model (Fig. 1 C) was developed that was constrained by biological single cell data as well as network data, via a three-level selection process. This enabled selection of random cells for the development of networks used for insights as outlined in later sections.

Active conductances necessary in neurite to reproduce experimental network responses

While the presence or role of active conductances in the neurite of the CG LC is not well understood, calcium (Ca²⁺) currents have been measured from ligated somata plus neurites of LCs, and voltage-gated Ca²⁺ channels have been localized with immunohistochemistry to both the somata and the neurites [27]. Further, although multiple potassium (K⁺) and persistent sodium (NaP) currents have been measured in ligated somata plus neurites of LCs [27], the sub-cellular localization of these channels is unknown.

Role of individual conductances.

We adopted a systematic procedure to determine a parsimonious set of active conductances in the neurite (Fig. S2A-H). For this, we first inserted only NaP in the neurite, and this helped counter the excessive leakage cited in the previous section, enabling the soma to generate a TEA response with spiking in the SIZ. With NaP, for every 1000 cells that passed selection level 1, about 35 passed selection level 2. Few cells passed because the spike height in the soma and LC spike frequency were both found to exceed the upper bounds in the control case. For instance, for a case with SC spike frequency of 22 Hz (which is low in the range of SC frequencies recorded by the Schulz Lab; unpublished data), the soma spike height was above the upper bound of 30 mV as was the spike frequency, in many of the control cases. The reason for this was that the cell was already close to excitable in the control case with the passive neurite. The key attribute that NaP provided was a TEA response that met the requirements of increased spike frequency and amplitude in nearly all cells. Thus, we experimented with current types to reduce the depolarization caused by NaP in the control case while retaining the TEA response. This led to the addition of BKKCa but that worked only for some models, even with maximal conductance of BKKCa exceeding the upper bound (see Table 1 in Methods). Furthermore, this manipulation did not provide the variability in TEA responses seen in experimental traces. Since NaP by itself was not sufficient, we then explored whether CaT and CaS channels could substitute for NaP. Even with values of conductances beyond the upper bounds for CaS and CaT channels, the TEA response was inadequate, and the peak spike height in the control case was also too high. So, we added BKKCa to this set of CaT and CaS, without NaP. Although this reduced the peak spike height to within permissible ranges, the TEA response was still inadequate. Probing deeper into why NaP was needed, an analysis of the time constants of the currents revealed that I_NaP_NEURITE had a time constant that was at least four-fold smaller than that of I_CaT_NEURITE and at least 14-fold smaller than that of I_CaS in the −50 to −20 mV range. This implies that I_NaP activates much faster than I_CaT, which in turn activates much faster than I_CaS. Since the membrane voltage of the soma never exceeded 0 mV, the time constant (during all model runs) of I_CaT_NEURITE was always at least twice as large as that of I_NaP_NEURITE, and that of I_CaS_NEURITE was always larger than that of I_CaT_NEURITE. This justified the inclusion of the following suite of active current channels to the neurite to reproduce the TEA responses: NaP, CaS, CaT, BKKCa. We note that the response of the SIZ changed minimally in the active vs passive neurite cases.

Table 1.

Ranges of maximal conductances

Current	G_min (S/cm2)	G_max (S/cm2)
gCaT	0.00016	0.00031
gCaS	6.50E-05	0.00013
gCAN	7.00E-05	1.50E-04
gNaP	3.50E-05	2.30E-04
gLeak (all segments)	6.20E-05	9.70E-04
gKA	0.000172	0.0019
gKd (Kd1)	0.000165	0.00127
gKCa	0.00079	0.0061
gSKKCa	0.00088	0.002
gKd2	0.000091	0.0005

To gain further insights into the process, we investigated the mechanism by which the TEA-sensitive BKKCa (conductance decreases with TEA) improved the soma TEA response together with NaP, and produced variable TEA responses seen in experiments, without disturbing the control responses. First, we found that BKKCa helped reduce the spike amplitude. However, there was little variability in TEA response waveforms of different cells. To explore why, we investigated the underlying current waveforms. For this cell type, CaT and CaS in the neurite produced less spiking and depolarization in the soma during TEA compared to NaP in the neurite. The waveform of BKKCa corresponds closely in time with those of Leak and NaP currents, all of which also closely follow the voltage waveform. This suggests that BKKCa has a greater impact on the voltage waveform than did CaT and CaS; note that there is a baseline concentration of Ca²⁺ in NEURON models (see Methods). However, the slow wave amplitude could, in general, be modified by CaT and CaS, allowing the model to exhibit varied TEA waveforms. With all these channels present, the waveform criteria for both control and TEA cases were met by larger numbers of cells and there was greater variability in the range of TEA responses. A typical set of TEA responses for five intact cells with all channels present is shown in Fig. S2-H. In summary, BKKCa (with CaT and CaS) reduced soma peak spike height and spike frequency in the control case. Although BKKCa did reduce the increased control response caused by NaP, it did not counter NaP’s facilitation of the TEA response. Furthermore, CaS and CaT channels were found to be important for the generation of varied TEA responses.

Validation of single cell model via ‘stimulus protocol’ waveform used in experiments.

To summarize the results in this section, the single cell model with a passive neurite did not produce realistic outputs. However, the inclusion of specific active channels produced appropriate control and TEA responses. To provide partial validation for the new single cell model, we used experimental data from our prior work. Specifically, in our prior experiments with the ligated soma, Ransdell et al. [27] recorded from intact networks and developed a current injection trace termed ‘stimulus protocol’ that, when injected into a ligated soma, resulted in a membrane potential profile that matched those from intact network recordings. Here, we found that the current entering the soma from the neurite in our model mimicked the experimental ‘stimulus protocol’ waveform with an active, but not a passive neurite (Fig. S3), providing support for our prediction of the presence of active conductances in the neurite.

Random sampling of neurite conductances and validation checks.

Based on the systematic initial trial-and-error investigation of the role of active conductances in the neurite discussed above, we decided to add the following current channels to the neurite using random sampling: CaS, CaT, NaP, BKKCa. The reader is reminded that maximal conductances of the soma currents were finalized in selection level 1, and so selection level 2 considers only the random selection of maximal conductances for the channels added to the neurite, i.e., selection level 2 of the protocol focused only on conductances in the neurite. Of the 20,000 intact cells tested in selection level 1, 15,161 were Level-1 cells. Of these Level-1 cells, 1264 distinct cells passed level 2 and are labeled as Level-2 cells.

Network responses with gap-junction coupling among SIZs

The 1264 distinct model cells that passed selection level 2 were grouped into 5-cell networks totaling 252 in number. Of these, 130 networks passed selection Level 3, comprising 650 individual cells. The ranges for three of the measured output features of these networks are shown in Figure 3A. The four colored circles in Figure 3A represent specific instances of one cell, LC5, at the edges of this output distribution in networks that passed all selection criteria. Figure 3B shows the soma membrane potential responses for all five LCs from these particular networks (LC1–5), colored to match the top panel. This illustrates the range of behavior of networks that include cells from the extremes of the space of the three electrophysiological features. While all five cells within a given network showed nearly identical output when coupled by gap junctions (Fig. 3B left), the dissimilar individual responses of LCs in the same network when gap-junction connectivity was removed (Fig. 3B right) demonstrates the conductance variability of individual cells even within a network that results from Level 3 selection.

We explored the mechanism by which the gap-junction coupling between the somata for cell pairs LCs1–2 and LCs 4–5 (shown to exist in biology [21]), and between the SIZ compartments of all cells (Fig. 1A) ensured synchrony among the LCs in the network. Focusing on one representative network, Fig. 3C provides a comparison of the magnitude of the gap junction current from LC2 soma to LC1 soma, and from the various SIZ compartments to the SIZ of LC1, labeled as ‘iGap SIZ’, to the synaptic current into the SIZ of LC1 due to SC input spikes. As can be seen from the traces, the magnitude of the total gap junction current into LC1 was found to be more than seven-fold smaller than that of the synaptic current into LC1 (Fig. 3C). Once the threshold is reached in the SIZ and spiking is initiated, the gap junction currents ensure that the spikes are synchronized among the cells. In summary, the analysis predicts a delineation of the primary functions of the two current types: synaptic (to increase SIZ membrane potential to threshold) and gap junction (which facilitates synchronization) and quantifies their magnitudes.

We also note that the observation of the gap junction current being considerably smaller than the synaptic current for each cell means that the gap junctions do not play the main role in ensuring the SIZ reaches threshold. Ensuring that the SIZ reaches threshold was the purpose of selection level 2, so the fact that the synaptic current is 7 times higher in magnitude than the gap junction current justifies the use of intact single cells in selection level 2 of the protocol. That is, without considering the gap-junction interaction effects from other cells and providing a validation check.

Parameter space variations following three levels of selection on model neurons

Following each of the levels of selection (Fig. 4A), we examined the overall distribution of membrane conductances in the soma and neurite compartments in the viable/passing neurons to determine whether the criterion at each selection level limited any conductance to a particular portion of the parameter space (Fig. 4). We visualized this in two distinct ways. Figure 4B uses a density plot to describe the conductances that passed a given level of selection. As selection continued, some conductances were increasingly limited to a portion of the parameter space, while others maintained a broader range of viable regions. For example, selection levels 2 and 3 result in skewed distributions of NaP_SOMA, NaP_NEURITE, Leak_SOMA, and Leak_NEURITE conductances (Fig. 4B). The other active conductances maintained a broad range of acceptable values through all levels of selection. However, these are not normally distributed. Rather, multiple peaks can be seen around conductance ranges that produced successful models for most of the conductances (Fig. 4B). Finally, because our goal ultimately is to understand the range and relationships of conductances underlying full network activity in these models, in Figure 4C we plot the conductance ranges for the 650 neurons that make up the 130 networks that resulted from the three levels of selection. These conductances are shown both as raw as well as log₁₀ transformed values to better appreciate the relative distributions across conductances that have large differences in their absolute magnitudes (Fig. 4C).

Figure 4.

Conductance distributions after selection of soma and neurite model conductances. (A) (Top) Number of neurons passing each level of selection and (Bottom) a histogram distribution of the number of cells from networks that passed selection level 3 using each SC input frequency. (B) Distribution of the conductances (x-axis in S/cm²) of models spared after each level of selection. Each selection level (0, 1, 2, 3) represents the conductances that passed that level with the level 0 being those that were initially generated and level 3 being those which satisfied all criteria. Each level’s density plot is scaled independently. Because selection level 0 is performed on isolated somata, there are no conductances represented at this level for the neurite compartment. (C) Violin plots for the raw (top) and log (bottom) conductance ranges for the neurons that passed all three levels of selection. Points in each plot represent the mean and the lines represent the standard deviation. Colors represent different conductances labeled on the x-axis.

Conductance parameter differences are larger between strongly gap-junction coupled cells.

Since the gap junction coupling between LC1 and LC2, as well as LC4 and LC5 is considerably stronger than between either of those and LC3 due to local coupling between cell somata (Fig. 1A, 1D; 0.65 μS [3] vs 0.067 μS; see methods), we hypothesized that the network would be able to support a larger variation in conductances between the strongly coupled neurons LC4 and LC5 compared to that between either of those and LC3. To test for this, we estimated the Euclidean distance between the parameter sets of each of the three cells. For example, because LCs 4 and 5 are tightly coupled, we also averaged the conductances between LCs 4 and 5, and then estimated the Euclidean distance between that average parameter set and that of LC3, for each network. These were then averaged across all 130 of the Level 3 selected networks. This yielded the following when averaged across all the networks: the Euclidean distance between parameter sets of LC4 and LC5 was 0.003106 and between averaged LCs 4 and 5 and LC3 was 0.002392; p<0.00001. The same calculations for LCs 1, 2 and 3 revealed that: the Euclidean distance between parameter sets of LC 1 and LC 2 was 0.002998 and between parameter sets of LC1&2 and LC3 was 0.002445; p=0.00012). Both these tests confirm the hypothesis that the LC network supports larger variations in intrinsic conductances between LCs 4&5 and LCs 1&2, compared to between the other combinations.

Intrinsic conductance covariations in the network model

Biological and modeling studies have suggested that it is important and/or necessary for pairs or “modules” of conductances to work in concert to control appropriate physiological output [15; 28; 29]. Therefore, we looked for relationships among the membrane conductances in our model data set – both within and across two compartments (soma and neurite) – at all three levels of selection.

Figure 5A shows membrane conductance relationships in the somata of our population of model cells. Correlograms (Fig. 5A, top) are shown to describe the relative strength of correlations within a given selection level. To better compare conductance relationships across selection levels, we also plotted each pairwise correlation coefficient at each level of selection as a barplot matrix (Fig. 5A, bottom). As a general observation, conductance correlations emerge and grow somewhat stronger over the four levels of selection. However, the overall correlation coefficients for any pairwise relationship are quite low, with the highest correlation coefficient only approaching 0.1. Therefore, there were no significant correlations identified among conductances in the soma compartment. Similarly, when comparing conductance relationships in the neurite compartment (Fig. 5B), the highest correlation coefficient is just over 0.1 (BKKCa_NEURITE versus Leak_NEURITE). The correlation coefficient for a given relationship did not change significantly between selection levels 2 and 3 (note, selection level 1 only involves Level-1 cells, and hence are not represented in Fig. 5B).

Figure 5.

Conductance correlations in the soma (A) and neurite (B) compartment of model neurons across selection levels. Top for A and B: Correlograms for four levels (0–3) of conductances in the somatic compartment of selected model neurons. Each pairwise correlation was calculated using Spearman’s correlation and reported as rho values. Each correlogram is scaled individually to maximize resolution, although all rho-values are overall quite low. Bottom for A and B: Bar plot showing the rho-value of each pairwise correlation across levels of selection. These are the same data plotted in the top row but allow for more precise determination of the more pronounced correlations.

Finally, we comprehensively compared conductance relationships across compartments, to determine whether there may be co-regulation of membrane conductances between the soma and neurite. Figure 6A provides the picture of the conductance data in both compartments after selection level 3. In this visualization, we can see the distribution of each conductance along the diagonal, as well as the raw scatterplots of the 650 neurons at this level of selection. Scatterplots reveal little coherence in terms of correlated levels with two exceptions of significantly correlated relationships across compartments: Leak_SOMA vs Leak_NEURITE (rho-value = −0.514; p <1e⁻¹²) and Leak_SOMA vs. NaP_NEURITE (rho-value = 0.405; p <1e⁻¹²) (red colored boxes).

Figure 6.

Conductance and output measure correlations in all cells of model networks passing Level 3 selection. All electrophysiological features are measured from the soma compartment. (A) Pairwise scatter- and parameter distribution plots of intrinsic conductances (red boxes indicate significant correlations across compartments). Scatterplots in gold show all cells as individual points, while the parameter distribution plots in blue represent the relative density of points across the distribution of the data (B) Top: scatterplots of output measures as a function of intrinsic conductances. Bottom: Correlograms to quantify correlations between intrinsic conductances and output measures. The size and color of the dot in a given pairwise relationship reflects the relative magnitude and direction (negative or positive) of the correlation as measured by the Rho-value.

Correlations between intrinsic conductances and electrophysiological features

To explore how intrinsic conductances affect specific electrophysiological features, we examined relationships between the following: individual membrane conductances in the soma and neurite compartments (Fig. 6A) and also the spike frequency of SC inputs and four distinct output features in the soma compartment: spikes per burst (SPB), area under the curve (AUC), spike height (Peak), and spike synchrony across LC3 and LC5 (determined by Pearson’s R), in both control and TEA conditions (Fig. 6B; circle diameter represents magnitude of the correlation).

The conductance that most affected these output features was Leak_SOMA. AUC and Peak in control, and the Peak in TEA were all negatively correlated with Leak_SOMA (p <1e⁻¹²), showing that a leakier soma will reduce total charge transfer to the soma. The results in the TEA condition may be impacted by a ceiling effect. Additionally, the SC input spike frequency was strongly correlated with SPB, but negatively associated with Peak in both control and TEA conditions (Fig. 6B). Active membrane conductances that were most strongly associated with these output parameters were SKKCa and CaS in the soma, which became moderately (relative to other correlations) negatively correlated with AUC in TEA. Additionally, there was a moderate negative correlation of NaP in the neurite with both AUC and Peak in the control, but not TEA, condition (Fig. 6B). Network Synchrony is not particularly associated with any single conductance in the control condition, but moderately positively correlated with Leak_SOMA in TEA. These conclusions are logical in that parameters from one section affect characteristics of the same section, i.e., AUC and peak are primarily affected by soma parameters, while SPB and synchrony by SIZ parameters.

Lastly, and perhaps not surprisingly, many of these output features were strongly associated with the SC input spike frequency. SPB is strongly positively correlated to SC input spike frequency, as cited earlier, and the SC inputs also similarly interact with SPB in TEA but to a somewhat lesser extent (Fig. 6B) due to the already high excitability of the cell in the TEA case (ceiling effect). Conversely, Peak is negatively associated with SC input spike frequency in both control and TEA (Fig. 6B).

Soma and neurite conductances are associated with SIZ spike characteristics

Our model suggests that the initiation of bursting activity in the SIZ is primarily controlled by the excitatory drive from the SCs, suggesting a forced as opposed to regenerative bursting output of the motor neurons in the C. borealis CG network. To investigate effects of the somatic conductances on the organization of SIZ bursts, we explored how the following SIZ characteristics were affected by the distinct suite of soma conductances present in the models that met level 3 selection criteria: (i) time to first spike, (ii) inter-spike interval or intra-burst frequency, and (iii) number of spikes per burst.

Across a range of SC input frequencies, we found that time to first spike in passing networks at a given input spike frequency varied by up to 300 ms (Fig. 7A). Note that there is more variation of the time to first spike among networks receiving lower SC input frequencies and this variance decreases as SC input spike frequency increases (Fig. 7A). When considering inter-spike interval, absolute variance is highest among the three characteristics measured (as determined by Coefficient of Variation), but more similar across frequencies, with a few exceptions (19 and 20 Hz higher, 28 and 29 Hz lower; Fig. 7B). Lastly, the number of spikes per burst was found to vary between 2 to 9 spikes for the same SC spike frequency (Fig. 7C) with a clear trend towards decreasing variability as SC frequencies increased (Fig. 7C).

Figure 7.

Spiking patterns in the SIZ are distinct across cells and networks with differing somatic and neurite conductances. Violin plots for the (A) Time to first spike, (B) Interspike interval, and (C) Spike number are shown for all 130 cells passing Level 3 selection across SC input frequencies ranging from 16 – 31Hz. Points in each plot represent individual cells. Colors correspond to the different input frequencies labeled on the x-axis. These same measures are provided for cells from Averaged Networks in panels (D), (E), and (F). Y-axes for the individual measures are scaled differently based on the range of the data. For each measure, a histogram shows the Coefficient of Variation across SC input spike frequency with consistent Y-axes ranges for comparison between All Cells and Averaged.

Because the CG network has five random LCs picked from single cells that passed the selection criteria, variations in SIZ spiking behavior could be due to the differences in the cells within the network rather than due to the specific behavior of the conductances, although both are linked. To focus only on somatic conductances, we repeated these analyses in our averaged networks. In an averaged network all LCs are identical (see next section for details), effectively eliminating gap junction currents between individual LCs, leaving the variation in SIZ spiking behavior explainable only by membrane conductances of the averaged LCs. That is, the variance is explained not by the conductance differences within a network but by the conductance differences across networks. Figs. 7 D–F provide the same results as in Figs. 7A–C, showing similar variability in output across averaged networks. For example, different averaged networks receiving the same SC spike frequency of 21 Hz can output 3–9 spikes, have ISI variations from 10–90 ms, and variations in time to first spike from 650–1150 ms. This suggests that conductance relationships might shape burst characteristics. Future tests can explore the causal effect of each conductance on SIZ spike characteristics.

To demonstrate the effect of soma and neurite conductances on the SIZ frequency, we performed a parameter sweep of all conductances for a particular cell (LC1 in Table 7) for a random sample of 50 networks from the 130 that passed LV3. Table 7 lists the SIZ spike numbers when the soma and neurite conductances listed are varied from the minimum to the maximum of their range (Table 1). Figure S4 shows an example of the changes in the SIZ spike numbers for a particular parameter. A comprehensive exploration of how soma and neurite conductances group the SIZ spikes will be a topic for future research.

Table 7.

Number of spikes in SIZ for changes in LC1 conductances in a random set of 50 networks that passed LV3. For each network, the SIZ spike numbers were beyond the range in Fig. 7C.

	low	med	high
soma_leak	0–11	0–11	0–11
soma_a2	1–9	0–9	0–9
soma_bkkca	0–10	0–10	0–10
soma_skkca	1–9	1–9	1–9
soma_g1kd2	1–9	0–9	0–9
soma_g2kd2	1–9	1–9	1–9
soma_cal	1–9	1–9	1–9
soma_cat	1–9	1–9	1–9
soma_Nn	1–9	1–9	1–9
soma_nap	1–11	0–11	0–11
neurite_gleak	0–14	0–14	0–14
neurite_CaT	1–9	1–9	1–9
neurite_CaL	1–9	0–9	0–9
neurite_NaP	1–15	0–15	0–15
neurite_Bkkca	0–12	0–12	0–12

Averaging of network parameter sets produces viable networks

The findings above led us to the hypothesis that the five LCs in a network function have the capability to function as a single LC due to gap junction coupling; however, to realize this capability, conductance correlations may be needed across cells in a network. To test this hypothesis, we averaged the maximal conductances of each parameter for the five Level-3 cells in each of the passing networks and then created new networks with 5 LCs that all contained the averaged levels of these conductances. Because averaging maximal conductance can produce results that are not biologically realistic [30], we ran these networks through all levels of selection, and retained only the networks which passed. We found that, of the 130 original networks, 129 passed all three levels of selection. We performed an additional test where we compared the outputs of the original, i.e., non-averaged network vs those of the averaged network, for each of the 129 networks. This analysis focused on the numbers of spikes/burst in the SIZ and the inter-spike-interval within the burst, which represent the functional output of the network. Model runs showed that the number of spikes/burst differed minimally between the two (average of 0.61 spikes) and so did the inter-spike-interval within the bursts (0.36 ms). These model observations suggest that the 5 LCs may function as a single LC.

We then probed deeper by using the correlogram across conductances in the averaged networks. Several correlations found among the membrane conductances in the non-averaged case changed appreciably (some increased and some novel) in the averaged case (Fig. 6A, 8A) or were appreciable in both cases (Fig. 6B and 8B; circle diameter in both represents magnitude of the correlation). Some examples of these changes were: NaP_SOMA vs. Leak_SOMA (increased from 0.096 to 0.24), Leak_NEURITE vs. NaP_NEURITE (−0.053 to −0.44), Leak_SOMA vs. NaP_NEURITE (increased from 0.45 to 0.71) and Leak_SOMA vs. Leak_NEURITE (increased from −0.54 to −0.81). However, there were few conductance correlations in the averaged case, similar to those in the individual cells (Fig. 6 vs. Fig. 8).

Figure 8.

Conductance and output measure correlations in model networks of Averaged neurons. (A) Pairwise scatter (gold) and parameter distribution (blue) plots of intrinsic conductances as described in Figure 6A. (B) Top: scatterplots of intrinsic conductances vs. output measures. Bottom: Correlograms to quantify correlations between intrinsic conductances and output measures. The size and color of the dot in a given pairwise relationship reflects the relative magnitude and direction (negative or positive) of the correlation as measured by the Rho-value.

DISCUSSION

We developed a morphologically enhanced biophysical model of the LC of Cancer borealis that included a new compartment (SIZ) that receives synaptic input and is informed by measured biological properties of these cells [21; 24; 27]. As mentioned in the results, biological reports have suggested but not confirmed the presence of active conductances in the ligated somata plus neurites of LCs [27], and the sub-cellular localization of these channels is unknown. There are no biological reports related to channels in the SIZ. The fact that the model included neurite and SIZ compartments, in addition to the soma, facilitated an investigation of structure-function relationships, including conductance covariations within and across compartments [14; 20]. Furthermore, we engaged in a selection regime based on biologically relevant network activity that simultaneously selects five motor neurons for inclusion in our final cell population. Using network level output as selection criteria, we generated a population of 1260 LC motor neurons in which we could characterize relationships among membrane conductances in two compartments (soma vs. neurite). In doing so, we identified relatively few correlations among ionic currents despite the fact that these relationships have been reported for both channel mRNA relationships [31–33] and membrane currents [5; 34] in biological networks. This is consistent with and extends similar modeling studies that demonstrate these correlations are not fundamentally emergent characteristics of these model populations [14]. These collective modeling data suggest that while functionally these relationships may not be necessary to generate appropriate output, nevertheless such correlations are prevalent in the biological population and therefore likely to confer a selective or functional advantage.

Biologically realistic framework for predicting function at single cell- and network-levels

Previous iterations of computational models of the crustacean cardiac ganglion [15; 16] relied on experimental data from distinct but related systems (e.g., stomatogastric ganglion [2]) and species (e.g., lobster [21]) for passive properties and conductance ranges. For the present model, the experimental data were obtained directly from the LCs of intact networks recorded from the Schulz lab [21; 24; 25; 27; 35] as well as SIZ recordings [26] to first match our model with the known data, and then to explore the functional characteristics in search of an explanation for the preservation of output across a wide variation range (see Methods Table 1). We started with a prediction of the intact single cell morphology that integrates information about structure and proposed a methodology to validate it. Specifically, although a SIZ has been conjectured as the site of SC input into the cardiac ganglion [36], reported single cell models of crab neurons have not explicitly considered such morphology. For instance, models of single LCs have typically considered only the soma compartment and possibly a neurite attached to it [21]. To validate our enhanced three-compartmental model that includes a SIZ in which synaptic input is localized, we estimated the SC spike frequency ranges from intact recordings and then found that a model that accounted for the biological data was able to successfully reproduce intact single cell responses (Fig. S2G). Validation is in terms of value for the model potentially representing the biological system, but not validating the presence of an SIZ compartment that is also the site for synaptic input.

As noted in the Results, the single cell model predicts the presence of four active conductances (NaP, CaS, CaT, and BKKCa) in the neurite, for functional reasons, and this awaits experimental validation. The focus here was largely on single-cell studies with only the issue of synchrony across LCs being considered at the network level. However, the framework can be readily used for network level studies such as potential co-variations of the synaptic or gap-junction conductances on the SIZs with the intrinsic conductances.

SC input spike frequency predicted to follow Gaussian distribution

The fact that networks passed at the same number of SC input frequencies when averaged as opposed to non-averaged, and that the distribution remained similar, indicates that the averaged networks contain the requisite functional properties. This outcome is additional evidence that the gap junctions allow a wide range of cells to be incorporated in a network, but the particular cells have ‘compensatory’ conductances. The results shown in Figure 4A (bottom) predict that with the given synaptic and gap-junction conductances (see methods), the frequency of the SC input to the SIZs should have a Gaussian distribution with about 50% of the SCs firing between 16 and 28 Hz. Future studies could explore the average firing rate across members of C. borealis, adding to the body of data that describe the cardiac ganglion and would suggest that the average conductances are optimized in nature for systems with an SC input around 24 Hz.

Exploring only the tails of the distribution, it was observed that the SC spike frequency had no correlation with Peak but had a large positive correlation with all other features. Most networks pass at 24 Hz, and fewer at the extremes. Failure at lower SC frequencies was primarily due to the failure of the network to produce spiking in the SIZ, whereas failure at higher frequencies was largely caused by loss of network synchrony between LC3 and LC5 (see Methods) due to the inability of the SIZ gap junction current between the two cells to synchronize. SC spike frequency had a weak negative correlation with synchrony in control (p <1e⁻¹²), and a strong negative correlation in TEA (p <1e⁻¹²), suggesting that a high SC spike frequency desynchronizes the network. This observation may explain how TEA causes the network to lose synchrony, because under TEA conditions the network will have higher activity than in control conditions (Fig. 2 in [21]) which is reflected in the model (Fig. 2 and Fig. 6B), and that explains the correlation between SC frequency and TEA. Furthermore, SC spike frequency interacts with conductances of the soma and neurite to generate variability in output spiking patterns across networks, both individual and averaged (Fig. 7).

Effect of conductance and input parameters on network output features

Three conductances, Leak_SOMA, Leak_NEURITE and NaP_NEURITE, had the strongest associations among all output features. This suggests that they are most responsible for ensuring the soma’s response to SC input, since Leak will compensate for high charge input and NaP will help increase charge from the SIZ to the soma. As Leak_NEURITE has all the same correlations as NaP_NEURITE but to a lesser degree, it can be said that the balance of Leak_NEURITE and NaP_NEURITE is the most important correlation for the network’s behavior, and that the network is less sensitive to Leak_NEURITE than to NaP_NEURITE. Their lack of correlation with synchrony suggests that the effect of these two conductances is primarily on the individual cell and has little impact on how well the network functions. As noted in methods, the gap junction conductance was fixed; the resulting gap junction currents were smaller compared to synaptic currents as shown in Fig. 3C.

The highest correlation of SC spike frequency is with SPB in both control and TEA. Given that most networks fail at the high frequencies of the input range provided due to lack of synchrony, we conclude that high spike rates increase desynchronization, and that the network is more sensitive to desynchronization under TEA conditions. This may be the reason why the model showed more correlations in TEA than in control, i.e., the additional correlations may be important in stabilizing the cell when disturbed.

We note that the network model provides an important test bed to explore relevant structure-function and parameter relationships. For example, SPB being associated most strongly with SC spike frequency is expected from a forced burster and supports the finding of few conductance correlations. And that, synchrony – particularly in the control condition – had little relationship to active membrane conductances and more likely was a result of the coupling across the network that provides currents comparable to intrinsic currents (Fig. 6B).

Minimal conductance correlations in the multicompartmental network model

One of the key hypotheses that this work tests is whether network level selection criteria will result in a population of models from which conductance correlations emerge. The collective literature in two similar crustacean networks (stomatogastric and cardiac ganglia) are somewhat inconsistent in this result. Previous computational models in the cardiac ganglion LCs from our group [15; 16] with a similar rejection sampling approach yielded strong correlations in the LC soma among two pairs of conductances: CaS_SOMA-A_SOMA and CaT_SOMA-Kd_SOMA. These data reveal that such relationships can emerge naturally from a selection process focused on output characteristics informed by biological data. However, those results were limited to models of the soma only and based on a different species (P. sanguinolentus) and different mode of activity (driver potentials) than the cells modeled in our study. Further, there was relatively little biological data available at the time for those experiments, and so ultimately our interpretation of these first models is that such relationships are theoretically possible to detect and quantify – but further study was needed, including more thorough grounding in biological data as well as model neurons that better reflect the morphological complexity (i.e., multiple compartments) of biological neurons.

Conversely, a thorough and extensive analysis of a multicompartment model employing a large population of LP neurons from the crustacean stomatogastric ganglion yielded a different outcome. Taylor et al. [14] utilized a large population selection approach, in a multicompartment model, and selected based on output characteristics that focused both on the single cell excitability as well as some features reminiscent of network level function (i.e., output as a result of synaptic input currents). In this study, they found only weak correlations among conductances – both within and across the compartments [14]. Thus, our current work employs a much more similar approach to Taylor et al., but in the system originally modeled by previous members of our group – the cardiac motor neurons – in which we had seen such correlations. When we combined a multicompartmental model, far more extensive first-hand biological data, and a more developed network level selection process, our results in this study were largely consistent with those of Taylor et al. [14]. That is, while we could detect some emergent correlations within and across model neuron compartments, these were overall weaker correlations (rho-values between −0.2 and +0.2).

There are (at least) two overall interpretations of these results. First, that the intrinsic conductance correlations found in biological neurons across a wide range of nervous systems [34] are not fundamentally necessary to generate baseline functional output of neurons. In other words, while these relationships may confer some adaptive advantage to neuron and network stability [8; 31; 33; 37; 38], and implicate compensatory relationships involved in homeostatic regulation [24; 39], they are not fundamental to the solutions capable of producing a given output of a neuron in a network. Taken together, our work shows that conductance correlations emerge in more narrowly constrained models with clear input-output relationships [15; 16]. However, as we add more free parameters to the system for which we have less knowledge of biological constraints – in this case multiple compartments and conductances therein, as well as synaptic inputs and network connectivity – we may lose the ability to detect fundamental relationships in biological neurons. If this is the case, then we predict that as more complex models become better informed by biological data, the possibility to recapitulate and interrogate these relationships may be more robust.

A second interpretation comes from the observation that in both detailed studies in which these selection protocols failed to result in correlated active conductances (our results and those of Taylor et al. [14]) were models derived from biological cell types that could be classified as “forced bursters.” Here we define a forced burster as a cell that produces a bursting output not as a result of a self-organized driver potential per se [16; 40], but rather due to receiving synaptic inputs that are themselves incoming as bursts. The LP neuron from the Taylor et al. study is identifiable as a tonic spiker when synaptic inputs are removed [41], but only fires as a bursting neuron when connected to inhibitory inputs from neurons that are intrinsic bursters [42; 43]. While this does not preclude neurons such as LP from manifesting such conductance correlations [44], the functional implications for such relationships are unknown and may be more related to synaptic parameters associated with phase relationships than the intrinsic bursts themselves [29]. Similarly, the LCs in our study are unable to generate self-organized driver potentials [45], but fire in bursts due to the periodic excitatory input from the SC pacemakers (see Fig. 1, [26]). Therefore, the most salient parameters necessary to constrain output in forced bursters would presumably be the input-output relationships of the synapses, which would be most directly influenced by the synaptic strength, the frequency of inputs, and the membrane resistance of the post-synaptic cell (i.e., the LC). This is consistent with our observation that the most tightly constrained parameters seemingly associated with output from these networks are the LC Leak and NaP in both the soma and the neurite compartments as well as the SC input spike frequency. Additional work to tease apart these (and other) hypotheses related to the nature of these conductance correlations is necessary but also likely to be greatly facilitated by these and similar modeling approaches.

Averaging in a network?

Lastly, we saw our models as an opportunity to follow-up on a provocative and elegant set of modeling experiments that showed that a selected population of neurons converging on an output can have a variable range of underlying maximal conductances. However, when the average of the conductances of those model neuron populations are used to render an output, this fails to recapitulate the desired output [30]. This “failure of averaging” has to our knowledge never been tested in a network context. When we did so, 129 of 130 networks passed output selection with the averaged conductances. This outcome suggests that either the failure of averaging that occurs at the single neuron level can be “buffered” by the network and its emergent properties, or once again this is a fundamental difference between endogenous bursters (as modeled in [30]) and forced bursters as featured in this and other studies [14]. Further experimentation to differentiate these (or other) hypotheses would add to this intriguing concept.

CONCLUSIONS

Our data provide multiple intriguing results for the crab cardiac network that support previous studies in related systems and for future experiments. First, we provide further evidence [14] that despite being prevalent in biological neurons, correlated membrane conductances are not a necessary feature of neurons in CPG networks to generate biologically realistic output. However, by examining the electrophysiological features correlated with conductances, we found that the soma and neurite do play a role in organizing the spikes in the SIZ, the ultimate output of the CG network that controls the crustacean heart. Second, we show that averaging conductances across the neurons in a network also resulted in maintaining biologically realistic output [30]. That said, both outcomes could be features specific to the network employed in this study: the crustacean cardiac ganglion. First, the crab LC may depend more on synaptic drive to determine network bursting output modes, rather than the intrinsic organization of individual neurons into endogenous bursters [16; 40]. Further, these results suggest that this absence of intrinsic bursting may be a feature of these networks that allow for robust activity across a range of parameter states. Thus, our results may be most relevant to similar networks, where “forced bursting” is a result of highly interconnected neurons with strong synaptic input organizing the main output features. Lastly, it must be noted that with the complexity of the model – including multiple compartments for which biological data are sparse – the biological constraints may simply not be appropriately captured by the free parameters we are able to include in our model.

METHODS

Experimental data to constrain single cell and network models.

Biological data used to constrain the LC model parameters (e.g., membrane currents) and outputs (of ligated and intact LCs) were collected as previously published and cited below (also see Tables 1,2,4).

Table 2.

Ranges of properties for valid LCs in SELECTION LEVEL 1

Parameter	Min	Max
Vrest	−53 mV	−39 mV
Rin	0.852 MΩ	13.3 MΩ

Table 4.

Model current parameters

Current	Gate	X∞	τx (ms)
INa	m3	0.3463 / (1 + 0.008685 e^−5.03324V) + 0.75187 / (1 + 1.11022 e^−9.510637V) + 0.162947	3.002 + 4.073 / (1 + exp((V+24.18)/2.592))
	h	0.93854475 / (1 + 144209.656 e^5.08660603V) + 0.02804584	9.434 + 11.7 / (1 + exp((V+1)/5.317))
ICaS	m2	1 / (1 + exp((V+24.75)/−5))	20 + 50.2 / exp((V+20.25)/1)
	h	45 / (40 + [Ca2+])	1 / 0.02
ICaT	m	1 / (1 + exp((V+20)/−1.898))	18.51 – 3.388 / (exp((V-6.53)/9.736) + exp((V+12.39)/−2.525))
	h	1 / (1 + exp((V+55.27)/6.11))	20.23 + 40 / (exp((V+23.48)/−9.976) + exp((V+5.196)/10.84))
IKd	m4	1 / (1 + exp((V+24.19)/−10.77))	25.049 + 25 / (1 + exp((V+25.84)/6.252))
	hi	0.3 + (1 – 0.3) / (1 + exp((V+15.87)/5.916))	550 + 954.9 / exp((V+10.8)/−15)
INaP	m2	1 / (1 + exp((V+23.32)/−10))	100 + 550 / exp((V+15)/12.46)
INaP	m3	1 / (1 + exp((V+32.7)/−18.81))	3.15 + 0.8464 / exp((V+0.8703)/−6.106)
ICAN	w	0.0002 [Ca2+]2 / (0.0002 [Ca2+]2 + 0.05)	40 / (0.0002 [Ca2+]2 + 0.05)
ISK(Ca)	w	0.0001 [Ca2+]2 / (0.0001 [Ca2+]2 + 0.1)	4 / (0.0001 [Ca2+]2 + 0.1)
IBK(Ca)	a	[Ca2+] / ((1 + exp((−15+0.08[Ca2+])/−15)) * (1 + exp((−5+0.08[Ca2+])/−9)) * (2 + [Ca2+]))	1 / 0.4
	b	7 / (5 + [Ca2+])	1 / 0.2

Membrane currents in the soma were measured in two-electrode voltage clamp while the network activity was silenced either with tetrodotoxin (TTX) or by severing the CG nerve trunk to remove the small cell (SC) inputs. The inward currents (all for soma and so subscripts omitted) I_CaS, I_CaT, I_NaP, and I_CAN were based on recordings and data as described in Ransdell et al. [27]. The outward currents I_A, I_Kd, I_BKKCa were based on recordings made in Ransdell et al. [24]. No biological characterization of I_SKKCa has been performed in crabs, and this work carries over SKKCa model currents as described in our previous CG modeling efforts [21]. Intracellular voltage follower recordings of ongoing network activity were made in all of the above studies, and from these we generated the biological parameters to constrain model network output.

Synaptic inputs (chemical) and connections (electrical) were also characterized from these recordings (see Table 3). EPSPs were characterized by measuring the amplitude and time constant characteristics from intracellular LC recordings in intact networks. Single SC action potentials that yielded clear (non-summating) EPSPs were used to generate a population of post-synaptic potential measurements that constrained the chemical synapse inputs. Electrical coupling was measured directly in two-electrode current clamp as described in Lane et al. [21]

Development of biophysical single cell models

The single cell model had three compartments (Fig. 1C): soma, neurite (neu) and spike-initiation-zone (SIZ). We have based the active conductances in the soma compartment based on our own experimental results, many of which are published [1, 2, 6]. The contents of the neurite compartment are not well determined biologically, and this was experimentally interrogated in this study as part of the model development (see below). The SIZ is modeled as a multifunctional compartment. First, we modeled it to serve as the action potential initiation site by including Na and K_d channels reliant on depolarization from the soma and neurite to activate these conductances. Ligature experiments combined with voltage-clamp and immunohistochemistry experiments for calcium channels [6] suggest that there are no other active conductances in the SIZ, but these experiments are incomplete and therefore we cannot confirm biologically that the SIZ does not itself contain such conductances. Indeed, a clean demarcation between “neurite” and “SIZ” is likely not an anatomical reality, and so we have separate functions of integration and spike initiation somewhat artificially in these compartments. Second, we chose this as the site of chemical synaptic input from the SC pacemakers, as the anatomical area of the ganglion corresponding to the SIZ in our model (see Fig. 1A) is known to contain the projections of the SC axons as well as rich synapsin-like immunoreactivity [46]. That said, it is known that syanpsin-like immunoreactivity is also present on the somata of these cells [46], but it is not clear whether these synapses would represent SC inputs or synaptic inputs from modulatory projection neurons (or both). More work is clearly needed to better understand the anatomy and subcellular compartments of these cells in the animal system, and specifically in Cancer borealis for which less is known relative to other crab species. Lastly, we made the SIZ the site of electrical coupling amongst the 5 LCs in the network, as anatomical data in other species [47] as well as our own dye fills (unpublished results) suggest this a site of gap junctions among these cells.

The soma compartment had a length of 120 μm and a diameter of 90 μm and contained 10 currents. The neurite had a length of 1,380 μm, and a diameter of 12 μm, and contained 5 currents. The SIZ had a length of 108 μm, and a diameter of 20 μm, and contained 3 currents. The Na and K_d channels in the SIZ were given fixed conductances of 0.2 and 0.4 S/cm², respectively, and we assumed a specific capacitance of 1.5 μF/cm² for all three compartments. The model for currents for each compartment followed the Hodgkin-Huxley equation formulation (Eqn. 1)

$$\begin{gathered} C_{\mathit{\text{soma}}}\frac{dV}{dt}=-I_A-I_{\mathit{\text{Kd}}}-I_{\mathit{\text{Nap}}}-I_{\mathit{\text{CaS}}}-I_{\mathit{\text{CaT}}}-I_{\mathit{\text{CAN}}}-I_{\mathit{\text{SKKCa}}}-I_{\mathit{\text{BKKCa}}}-I_{\mathit{\text{leak}}}-I_{\mathit{\text{soma}}-\mathit{\text{to}}-\mathit{\text{neu}}}\left(\text{Soma}\right) \\ {C}_{\mathit{\text{neu}}}\frac{dV}{dt}=-I_{\mathit{\text{CaT}}}-I_{\mathit{\text{CaS}}}-I_{\mathit{\text{BKKCa}}}-I_{\mathit{\text{NaP}}}-I_{\mathit{\text{leak}}}-I_{\mathit{\text{neu}}-\mathit{\text{to}}-\mathit{\text{soma}}}\left(\text{Neurite}\right) \\ {C}_{\mathit{\text{siz}}}\frac{dV}{dt}=-I_{\mathit{\text{Na}}}-I_{\mathit{\text{Kdr}}}-I_{\mathit{\text{leak}}}-I_{\mathit{\text{siz}}-\mathit{\text{to}}-\mathit{\text{neu}}}+I_{\mathit{\text{syn}}}\left(\text{SIZ}\right) \end{gathered}$$ (1)

where the currents on the right-hand side of the first equation are: A-type potassium (I_A), delayed rectifier (I_Kd), persistent sodium (I_Nap), slow persistent calcium (I_CaS), transient calcium (I_CaT), calcium-dependent non-selective cation (I_CAN), two calcium-dependent potassium currents (I_SKKCa and I_BKKCa), leak (I_Leak) and the current from soma to neurite. The individual currents were modeled as $I_c=g_{\mathit{\text{max}},c}{m}^p{h}^q(V-E_c)$, where $g_{\mathit{\text{max}},c}$ is its maximal conductance of compartment c, m its activation variable (with exponent p), h its inactivation variable (with exponent q), and $E_c$ its reversal potential (a similar equation is used for the synaptic current but without m and h). The kinetic equation for each of the gating functions x (m or h) takes the form

$$\frac{dx}{dt}=\frac{x_{\infty }\left(V,{\left[{Ca}^{2+}\right]}_i\right)-x}{{\tau }_x(V,{\left[{Ca}^{2+}\right]}_i)}$$ (2)

where $x_{\infty }$ is the steady state gating voltage- and/or Ca²⁺- dependent gating variable and ${\tau }_x$ is the voltage- and/or Ca²⁺- dependent time constant. The equations for the active channels in the soma compartment were fit using biological recordings for these currents from the cardiac ganglion of Cancer borealis in our Lab (Table 4). These currents were fit as follows: Voltage clamp data obtained with Clampfit were imported into MATLAB and fit using the MATLAB curve-fitting toolbox. Current data were converted to conductance data by dividing by (V_m – E_Rev), where E_Rev was calculated as follows: E_Na = +55 mV, E_K = −80 mV, E_Ca = +45 mV, E_Leak = −50 mV, and E_CAN = −30 mV. The time axis was adjusted to start from 0 for the beginning of the clamp. The following parametrization was used:

$$g\left(t\right)=\sum _{i=1}^n{A}_i\left(1-\text{exp}\left(\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau }_{m,i}$}\right.\right)\right)\left(h_i-\left(h_i-1\right)\text{exp}\left(\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau }_{h,i}$}\right.\right)\right)$$ (3)

In this equation, ${\text{A}}_{\text{i}}={\text{G}}_{\text{i},\text{max}}\times {\text{m}}_{\text{i}}$ was the maximal conductance of the current i multiplied by its voltage-dependent steady-state activation (m_i), h_i was the steady-state inactivation value, and τ_m,i and τ_h,i were the time constants with which activation and inactivation reached steady-state, respectively. This fitting procedure assumed that ionic currents were completely deactivated m=0 and deinactivated (h=1) prior to the onset of the voltage clamp. This was fit to each trace in voltage clamp experiment, giving values of each of the four parameters for each test clamp voltage (V_c). These values were then fit for each current as functions of V_c using the general forms as stated below. This procedure yielded equations for the currents recorded in voltage clamp that could be used in the Hodgkin-Huxley formalism.

$$\begin{gathered} {A\left(V_c\right)=G}_{\mathit{\text{max}}}\times m\left(V_c\right)=G_{\mathit{\text{max}}}\times {\left(1+\text{exp}\left(\left(V_c-V_{m,1/2}\right)/k_m\right)\right)}^{-1} \\ h\left(V_c\right)={\left(1+\text{exp}\left(\left(V_c-V_{h,1/2}\right)/k_h\right)\right)}^{-1} \\ {\tau }_m\left(V_c\right)={\tau }_{\mathit{\text{base}},m}+{\tau }_{\mathit{\text{amp}},m}{\left(\text{exp}\left(\left(V_c-V_{\tau 1,m}\right)/k_{\tau 1,m}\right)+\text{exp}\left(\left(V_c-V_{\tau 2,m}\right)/k_{\tau 2,m}\right)\right)}^{-1} \\ {\tau }_h\left(V_c\right)={\tau }_{\mathit{\text{base}},h}+{\tau }_{\mathit{\text{amp}},h}{\left(\text{exp}\left(\left(V_c-V_{\tau 1,h}\right)/k_{\tau 1,h}\right)+\text{exp}\left(\left(V_c-V_{\tau 2,h}\right)/k_{\tau 2,h}\right)\right)}^{-1} \end{gathered}$$ (4)

All the maximal conductances (G_i,max) were in μS, time constants in ms, and voltages in mV.

Calcium dynamics.

Intracellular calcium modulates the conductance of the calcium-activated potassium currents (I_BKKCa and I_SKKCa), and calcium-activated non-selective cation current (I_CAN) influences the magnitude of the inward calcium current in the LC [48]. A calcium pool was modeled in the LC with its concentration governed by the first-order dynamics [2; 3] below:

$${\tau }_{Ca}\frac{d\left[{Ca}^{2+}\right]}{dt}=-F\times I_{Ca}-(\left[{Ca}^{2+}\right]-{\left[{Ca}^{2+}\right]}_{\mathit{\text{rest}}})$$ (5)

where F = 0.256 μM/nA is the constant specifying the amount of calcium influx that results per unit (nanoampere) inward calcium current; τ_Ca represents the calcium removal rate from the pool; and [Ca²⁺]_rest = 0.5 μM. Voltage-clamp experiments of the calcium current in our lab showed the calcium buffering time constant to be around 690 ms (τ_Ca) [4]. We note that NEURON has a baseline Ca2+ concentration in all single cell models [49].

Searching for viable biophysical LC neurons within the model parameter space

We used a three-level selection protocol to select viable networks with five distinct LC cells, satisfying biological single cell and current injection response criteria. For the single cell model search, we started with a set of 20,000 three-compartment cell models with the 14 conductances selected randomly from uniform distributions of values between their respective minimum and maximums given in Table 1. The model SC frequency range of 16–32 Hz was determined from SC recordings in our lab. The recordings showed that the combined small and medium spike frequency averages had a minimum and maximum frequency of 16 Hz and 32 Hz, respectively. Since the method we used is new, we have summarized the three-level selection protocol in Results.

In SELECTION LEVEL 1 of the rejection protocol, ligated soma (soma + 600 μm neurite) models were tested for passive properties of resting membrane potential and input resistance. A 600 μm of passive neurite was included because we assume that in ligation some of the neurite remains and the damage done to the neurite during ligation reduces the functionality of the active channels. Cells for which these values were within ranges in Table 2 were retained. In SELECTION LEVEL 2, SIZ was added to the passing cells and a synaptic input (spike train mimicking SC input) was provided. The neurite length was also increased to 1380 μm, using estimates from an electron micrograph picture of the cardiac ganglion [50]. Based on biological recordings from our lab, the average values for an SC burst were as follows: a total period of 1000 ms, and the synaptic input frequency started at 300 ms and increased for a duration of 600 ms, until 900 ms, after which it fell back to the original frequency and terminated at 1000 ms. The change in frequency that occurs mid-burst was designed to start at an equal distance from the first. For example, if the initial burst was spaced such that each burst was 40 ms apart, the higher frequency burst would start 40 ms after the previous one, rather than exactly at 300 ms. (Fig. S1). The synapse had a fixed gain of 0.09 μS in LV2. The reversal potential was chosen as −15 mv, from STG studies [51], and did not incorporate short term plasticity. This was chosen from a range of values as the one that produced the most passing cells in the tested sample. Values were picked from a range between 0 −0.2 uS, since 0.2 uS restored synchrony in TEA conditions in [3]. The frequency of SC input for the 600 ms duration is randomly picked betweeen 16 and 32 in 1 Hz interval. For example, for the first frequency interval, if a number between 16 and 17 is chosen randomly, say 16.4, then the input for the first 300 ms would be 40% of 16.4, which is 6.56 Hz. then for 600 ms it would be 100%, i.e., 16.4 Hz, and then it would drop to 6.56 Hz for the remaining 100 ms. The current was delivered to the SIZ using Neuron’s VecStim. Events were delivered to VecStim from a csv which was generated based on the random SC frequency chosen. The protocol for SELECTION LEVEL 3 (waveform and synchrony) involved testing every cell at each frequency interval between 16 and 32 Hz, i.e., a random frequency between 16–17, then a random frequency between 17–18, etc. Similarly, level 3 selection protocol for each network involved testing it at each frequency interval. The reason for using randomly chosen intervals rather than fixed numbers was to ensure that more frequencies would be used and the distribution of passing cells across frequencies would be more even. The network model uses gap junctions between LC1 and LC2, and between LC4 and LC5, and had gap junctions between all SIZs. The coupling conductance between the large cells was selected from our prior study as 0.65 uS [21]. And the coupling conductance between the SIZs was identical and selected as 0.067 uS since this provided reasonable numbers of passing networks.

Both control and post-TEA case are considered in intact cells whose resulting membrane potential waveform characteristics were within the ranges shown in Table 5 were retained. In a third level of the selection protocol, we assembled networks and selected viable ones as described next.

Table 5.

Ranges of waveform properties for selection of valid intact cells

Parameter	Min	Max
Spike number per burst	4	8
Avg LC spiking frequency	4 Hz	8 Hz
VPeak Control	7 mV	30 mV
AUC Control	2867 mV.ms	18373 mV.ms
Spike number per burst TEA	>1.13 × Control SPB	none
AVG LC spiking frequency TEA	>1.21× Control frequency	none
VPeak TEA	>1.3× Control Vpeak	none

Determination of viable network models using the selected single cell models

First, we randomly selected five distinct LCs for the network from the pool of intact LCs that pass selection protocol level 2. Then we connected an SIZ to the distal end of the neurite of each LC with fixed sodium, potassium, and leak conductances, and attached a synapse to the SIZ. The SC input drive was delivered as a spike train to the five excitatory synapses on their SIZs. The SIZ compartment and the excitatory SC-SIZ synapse configuration was identical for each LC, i.e., we did not vary model parameters for these compartments and synapses. However, the SC drive was the same for all SIZs.

Experimental TEA block was simulated by reducing the conductances G_BKKCa, G_Kd and G_A by 97% in the LCs and neurites [27]. Synchrony scores were computed by using Pearson’s R² between the large cells 3 and 5 to compare with our biological findings. To determine the range for synchrony scores, we examined five experimental recordings for LC3 and LC5 (unpublished data). Taking the first two minutes of TEA exposure (acute), we measured the R² between LC3 and LC5 recordings. We chose the maximum TEA synchrony to be the lowest control synchrony score minus 1.5 times the control interquartile range. The lowest control synchrony score was 0.9425, and the control interquartile range was 0.0318, therefore the maximum of TEA synchrony was taken to be 0.8948. The cells were considered desynchronized if the synchrony score was below 0.9425. For the synchrony score between LC3 and LC5, we considered anything below 0.89 to be asynchronous, and anything above 0.9425 to be demonstrating synchrony.

After performing control and TEA runs using these networks, in the third level of our selection protocol, we rejected networks that had waveform characteristics outside the ranges shown in Table 6. We rejected networks that showed excessive TEA synchrony between LC3 and LC5, or insufficient spikes per burst since neither behavior was observed in biological traces. This resulted in 130 networks that reproduced the biological trends, and these were used in subsequent analyses to explore potential conductance changes that could restore network synchrony.

Table 6.

Ranges of waveform and synchrony properties for selection of valid networks

Parameter	Min	Max
Spike number per burst	4	8
Avg LC spiking frequency	4 Hz	8 Hz
VPeak Control	7 mV	30 mV
Spike number per burst TEA	>1.13 × Control SPB	none
AVG LC spiking frequency	>1.21 × Control frequency	none
VPeak TEA	>1.3x Control Vpeak	none
R2 synchrony LC3 to LC5 control	0.95	1.0
R2 synchrony LC3 to LC5 in TEA	0	0.89

Repeated parameter sets

Figures 3 and 4 provide visualization of the parameter sets that pass the selection criteria. The parameter sets are included for all frequency values for which they passed. For example, if a parameter set for one network passed at 6 different SC frequencies, it was included 6 times. This allows us to look for correlations between individual parameters and SC frequency, as well as to check if a particular parameter set could pass at many frequencies.

Construction of averaged networks

To form the averaged networks, we calculated the average of each LC conductance for each network that passed LV3 rejection criteria. Then we constructed one ‘averaged network’ for each of the passing networks, resulting in a pool of 130 averaged networks in our case.

Statistical Analyses

Statistical analyses and data visualizations were performed using R version 3.5.3 (2019-07-05) “Action of the toes.” To examine pairwise relationships among conductances we used Spearman correlation tests, as most data were not normally distributed (Shapiro-Wilks test for normality). Estimates of the Euclidean distance between the parameter sets were performed by SciPy’s Euclidean function in the spatial distance module. Euclidean distance was estimated as follows, $\text{Euclidean}\text{distance}={\left(\sum {\left|{(u}_i-v_i)\right|}^2\right)}^{1/2}$ [52]

Model availability

As with our prior models, the computational model for the CG ganglion network will be made available at ModelDB upon publication.

Supplementary Material

Supplementary Materials are available at the link: https://doi.org/10.6084/m9.figshare.30983881