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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Neurosci. Author manuscript; available in PMC 2010 August 3.
Published in final edited form as:
PMCID: PMC2851246
NIHMSID: NIHMS176368

Model calcium sensors for network homeostasis: Sensor and readout parameter analysis from a database of model neuronal networks

Cengiz Günay, Postdoctoral Fellow and Astrid A. Prinz, Assistant Professor

Abstract

In activity-dependent homeostatic regulation (ADHR) of neuronal and network properties, the intra-cellular Ca2+ concentration is a good candidate for sensing activity levels because it is correlated with the cell's electrical activity. Previous ADHR models, developed with abstract activity sensors for model pyloric neurons and networks of the crustacean stomatogastric ganglion (STG), showed that functional activity can be maintained by a regulation mechanism that senses activity levels solely from Ca2+. At the same time, several intracellular pathways have been discovered for Ca2+-dependent regulation of ion channels. To generate testable predictions for dynamics of these signaling pathways, we undertook a parameter study of model Ca2+ sensors across thousands of model pyloric networks. We found that an optimal regulation signal can be generated for 86% of model networks with a sensing mechanism that activates with a time constant of 1 millisecond and that inactivates within 1 second. The sensor performed robustly around this optimal point and did not need to be specific to the role of the cell. When multiple sensors with different time constants were used, coverage extended to 88% of the networks. Without changing the sensors, it extended to 95% of the networks by letting the sensors affect the readout non-linearly. Specific to this pyloric network model, the follower pyloric constrictor cell's sensor was more informative than the pacemaker anterior burster cell for producing a regulatory signal. Conversely, a global signal indicating network activity that was generated by summing the sensors in individual cells was less informative for regulation.

Keywords: calcium sensor, activity-dependent homeostatic regulation, pyloric network, stomatogastric ganglion, lobster, neuronal network model

Introduction

Central pattern generating (CPG) neuronal networks produce rhythmic motor activity patterns, which are vital for the survival of the organism. Although the constituent cells of CPG networks are faced with changes in environmental conditions and constant molecular turnover, the ability to generate these activity patterns is always maintained (Marder and Calabrese, 1996; Marder and Bucher, 2001). An activity-independent component of this regulation is controlled inherently (MacLean et al., 2003, 2005) or by neuromodulators endogenous to the STG (Thoby-Brisson and Simmers, 1998; Khorkova and Golowasch, 2007). In contrast, in the pyloric CPG network of the crustacean stomatogastric ganglion (STG) that controls the dilation of the pylorus, long-term changes in isolated neurons can result from external stimulation (Turrigiano et al., 1994; Thoby-Brisson and Simmers, 1998; Golowasch et al., 1999a; Zhang et al., 2009). This activity-dependent homeostatic regulation (ADHR) affects ion channel properties (Turrigiano et al., 1994, 1995; Golowasch et al., 1999b; Baines et al., 2001; Nelson et al., 2003) that govern the neurons' electrical activity (Foster et al., 1993; DeSchutter and Bower, 1994; Prinz et al., 2003; Günay et al., 2008). ADHR also affects synapses in CPG networks (Soto-Treviño et al., 2001; Thoby-Brisson and Simmers, 2002) and elsewhere (Turrigiano et al., 1998; Stellwagen and Malenka, 2006; Wilhelm and Wenner, 2008; Ibata et al., 2008). In both isolated and networked cases, ADHR requires an activity sensing mechanism.

A cell's electrical activity is correlated with the intracellular Ca2+ concentration (Ross, 1989), which was found to play a role in ADHR (Turrigiano et al., 1994) by possibly affecting ion channels (Linsdell and Moody, 1995; Golowasch et al., 1999a,b) through Ca2+ sensing proteins (Carrión et al., 1999; An et al., 2000; Mellström and Naranjo, 2001; Gomez-Ospina et al., 2006). For instance, the protein frequenin in Xenopus oocytes (Nakamura et al., 2001) and STG cells of the spiny lobster, Panulirus interruptus, (Zhang et al., 2003) affects the inactivation gate of the voltage-gated K+ transient outward current (IA). IA, by modulating the interspike interval, is crucial in regulating activity (Tierney and HarrisWarrick, 1992; Golowasch et al., 1999a).

Such a sensing mechanism was previously modeled (Abbott and LeMasson, 1993; Liu et al., 1998) and succeeded in maintaining a predefined target activity pattern. Also, a model of regulation in the pyloric network showed that stable network states can be reached by modulating ion channels in individual cells (Golowasch et al., 1999b). However, these models only demonstrated regulation with few, fixed sensor parameters. For biological systems, which often exhibit redundant, non-linear mechanisms, it has proved beneficial to explore the model parameter space (Foster et al., 1993; Golowasch et al., 2002) and store exploration results in databases (Prinz et al., 2003, 2004; Calin-Jageman et al., 2007; Günay et al., 2008). Here, we built a model Ca2+ activity sensor database to test the robustness of sensor performance by using an existing database of 20 million pyloric network models (Prinz et al., 2004) as a repository of different network configurations. From these networks, we tested whether the sensors can indicate (1) that a given network is producing functional activity patterns, and if not, (2) the direction into which network parameters need to change to reach a functional state. We achieved this by estimating the network activity from a statistical classifier that is trained as a sensor readout.

Methods

Model pyloric network database

This study used a previously described database of 20,250,000 model pyloric networks (Prinz et al., 2004). In the database, each model network consists of three single-compartment conductance-based model neurons—constructed with the Hodgkin and Huxley (1952) formalism—of a combined anterior burster and pyloric dilator (indicated as AB/PD or just PD throughout this paper) neuron, a lateral pyloric (LP) neuron, and a pyloric constrictor (PY) neuron

(Fig. 1). Different network models have model neurons selected from pools of five AB/PD, five LP and six PY models, whose parameters were described earlier (Prinz et al., 2003). The model neurons contain eight currents: a fast sodium current, INa; slow and fast transient calcium currents, ICaS and ICaT; a transient potassium current, IA; a calcium-dependent voltage-gated potassium current, IK(Ca); a delayed rectifier current, IKdr; a hyperpolarization-activated mixed-ion inward current, Ih; and a leak current, Ileak. The voltage-dependence and kinetics of these currents are based on recordings from STG neurons in the spiny lobster (Turrigiano et al., 1995)—except Ih, which is modeled after that found in guinea pig lateral geniculate relay neurons (Huguenard and McCormick, 1992). These conductance parameters are identical in all model neurons (Prinz et al., 2003). The selected maximal conductances for each current and their selection criteria were described in detail earlier (Prinz et al., 2004). The synapses between the model neurons are modeled with the dynamics used in Prinz et al. (2004), in which glutamatergic and cholinergic synapses were modeled differently. Each synapse was varied to five or six different maximal synaptic conductance values in different networks. The networks were simulated using a custom C++ program.

Figure 1
Biological pyloric rhythmic activity pattern and the model circuit architecture

Model calcium sensors

The calcium sensors explored here are inspired by an ADHR model for STG neurons (Abbott and LeMasson, 1993; LeMasson et al., 1993; Siegel et al., 1994; Liu et al., 1998) and measure the model cell's electrical activity from the Ca2+ inflow, ICa (Fig. 2A). Here, ICa = ICaT + ICaS and affects the Ca2+ dynamics from the same model, which describes the intracellular Ca2+ concentration ([Ca2+]) by

τCad[Ca2+]dt=fICa[Ca2+]+[Ca2+]0.
(1)

In this model, f = 14.96 μM/nA is a factor that relates ICa to the derivative of [Ca2+], τCa = 200 ms is the time constant for Ca2+ removal from the cytosol, and [Ca2+]0 = 0.05 μM is the intracellular equilibrium value of [Ca2+]. This model successfully generates Ca2+ transients similar to those measured in pattern-generating neurons (Di Prisco and Alford, 2004) for different types of electrical activity (Liu et al., 1998), although it assumes that the rate of removing, sequestering and buffering of Ca2+ is proportional to [Ca2+].

Figure 2
The sensor and readout components within the context of ADHR

Homeostatic regulation, for example through gene transcription, works on time scales much slower (minutes to hours) than the network rhythm period, which is why the readings of these ICa-dependent sensors were time averaged (Fig. 2A). The sensors were averaged over a network rhythm period to simulate an integrative process with a very long time constant, as opposed to Liu et al. (1998), where they were averaged over a fixed period of 5 s. The disadvantage of a fixed integration period is that it interacts with the length of the rhythm period, which varies across networks. In either case, averaging loses information about the temporal patterns of ICa, which are found to be important for influencing regulatory signal transduction pathways in experiments (Gallin and Greenberg, 1995; Bito et al., 1997). We follow the proposal of Liu et al. (1998) that the temporal information can be retained by averaging with different types of sensors, X, each of which is defined by a product of activation, M, and inactivation, H, variables (Fig. 2B) that depend on ICa with different time-scales and sensitivities:

X=M2H.
(2)

For non-inactivating sensors, H is omitted. Both variables, y [set membership] {M, H}, obey the formula,

τydydt=y(ICa)y,
(3)

where τy is the activation or inactivation time constant and the function y (ICa) determines the steady-state activation or inactivation, which is defined as

y(ICa)=11+exp(k(Zy+ICa)).
(4)

Here, k respectively becomes +1, −1 for defining [M with macron] and [H with macron] variables. The Zy parameters indicate current density thresholds (in nA/nF). In addition to the temporal averages of sensors, we also investigate the hypothesis that regulation may employ the minimal and maximal values of a sensor.

To study the performance of these sensors, we varied their τm, τh, Zm, and Zh parameters according to rules consistent with earlier work (Liu et al., 1998): τm < τh, because activation should be faster than inactivation (Eq. 3), and Zm > Zh, such that the sensor activates with larger currents (Eq. 4). The time constants are chosen across orders of magnitude from a logarithmic scale of 100 μs, 1 ms, 10 ms, 100 ms, 1 s, and 10 s; and current thresholds were chosen from the values of 0, 5, 10, 15, 20, 30, 40, and 50 nA/nF. This yielded 330 inactivating and 36 non-inactivating sensors. The inactivating sensor parameters were limited by the rules for the fast and slow sensors below.

To detect different types of activity, we also employed a case with multiple sensors in each cell (Liu et al., 1998). The fast (F), slow (S) and non-inactivating (D) sensors in this case (Fig. 5E) were defined by,

F=MF2HF,S=MS2HS,D=MD2,
(5)

where the M and H activation and inactivation variables were defined as above in the single-sensor-per-cell case. The FSD sensors were selected from the available single sensors using rules consistent with Liu et al. (1998):

τm,D>τm,S>τm,F,τh,S>τh,F,Zm,D<Zm,S<Zm,F,Zh,S<Zh,F.
(6)

These rules reduced the number of possible FSD sensor combinations from 254,803,968 to 85,750.

Figure 5
Activity sensors provide more information than activity characteristics such as bursting duty cycle, and activation and inactivation variables improve the estimation from sensors

During simulation, the time averaged readings, maxima and minima of sensors with varying parameters were saved into a file corresponding to the network number. In this fashion, a sensor database was generated of sensor average and extremum values from each of the three model cell types in the 20,250,000 networks. This sensor database was then read into a relational database management system (Codd, 1970), MySQL (MySQL AB, Uppsala, Sweden). The database occupied 285 gigabytes (GBs) of space in a table and another 90 GBs for its index structure that enabled optimized access and query to the table for the analysis of the results. The sensor database is available on request from the authors. File management and analysis were performed with scripts in PERL (The PERL Foundation, Holland, Michigan) and Matlab (Mathworks, Natick, MA); those scripts are also available upon request. The Matlab analysis scripts took advantage of the database analysis functions of the Pandora Toolbox (http://software.incf.org/software/44/view/PANDORA; Günay et al., 2009). The sensor database was used to separate networks with functional and non-functional activity patterns using a classifier algorithm.

Using a classifier

Out of the networks in the database, only 2% were previously categorized as functional because they were within ±2 standard deviations of recorded data from the lobster in terms of 15 salient features (such as burst period, durations, and phases) of their activity patterns (Prinz et al., 2004). The difference between functional and non-functional networks that was reflected in their activity patterns was also captured in their sensors readings (Fig. 3B,C). Based on the average of these sensors, we detected whether a network is functional by training a custom classifier (Fig. 2C).

Figure 3
Functional versus non-functional activity patterns of the model network are reflected in the sensor readings

To find a classification solution that is simple and easily interpretable, we first employed a linear classifier. We employed the perceptron classifier (Rosenblatt, 1958) using the newff function of Matlab (Mathworks, Natick, MA). The perceptron is a single-layered, feed-forward artificial neural network (ANN) classifier (Rumelhart et al., 1986b) that defines an optimal hyperplane to linearly separate its inputs. Such optimal separating hyperplanes were used before in similar work to separate neural activity types (Goldman et al., 2001; Taylor et al., 2006).

In the space of multiple sensor readings from the same or different cells, the classifier finds a hyperplane defined by the weight vector, w, and the offset, b, parameters, w × Xj + b = 0, that best separate functional networks (Fig. 3B) from non-functional networks (Fig. 3C) using the sensor vector, Xj, across all networks 1 ≤ jN (Fig. 2C). It does this by minimizing the sum of squares, S, of the differences between the classifier score, cj = h (w × sj + b), and the labels, ĉj, which are 0 for non-functional or 1 for functional for each network j. To match these labels, we normalized the inputs and output of the function, h. The function h is a linear function, which provides a low or high output based on which side of the hyperplane the given sensor values lie. The classifier was initialized with random weights and optimized with the Levenberg-Marquardt training algorithm (Levenberg, 1944; Marquardt, 1963). We judged the success of the classification by the average percentage of correctly and incorrectly classified networks,

r=100(cFnF+cNnN)2,
(7)

where c and n respectively denote the correctly classified and total number of networks, and the subscripts F and N respectively stand for functional and non-functional networks (Taylor et al., 2006).

Training the classifier would suffer from the imbalance in the database between positive and negative samples (2% functional vs. 98% non-functional networks), which was also preserved in its selected smaller subsets (see Results below). To overcome this, we presented each set equally by randomly repeating the fewer functional networks (Lawrence et al., 1998; Günay and Prinz, 2009).

The ANN classifier reached a different sub-optimal local minimum for repeated training even with the same sensor inputs because of the random initial conditions, yielding variable success rates. However, the variability was low: the success rate obtained from training a linear classifier 20 times from the readings of the best sensor varied less than 1% (Fig. S1). Consistent with our search for an optimal classifier, here we always report the maximum success rate obtained across ten repeated classifier training runs. We also verified ANN training using a 50% cross validation set (data not shown).

As a more complex classifier, we used a multi-layered perceptron (MLP), or multi-layered ANN (Rumelhart et al., 1986a). In this classifier, an output layer unit linearly weights the outputs of several single-layer perceptrons, as described above, with ej = h (∑i vicij + d), where h is a sigmoidal function, cij are the outputs of perceptrons, vi are linear coefficients, and d is another offset. In the multi-hyperplane case, the weights of sensors within each hyperplane, w, as well as the hyperplane weights, vi, vastly varied across different training runs of the classifier, preventing extraction of consistent rules (Fig. S5). Two factors contributed to getting different weights every time: (1), sensor weights are multiplied with hyperplane weights, allowing them to switch signs to achieve the same result; and (2), hyperplanes need not be in the same order to achieve the same result. We eliminated these two factors that added to the variation of the resulting weights by: (1), switching all output hyperplane weight signs to positive by inverting their sensor weights if necessary; and (2), sorting hyperplane positions according to the magnitude of their output weights (Guha et al., 2005; Günay and Prinz, 2009).

Training for neither perceptrons or MLPs required manually specifying any parameters such as learning rate or momentum.

Results

For activity-dependent homeostatic regulation (ADHR), the feedback element should be capable of sensing network activity levels. Here, we show that the proposed calcium sensors perform activity sensing as needed.

Activity sensors are sensitive to firing rate and duty cycle

It was shown previously in individual and networks of model cells (Liu et al., 1998; Golowasch et al., 1999b) that homeostasis can be maintained with activity sensors based on calcium current, ICa. When we place sensors of this type in all model cells of our pyloric network (Fig. 3A), firing activity is apparent in sensor readings. In particular, sensor readings look vastly different between example networks with functional (Fig. 3B) and non-functional (Fig. 3C) activity patterns.

However, it is not known which features of these activity patterns are important for homeostatic regulation. For instance, the duration of membrane depolarizations in cultured hippocampal neurons affects which gene expression targets are triggered by ICa (Mermelstein et al., 2000). Assuming that our ICa-based sensors serve a regulatory function also, we asked which activity pattern features are reflected by the sensor outputs. We achieved this by looking for correlations between activity characteristics and sensor averages.

Among all activity characteristics, we found significant correlations between burst duration, number of spikes per period (i.e., firing rate) and bursting duty cycle. It is straightforward to assume that a longer burst duration will increase the sensor average. Traces from example model cells (Fig. 4A) confirm that this relationship is true; however, it is only true when only the burst duration increases and the period is fixed. Without fixing the period, the relation is still true for some model cell types across the networks in the database (regression p < 10−4 for PY cell; Fig. 4E), but does not generalize to others (regression p = 0.7 for the AB/PD cell; Fig. 4C) because the AB/PD cell maintains a fixed duty cycle—the ratio of the burst duration to the cell period (Fig. S3)—which is consistent with recent experimental evidence from spiny lobsters (Reyes et al., 2008).

Figure 4
Activity sensors represented both the cell's bursting activity and its rhythm period, being a good indicator of the bursting duty-cycle. The sensor used is the same as in Fig. 3

A measure of activity that correlates better with the sensor average is the number of spikes in a period (Fig. 4D). However, the number of spikes does not generalize as an activity measure because it does not take into account the effect of the network period on the sensors (Fig. 4B). Duty cycle, which accounts for the period, is consistently and positively correlated with sensor averages from all three model cells (p < 10−4, Fig. 4F–4H). We next investigated what this means for homeostatic regulation.

An optimal sensor readout can estimate success of homeostatic network regulation

The original networks in the database were categorized as functional if they were producing activity patterns that are within 2 standard deviations of 15 electrophysiological criteria; including the burst duration, number of spikes in burst, and burst start time (Prinz et al., 2004). Based on this categorization, we trained an optimal classifier to quantify the sensor performance independent of a specific readout or regulatory mechanism (Fig. 2A). If this optimal readout can accurately estimate the functional network state across many network configurations, based alone on the activity sensors, it demonstrates that the sensors provide sufficient information for ADHR. The accuracy of this estimate provides the qualitative measure of goodness for the sensors used, which is calculated as a success rate (see Methods). Using this readout, we evaluated different parameters of previously proposed sensors (Liu et al., 1998) and their placement in the model networks (Prinz et al., 2004).

Sensors were individually weighted to optimally separate a randomly selected 10,000-network subset of the original network database into functional and non-functional networks. Sensor data from all cells were used at the same time, as a global signal, to find the optimal sensor properties. We addressed more realistic cases with sensor data from single cells further below. With a single activity sensor in each model cell (Fig. 3), each sensor weight indicated the importance of that cell's sensor readings (Fig. 3D). The LP and PY cell sensors were almost twice as influential as AB/PD in determining the functionality of the sensed activity pattern. From the weighted sensors, we calculated a classifier score that successfully distinguished 86.40% of the networks with triphasic bursting activity (Fig. 3E). In the classifier score distribution, most non-functional networks are grouped around zero, which also contains tonic-firing networks (Fig. 3C). Some of the non-functional networks were estimated to be functional (i.e., classified as false-positive), such as ones showing depolarization block during bursts (Fig. 3F). Most functional networks (95%) produced scores above 0.5 (Fig. 3E).

To test whether the results obtained from this random selection of model networks can predict the rest of the model network database, we used the classifier obtained from the selected random subset to classify other sensor subsets from up to 100,000 networks and consistently obtained maximal success rates >85% (Fig. S2). Next, we tested whether this prediction performance is specific to our sensors.

Sensors are more informative than activity characteristics

If the sensor readings are highly correlated with specific activity characteristics, we asked if, in theory, these characteristics can provide sufficient information for homeostatic regulation and replace the sensors completely. To answer this, we compared the estimation power from the optimal readout strategy using the activity characteristics versus the sensors. We found that the duty cycle characteristic has the most estimating power (70%) compared to other activity characteristics, such as spike rate (Fig. 5A). However, although the duty cycle is highly correlated with the sensors, the sensors were +16% better in estimating the network output. Thus, the sensors convey more information than any of these activity characteristics alone or even all combined together (Fig. 5A). To address exactly what features of the sensors are responsible for this added information, we examined the estimation power of the underlying observed quantity, ICa.

Activity sensing using ICa is as informative as using [Ca2+]

For ADHR, ICa may be more informative than the intracellular [Ca2+] (Turrigiano et al., 1994) because the mode of Ca2+ entry into the cell matters for gene regulation targets (Murphy et al., 1991; Gallin and Greenberg, 1995; Tadross et al., 2008); however, it is not known whether this makes a difference in a model of ADHR. To answer this question, we used a testbed identical to the one used above to make estimations from the three model cells, but instead of sensors, we compared the estimating power of total ICa and of [Ca2+]. To establish a control case, we shuffled [Ca2+] values by assigning them to random networks, which disrupted the classifier and caused it to estimate at close to chance levels (53.37%). Without shuffling, we achieved success rates above the control level and above success obtained from activity characteristics, but also found that using either ICa or [Ca2+] resulted in very similar rates at 77% (Fig. 5B; Table S3). This suggests that ICa is as informative as [Ca2+] for use in homeostatic regulation. But since the sensors can achieve +9% better success than using ICa alone, we investigated the benefits of the activation and inactivation mechanisms in the sensor model.

Sensor inactivation is essential for detecting functional activity

The activation and inactivation of sensors roughly correspond to the production and removal of second messenger proteins in the cell and it was shown that inactivation is necessary for detecting different types of activity patterns (Liu et al., 1998). Compared to the 77.14% estimation success obtained with using ICa directly, using sensors with only an activation variable increased the success to 82.57% (Fig. 5B; Table S3). Adding an inactivation variable further increased the estimation success to 86.40%. To find the optimal sensor from each type of sensor compared, we tested 366 sensors with different activation and inactivation parameters (Fig. 5C). The optimal sensor activated with a time constant of 1 ms and a half-activation threshold of 5 nA/nF of calcium current and inactivated with a time constant of 1 s and a half-inactivation threshold of 0 nA/nF (sensor #87 in Table S2).

Sensor minimum and maximum are also informative for regulation

In the slow time-scale of homeostasis, ADHR may use activity sensor features other than the average. Adding the sensor minimum and maximum to the sensor average in the estimation by using their weighted sum—combined the same way as in FSD sensors (see Methods)—increased the success rate to 87.69% (Fig. 5B,D and Table S3). Compared with the optimal sensor, the sensor that had the most information in its minimum and maximum has different parameters (sensor #268 in Table S2); it is a non-inactivating (DC) sensor with Zm = 0 nA/nF and τm = 0.1 s. This suggests that inactivation could be replaced without loss of information by the minimal and maximal values of a slowly activating sensor. Because prediction using multiple values derived from a single sensor improved classification, we also tested using multiple different sensors in each cell.

Using fast, slow and DC sensors in each cell

For ADHR of conductances in a model neuron, Liu et al. (1998) suggested that a fast, a slow and a non-inactivating DC sensor (FSD sensors) that correspond to homeostatic regulation pathways operating at different speeds can be combined to detect activity at different time scales. We used these FSD sensors, but they barely improved our success to 88.17% (see Table S3). We generated 85,750 combinations of FSD sensors following previous rules (see Methods; Liu et al., 1998). The distribution of classification success obtained with these FSD sensor combinations was unimodal (Fig. 5E), similar to the success distribution obtained from networks with a single, same sensor in each cell (Fig. 5C).

Functional sensor parameters are broadly tuned

It is important to understand the parameters of successful activity sensors to make predictions about possible biological mechanisms that may underlie them. By comparing the maximal success rates obtained for different parameter configurations, we found optimal values for each of these parameters for the non-inactivating (Fig. 6A) and inactivating (Fig. 6B) sensors types separately. The most successful sensor is an inactivating-type sensor with low current threshold values, Zm = 5 and Zh = 0 nA/nF, and time constant values of τm = 1 ms and τh = 1 s (same as sensor #87 above in Figs. 2B, ,3,3, and and5A).5A). The tendency to select a low activation current threshold implies that the ideal sensor is sensitive to small Ca2+ currents, whereas an inactivation threshold of zero indicates that the sensor never inactivates completely, since the calcium current is always negative due to its reversal potential (see Methods).

Figure 6
Sensor time constant (τ) and calcium current sensitivity (Z) parameters are broadly tuned. In the bar plots, the inner bars show the maximal success and blank bars show the mean success. The dotted horizontal line shows the mean success from all ...

In general, lower current threshold values yielded higher success, except in the non-inactivating sensors as observed from the success rate distributions. The distributions were unimodal: the success rate diminishes smoothly as parameters vary away from the optimal point, indicating that the sensor is tuned broadly. The parameters have no critical values beyond which the sensors failed—except the DC sensors with Zm > 0, which are significantly worse. The optimal points found by the mean success value are generally consistent with the points found from maximal success, except for few parameters (e.g., τm of inactivating sensors and Zm on non-inactivating sensors). When assessing optimal values from sensors with multiple varying parameters, the mean and maximal values may be artifacts from dependencies between two or more parameters (Golowasch et al., 2002). However, independently selecting each parameter's value at the peak of its maximal success corresponds to the parameters of the best sensor (#87) found earlier, suggesting that parameters are not interdependent. We completely ruled out the possibility of interdependency by inspecting interactions among all possible parameter pairs for inactivating and non-inactivating sensor types (Fig. S4).

The parameters of the FSD sensors are also broadly tuned (Fig. 6C). The peaks of the maximal success tuning curves are consistent with the optimal FSD sensor combination (#34,457) we found, whose parameters slightly differ from those used in previous work (Table 1). Although the FSD sensor current thresholds are in decreasing order and time constants are in increasing order (see Methods), each of the three sensors we found are specialized to different roles (Fig. 6C). Specifically, we observed that:

  1. The activation time constant, τm, of the fast and slow sensors tends to be as fast as possible, while the DC sensor has no preference.
  2. The inactivation time constant, τh, of the slow sensor prefers a long value of about 1 s, which is similar to the case of a single sensor (Fig. 6B), except that the fast sensor produced high success rates irrespective of the selected τh.
  3. The activation current threshold, Zm, tends to be higher for fast sensors (20–50), consistent with the idea of DC and slow sensors detecting the bursting envelope and the fast sensor detecting the spike firing activity, which causes larger calcium currents.

Once we found the most effective sensor type and its parameters, we investigated the advantages of local versus global placement of sensors in achieving the best separation between functional and non-functional network patterns.

Table 1
Comparison of the parameters of the optimal FSD sensor combination (#34457) with those of the previously published FSD sensor from Liu et al. (1998). All Z values are in nA/nF and all τ parameters are in seconds.

Local sensors can estimate network activity

In above results, we allowed the readout to access sensors from all three cells. However, readouts in living cells are most likely to have access to the local activity sensors only. To test if local sensors from a model cell can estimate the outcome of the network activity, we measured the estimation power of sensors from each individual model cell (Fig. 7A). From individual cell sensors, we obtained estimation success levels well above chance (Fig. 7A). In particular, different model cells provided different levels of estimation power: the PY cell sensor output is most informative, yielding an 83.34% classification success, while the sensors in the LP and AB/PD model cells are less informative, both yield a success of about 76%. As opposed to the limited information given by a local sensor, a global sensor may give a more accurate estimation of network activity.

Figure 7
Local sensors can estimate network activity and sensor variety between cells does not yield better estimation success

A global sensor can estimate network activity

To test if a global activity sensor can be more informative, we built an abstract model of a global sensor by summing the activity sensors from each model cell into a single value. First, we summed the optimal sensors of each model cell found above, which resulted in a estimation success of 82.46%. Second, we summed the same sensor in all three cells to find the success distribution across all possible sensor parameters, which reached a higher success of 83.66% (Fig. 7B). Considering that the standard deviation (SD) of the success rate is 0.15% (Fig. S1), the global sensor estimations were not better than, but at about the same level with, estimations obtained from individual model cell sensors.

Cells with different roles in the network need not have different activity sensors

The different optimal sensors from each cell estimated the network outcome well independently, but when they were summed before classification, the estimation success did not increase. Can they generate a better estimation if all of their information is passed to the readout; that is, when their sensors are individually weighted and summed by the classifier? To answer this question, we classified the networks using the optimal sensors from all model cells. If the information given from the PY sensors would add to the information from the other two cells, we would predict reaching a success rate higher than 83.34%. However, this was not the case and the best three-sensor classifier reached a similar, 83.33%, success rate—almost identical considering an SD of 0.15%. This indicated that: (1) the sensors in the three cells contain redundant information that does not add up when combined; and (2) using the optimal sensors for each cell is less informative (83%) than finding one general sensor that maximizes the estimation in the whole network (86%).

Optimal sensors found by independent estimation also gave insights about the sensor readout: the classifier weights for the best sensors for each cell (Fig. 7C) were different than the weights found when the same optimal sensor was employed in all cells (Fig. 3D). Specifically, the weight and offset for the LP cell sensor was the reverse of the AB/PD and PY cells. This indicated that the role of the LP cell in affecting the outcome of the network activity is different than the two other cells. This relationship between sensors from the different cells was maintained in the weights obtained from the classifier trained on the combined sensor outputs from the three cells (Fig. 7D), but was inconsistent with weights of the optimal sensor readings (Fig. 3D). The difference between weights may mean that sensors are tuned to best read a specific neuron's activity based on its function in the network. To address each cell's function in the network, we looked for a relation with its sensor outputs.

Sensor readings in each cell contribute differently to classifying functional network activity

The classifier weights trained to the best sensor (#87) indicated that the AB/PD activity is about half as important as the LP and PY activity (Fig. 3D and and8A).8A). The classifier weights obtained from the next-best ten sensors (Fig. 8A), as well as the statistics of all sensors performing better than a 80% success rate (Fig. 8B) were consistent with this distribution of weights. The weights found were significantly different from cell to cell for the most successful sensors (p < 10−4 with one-way ANOVA; Fig. 8B).

Figure 8
Classifier weights of best performing sensors consistently selected specific proportions of each cell's sensor for classifying functional network activity

One explanation for the imbalance between contributions of AB/PD cell's activity versus the other cells can be found by inspecting the average sensor readings of the networks: the readings of the AB/PD cell were generally higher than those of the other two cells, PY producing the lowest readings (Fig. 8C). This means that generally the AB/PD cell has a larger duty-cycle than the LP cell, and LP has a larger duty-cycle than the PY cell. When all networks (n = 9, 915) are considered (Fig. 8C), activity from non-functional networks may confound these results. However, the ranking between the sensed activity of the three cells is preserved for the smaller subsets of networks with functional (n = 221, Fig. 8D) and networks that were found functional by the classifier (“classified functional”, n = 2, 834, Fig. 8E), with the exception of high activity in the PY cell sensors in the functional networks. From these sensor reading distributions, we were also able to predict types of activity patterns produced by the network (see Fig. S6 and the Supp. Mat.).

Linear readout failed to correctly classify specific activity patterns

With the linear classifier, although using FSD sensors from each cell improved the estimation of functional network patterns, about 12% of the networks were still misclassified. This may point to a limitation of either the model calcium sensors or their readout, the linear classifier. To distinguish between these two possibilities, we investigated the types of activity changes that were misclassified. To find hard-to-classify examples, we searched the sensor database subset for examples of similar network activity at the decision threshold of the classifier score: to find two networks that produce activity patterns as similar as possible, while one of them is misclassified. Such an example network pair illustrates that our estimation testbed can confuse two similar types of activity (Fig. 9). In this example, both network activity patterns were estimated as functional, although one network's activity was previously categorized (Prinz et al., 2004) as non-functional (Fig. 9A). The non-functional network exhibited an excessive activity in the LP cell which was reflected as an increase in its DC sensor average (Fig. 9B). In turn, this sensor had a negative classifier weight (Fig. 9C), translating the increase in the sensor average into a decrease in the classifier score. This reduced the non-functional network's score down to 0.7 from an initial score of 1.1 in the functional network. However, this decrease was insufficient to cross the decision threshold at 0.5. Therefore, this example demonstrates that the calcium sensors can correctly detect slight changes in activity but the linear classifier used to interpret the sensor readings is too simple to produce the correct output. This suggests that to improve estimation from sensors it is necessary to have a better readout mechanism than a classifier that is linear.

Figure 9
The calcium sensors misclassified about 12% of the networks

Using a nonlinear readout increased success to 95%

A linear classifier fails when the decision boundary between functional and non-functional networks are linearly non-separable in the sensor space (Fig. 2C). Linearly non-separable sensor inputs require multiple hyperplanes to correctly separate the two network populations (Fig. 2D). When we increased the number of hyperplanes in classifying from the FSD sensor readings, we reached success rates of 95% correct (Fig. 10A). The success rate saturated after 5–10 hyperplanes, indicating that a small number of decision boundaries is sufficient to make accurate estimations.

Figure 10
Increasing the number of decision boundaries (hyperplanes) of the classifier allowed to make finer and more accurate estimations of functional network patterns

Solutions provided by the multi-hyperplane classifier contained more weights, which were harder to interpret than solutions of the linear classifier. The magnitude of contribution to the classifier estimation by each hyperplane is revealed by sorting hyperplanes by their weights (Fig. 10B). The sensor weights within each sorted hyperplane were similar across training runs (Fig. 10C), which indicated that specific sensor rules carried a specific importance in estimating if the model network activity pattern is functional. Consistent with the linear classifier results showing that PY and LP sensors contain more information in making estimations (Figs. 3D, ,7A7A and and8),8), the sensor weights in the most important hyperplane (H3) weighed higher the sensors of PY exclusively. The next important hyperplane (H2) weighed both LP and PY sensors high, and the least important hyperplane (H1) weighed sensors from all cells but employed a rule that complements the other two hyperplanes.

Discussion

We generalized and explored an existing, biophysically-inspired, abstract Ca2+ sensor model (Abbott and LeMasson, 1993; LeMasson et al., 1993; Siegel et al., 1994; Liu et al., 1998) to make predictions about dynamics of Ca2+ sensing mechanisms.

A classifier can be used as an optimal readout for regulation

Instead of explicitly defining the readout mechanism as done before (Golowasch et al., 1999b), we assumed that there is an optimal mechanism by replacing it with a classifier (using a similar approach to Poirazi and Mel, 2001 and the “ideal observer” in Felsen and Mainen, 2008). This classifier generates a single score for each network that indicates on which side of the functional boundary the network lies. It has been shown that such a single error measure can be sufficient, under certain conditions in ADHR modeling studies, to regulate ionic conductances along a line in the parameter space (Liu et al., 1998; Golowasch et al., 1999b; Olypher and Calabrese, 2007; Zhang et al., 2009; A.V. Olypher and A.A. Prinz, unpublished observation). Linear correlations found between conductances in experimental and modeling studies also support this result (MacLean et al., 2005; Schulz et al., 2006, 2007; Khorkova and Golowasch, 2007; T. Smolinski, P. Rabbah, C. Soto-Treviño, F. Nadim and A.A. Prinz, unpublished observation; A.E. Hudson and A.A. Prinz, unpublished observation; but also see Taylor et al., 2009).

Cell activity characteristics correlate with model Ca2+

The value of the Ca2+ sensors in this study represents the average firing activity and correlates highly with the cell bursting duty cycle (Fig. 4). However, when we used activity characteristics like duty cycle to make predictions about the network activity, we found that the Ca2+ signal and sensors were better predictors (Fig. 5A,B). This indicates a limitation to modeling studies that use spike time-based activity characteristics that disregard the sub-threshold voltage dynamics of the cell (Prinz et al., 2003, 2004; Günay et al., 2008; Taylor et al., 2009).

Both Ca2+ concentration and current are informative for homeostatic regulation

In general, our methodology is not limited to comparing only one type of sensor, as we showed that the controlling quantity in the sensors is interchangeable between the total Ca2+ current (ICa) and concentration ([Ca2+]) (Fig. 5B)—we could as well have chosen to use the membrane voltage as in Rabinowitch and Segev (2006). In this comparison, our [Ca2+] dynamics in Eq. (1) was limited to only one set of fixed parameters (Prinz et al., 2003). Although varying these parameters would have given a more realistic range of Ca2+ dynamics, it would have also increased the parameter space significantly and thus was kept outside the scope of this work. Our use of ICa for activity sensing is consistent with experiments showing that ADHR is modulated not by the steady state of the activity level, but by its change (Thoby-Brisson and Simmers, 2000). We assume that the change in activity reflects in [Ca2+] close to the membrane where it is more likely to affect ion channels (see Methods). Although this assumption may exclude processes that involve transport and signaling between the nucleus and the cell membrane, it is consistent with the localization of some Ca2+sensing proteins (e.g., frequenin) near the plasma membrane (O'Callaghan et al., 2003).

In spite of the experimental evidence connecting Ca2+ to regulatory processes (Turrigiano et al., 1994; Linsdell and Moody, 1995; Golowasch et al., 1999a), another modeling study suggested that the Ca2+ concentration across different cells may be too variable to be useful for homeostatic regulation (Achard and De Schutter, 2006, 2008). This is inconsistent with our results that correctly predict network regulation targets from readings of [Ca2+], ICa and ICa-based activity sensors across the vastly different pyloric network models found in our database (Fig. 5A,B). The discrepancy can be explained if the lobster CPG pyloric network employs a very different regulation rule than mouse cerebellar Purkinje cells (PC) considered in Achard and De Schutter (2008).

Single, same Ca2+ sensor in each cell can indicate if a network is functional

The use of multiple FSD sensors by Liu et al. (1998) was justified because Ca2+ affects ion channel properties through multiple parallel pathways, triggered by proteins such as frequenin, the downstream regulatory element antagonist modulator (Carrión et al., 1999; Mellström and Naranjo, 2001), and a transcription factor encoded by the C terminus of the L-Type voltage-gated calcium channel Cav1.2 (CCAT) (Gomez-Ospina et al., 2006). Consistent with the specialization of FSD sensors for a division of labor in parallel pathways, our results showed that FSD sensors increased estimation success rates. However, this increase was too small to be important in our simpler task of separating functional from non-functional network activity (Fig. 5B and Table S3), and rather a single sensing mechanism with activation and inactivation variables in each cell was able to estimate the network activity pattern up to 86% correct (Fig. 5C), which is consistent with other recent pyloric network models (Zhang et al., 2009). Furthermore, this sensor need not be specific for each cell in the network (Fig. 7), thus it could be one of the possible Ca2+ sensing proteins. This prediction can potentially be experimentally tested by blocking these candidate sensor pathways (Carrión et al., 1999; An et al., 2000; Zhang et al., 2003; Gomez-Ospina et al., 2006).

In the search for the optimal sensor, if access to minimal and maximal values of the sensor are also used for the estimation, the success increases to 87% (Fig. 5D). However, this sensor need not have an inactivation variable (see Tables S3 and S2), which suggested that inactivation could be replaced without loss of information by the minimal and maximal values of a slowly activating sensor. These peak values were found to be important in the control loops of signaling pathways in the living cell (Pouvreau et al., 2006).

Local activity sensors are capable of network homeostasis

Can global features of network activity be detected by local sensors? Consistent with other models (Golowasch et al., 1999b; Zhang et al., 2009), in this pyloric network model, local sensors in the PY cell were successful estimators across 83% of networks in the database with varying connection topologies (Fig. 7). This follower cell produced higher success rates than the other two cells because of the asymmetric connections specific to the pyloric network (see Fig. 1B; no feedback connection from PY to AB/PD cell), where observing PY indicates the state of both AB/PD and PY; PY would not burst if AB/PD was not properly bursting. The same asymmetry is confirmed in the weights of the classifier (Fig. 8A) that performs better than local sensors (up to 95% in Fig. 10A) using all global features. This classification paradigm was only used to find optimal sensor configurations for ADHR, since it may be unrealistic for sensors to have global access.

One such global signal appears in mice outside the neurons—in the glial cells. These glial cells were found to regulate the levels of the TNF-α (pro-inflammatory cytokine tumour-necrosis factor-α) to mediate homeostatic synaptic scaling in an activity-dependent manner (Stellwagen and Malenka, 2006). In the pyloric network, although a similar global, activity-dependent homeostatic regulation mechanism is not known, global neuromodulatory inputs to the network modulate ion channel conductances independent of activity (Thoby-Brisson and Simmers, 1998; Khorkova and Golowasch, 2007). However, a single global sensor that we simulated with a simple, instantaneous model by summing individual cell sensors was only as informative as the PY sensors (Fig. 7B). These results together suggest that more complex, network-level communication is necessary between the cells to achieve better ADHR (e.g., through synaptic or intracellular signaling mechanisms).

Optimal sensor parameters are consistent with biological data

The optimal parameters of the calcium sensors that we found predicted suitable ranges of operational parameters for Ca2+ sensing proteins that can be tested experimentally by blocking specific proteins and by applying artificial stimulation protocols to drive the regulation. The optimal parameters were consistent with observed biological properties of the pyloric network (Fig. 6), which indicated that homeostatic regulation is insensitive to unpysiologically long bursts but is sensitive to single spike events in the bursts at the millisecond-range. This is too fast for many Ca2+ sensing proteins that operate a Ca2+/myristoyl switch and need to translocate from the cytosol to membranes, except NCS-1/frequenin, whose N-terminal myristoylation targets it at the plasma and trans-Golgi membranes, allowing it to respond quickly to local changes in Ca2+ (Burgoyne, 2004).

Similar tuning properties of the parameters were found also for the case of three FSD sensors in each cell (Fig. 6C). For instance, the fast sensor did not prefer an inactivation time constant, because a fast activation variable can follow the ICa, making the inactivation unnecessary. This is confirmed by the increasing success of faster noninactivating sensors (Fig. 6A).

Sensor readout approximates a simple non-linear boundary

The architecture of the optimal classifier gave us information about the readout mechanism reacting to the calcium sensors. The specific predictions about readout mechanism can be tied to how Ca2+ sensing proteins affect their downstream targets, such as ion channel voltage dependence and surface expression (Zhang et al., 2003). Since our results did not improve for more than 5–10 hyperplanes, we can speculate that a non-linear, biologically-feasible decision boundary would be sufficient for the regulatory mechanism to be correct for 95% of the networks (Fig. 10).

Together with the results that a single sensor with a low-dimensional readout decision can perform well for estimation, we can hypothesize that a single optimal sensing pathway (e.g., frequenin) that produces an informative quantity can be used by multiple downstream targets to push the cell toward a functional activity regime.

Supplementary Material

Supp1

Acknowledgments

This work is supported by 1 R01 NS054911-01A1 from NINDS and a Career Award at the Scientific Interface from the Burroughs Wellcome Fund awarded to author AAP. R.M. Hooper and K.R. Hammett contributed to preliminary results. Thanks to A. Hudson for help with intracellular pathway literature search. We thank Tomasz Smolinski, Amber Hudson, and Andrey Olypher for comments on earlier drafts.

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