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Temperature compensation in a small rhythmic circuit
Leandro M Alonso ,
Eve Marder,
Volen Center and Biology Department, Brandeis University, United States;
Leandro M Alonso
Eve Marder
Temperature affects the conductances and kinetics of the ionic channels that underlie neuronal activity. Each membrane conductance has a different characteristic temperature sensitivity, which raises the question of how neurons and neuronal circuits can operate robustly over wide temperature ranges. To address this, we employed computational models of the pyloric network of crabs and lobsters. We produced multiple different models that exhibit a triphasic pyloric rhythm over a range of temperatures and explored the dynamics of their currents and how they change with temperature. Temperature can produce smooth changes in the relative contributions of the currents to neural activity so that neurons and networks undergo graceful transitions in the mechanisms that give rise to their activity patterns. Moreover, responses of the models to deletions of a current can be different at high and low temperatures, indicating that even a well-defined genetic or pharmacological manipulation may produce qualitatively distinct effects depending on the temperature.
Biological systems depend on many interacting nonlinear processes that together produce complex outputs. In the nervous system, neuronal activity requires the coordinated activation and inactivation of many inward and outward currents. Temperature influences all biological processes, to a greater or lesser degree. This poses an inherent difficulty for neuronal signaling: if the currents involved in neuronal and network dynamics are differentially temperature-dependent a system that is well-tuned to work at one temperature may not function at a different temperature (Caplan et al., 2014; O'Leary and Marder, 2016; Tang et al., 2010; Tang et al., 2012). Nonetheless, many ectothermic animals have neurons and circuits that function well over an extended temperature range (Robertson and Money, 2012). It then becomes important to understand how and to what extent this can occur.
The effects of temperature on channel function and neuronal activity have been studied extensively for many years (Taylor and Kerkut, 1958). Increasing temperature generally results in increases of channel maximal conductances and faster activation/inactivation rates (Frankenhaeuser and Moore, 1963). But, importantly ionic channels of different types are affected by temperature to different extents: each of these processes has a different Q10 (Schauf, 1973; Kukita, 1982; Ruff, 1999; Tang et al., 2010; Ranjan et al., 2019). The effect of temperature on neuronal intrinsic excitability such as voltage and current thresholds can be diverse (Sjodin and Mullins, 1958; Guttman, 1962; Guttman, 1966; Fitzhugh, 1966). For example, identified neurons in locust showed reversible changes in spike amplitude and duration of spikes as temperature varies from 18 to 35°C (Heitler et al., 1977). Studies in the jump neural circuit of grasshopers showed that neural excitability is differently affected by temperature across neuronal types (Abrams and Pearson, 1982).
There are also examples of neuronal and circuit processes that are relatively temperature-compensated. The frequency/current (f/I) curves in both molluscan and locust neurons are not substantially affected by small temperature changes (6–8°C) (Connor, 1975; Heitler et al., 1977). The f/I curves of auditory sensory neurons in grasshoppers remain largely unaffected by changes in temperature between 21°C and 29°C (Roemschied et al., 2014). Temperature compensation also takes place across the behavioral level. For example, a recent study in hunting archerfish showed that the duration of the two major phases of the C-start—a fast escape reflex which follows prey release—were temperature compensated (Krupczynski and Schuster, 2013). The question then arises of how these behaviors are preserved when the currents in the cells and their intrinsic excitability properties are differentially modified by temperature.
Here, we explore these issues using computational models of the pyloric network—a subnetwork within the stomatogastric ganglion (STG) of crustaceans (Marder and Bucher, 2007; Maynard, 1972). The pyloric rhythm is a triphasic motor pattern that consists of bursts of action potentials in a specific sequence. This behavior is robust and stable, and the cells and their connections are well-characterized. Robustness to temperature in the pyloric network has been explored experimentally by Tang et al., 2010; Tang et al., 2012; Soofi et al., 2014 and Haddad and Marder, 2018. These studies show that as temperature increases the frequency of the pyloric rhythm increases but the phases of the cycle at which each cell is active remain approximately constant. Additional work on the pacemaker kernel of the pyloric network (three cells connected by gap junctions that burst synchronously) showed that temperature increases the frequency of these bursts (Q10≈2), but their duty cycle (the burst duration in units of the period) stays approximately constant (Rinberg et al., 2013).
Temperature robustness was explored in computational models of the pacemaking kernel (Soto-Treviño et al., 2005) by Caplan et al., 2014 and O'Leary and Marder, 2016. They showed that it was possible to find multiple sets of Q10 values for different membrane conductances processes, so that the duty cycle of the cells remained constant as their bursting frequency increased. In this work, we build on the results in Caplan et al., 2014 and implemented temperature sensitivity in a model of the pyloric network (Prinz et al., 2004). We show that in these models, there are multiple sets of maximal conductances and temperature sensitivities that reproduce much of the experimental phenomenology previously reported (Tang et al., 2010; Tang et al., 2012; Soofi et al., 2014; Haddad and Marder, 2018). In addition, we explored how the dynamics of the currents are modified to sustain the correct activity at each temperature. We performed this study for 36 different models and found that in all cases, the contributions of the currents to the activity can be significantly different across temperatures. The currents are not simply scaled up but instead become reorganized: a current that is important for burst termination at 10°C may no longer play that role at 25°C. Because the contribution of a given current to a neuronal process can be replaced by another at different temperatures, deletion or blockade of a current can produce qualitatively different effects at high and low temperatures. These results provide a plausible hypothesis for why interactions between temperature and a second perturbation can be observed experimentally (Haddad and Marder, 2018; Ratliff et al., 2018).
Duty cycle and phase maintenance in model pyloric networks
The triphasic pyloric rhythm is produced by the periodic sequential activation of the pyloric dilator (PD) neurons, which are electrically coupled to the anterior burster (AB) neuron forming a pacemaking kernel, the lateral pyloric (LP) neuron and five to eight pyloric (PY) neurons. We modified the model of the pyloric circuit in Prinz et al., 2004 to include temperature sensitivity. Figure 1A shows a schematic representation of the model pyloric network studied here. The model network consists of three cells, each modeled by a single compartment with eight currents as in previous studies (Golowasch and Marder, 1992; Buchholtz et al., 1992; Goldman et al., 2001). The synaptic connections are given by seven chemical synapses of two types as in Prinz et al., 2004. In this model the pacemaking kernel is aggregated into a single compartment AB-PD (here we refer to this compartment as PD). Following Liu et al., 1998, each neuron has a sodium current, INa; transient and slow calcium currents, ICaT and ICaS; a transient potassium current, IA; a calcium-dependent potassium current, IKCa; a delayed rectifier potassium current, IKd; a hyperpolarization-activated inward current, IH; and a leak current Ileak. The traces in Figure 1A show a solution of this model for one set of maximal conductances G. The traces exhibit a triphasic pyloric rhythm that consists of the sequential bursting of the PD, LP and PY cells. This pattern remains approximately periodic so we can measure the phases of the cycle at which each burst begins and terminates using the burst start of the PD cell as reference (indicated as BREF in magenta in Figure 1A), as indicated by the color labels.
Effects of temperature in a model pyloric network.
(A) Schematic diagram of the model pyloric network in Prinz et al., 2004. The three groups interact via seven inhibitory chemical synapses. The red synapses are cholinergic from the PD neurons and all others are glutamatergic. The traces below show a representative solution that exhibits a triphasic rhythm: the activity is approximately periodic and the cells burst in a specific sequence: PD-LP-PY. (B) Activity of a temperature robust model network at 10°C - 25°C. As temperature increases the frequency of the rhythm increases but the duty cycle of the cells (the burst duration in units of the period) remains approximately constant. (C) Top: average burst frequency of each cell over temperature (values are nearly identical so dots overlap). Middle: average duty cycle of each cell (cell type indicated in colors). Bottom: average phases of the cycle at which bursts begin and terminate using the start of the PD burst as reference (indicated by label BREF in magenta). As temperature increases these phases remain approximately constant. The panels show average values over 30 s for 16 values of temperature between 10°C - 25°C.
Temperature sensitivity was modeled for each conductance as in Caplan et al., 2014 and O'Leary and Marder, 2016 (see Materials and methods). We increased each conductance by an Arrhenius type factor R(T)=Q10T-Tref10 (with Q10≥1 and Tref = 10°C) and also increased the rates of the channel kinetics τ in similar fashion. We assume that Q10G∈[1,2] for the maximal conductances and Q10τ∈[1,4] for the timescales as these ranges are consistent with experimental measurements of these quantities (Tang et al., 2010). The Q10 values are a property of the channel proteins, so in a given model we use the same Q10 for a given current (and synapse) type in all cells. Different values of the maximal conductances G can produce an innumerable variety of activities. The space of solutions of the model—the different patterns produced for each set of maximal conductances G—is formidably complex. This is in part due to the nonlinear nature of neuronal dynamics but also because the number of conductances that need to be specified is large (31 in total, 8 intrinsic conductances ×3 cells +7 synaptic conductances). The picture is more complicated as we consider the effect of temperature because it affects both the conductances and the time scales of the different processes (14 in total: 12 intrinsic, 2 synaptic). Therefore in the models, changes in temperature correspond to a coordinated change or a path in a 45-dimensional parameter space. The complexity of this problem led us to confine this study to specific questions suggested by biological observations over a permissible temperature range.
In the crab, as temperature is increased pyloric activity remains triphasic—the cells fire in the same order and at the same relative phases—while the pyloric network frequency increases by a two- to three-fold factor over a 15°C range (Tang et al., 2010). For this to happen in the model, there should be many regions in the parameter space for which the activity is triphasic at different frequencies and paths in the parameter space (sequences of values of G and τ) that interpolate these regions. One of the main results in this work is that such paths do exist and that temperature compensation is possible in this model. Finding these temperature compensated networks is nontrivial: it requires the specification of 31 maximal conductances G and 24 Q10 values (see Materials and methods). Because it is virtually impossible to find these sets of parameters at random we employed a landscape optimization scheme described previously (Alonso and Marder, 2019) and adapted it to this particular scenario (see Materials and methods). We first searched for sets of maximal conductances G that displayed triphasic rhythms at control temperature (10°C). Inspired by physiological recordings of the pyloric rhythm of crabs and lobsters we targeted control activities with a network frequency of 1Hz, duty cycle ≈ 20% for PD, and duty cycle ≈ 20% for LP and PY (Bucher et al., 2006; Hamood et al., 2015). We then selected 36 of these models and for each of them we searched for Q10 values so that the activity is preserved over a temperature range. We found that, regardless of the maximal conductances G, it was always possible to find multiple sets of Q10 values that produced a temperature robust pyloric rhythm.
Figure 1B shows the activity of one example network at several temperatures, similar to the experimental results reported by Tang et al., 2010. Figure 1C shows the same analysis as in Tang et al., 2010 performed on this particular model. We simulated the model at several temperatures over the working range (10°C to 25°C) and computed the average duty cycle, frequency and phases of the bursts. While the burst frequency increases with Q10≈2, the duty cycle of the cells, and the phases of the onsets/offsets of activity, remain approximately constant. Figure 1—figure supplement 1A shows the same analysis performed on three different models. In all cases, the duty cycle and the phases stay approximately constant while the frequency increases. Figure 1—figure supplement 1B shows the values of the maximal conductances G and the Q10 values for each of these models.
While the duty cycle remains approximately constant on average, the spiking patterns of the cells show discernible differences across temperatures. We subjected the models to temperature ramps from 10°C to 25°C over 60 min and recorded the spike times (Figure 1—figure supplement 2). As temperature is increased the spiking patterns change in complicated ways and the largest ISI decreases monotonically, consistent with the overall increase in bursting frequency of the cells. Different values of temperature result in slightly different spiking patterns which in turn result in different values of the duty cycle. For some temperatures, the duty cycle is nearly identical in every burst resulting in a single point in the y-axis (Figure 1—figure supplement 2D, blue box in PD cell). There are temperatures for which the duty cycle takes two values preferentially (Figure 1—figure supplement 2D, pink box in PD cell) and there are temperatures for which the duty cycle differs noticeably from burst to burst. Although in all models, the average duty cycle and phases stay approximately constant, as temperature is changed the precise way in which the spiking patterns change show marked differences across models: different maximal conductances and different Q10 values produce different patterns. Figure 1—figure supplement 3 shows the temperature dependence of the duty cycle in the LP cell over the working range for all 36 models and provides a notion of how variable the bursting patterns of the cells can be across temperatures and individuals.
Changes in membrane potential over temperature
Experimental studies on other systems reported measurable effects of temperature on resting membrane potentials (Klee et al., 1974), spiking thresholds, and amplitudes of spikes (Heitler et al., 1977). In the models studied here, some temporal properties of the activity remain approximately constant over a temperature range but the precise shape of the waveforms can also change. To inspect how the membrane potential changes over the working temperature range (10°C - 25°C), we computed the distribution of V of each cell for 101 values of temperature. Temperature changes features of the voltage waveforms, resulting in changes in these distributions. Figure 2 shows example traces and the result of this analysis for the LP cell in one model. The membrane potential changes with temperature in several ways: the spike amplitude decreases and the spike threshold depolarizes as temperature increases. At each temperature, we computed the distribution of V, and used a gray scale to indicate the fraction of time that the cell spends at each voltage value (y-axis). The color lines match features of the voltage waveforms at each temperature, with features in the distribution on the right. The distributions permit visualizing changes in the waveform as the control parameter is changed (Alonso and Marder, 2019). The effect of temperature on membrane potential is not consistent across models and depends on the precise values of conductances and temperature sensitivities. Figure 2—figure supplement 1 shows the membrane potential distributions of each cell for three example models. In all cases, the waveforms show visible differences across temperatures and there is considerable variability in the amplitudes of the oscillations across models. In some models, the peak voltage of spikes, as indicated by the upper envelope of the distributions, decreases (left), increases (middle), or remains relatively constant (right). In addition, we found that cells can either become more depolarized or more hyperpolarized as temperature increases.
Changes in membrane potential over temperature.
(left) Representative traces of the LP cell in one model at different temperatures. (right) Membrane potential distribution at each temperature. The gray scale indicates the proportion of time the cell spends at that potential. The distribution facilitates inspecting how features such as the total amplitude of the oscillations and spiking thresholds change with temperature.
Dynamics of the currents at different temperatures
Neuronal activity is governed by currents that result from the precise activation and inactivation of ion channels with different kinetics. Experimental access to these quantities is limited because it is hard to measure currents individually without blocking all other currents, and thus changing the activity. Here, we employ models to explore how these currents are differentially altered by temperature, and yet are able to remain balanced in such a way that the network activity speeds up while preserving its behavior and phase relationships.
We first compared the dynamics of the currents in two model PD neurons with different maximal conductances G and temperature sensitivities Q10 by direct inspection of the currents' time series. Figure 3A shows the time series of each intrinsic current in PD at 10°C and 25°C for one model. The corresponding membrane potential activity V is shown in the blue traces on top. The red horizontal dashed lines indicate the peak amplitude of the currents at control temperature (10°C) to visualize whether it increases or decreases at 25°C. This is indicated by the arrows in each row (in colors: ⇑ red increases, ⇓ green decreases). In this example, the Na current increases its amplitude by more than two fold over the 15°C range. Despite the fact that gCaT increases with temperature, the amplitude of the current decreases to about half its control value at 10°C. Similar observations can be made for other currents (Figure 3A). Note that all Q10 values in these models are greater than 1. Therefore, when the current decreases it is because the relative timing of the currents' activation is changed by the neurons' firing properties. The way currents are modified by temperature is different across models. Figure 3B shows the currents in the PD cell at two temperatures for another model. In this case, the Na current decreases by a small amount despite the fact that the maximal conductance and timescales of this current increase with temperature. The KCa current which in Figure 3A grows by a factor of six, is moderately decreased by temperature in Figure 3B.
Dynamics of the currents at two temperatures.
Currents of the PD cell for two different models (A-B). Temperature increases the peak amplitudes of some currents and it decreases it for others, as indicated by the ⇑ and ⇓ arrows in red and green. The way these currents are modified can vary substantially across models.
Because the total number of currents in the circuits is 31, it becomes cumbersome to compare them separately across temperatures (and models). For this reason, we computed and inspected their currentscapes (Alonso and Marder, 2019). The currentscapes use colors to show the percent contribution of each current to the total inward (or outward) current and are useful to display the dynamics of the currents in a compact fashion. Figure 4 shows the currentscapes of each cell in one model at 10°C and 25°C. The shares of each current of the total inward and outward currents are displayed in colors over time. The total inward and outward currents are represented by the filled black curves on a logarithmic scale at the top and bottom. The currentscapes are noticeably different at 10°C and 25°C indicating that the currents contribute differentially across temperatures. For example, in the PD cell the Leak current contributes a visible share of the inward current both at the beginning and the end of the burst at 10°C, but at 25°C its contribution is mainly confined to the beginning of the burst. The A current is evenly distributed during the burst at 10°C but at 25°C its contribution is visibly larger at the beginning of the burst. In all cells, but most notably in the LP and PY neurons in this example model, temperature substantially increases the contribution of the KCa current toward the end of the bursts.
Currentscapes at different temperatures.
The currentscapes use colors to display the percent contribution of each current type to the total inward and outward currents over time. The filled lines at the top and bottom indicate the total inward/outward currents in logarithmic scale. The panel shows the currentscapes for the three cells in the model at two temperatures. The panels show 2 periods τ of the oscillation.
The currentscapes show that in all models and at all temperatures, there are periods of time when the activity is dominated by one or two currents, and periods of time when several currents contribute to it by similar amounts. During spike production, the activity is first driven by INa and later by IKd. During these periods, the total inward and outward currents differ by orders of magnitude and are entirely dominated by one current type. Similarly, when the cells are most hyperpolarized the dominant currents are IH and the synaptic currents ISyn. There are periods of time—such as the beginnings and ends of bursts—when the total inward and outward currents are composed of comparable contributions of several current types. It is during these periods, when multiple currents act together to change the membrane potential, where we observe the most dramatic changes with temperature.
Figure 5 shows 20 ms after the burst in a PD neuron over temperatures between 10°C and 25°C. Almost all of the currents change over this temperature range, most notably there is almost a tradeoff between the contributions of the A and KCa currents, with A decreasing smoothly and KCa increasing smoothly. The specifics of how the contributions change is different across models, but in all cases there is a smooth transition between different relative fractions of currents. This allows the neuron to slide smoothly through changes in burst termination mechanisms.
Current contributions at the end of bursts across temperature.
The figure shows the currentscapes of the PD cell for the 20 ms following burst termination, and how these change over temperature (same model as in Figure 7A).
Different compensation mechanisms fail in different ways. Figure 6 shows full currentscapes for two network models as they 'crash' at high temperature. Note that the complete loss of activity in the PD neurons results from different currents being active in the two examples. Further examples of high temperature 'crashes' are shown in Figure 6—figure supplement 1. These examples show that the networks are crashing by different mechanisms in each case.
Currentscapes reveal that 'crashes' occur by different mechanisms.
Currentscapes of two networks at 10°C and 25°C showing phase compensation, and 'crashes' at 30°C.
High temperatures produce disordered circuits or 'crashes'
For temperatures higher than a critical temperature, the biological network displays anomalous regimes characterized by the emergence of slow time scales and intermittency between what appear to be metastable states (Tang et al., 2012; Haddad and Marder, 2018). At temperatures higher than 25°C some models do not display irregular behavior but just become quiescent, or produce subthreshold oscillations. Other models display irregular and seemingly aperiodic regimes. As expected, irregular regimes typically take place at values of the temperature near a transition between qualitatively different patterns of activity. Figure 7 shows the activity of five models at temperatures higher than their working regimes. The left panels show 30 s of data and the right panels show 5 s. In all cases, there are timescales in the behavior that are much longer than the period of the rhythm at 10°C (≈ 1 s). The dynamics of the models in these regimes is daunting and their characterization is beyond the scope of this work. However, qualitative features of these states such as the appearance of slow timescales or absence/presence of activity in one or more cells, are captured by these models. This, together with the multistable regimes shown before plus a source of noise, may provide a reasonable model to account for the irregular regimes in the biological network (Tang et al., 2012; Haddad and Marder, 2018).
Disordered circuits at high temperatures.
The models become dysfunctional—or crash—at temperatures higher than 25°C and do so in different ways. (A-E) Membrane potential of five models. The traces on the left show 30 s of data and the traces on the right show an expanded trace of 5 s. Note the emergence of time scales much longer than that of the original rhythm. Temperatures in the simulations are (A–E): 34°C, 34°C, 33°C, 29°C, 30°C.
Hysteresis and multistability
It is natural to ask whether these models show hysteresis when the temperature is returned to control levels. Moreover, multistability is also often observed in conductance-based models of neuronal activity and biological neurons (Cymbalyuk and Shilnikov, 2005; Malashchenko et al., 2011; Marin et al., 2013; Lechner et al., 1996; Paydarfar et al., 2006), and the network studied here also exhibits such features. We explored hysteresis in the models by increasing temperature linearly from 10°C to 35°C in 30 min and then decreasing it back to 10°C at the same rate. Figure 8 shows the spiking patterns for two PD neurons during such temperature ramps (schematized on top). In Figure 8A the ISI distributions show bursting activity up to a critical temperature at which the model becomes quiescent. The quiescent state remains stable as temperature continues to increase and eventually recovers activity during the decreasing ramp, but at a lower temperature than when it stopped spiking during the increasing ramp. This implies that there is a range of temperatures where the model can be either quiescent or spiking, and thus is multistable. In most models (34 of 36), the spiking patterns over the working temperature range are similar across the up and down ramps but are not identical, suggesting there are multiple stable attractors that correspond to the triphasic rhythm but differ slightly in their spiking patterns (Figure 1—figure supplement 2E). This is consistent with the findings in Tang et al., 2010 where temperature was ramped slowly and no significant differences were found when ramping up and down in temperature.
Hysteresis and multistability.
Temperature was increased from 10°C to 35°C and then decreased back to 10°C symmetrically. (Top) Schematic representation of the temperature ramp. Panels A and B show the ISI distributions of the PD cell over time for two different models. (A) The cell ceases to produce spikes before 600 s, which is also before reaching 35°C, and remains quiescent until temperature is ramped down. The temperature at which the cell resumes spiking is ≈ 2°C lower than the temperature at which it became quiescent, indicating that the system is multi-stable over that temperature range. (B) Hysteresis is more evident in this model. The spiking patterns during the down ramp differ visibly from those during the up ramp over a wide temperature range.
We also found models with pathological stable attractors that coexist with the pyloric rhythm at working temperatures. Figure 8B shows a model in which hysteresis is salient. At high temperatures (times between 25 and 35 min), this model produces ISI values as long as ≈ 1 s and remains spiking at all temperatures. The spiking patterns differ noticeably on the up and down ramps. The spiking pattern of the PD cell changes from regular bursting to a pathological state that remains stable until the model is ramped back down to ≈ 20°C. In our simulations, hysteresis mainly occurs at high temperatures indicating that states such as those in Figure 7 can be multistable. The extent such hysteresis can be revealed in the biological experiments depends on the details of the perturbation protocol, and its currently under study.
Response to perturbations at different temperatures
In all models, the dynamics of the currents are different at different temperatures and their contributions become reorganized in complex ways. The combined activity of all the variables in the model—the dynamical attractor—is therefore expected to have different stability properties at different temperatures. This means that an extreme perturbation can have qualitatively different effects at different temperatures. The responses of different models to extreme perturbations such as partially or completely removing a current can be diverse at any temperature (Alonso and Marder, 2019). While this is expected due to the complexity of these models, the fact that these responses can also be qualitatively different across temperatures indicates that changes in the current contributions are meaningful and affect the stability of the network. To shed light on this issue, we performed several simple perturbations.
Figure 9 shows one perturbation in two models. In Figure 9A, we explored the interaction between temperature and the effect of removing the A-type K+ channel at two temperatures: the models were simulated for 20 s in control conditions and then IA was completely removed for the next 20 s. Figure 9A (top) shows that removing the A current at 10°C has catastrophic consequences for the network activity which becomes irregular. When the same perturbation is performed at 25°C (bottom), the triphasic rhythm is almost normal except for the irregular bursting of PY. Figure 9B shows a similar pertubation in which we remove the KCa current. This model displayed normal triphasic behavior at 10°C (top) and when the KCa current was removed its activity slowed but the triphasic rhythm was maintained. The same perturbation at 25°C results in quiescence. In this model, the KCa current is necessary for the activity at 25°C but not at 10°C.
Response to perturbations at different temperatures.
The response of the models to extreme perturbations can be different at different temperatures. The figure shows the responses of models to the same perturbation at two different temperatures. (A) (top) Membrane potential over time at 10°C. The first 4 s correspond to the control condition with all currents intact. At T=4sec we removed the A current. (bottom) Same perturbation performed at 25°C (same model as in Figure 4). (B) Same protocol as in A but removing the KCa current in a different model (Figure 7A).
To further characterize the differential response to perturbations at different temperatures, we performed a second assay in which we gradually decreased a current from control (100%) to complete deletion (0%) in five steps. We performed these simulations for all 36 models and all current types at 10°C and 25°C and compared their responses. In general, gradually removing a current can result in qualitatively different states depending on the temperature at which this perturbation is performed. The responses are so diverse that a coarse classification scheme is sufficient to highlight this observation. We classified the network activity based on the spiking patterns and waveforms of each cell, and also by their relative activities. The activities of each cell were classified as: regular bursting, irregular bursting, tonic spiking, irregular spiking, single spike bursting, quiescent and other, while the network activity was classified as triphasic or not triphasic. The classification scheme consists of a simple decision tree based on the statistics of spiking and is sufficient to tease apart the different regimes we observe at a coarse level (Materials and methods).
Figure 10A shows the result of classifying the responses of one model to gradually decreasing a conductance in five steps, at two different temperatures, for four conductances. We employed 5 × 4 response grids to summarize the effects of partially or completely removing a conductance at 10°C and 25°C. The rows in the grids correspond to different values of conductance removals, with the control condition (100%) on top, and the completely removed condition (0%) at the bottom. The first three columns correspond to each of the cells (PD-LP-PY) and the fourth column corresponds to the network (NET). The colors indicate the type of activity of each cell and the network color coded for condition as indicated by the labels in the figure. The top left panel in Figure 10A shows the responses of one model to removing the sodium current INa. At 10°C, when gNa→75%gNa the LP cell bursts irregularly and the activity of the network is not triphasic, but if the same perturbation is performed at 25°C the activity remains triphasic. Removing the KCa current in this model has little effect at 10°C as the activity remains triphasic. However, this current plays an important role at 25°C and the model becomes quiescent upon complete removal of IKCa. The same observation that the responses to perturbation are different at each temperature holds regardless of the perturbation type. In the cases of the CaT and CaS currents complete removal results in different non-triphasic activity at each temperature. The responses of the model are in general qualitatively different at different temperatures, and they are also diverse across models (not shown). The fact that the models 'break-down' in different ways is consistent with the observation that these network mechanisms are diverse. In all models, the responses to extreme perturbation are temperature dependent for most perturbation types. In Figure 10B we quantify how many models respond differently to complete removal of a current at 10°C and 25°C. This corresponds to comparing the bottom row of the response grids at each temperature, for each conductance type. In most cases, about half of the models respond to complete removal in different ways at each temperature. The exceptions are the CaT current for which most models respond in different ways, and the H current where complete removal results in equivalent states (quiescence in this case) at both temperatures.
Classification of responses at different temperatures.
The models' responses to perturbations are diverse and are in general different at different temperatures. We classified the responses of the models at two temperatures for five values of gradual decrements of each current. (A) Response to removal of Na, KCa, CaT and CaS currents for one model. (B) The pie charts show the number of models for which the responses to complete removal (0%) are classified as the same (green) or different (blue) at 10°C and 25°C.
Enzymes and ion channels show altered behavior when temperature is changed, as temperature influences their conformation. Yet many animals live successfully over wide temperature ranges. For example, North Atlantic lobsters and crabs routinely experience more than 25°C temperature swings during a year, and may see 8 - 10°C changes within a short time period (Haefner, 1977; Stehlik et al., 1991). This raises a series of fascinating questions of how behavior and circuit performance can be maintained while all biological molecules are, to a greater or lesser degree, differentially temperature dependent (Robertson and Money, 2012).
It is important to distinguish between altered activity that occurs rapidly in response to acute, or immediate changes in temperature and those changes that occur during slow adaptation and acclimation, processes that can take days, weeks or months. In this study, we have modeled the fast changes that we assume would occur 'automatically' as proteins change conformation and properties rapidly, and as are usually measured in their Q10's. This is to be contrasted with the slower, long-term effects that result in acclimation in which a variety of molecular processes are triggered that may in turn change the proteins present in the membrane. These mechanisms, and much adaptation to long-term changes in temperature depend on the activation of heat shock proteins, RNA editing and a whole array of molecular mechanisms (Garrett and Rosenthal, 2012; Rosenthal, 2015; Martin Anduaga et al., 2019). We will first discuss the issues that arise from the examination of robustness to acute temperature changes, as modeled here, and then offer some speculations for additional processes that may be important to enhance robustness in response to long-term temperature changes.
Unfortunately, there are relatively few data relevant to whether the Q10 of a channel in a given animal changes in response to acclimation or seasonal conditions. Therefore, although it is often assumed that a Q10 for a given protein is invariant over the lifetime of the animal this may not necessarily be the case. Another difficulty is that there are few, if any, studies of many ion channel Q10s in the same animal in the same temperature range. Computational modeling will not answer what the 'real' conductance and Q10 combinations are.
The assumption that these quantities are stationary throughout the life of an animal (and/or the same across species) may not be warranted. For example, long-term acclimation over months resulting in RNA editing, could alter some Q10s. Moreover, it is possible that minor genetic variations in channel proteins exist in the population so that different animals in the population could show modestly different Q10s. Therefore, the point of this study is to ask generically how compensation occurs acutely assuming different temperature sensitivities for the processes, within the experimental range. For these reasons, in the absence of data, it is valuable to explore the Q10 space to have on overview of how compensation takes place acutely at any point during the lifetime of an animal.
One of the most important and non-intuitive results from this work is seen in Figure 5, which demonstrates that a neuron can smoothly slide through mechanisms that govern a physiological process as temperature is changed. In the case of Figure 5, the repolarization of the last spike in the burst shows a large contribution from IA at 10°C but becomes small at 25°C while IKCa contributes little at 10°C and becomes dominant at 25°C. This shows that while both of these K currents can contribute to spike and burst repolarization, their shares in these processes change, and the neuron is robust as it smoothly slides through these mechanisms. This smooth transition can occur precisely because these three K+ currents depend on voltage, time, and temperature differently.
We speculate that one advantage conferred by the expression of many different ion channels that differ in their properties is to provide many sets of such 'sliding stability mechanisms'. Individual neurons in all species have large numbers of ion channel genes that encode proteins that give rise to the membrane currents that are usually measured and characterized in voltage-clamp. The model neurons used are loosely based on voltage-clamp measurements from crab and lobster stomatogastric ganglion (STG) neurons (Golowasch et al., 1992; Turrigiano et al., 1995) and have eight currents. That said, we now know that there are at least twenty to thirty and probably more voltage-gated ion channel genes in individual STG neurons (Northcutt et al., 2016). In some cases, several of these genes may contribute to one of the currents that we model here. For example, the delayed rectifier current modeled here may have contributions from at least three different genes, and IA, modeled here, also probably has contributions from at least two different genes. Given that these genes encode similar but not identical protein subunits, it is not unreasonable to expect that these proteins may respond differently to temperature, and that the actual composition of the activated K currents may reflect a change in these contributions as a function of temperature. This again, could provide still another 'sliding stability mechanism', and it would be interesting in the future to create models using data from currents that have been measured in isolation, such as in Ranjan et al., 2019.
The same principle of sliding circuit mechanisms is likely to hold when we approach network dynamics. The phase relationships of the pyloric rhythm are maintained over a considerable temperature range (Tang et al., 2010; Soofi et al., 2014). Likewise, the pyloric phase relationships are also maintained over a considerable frequency range (Hamood et al., 2015). In this latter case, it has been argued that the phase compensation depends on the conjoint action of a number of different cellular mechanisms, including synaptic depression (Manor et al., 1997; Hooper, 1997), and the activation and inactivation of IA and IH (Nadim et al., 1999; Nadim and Manor, 2000; Tang et al., 2010; Harris-Warrick et al., 1995). So, here the principle also holds: resilience and robust function may require smoothly moving between a variety of different cellular mechanisms.
Long-term mechanisms for temperature adaptation and acclimation may extend robustness by changing the functional properties of ion channel proteins. In biological systems, sustained exposure to temperature often activates heat shock proteins and has effects on many cellular and molecular functions (Sharp et al., 1999). Interestingly, changes in ion channel composition can result from RNA editing, or changes in splice variants (Johnson et al., 2011; Lin and Baines, 2015). It is easy to imagine that these kinds of molecular mechanisms can result in shifts in the temperatures at which neurons and circuits may function. Acclimation to high temperatures usually requires at least 3–4 weeks of sustained alteration of the environmental temperature, and this is enough time to replace a large subset of ion channel proteins in the membrane. This raises a series of fascinating questions for how homeostatic regulation of excitability may be preserved if the molecular mechanisms that are engaged in homeostasis are also influenced by temperature.
Hysteresis in response to perturbation can result from a variety of time-dependent mechanisms. Figure 8 shows examples of hysteresis that can occur in response to temperature ramps, even in models in which the effects of temperature on the properties of each ion channel are viewed as essentially instantaneous. This hysteresis can occur because at high temperatures (> 25°C) the network can produce several different patterns of activity or dynamical states (multistability). The activation and inactivation rates that characterize ion channel function are different for each of these states and therefore, as temperature is lowered, they lose stability in different ways. Of course, any long-term molecular mechanisms (not modeled here) that alter channel expression or splicing or phosphorylation, could produce long-lasting hysteresis.
In this work, we used a genetic algorithm (Alonso and Marder, 2019) to find temperature robust networks. It is important to reiterate that temperature robust neurons and networks are difficult to find doing random searches (Caplan et al., 2014). Therefore, despite the significant animal-to-animal differences in conductance densities seen across the population, randomly sampling conductance values is unlikely to result in successful solutions for individual animals. This argues that evolution has arrived at a series of biological mechanisms that give rise to successful temperature compensation. These are likely to include rules by which correlated values of conductances are produced, such as those found in homeostatic models (O'Leary and Marder, 2016; O'Leary et al., 2013).
Taken at face value, models suggest that conductance densities drift throughout the life of an animal as a result of homeostatic processes (LeMasson et al., 1993; Liu et al., 1998; Golowasch et al., 1999a; O'Leary et al., 2014). Conductance densities can also change in an activity-dependent manner as a result of perturbations (Turrigiano et al., 1994; Golowasch et al., 1999a; Golowasch et al., 1999b; Golowasch et al., 1999a; Santin and Schulz, 2019; Golowasch, 2019). The temperature sensitivities of ion channels are dictated by their molecular structures, and for this reason it is tempting to assume that the temperature sensitivity of a given channel is similar throughout the lifetime of the animal. Nonetheless, as all channels are subject to splice variants and other mechanisms of molecular regulation, it is possible that environmental temperature results in changes in Q10 that are consistent with acute robustness, even if there have been changes in conductance densities. This suggests that conductance densities and Q10 regulation may be coordinately controlled in biological systems.
One of the most significant results of this paper is that temperature robustness of network function can take place in multiple different ways. In these models, both the membrane potential and the spiking patterns are affected by temperature changes and these changes appear to be different in every model we inspected. Across the models, different values of the maximal conductances G and temperature sensitivities Q10 result in consistent differences in the duty cycle distributions. Measurements of the values of the maximal conductances in the STG show large variability across individual animals (Goaillard et al., 2009; Schulz et al., 2006; Schulz et al., 2007; Temporal et al., 2014; Northcutt et al., 2016) so our expectation is that the duty cycle distributions of the biological cells will also display intricate dependencies with temperature, and that these distributions will be different across individuals.
Temperature is not the only perturbation that crabs and lobsters experience. As with temperature, the responses of the STG to changes in pH are diverse across individuals (Haley et al., 2018), again consistent with the large amount of animal-to-animal variability in the expression of ion channels (Schulz et al., 2006; Temporal et al., 2014; Tran et al., 2019). It is therefore reasonable to assume that the responses to a global perturbation are diverse across individuals because different compositions of channel densities—which produce similar pyloric rhythms—are differentially resilient to any given perturbation. By the same token, as currents change as a function of temperature we expect that a second global perturbation of an individual may have qualitatively different effects at different temperatures, as is illustrated with the models in Figure 10. The interaction between temperature and a second perturbation is the subject of recent experimental studies (Haddad and Marder, 2018; Ratliff et al., 2018). These studies are consistent with the interpretation that different current configurations have different stability properties and that temperature changes these configurations.
One of the general take-home lessons from this work is that models can be resistant to a pharmacological perturbation at one temperature but the same model can be sensitive to the same perturbation at another temperature (Figure 10), and that different individuals may differ in their responses to the same perturbation. The potential implications of this for therapeutics are evident. But also evident, is that the search for therapeutic agents can be fraught if the assay itself is differentially sensitive to temperature or another perturbation. Moreover, the 'sliding mechanisms' provided by multiple ion channel genes and their molecular regulation suggests that over time therapeutic sensitivity to a given pharmacological intervention could be quite variable. These findings may contribute to our understanding of individual variable therapeutic efficacies within the population and across time within an individual.
The activity of the cells was modeled using single-compartment models similar to those described previously (Turrigiano et al., 1995; Liu et al., 1998; Goldman et al., 2001; Alonso and Marder, 2019). Each neuron has a sodium current, INa; transient and slow calcium currents, ICaT and ICaS; a transient potassium current, IA; a calcium-dependent potassium current, IKCa; a delayed rectifier potassium current, IKd; a hyperpolarization-activated inward current, IH; and a leak current Ileak. The number of state variables per cell is 13. The units for voltage are mV, the conductances are expressed in μS and currents in nA. The pyloric network consists of three cells with the same ion channel types but different conductance densities. The interactions in the network consist of seven chemical synapses and are similar to Prinz et al., 2004. The synaptic current is given by Is=gss(Vpost-Es), where gs is the synapse strength, Vpost is the membrane potential of the postsynaptic neuron and Es is the reversal potential of the synapse. The activation of a synapse s(t) is given by
(1) dsdt=s∞(Vpre)-sτr+τs
with,
(2) s∞(Vpre)=11+exp((Vth-Vpre)/Δ),
(3) τs=1-s∞(Vpre)k-.
These equations are identical to Prinz et al., 2004 except for the inclusion of a bound for the timescale of activation τr=20msec (we want to avoid the case that as s∞→1 then τs→0 and s˙→∞). All other parameters (except gs) are identical to Prinz et al., 2004. Following Prinz et al., 2004, we set Es=−70mV and k−=140ms for glutamatergic synapses, and Es=−80mV and k−=1100msec for cholinergic synapses. We set Vth=−35mV and Δ=5mV for both synapse types.
Temperature effects were included in this model as done previously by Caplan et al., 2014 and O'Leary and Marder, 2016. Temperature dependence was introduced in the time constants of the channel-gating variables τmi and τhi, the maximal conductances gi, the time constants of calcium buffering τCa, and the maximal conductances and time constants of the synapses. This was done by replacing all conductances gi by Ri(T)gi and all time scales τi by Ri(T)-1τi where
(4) Ri(T)=Q10iT-Tref10.
Here T is the temperature, Tref=10∘ is the reference temperature and Q10i is defined as the fold change per 10°C from the reference temperature (i indicates the process type). Finally, the calcium reversal potential ECa depends on temperature through the Nernst equation.
The total dimension of the model is 46 = 3 × 13 + 7 with 13 state variables per cell, and 7 variables for the state of the synapses. In this work some parameters were considered fixed while others were allowed to take values in a range. The number of parameters we varied per cell is 9: the 8 maximal conductances plus the calcium time constant τCa. The temperature sensitivities were assumed to be the same for all cells and amount to 24 additional parameters. We need to specify 8 Q10 values for the intrinsic maximal conductances (one per channel type), plus 11 Q10 values for the time scales of each intrinsic process (not all currents inactivate), plus the Q10 of the calcium time constant. The number of Q10 parameters for the synapses is 4:2 for the maximal conductances (of each synapse type), and 2 for the timescales of activation. The models were simulated using a Runge-Kutta order 4 (RK4) method with a time step of dt=0.05msec (Press et al., 1988). We used the same set of initial conditions for all simulations in this work V=−51mV, mi,hi=0 and [Ca+2]=5μM.
Finding parameters
Each model is specified by the maximal conductances G and calcium time constants of each cell (9 × 3 parameters), and the temperature sensitivities Q10 of each process (24 parameters). We recently introduced a function that upon minimization, results in values of the maximal conductances for which the activity of a single compartment corresponds to periodic bursting with a target frequency (ftg) and duty cycle (dctg). The function uses thresholds to obtain temporal information of the waveform, such as spike times, and then uses it to assign a score or error that measures how close the solutions are to a target activity. This function is described in detail in Alonso and Marder, 2019 and was used here to find temperature robust networks.
For any given set of conductances we simulated the model for 20 s and dropped the first 10 s to minimize the effects of transient activity. We then computed the average (<>) burst frequency <fb>, the average duty cycle <dc>, the number of crossings with a slow wave threshold #sw=−50mV, the number of bursts #b, and the average lags between bursts <ΔPD−LP> and <ΔPD−PY>. To discard unstable solutions we checked if the standard deviation of the burst frequency and duty cycle distributions was small; a solution was discarded if std({fb})≥<fb>×0.1 or std({dc})≥<dc>×0.2. If a solution is not discarded we can use these quantities to measure how close it is to a target behavior,
(5) Ef=∑i=cell(ftg−<fb>i)2Edc=∑i=cell(dctg−<dc>i)2Esw=∑i=cell(#sw−#b)2Eph=(ΔPD−LPtg−<ΔPD−LP>τb)2+(ΔPD−PYtg−<ΔPD−PY>τb)2.
Here Ef measures the mismatch of the bursting frequency of each cell with a target frequency ftg and Edc accounts for the duty cycle. Esw measures the difference between the number of bursts and the number of crossings with the slow wave threshold tsw=−50mV (we ask that #sw=#b). Eph compares the lags between bursts (in units of the bursting period τb=1<fb>) to a target lag Δtg. These measures are discussed in more detail in Alonso and Marder, 2019.
Let G denote a set of maximal conductances (and τCa), we can then define an objective function
(6) E(G)=αEf+βEdc+γEsw+ηEph,
where the coefficients (α,β,γ,η) define how the different sources of penalties are weighted. In this work we used α=10, β=1000, γ=1, η=10. These weights were found by trial and error and kept fixed. The penalties Ei were calculated using T=10 secs with dt=0.05 msecs. The target control behavior was defined as having all cells bursting with 20% duty cycle for PD (dctgPD=0.2) and 25% for the LP and PY cells (dctgLP,PY=0.25). The lag between bursts was targeted to be ΔPD-LPtg=0.5 and ΔPD-PYtg=0.75. The target burst frequency of all cells was set to ftg=1Hz (Bucher et al., 2006; Tang et al., 2010; Hamood et al., 2015).
Minimization of the objective function produces sets of maximal conductances G for which the resulting circuit activity mimics the pyloric rhythm. The minimization was performed over a search space of allowed values listed here: for each cell we searched for gNa∈[0,103] ([μS]), gCaT∈[0,102], gCaS∈[0,102], gA∈[0,102], gKCa∈[0,103], gKd∈[0,102], gH∈[0,102], gL∈[0,10], τCa∈[0,2×103] (msec). All synaptic conductances were searched in the range gsyn∈[0,5×10-2] ([μS]). The choice of search space contains our knowledge that some conductances like gNa and gKd are significantly larger than gleak or the synaptic conductances. This choice makes optimizations faster and produces diverse solutions. We discretized our search space by taking 103 equally spaced values for each parameter. We minimized the objective function (Equation 6) using a custom genetic algorithm (Holland, 1992) on a desktop computer with 32 cores. For each network, we ran the algorithm using 1000 individuals taken at random in the search space for 10,000 generations. Finding a successful population takes several hours.
Temperature robustness is achieved in our models by searching sets of Q10 values that produce the target pyloric rhythm at each temperature. As in Caplan et al., 2014 and O'Leary and Marder, 2016 we searched for values of Q10i between 1 and 2 for conductances and between 1 and 4 for activation/inactivation time scales. We built a new objective function by evaluating the previous objective function (Equation 6) over a set of control temperatures Ti taken at 15°C, 20°C, 25°C and 35°C,
(7) EC(Q10)=E(G,T=15∘)+E(G,T=20∘)+E(G,T=25∘)+Ecrash(G,T=35∘).
Here, E(G,Ti) is the score of the solutions of the set of conductances G at temperature Ti and Ecrash is the total number of spikes in all three cells. This last term is introduced to enforce that when the temperature is close to 35°C the network stops working as in experiments. Evaluation of the objective functions (Equation 6) and (Equation 7) requires that the model is simulated for a number of seconds and this is the part of the procedure that requires most computing power. Longer simulations will provide better estimations for the burst frequency and duty cycle of the solutions, but it will linearly increase the time it takes to evaluate them. If the simulations are shorter, evaluations of the objective function are faster but its minimization may be more difficult due to transient behaviors, and its minima may not correspond to stable pyloric rhythms.
Because we only target the activity at three temperature values, it is not guaranteed that the models will behave properly for temperatures in between. We found that if the system meets the target activity at the control temperatures, in the vast majority of the cases, the models also displayed the correct activity for temperatures in between, but there were exceptions (2 in 40 cases). This means that even if a solution achieves low scores, we are still required to screen the temperature values in between to make sure it is indeed temperature compensated over the full range. Increasing the number of control temperatures makes the estimation procedure slower and we met a good balance with 4 control temperatures. All models were subject to the same temperature perturbations and temperature compensation was confirmed in all cases by recording spikes and phases of bursting (Figures 1 and 3), membrane potential (Figure 2), and currents' dynamics (Figures 7, 8 and 9).
Membrane potential and current shares distributions
The membrane potential distributions were computed for 101 values of temperature between 10°C and 25°C. For each temperature, the models were simulated from identical initial conditions for 30 s with high numerical resolution (dt=0.001). The distributions were computed using the last 10 s of each simulation. The number of samples of the distributions at each temperature is 5 × 105.
We classified the activity of each cell into several categories by inspecting their spiking patterns and features of their waveforms. For this we first compute the spike times and their ISI distributions. If the coefficient of variation CVISI<0.1, we declare the activity as spiking. We then measure the lag δ between the spike onset and the crossing times with a slow wave threshold at −50mV. If this δ<20msec we label the trace as tonic spiking and else, we label it as single-spike bursting. If CVISI>0.1 then we ask if the cell is bursting. For this we group spikes into bursts using a temporal threshold of 200 ms and compute the distributions of duty cycle dc and instantaneous burst frequency fb. If CVdc<1, CVfb<0.1 and mean(dc)>0 we label the trace as regular bursting. If mean(dc)>0 and either CVdc>1 or CVfb>0.1 we label the trace as irregular bursting. If mean(dc)=0 we label the trace as irregular spiker. If none of these conditions is met we label the trace as other and finally, if the total number of ISI values is not greater than 1 we label the trace as quiescent. The network state was classified as triphasic or not triphasic. We declared the network (NET) activity as triphasic if the frequency of the cells was similar (within 5%) and they fired in the right order, and not triphasic elsewhere.
Parameters used in this study
Each model was assigned a six-character name. Here, we list which model was used in each figure. Code to simulate the network and the parameters of each model are supplemental to this work.
Figure 1 4 × 6W66. Figure 1—figure supplement 1: from left to right, WWZ3CN, FGFKQS, 71G6LA. Figure 1—figure supplement 2: PLKTEM. Figure 1—figure supplement 3: all models. Figure 2: G7P2WE. Figure 2—figure supplement 1: from left to right, G7P2WE, 4 × 6W66, MAXTTP. Figure 3: 4 × 6W66, SJR46Y.Figure 4: WWZ3CN. Figure 5: FGFKQS.Figure 6: FGFKQS, 71G6LA. Figure 6—figure supplement 1: in lexicographical order, B7SDOL, J9U8SQ, 4R2719, R3VIEW, Q8FBN1, RMZ8KZ, EF84RN, SJR46Y. Figure 7: (top to bottom), J9U8SQ, G81PUL, G7EOZ8, SAXT8Y, KNMLBC. Figure 8: OSHNMD, H5AIZR. Figure 9: WWZ3CN, FGFKQS. Figure 10: 4 × 6W66.
Code and parameters to reproduce the results in this work was uploaded to Dryad, http://doi.org/10.5061/dryad.m31b1h7.
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Frances K Skinner
Reviewing Editor; Krembil Research Institute, University Health Network, Canada
Gary L Westbrook
Senior Editor; Oregon Health and Science University, United States
Farzan Nadim
Reviewer; New Jersey Institute of Technology, United States
Alexandre Guet-McCreight
Reviewer; Krembil Research Institute, University Health Network, Canada
Taking advantage of the pyloric network in crustaceans, this computational study beautifully shows how circuits and behaviour can adapt to the effect of temperature changes as a result of smooth adaptations in cellular and channel protein mechanisms, thus preserving the overall function of the circuit.
[Editors' note: the authors submitted for reconsideration following the decision after peer review. What follows is the decision letter after the first round of review.]
Thank you for submitting your work entitled "Temperature compensation in a small rhythmic circuit" as a Research Advance for consideration by eLife. Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by a Senior Editor. The reviewers have opted to remain anonymous.
Our decision has been reached after an extensive discussion between the reviewers. Based on these discussions and the individual reviews below, we regret to inform you that your work will not be considered further for publication in eLife. As you will see, this was not an easy decision, but one in which the reviewers and Reviewing Editor reached consensus.
We viewed this work as a thorough demonstration of the complex effects of temperature, but the results did not represent enough of a conceptual or mechanistic advance as presented. In essence, although all the authors found the work interesting and visually stunning, several concerns were raised, as detailed below in the reviewers' comments. Further, even though it was felt that the authors could potentially address some of the concerns raised (e.g., re Q10 variation assumptions etc.), all of the reviewers felt that something more besides a descriptive presentation was warranted to justify publication in eLife.
Since the paper that this work is building on is a tools and methods paper, we do not think that it is necessarily appropriate to be considered as a research advance paper, because A) it does not appear to be building on any of the tools and modeling techniques established in their previous publication, and B) it is using those techniques to explore a research topic that was not explored in the original publication. In other words, it would seem to be more appropriate to considered novel research, and not a research advance on the previous tools and methods paper. In their cover letter, the authors also considered this as an option. Our review is considered in the context of a research article.
In this manuscript, the authors use specialized computational methodologies and plotting techniques to uncover how current contributions in a set of models of the pyloric network change with temperature while still robustly generating stereotypical electrophysiological outputs. In selecting the models using their landscape optimization + a screening process approach, they ensured that the models were first capable of robustly generating triphasic spiking in the right order across different temperatures before analyzing current mechanisms through which this output emerges. For these models, they highlight a gradual change in current contributions as temperature is altered, showing that these current changes are complex and do not follow the same trends across models. In essence, they are able to show the existence of multistabilty, hysteresis and in general the non-trivial responses to changes in temperature.
The Marder lab has produced several interesting experimental and modeling studies examining temperature perturbations in previous works and this computational study adds to it. While the modeling work is robustly performed, it was thought that certain aspects of the modeling could be better contextualized in their connections to experimental literature. For example, when looking at hysteresis or removal of certain conductances during temperature changes, do the models – which were not designed to replicate any experimental electrophysiology outputs in these contexts – exhibit changes that are in line with what is seen experimentally (i.e. have there been any comparable experiments performed)? Could the authors consider/describe and/or design experiments that could be performed in light of their findings? For example, the finding that h-currents are not sensitive (and CaT are sensitive) to temperature (Figure 14C). Or perhaps a take-home finding would be that some types of currents (and not others) are sensitive? Is this something that the authors think appropriate in their context?
This study compares models that reproduce operation of the pyloric network across a range of temperatures based on different sets of parameter values. Rather than identifying models that reproduce network operation using different sets of conductance densities (G), they identify models with different sets of G values and Q10 values. They then use their currentscape visualization approach to ascertain that "temperature changes the relative contributions of the current to neuronal activity so that rhythmic activity smoothly slides through changes in mechanisms." Though interesting, there has been a lot of work in this area (i.e. temperature compensation in the pyloric network in particular, and more generally about degeneracy in the STG), which impacts the novelty of the current study. Moreover, I have significant reservations about how Q10 values are treated.
Main Concerns:
1) To my knowledge, the Q10 value for a given process (single channel conductance or gating) in a certain channel type tends to be quite consistent, which means it shouldn't differ much between two individuals of the same species. But if I understand the current study, the Q10 value for a given process is constant across the cells of a given network, but varies between networks. Is that correct? If so, is that justified? This comes up in the Discussion, where a handful of papers about RNA editing are cited. But in Garrett and Rosenthal, 2012, the difference is observed in different (arctic vs tropical) species of octopus. Reenan et al., 2015, address acute temperature effects, but don't report any physiological (Q10) changes. Is there any evidence for this sort of RNA editing in crabs/lobsters? Variation in Q10 values across individual is a critical yet dubious assumption. At the very least, the authors need to be transparent about this at the start of the paper.
2) On the flip side, conductance densities within a network are fixed (i.e. do not vary with temperature). Is that correct? I thought that compensation was mediated via homeostatic changes mediated by changes in ion channel expression (O'Leary and Marder, 2016).
3) There are 37 conductance densities in the model, plus 14 different Q10 values. Even before variations in Q10 values are considered, I suspect that there are many different G values that can produce models that behave appropriately across the full temperature range. Indeed, the solution space is high dimensional, which allows for a variety of solutions. It is not clear to me if the even higher dimensionality afforded by variations in Q10 is really necessary. For instance, if the analysis presented in Figure 4 of this paper (to models with different sets of Q10 and G values) was applied to models with the same Q10 values but different G values, would the results be fundamentally different? Wouldn't the currentscapes still look different because of differences in G values (especially if G values were subject to homeostatic regulation)? I suspect there would still be smooth transitions between different relative fractions of currents. Showing that this doesn't happen without variation in Q10 values is important if a conclusion of this paper is that Q10 variation is critical.
This work uses simulations of a reduced computational model of the crustacean pyloric network to make the important points that (a) neuronal networks that generate multi-phase bursting rhythms may be able to do so robustly across temperature changes despite the fact that these changes have different effects on different currents, and (b) the mechanisms underlying rhythmogenesis, including relative importance of currents, may change with temperature. The work mostly uses methods first presented by the authors in a 2019 eLife paper but looks at new issues.
Given the past works by these authors and others, I do not find these points to be highly surprising, and the paper also has some flaws that I will discuss in more detail below. My general assessment is that the authors' vivid illustration of these points in a computational model is a worthy contribution to the literature, but not a major advance.
Substantive concerns:
1) Much of the paper is written as a show and tell rather than as a flowing narrative. The authors tend to start sentences and paragraphs with phrases like "Figure X shows…" instead of presenting ideas, with figures shown in support of these ideas. I think some significant rewriting is needed to rectify this issue.
2) One of the issues that the authors consider is the relative roles of different currents in rhythms associated with different models and temperatures. I am not sure about the logic here: if a current has a larger magnitude in one regime than another, can we necessarily conclude that it has become more important? It may be that current A has grown larger, but current B has grown larger still; or perhaps current C, which is redundant with A, has also grown larger, such that A is no longer essential for rhythm generation. The authors need to be more careful with how this issue is presented; ideally, they would demonstrate that the changes are not just there, but are also meaningful. In a similar vein, they are making a choice to emphasize features that change between regimes, but there are also features that change very little, and this invariance could be as important as – or more important than – the changes. Finally, the extent of the changes in currents is not quantified – it is just displayed visually across Figures 8-10 and Figure 11. I would like to see some quantification of the changes, and I am not convinced that Figures 8-10 are needed at all since, to me, Figure 11 (which is beautiful) is much more illustrative.
3) I think the Materials and methods section needs some improvements and additions. The authors should clarify how they set up and check their target features for a rhythm to be considered valid – are there specifications for each cell? for the network as a whole? what if activity is rhythmic but not every cell bursts on each cycle? how are temperatures between the 4 main values "surveyed", to ensure that rhythmicity persists? Also, the authors should provide experimental citations for target values used for duty cycles and conductance ranges. I would like some justification for the use of the non-standard reversal potential of -70 mV for glutamate and for why the leak conductance is varied over a much smaller range than other conductances. For those interested in reproducing the results, the authors should indicate what computing resources (desktop? cluster?.…) and coding environment were used and how long simulations took (roughly – are we talking hours, days, weeks?). Finally, how were weights chosen for Equation 6?
4) Although the figures overall are informative and aesthetically pleasing, some revisions and clarifications are needed:
a) Figure 1A: I don't understand the rationale for the phase labels used. Why does the "PD off" box enclose the time when the PD cell is active? Why is there overlap between "LP on" and "LP off"? And so on! Some explanations are needed. Also, I don't see 3 curves in the top panel of Figure 1C, and the "PD, LP, PY" labels are too far removed from the data in the middle panel.
b) Figure 2: Panel B shows 6 examples. It would be nice to have some sort of summary of the findings over many models, not just these 6.
c) Figure 3D: the caption erroneously suggests that the duty cycle is multi-valued in the pink box.
d) Figure 6 and associated text: The authors should indicate how they define "failure" and how they detect it computationally (see (3) above).
e) Figure 11 supplement is lovely but clarification is needed about what p(V) (probability of voltage) means here. I am not seeing any dramatic effects, which makes me wonder if there is any scientific reason to keep this figure.
f) This is a very long paper. I am not convinced that the PCA adds much, and it's especially not clear to me how to get solid information from Figure 12C.
[Editors' note: further revisions were suggested prior to acceptance, as described below.]
Thank you for submitting your article "Temperature compensation in a small rhythmic circuit" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Gary Westbrook as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Farzan Nadim (Reviewer #2); Alexandre Guet-McCreight (Reviewer #3).
This manuscript uses computational modeling to examine how neural circuits can produce consistent activity across a wide range of temperatures. The authors use a family of models of the crustacean pyloric network, based on prior work from the Marder lab and a recent paper of the same authors. The main finding is that similar activity in the network, at different temperatures, can arise from the dynamic contributions of different currents. Additionally, the study shows that different currents play a compensatory role for one another at different temperatures, and that strong perturbations to the system at different temperatures could result in very different outcomes. It is noteworthy that this detailed level of analysis of current contributions is not possible to do experimentally. Therefore, this work also offers a glimpse at the coordination of ion channel mechanisms that could be occurring beyond the scope of current-day experimental apparatuses.
The reviewers found significant improvement over the original version, and the work was found to be comprehensive and worthy. However, concerns still remain. Overall, it was felt that the work as presented was too much on the descriptive side, the current narrative did not reflect what the interesting and important points of the study are. Some insight into why/how the compensation works needs to be provided. Specifics regarding these points are detailed here, and the various minor points raised by the reviewers are given below.
1) The narrative of the results should be changed and arranged around the main points.
That is, the Results narrative should draw the reader's attention to the highlights of the findings and (when allowed) the details should be put in the supplemental figures and tables.
In particular the many details of all the models voltage ranges, duty cycles etc. could be put in figure supplements. For instance, Figure 8, is quite beautiful with its currentscapes, but it is unclear what the point of the figure is? What information is the reader is supposed to take from this? There's no useful information even in the legend. The one point they bring up in the Results is rather subtle (they could've at least put arrows or circles) and the main points they make in the Results are actually in the figure supplements.
2) A problematic point here is that temperature changes are modeled in such an unconstrained manner (with 31 free parameters) that it is difficult to know whether any of the trajectories along different temperatures correspond to real biological changes. Does an ionic current respond to the same temperature change with widely different Q10s across preparations? If every single parameter is variable, independent of the conditions imposed, then what is the point of even measuring it? Would solution trajectories across a range of temperatures show more consistent underpinnings if the Q10s were consistent for each ionic current? In short, I find the unconstrained fitting of the parameters until the right solution is found akin to overfitting a curve to a few data points, and then trying to glean information from that fit. Please speak to this issue in some capacity.
3) There are interesting and important points in the study: 1. compensation of currents for one another at different temperatures (which I presume is a form of homeostasis), 2. Smooth transitions of these currents for one another. 3. Different responses to the same perturbation at different temperatures. 4. Distinct ways that the system crashes at high temperatures and 5. The presence of temperature-dependent hysteresis. However, the narrative in the presentation of the results (especially in the first half) is mainly descriptive and detail oriented, rather than driven by questions.
4) If one point of the study is that getting similar solutions at different temperatures requires ionic currents to assume different levels of contribution, this could be set as a central question and shown more directly. What is learned from how the models "crash" at high temperature? There may be some lessons there, but this point is not properly analyzed or explored.
Similarly, with the hysteresis. What is the lesson and what does it have to do with temperature or temperature compensation, rather than simply that a set of parameters may result in multistability?
5) Subsection "Spiking patterns during temperature ramps" and Figure 3E: Although clearer in this updated manuscript, can the authors add something to the figure to make this more obvious (e.g. adding the number of spikes in each burst above each burst in the trace or adding an inset plot showing superimposed traces of a cycle with and a cycle without the extra spike). Without actually counting the number of spikes in each cycle it's not obvious to me just from looking at the trace.
6) Paragraph three subsection "Spiking patterns during temperature ramps": If these spiking patterns can be compared to experimental recordings, why is it the case that they are not compared here? In fact, this paragraph appears to end abruptly without much dissemination of Figure 4. I'm also hesitant on calling the duty cycle "temperature invariant", since it seems clear from Figure 4 that duty cycle definitely does vary with temperature to some degree – how the authors seem to define temperature-invariant strikes me as a bit too qualitative.
7) Paragraph six of subsection "Dynamics of the currents at different temperatures": There are a lot of qualitative observations made regarding changes in currents across temperature and models. I feel as though this is an easy trap to fall into when there are many qualitative observations that could be made – for the reader, I find it becomes hard to follow since I end up going through the figures and squinting to try and see exactly what point the authors are trying to make for each observation. The main point is stated at the end and is very simple and much less specific: "Together, these examples illustrate how a current can play different roles at different temperatures, and how diverse these mechanisms can be across individual solutions". To bring this point forward better, I suggest the authors cut down on specific qualitative observations that individually do not carry much weight and speak more broadly. For example, you could replace the third sentence with something like "the current contributions profiles in model 2 are different leading up to and following bursts". Though I highlight this paragraph, I also feel that the writing is like this at various other points in the results as well.
8) Final paragraph of subsection "Dynamics of the currents at different temperatures": Is there a case where current contribution transitions across temperature are not smooth? And is there any reason to believe that they would not be smooth in the first place? If not, I would argue that this does not seem like a very surprising finding given that many e-phys features are preserved and change gradually with temperature themselves. Of course, it is still informative as to why the e-phys features change gradually – just not really surprising. Perhaps some of the points put forth in the discussion should be first mentioned at this stage in the results to help the reader better understand why this finding is both important and non-intuitive.
Thank you for submitting your article "Temperature compensation in a small rhythmic circuit" for consideration by eLife. Your revised article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Gary Westbrook as the Senior Editor The following individuals involved in review of your submission have agreed to reveal their identity: Farzan Nadim (Reviewer #2); Alexandre Guet-McCreight (Reviewer #3).
We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus, the revisions requested below only address clarity and presentation.
This manuscript uses computational modeling to examine how neural circuits can produce consistent activity across a wide range of temperatures. The authors use a family of models of the crustacean pyloric network, based on prior work from the Marder lab and a recent paper of the same authors. The main finding is that similar activity in the network, at different temperatures, can arise from the dynamic contributions of different currents. Additionally, the study shows that different currents play a compensatory role for one another at different temperatures, and that strong perturbations to the system at different temperatures could result in very different outcomes.
All the reviewers found that the work was improved with better flow and was less descriptive. There are two aspects that the reviewers think should be addressed
1) Although the hysteresis result is very interesting, the motivation for looking at it in the context of temperature sensitivity/robustness still isn't provided, nor is the biological link. This aspect of the paper would benefit from providing more biological context about the significance of multistability with respect to temperature sensitivity.
2) Clarification of the authors' treatment of Q10s with a reasonable justification in the discussion, rather than hand waving. Specifically, although it is safe to assume that not all of the Q10 values are known and the authors set reasonable bounds when fitting the Q10 values, the concern is not that the Q10 for membrane current X sits somewhere between 1 and 4, but that the Q10 for membrane current X differs between models A, B and C when, in fact, the molecular identify of current X is the same in all of those models, because a Q10 value is not something that is regulated. Thus, if the authors took the set of Q10 values from model A or B or C, and then, using that set of fixed Q10 values, re-ran their genetic algorithm to determine new sets of conductance combinations that produce an acceptable triphasic rhythm, one might expect that successful conductance combinations will be far less diverse than when Q10 values are allowed to differ (as shown in Figure 1—figure supplement 1B). Without knowing the real Q10 values, we still won't know what the real conductance combinations could be, but we'd have a clearer picture of how diverse those combinations could be under biologically realistic conditions.
Although we encourage such additional simulations, it is not a requirement. However, an expanded discussion of how Q10's are treated is warranted.
[Editors' note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors' response to the first round of review.]
[…] The Marder lab has produced several interesting experimental and modeling studies examining temperature perturbations in previous works and this computational study adds to it. While the modeling work is robustly performed, it was thought that certain aspects of the modeling could be better contextualized in their connections to experimental literature. For example, when looking at hysteresis or removal of certain conductances during temperature changes, do the models – which were not designed to replicate any experimental electrophysiology outputs in these contexts – exhibit changes that are in line with what is seen experimentally (i.e. have there been any comparable experiments performed)? Could the authors consider/describe and/or design experiments that could be performed in light of their findings? For example, the finding that h-currents are not sensitive (and CaT are sensitive) to temperature (Figure 14C). Or perhaps a take-home finding would be that some types of currents (and not others) are sensitive? Is this something that the authors think appropriate in their context?
H-currents are sensitive too. Old Figure 14C in the original manuscript (now Figure 11) shows that removing this current results in similar activities at both 10 °C and 25 °C. Most models become quiescent upon removal of the H-current, and thus, removing it results in qualitatively similar activity.
Although we do observe some behaviors in common across models, the cohort is too small to make statements about which perturbation is most likely to result in a given behavior.
For the past year we have been trying to do experiments on hysteresis to temperature changes. These experiments are quite challenging because, as expected, the hysteresis depends sensitively on protocol, and of course, experimental preparations are never stationary. We are still struggling with these data, but have benefited by the guidance of these theoretical studies.
Despite extensive experimental work this is the first theoretical paper to address the potential mechanisms of phase compensation in response to temperature in the network. This is a fascinating issue and is present biologically, but non-trivial to capture in models.
We are implicitly assuming that the Q10 (which is a function of the protein structure) could vary across individuals if they show different isoforms of a channel protein, but within an individual it might be constant. That said, we do not assert that Q10 values are different across individuals, and we don't know the experimental Q10 values of all the processes in our models. We show the existence of multiple mechanisms by which this can happen by finding Q10 values that permit the activity.
This is not completely correct. Conductance and the time scales of the processes both change acutely with temperature. This is our starting assumption. We show that there are many ways in which these quantities can change (Q10) without affecting the activity.
Homeostatic changes take place over a much longer time-scale. This work is about acute temperature perturbations that produce an instantaneous effect on the ion channels. This is now explicitly dealt with in the new Discussion.
As we described above and in the new Discussion, we are not dealing with homeostatic compensation in this manuscript.
We have majorly rewritten the paper to make its new principles more evident.
Significant rewriting took place with special emphasis on the reviewer's suggestion. Hopefully it reads much better now, thank you.
We do not conclude that larger magnitude implies more importance. Moreover, we focus on the changes in percent contribution -not absolute value- to compare one current against the other and highlight changes.
It is true that even if a current contributes with only 1% of the total it could still be more important to the activity than another current that contributes more, and this is regardless of temperature. We do not attempt to establish a link between the magnitude of a currently and its importance by any definition. We refer as a current as "important" in the final paragraph of the Results where we explicitly show that deletion of a specific current is critical at one temperature but not the other.
The first half of the paper is about things that stay the same in the system, like duty cycle and phases of bursting. The second half is about features that change, like current contributions and so on. To demonstrate that the changes are not just there, but are also meaningful, we perform channel deletions and find that the responses are qualitatively different at different temperatures. If temperature scaled the currents by similar ratios then the activity should break down in a qualitatively similar way (only faster) at all temperatures. We find that in general this is not the case, and instead different current contributions result in different stability properties.
Old Figures 9 and 10 from the previous manuscript are now supplemental to Figure 8 of. The current version and Figure 11 (now Figure 9) was simplified to show only one model.
In subsection "Finding parameters" we state the target values for each cell and the network. We missed stating that the target frequency is 1 Hz. This is now fixed.
If the activity is rhythmic but one cell bursts every cycle or so, then the standard deviation of the duty cycle distribution is large and the solution is discarded. Minimization of Equation 6 produces sets of conductances for which the activity is rhythmic and all cells burst periodically as shown across this work.
The models are surveyed for temperatures in between the control temps. (10,20,25) in most figures. We performed the same analysis as in Figure 1 and Figure 1—figure supplement 1 for all models. We computed at the membrane potential distributions as in Figure 3 for all models taking 101 values between 10 °C and 25 °C. We subjected all models to continuous temperature ramps (see Figures 4 and 5) of different durations (see up ramps in Figure 5). We computed the distribution of currents over 101 values of temp. between 10 °C and 25 °C (current distributions not included in current version)
We included a citation to experimental values for the targets (Tang et al., 2010).
Glutamate is an inhibitory transmitter in the STG, and the glutamate synapses have reversal potentials of -65 mV to -70 mV (Marder and Eisen, 1984, Prinz et al., 2004). This is standard in this preparation.
We discretize our search space by taking 1000 equally spaced values between the bounds for each conductance. Thus, a larger domain is sampled with less resolution. The search space for the leak conductance is smaller because this conductance is typically much smaller than others.
We chose a search space that is large enough to contain variable solutions and that also contains some knowledge of where good solutions may be found to speed up our optimizations.
We used a commercially available desktop computer with 32-cores. Finding a control network from random initial parameters takes several hours. Finding Q10 values for a network takes another few hours.
Weights for Equation 6 were chosen by trial and error and were kept fixed throughout the work. Notice that duty cycle is on the order of 10-1, and this is squared so 10-2. Factor β=1000 makes it comparable to the other quantities, and maybe weighs burst duration more than other features.
Most of this information was included in the Materials and methods section of the original manuscript and is present here. Further information has been added in the current manuscript.
We added additional labels to make the figure clearer.
The reason there are not three curves in C-top is that the bursting frequency of the three cells is almost identical (as desired) and the dots overlap. This was clarified in the figure caption.
The figure was simplified to show 3 models where changes in overall amplitude are substantially different across them.
Caption was rewritten. This figure was modified.
Failure is loss of triphasic rhythmic activity. We detected it by surveying the traces at different temperatures higher than 25 °C.
Probability of V means the probability that you will measure V if you observe the system at a random instant. (blue is low probability, for example during spikes. It is more likely one will observe the system in between bursts than during a spike).
Changes across temperature (x-axis) are clear (panels B and C).
We concluded that this figure is too complicated and obscures the main points of the manuscript, and removed it for these reasons.
We agree with the reviewer. We removed several figures and the text that goes along with them. Figure 12 and the corresponding subsection were removed. We included a panel in Figure 8 showing how the contributions change across models.
[Editors' note: what follows is the authors' response to the second round of review.]
The Results section was heavily modified and rearranged. Much material was moved to the supplemental figures. The narrative was changed and several details are omitted.
Our study is motivated by the experimental observation that neuronal circuits can remain performing their functions over a range of temperatures, at the same time that each of their molecular processes are affected differentially by temperature. Here we employ computational models to ask qualitatively how this can potentially take place in a network of neurons with multiple different types of ionic channels. For this we assumed a traditional Hodgkin and Huxley approach and asked if there are temperature robust solutions under the assumption that the temperature sensitivities Q10 of the different processes are different, and that the Q10 values are within experimental ranges (1 to 4 for timescales, 1 to 2 for conductances). We then employed the models to obtain qualitative insight into how compensation works and to formulate testable hypothesis. We are not aiming at extracting quantitative information about the pyloric network. Rather, we use the pyloric network to ask in general how this is possible.
Homeostasis as currently described in the models produced by this lab (Liu et. al, 1998, O'Leary et. al, 2014) corresponds to changes in maximal conductances over time. In this study we are assuming that maximal conductances are fixed, so there is no homeostatic mechanism of any kind.
We assume this confusion stems from our previous past use of the term
"compensation" which evokes active changes in conductance densities. We now replaced the word "compensation" in most places by "robustness", to highlight that here we are looking at automatic effects of neurons and networks that arise from the direct actions of temperature on ion channel proteins.
The Results section was modified to address this point. The "crashes" are now presented after we show currents. Hysteresis is also presented in that context.
This figure was modified to highlight differences in the bursts. The figure was moved to the supplements (it is now Figure 1—figure supplement 2).
We decided to leave data out of this study as we are still working on these experiments.
The duty cycle changes with temperature but on average it stays close to a fixed value (red line in the plots). The y-axis scale in the plots was chosen to highlight these departures. We have a partial dataset which suggests the biological network also displays such departures and temperature dependence, but the lab was shut and will be many months until we can collect such data. In the context of eLife policy regarding COVID19 we want to go without data at this point and in the future we will come up with a full dataset.
Figure 4 has a thicker red line so small departures from the average value are now more evident. Also, this figure was moved to the supplements.
We modified this section and removed much detail.
We did not see cases where the transitions are not smooth. However, we do not outrule that sharp transitions can take place in the working range, since maybe they are sharp but small. We do observe sharp transitions (possibly global bifurcations) when the models crash. We describe them as smooth because that's how they are. It is unclear whether this is surprising or not.
Hysteresis mainly occurs when the models "crash" at high temperatures. The states such as those in Figure 7 in the previous version (now Figure 8) are the ones for which the system can display multistability and therefore hysteresis as temperature is ramped back and forth. Tang et al., 2010, did not report multistability in the working range (10 to 25 oC), which is consistent with the models. That hysteresis occurs upon repeated "crashes" is a prediction of the models. This information was now added to the text, and is the subject of ongoing experiments in the lab. We are still struggling with the exact protocols to use to extract the most reliable insights from these experiments, as they are more technically difficult than might be expected.
Computational modeling will not answer what the "real" conductance and Q10 combinations are. Even the assumption that these quantities are stationary throughout the life of an animal (and/or the same across species) may not be warranted. For example, long-term acclimation over months resulting in RNA editing, could alter some Q10s. Moreover, it is possible that minor genetic variations in channel proteins exist in the population so that different animals in the population could show modestly different Q10s. Therefore, the point of this study is to ask generically how compensation occurs acutely assuming different temperature sensitivities for the processes, within the experimental range. We are not attempting to make quantitative observations about the pyloric network. We included a paragraph in the Discussion that makes this point more explicit.
We agree with the reviewer that if fixes one set of Q10s and asks for conductances that will produce temperature robust activity, this will be less diverse than if the Q10s are allowed to be different. We did perform some of these simulations, and it appears that the reviewer's intuition is likely correct. However, quantifying this observation is tricky and beyond the scope of our work, and it overweighs the benefit of making such claim more precise.
In summary, we are not trying to find the one best solution. We want to have an overview of how these solutions can be, how diverse they can be, and what are the general features in common among them (such as differential stability, changes in contributions, etc.). Given all the compensated solutions we found share these qualitative features, we believe it is likely that the 'true' biological mechanism also display these features. Our models provide testable predictions.
Volen Center and Biology Department, Brandeis University, Waltham, United States
Conceptualization, Software, Formal analysis, Investigation, Visualization, Writing - original draft, Writing - review and editing
lalonso@brandeis.edu
Conceptualization, Resources, Supervision, Funding acquisition, Writing - original draft, Writing - review and editing
National Institutes of Health (T32 NS07292)
National Institutes of Health (MH046742)
National Institutes of Health (R35 NS097343)
Swartz Foundation (Swartz Foundation 2017)
LMA acknowledges useful conversations with Hillary Rodgers.
Gary L Westbrook, Oregon Health and Science University, United States
Frances K Skinner, Krembil Research Institute, University Health Network, Canada
Farzan Nadim, New Jersey Institute of Technology, United States
Alexandre Guet-McCreight, Krembil Research Institute, University Health Network, Canada
Accepted: May 31, 2020
Accepted Manuscript published: June 2, 2020 (version 1)
Version of Record published: July 2, 2020 (version 2)
© 2020, Alonso and Marder
neuronal oscillators