Minimal requirements for a neuron to coregulate many properties and the implications for ion channel correlations and robustness
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Evaluation Summary:
This paper examines model neuron solution sets - combinations of ionic membrane conductance parameters that allow the model to produce functional output properties - and how the extent and shape of these solutions sets depends on ion channel pleiotropy, i.e., that ion channels can influence multiple outputs simultaneously. The work provides an organizing framework for previous experimental and modeling studies related to degeneracy of solutions, ion channel correlations, and homeostatic regulation, and should therefore be of interest to researchers in this area.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 agreed to share their name with the authors.)
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Abstract
Neurons regulate their excitability by adjusting their ion channel levels. Degeneracy – achieving equivalent outcomes (excitability) using different solutions (channel combinations) – facilitates this regulation by enabling a disruptive change in one channel to be offset by compensatory changes in other channels. But neurons must coregulate many properties. Pleiotropy – the impact of one channel on more than one property – complicates regulation because a compensatory ion channel change that restores one property to its target value often disrupts other properties. How then does a neuron simultaneously regulate multiple properties? Here, we demonstrate that of the many channel combinations producing the target value for one property (the single-output solution set), few combinations produce the target value for other properties. Combinations producing the target value for two or more properties (the multioutput solution set) correspond to the intersection between single-output solution sets. Properties can be effectively coregulated only if the number of adjustable channels ( n in ) exceeds the number of regulated properties ( n out ). Ion channel correlations emerge during homeostatic regulation when the dimensionality of solution space ( n in − n out ) is low. Even if each property can be regulated to its target value when considered in isolation, regulation as a whole fails if single-output solution sets do not intersect. Our results also highlight that ion channels must be coadjusted with different ratios to regulate different properties, which suggests that each error signal drives modulatory changes independently, despite those changes ultimately affecting the same ion channels.
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Author Response
Reviewer #1 (Public Review):
Yang. et al. use computational modeling to explore how neurons can co-regulate different properties (firing rate, excitability thresholds, energy consumption) by adjusting ion channel expression. To do so, they rely on the activity-dependent channel expression model introduced in O'Leary et al. 2014 and assume that any regulation loop can be boiled down to such a model. They thus propose a parallel feedback loop regulation model, in which each loop regulates a property by chasing its target value and regulating ion channel expression according to an integral control law.
The authors start by proving experimentally and computationally ion channel pleiotropy. This preliminary analysis is clearly developed and confirms/rediscovers in CA1 neurons known facts originally observed in …
Author Response
Reviewer #1 (Public Review):
Yang. et al. use computational modeling to explore how neurons can co-regulate different properties (firing rate, excitability thresholds, energy consumption) by adjusting ion channel expression. To do so, they rely on the activity-dependent channel expression model introduced in O'Leary et al. 2014 and assume that any regulation loop can be boiled down to such a model. They thus propose a parallel feedback loop regulation model, in which each loop regulates a property by chasing its target value and regulating ion channel expression according to an integral control law.
The authors start by proving experimentally and computationally ion channel pleiotropy. This preliminary analysis is clearly developed and confirms/rediscovers in CA1 neurons known facts originally observed in invertebrate systems like the STG. They subsequently use their regulation model to provide elegant geometric explanations for the emergence of ion channels correlations and for the success or failure of homeostatic regulation, as a function of the number of regulated properties and the number of ion channels.
The model they rely on exhibits two basic characteristics. Suppose the model possesses N different types of ion channels.
1.As in the model in O'Leary et al. 2014, when taken in isolation, the regulator of each neural property possesses an N-1 dimensional submanifold of steady-states in the N-dimensional maximal conductance space where the target for the regulated property is reached.
2.Regulation loops are independent of each other.
Given these two characteristics, in the presence of M regulated neural properties, the regulation target is reached simultaneously for each of them on an (N-M)-dimensional joint target submanifold of homeostatic steady states, i.e., the intersection of M (N-1)-dimensional target submanifolds. Depending on initial conditions, homeostatically regulated maximal conductances will spread out along this submanifold, thus creating N-M dimensional correlations. This (mathematically) elementary observation leads to the various general conclusions of the paper, the main of which are i) that increasing the number of regulated properties increase ion channel correlations (by reducing the dimension of the joint target submanifold) and ii) that increasing the number of regulated properties makes homeostatic regulation more likely to fail (when the intersection between the target submanifolds is empty). This simple geometric view on multi-property regulation is neat and, most importantly, experimentally verifiable/falsifiable.
The main drawback of this approach is that characteristics 1 and 2 (above) of the model used in the paper are non-generic and makes the model a useful but maybe oversimplified (and fragile) testbed. Let me develop.
Property 1 reflects one of the main limitations of the model in O'Leary et al. 2014, namely, that this model provides biologically meaningful results and predictions only when initial conditions are small and in the absence of any disturbance to its regulation dynamics (including the presence of multiple master regulators).This follows exactly from the fact that, in that model, the homeostatic regulator has zero effect once the state reaches the N-1 dimensional submanifold of target steady states. Thus, different initial conditions or exogenous disturbances will arbitrarily spread conductances on this submanifold, making it unrealistically unrobust to both types of perturbations.This limitation was overcome in a simple, biologically plausible way in Franci et al. 2020 by adding a molecular regulatory network between the homeostatic sensor and the regulated conductances. The revised model still exhibits the same correlated variability between maximal conductances as the original model. But only in the revised model the correlation ray exhibits biologically-meaningful levels of robustness both to disturbances and initial conditions. In dynamical systems terminology, the model in O'Leary et al. 2014 is structurally unstable (non generic) whereas its 2020 revision is structurally stable (generic). Crucially, in the 2020 model the homeostatic target is approximately reached at a unique steady state (i.e., the target submanifold is zero-dimensional) but the presence of a slow direction v_slow in the regulation space amplifies heterogeneities and disturbances, which leads to correlated variability (along v_slow).
We have identified several issues, and will address each in turn.
Initial conditions: Our results do not rely on initial conditions being small. This is first illustrated in Fig. 4 with the conditions immediately post-perturbation. Whereas O’Leary et al. (2014) presented results in the context of development (starting from nothing), we have focused on recovery from an acute perturbation (recovering from something). The statement that our model provides “biologically meaningful results and predictions only when initial conditions are small” is inaccurate – this depends on the question being asked. We use (a simplified version of) the O’Leary regulation mechanism to show that correlations arise in different ways depending on the dimensionality of solution space. This does not depend on initial conditions and, as explained in a new figure (Fig. 8), the dimensionality of solution space still has an important impact on the correlations that emerge under noisy conditions. We believe these to be meaningful results.
Unrobustness / structural instability: Noise is an important, ongoing perturbation that we did not consider in the original version of our paper. Franci et al. (2020) show that noise disrupts solutions found by the simple regulation mechanism used by O’Leary et al. (2013, 2014). The basis for this – that the simple regulation mechanism brings conductance densities to the solution manifold but does not control their spread across the manifold – is now highlighted in our revised text (lines 208-210). As now shown in a new figure (Fig. 8), ion channel correlations are disrupted by noise, depending on solution space dimensionality, but other correlations arise. Importantly, our regulation mechanism successfully maintains regulated properties near their target values despite noise; in that respect, regulation is robust.
Conductance densities do not remain / return to a restricted location on the solution manifold. In that sense, the solution set is not attractive, which, if we understand correctly, is equivalent to being unrobust, non-generic, and structurally unstable. Being unstable sounds bad, but from our perspective, so long as the system can regulate properties to their target values, the conductance density combinations used to do so are not a main concern (see Discussion, lines 310-314). Indeed, we highlight that conductance densities do not converge on the same restricted space if we control different ion channels (Fig 4) or if we control the same ion channels with different regulation rates (Fig 7). In fact, being too stable would reduce the hidden variability required to explain why neurons that appear similar respond differently to a perturbation, which is a concern. Furthermore, even if there is an attractive subspace, this could arise from regulation of other unaccounted for properties (which would reduce solution space dimensionality) rather than because of cooperative molecular interactions, which is to say that there are alternative explanations for attractive solution sets. Addressing such issues arguably extends beyond the scope of our study.
Unbounded spread: Noise-induced spread would be a problem if it caused conductance densities to spread infinitely, but there are biological mechanisms (apart from cooperative molecular interactions) that prevent this. (i) Conductance densities cannot become negative. One conductance density hitting zero prevents other conductance densities from continuing to rise when solution space dimensionality is low (see Fig. 8C). In other words, a lower bound is sufficient to prevent infinite spread under certain conditions. (ii) Upper bounds undoubtedly emerge from saturation of rate-limiting steps controlling the transcription, translation and/or insertion of ion channels (lines 657-659, including reference 89). With upper and lower bounds on conductance densities, the solution space is bounded regardless of its dimensionality. When the solution manifold is bounded, conductance densities cannot spread infinitely. Newly added simulations also revealed, under noisy conditions, that conductance densities drift in different preferred directions depending on regulation rates (see Fig. 8–figure supplement 1). The direction of that drift can reduce the likelihood of conductance densities ever reaching an upper bound.
In summary, we have verified that noise impacts the (stability of) solutions found by our simple homeostatic regulation mechanism. We have clarified how ion channel correlations in our model are affected. Unbounded spread is not a problem that necessitates cooperative molecular interactions. Other issues such as the initial conditions and attractiveness of the solution set (which are in fact connected) are interesting but not directly relevant to our study. We have struggled to grasp all the mathematical arguments presented in Franci et al. (2020) and were unable to implement that regulation mechanism in our model (see below), but we hope that our newly added simulations and edits to the text address concerns about Property 1.
Characteristic 2 of the model used in this paper is also non-generic. Consider the simple homeostatic parallel control scheme,
output = x+y
tau_x*dx/dt = tgt - (x+y)
tau_y*dy/dt = tgt - (x+y)
which (in line with the present paper) reaches the desired target output tgt on the 1-dimensional subspace of steady states x+y=tgt. Let's now introduce small coupling between the two variables as follows
output = x+y
tau_xdx/dt = tgt - (x+y) - epsilony
tau_ydy/dt = tgt - (x+y) - epsilonx
where epsilon>0 is small. It is easy to verify that the new model has a unique exponentially stable steady state given by
x = y; x+y= 2/(2+epsilon)*tgt ~ tgt (for epsilon sufficiently small)
Thus, introducing an arbitrary small coupling between the two regulation loops is sufficient to change the dimension of the target subspace from 1 to zero (without introducing new regulated properties!) while only leading to a small (O(epsilon)) error in the regulated property. It is also easy to show that the uncoupled/parallel model is not structurally stable, while the weakly coupled model is.
These observations lead to the question of whether the mathematical/computational results of the paper are realistic or whether they are artifacts of the non-generic modeling assumptions used for the regulation loops.
We do not understand the rationale for coupling the feedback loops. Is the motivation for doing this (i) because of experimental evidence that feedback loops are coupled, or (ii) because coupling solves some problem? We suspect it is the latter, but we will try to address each issue in turn.
(i) With respect to experimental evidence, firing rate and energy efficiency have different targets values (and different units) and the difference from target ought to be calculated separately. This relates to whether both error signals are encoded by calcium. As addressed at some length in our Discussion (see lines 332-351), we do not believe that all error signals are encoded by calcium and that feedback loops are coupled because of a common feedback signal. Even if multiple error signals are encoded by calcium, we suspect that such signals need to be functionally independent, e.g. spatially segregated and independently sensed (see lines 336-339), which would preclude the feedback loops from being coupled. We highlight evidence of other ways to encode error signals. If different error signals are encoded differently, it seems unlikely that feedback loops are coupled in the manner proposed here.
(ii ) With respect to solving some problem by coupling the feedback loops, we return to points raised above in connection with Property 1. Our main concern in this study is whether outputs can be regulated to their target value. Newly added simulations demonstrate that this occurs with uncoupled feedback loops even in the presence of noise (Fig. 8). Accordingly, we really do not see the impetus for coupling the feedback loops, especially since the experimental evidence (based on our interpretation of it) points away from this (see above).
The comment “without introducing new regulated properties” suggests that we considered additional regulated properties to stabilize the solution for property X. On the contrary, we considered additional regulated properties because we believe that neurons simultaneously regulate many properties. That regulating property Y stabilizes the solution for property X is an interesting observation, not a contrived explanation, and might obviate the need for cooperative molecular interactions (see above) if such a need exists. Insofar as uncoupled feedback loops can regulate multiple outputs, even in the presence of noise, coupling is unnecessary from our perspective and might actually compromise co-regulation. We are not starting from the assumption that the target subspace has dimension 0, and we are uncomfortable with that assumption (see above re. solutions being “too stable”). We nevertheless tried to adapt the suggested implementation, but it opened up a can of worms (see below).
If the question is ultimately whether coupling is required to enable co-regulation of multiple properties under noisy conditions, the answer is no, as now shown in Figure 8C.
Reviewer #2 (Public Review):
Yang, Shakil, Ratté, and Prescott conducted a combined dynamic clamp and modeling exploration on the geometry and dimensionality of single-output and multi-output solution sets embedded in the cellular parameter spaces of single neurons, i.e. CA1 pyramidal neurons for the dynamic clamp studies, and a generic single-compartment model with several varieties of sodium and potassium channels for the computational studies. Both types of neurons were stimulated with a randomly fluctuating current, and output measures (termed 'properties') of the neurons were measured or calculated, including rheobase, firing rate, energy consumption, and energy efficiency per spike. Ion channel maximal conductances were then varied, and the dependence of the output properties on the conductance values and their combinations were explored.
The authors define as a single-output solution set the subset of maximal conductance space containing conductance combinations that produce a value of one of the output properties within a tolerance range around a target value. These single-output solution sets can take the shape of points, curves, surfaces or volumes in parameter space. A multi-output solution set is then defined as the subset of parameter space that produces values within a tolerance range for multiple properties simultaneously, and thus lies at the intersection of several single-output solution sets (if such an intersection exists).
A major focus of the work is on the effect of channel pleiotropy - the impact of one ion channel type on more than one cellular output property - on the shape of solution sets, and whether homeostatic regulation schemes that adjust ionic membrane conductances to maintain and restore output properties in a target range can "find" the solution sets, and maintain the neuron within the solution set. The regulation schemes explored in this work directly use error signals of the output properties (the difference between an output property produced by a neuron and its target value) to increase or decrease maximal membrane constants, with pre-assigned regulation time constants.
The study is systematically executed, and results support the main conclusion, that successful regulation of n independent neuronal output properties requires that at least n ion channel densities be adjustable for a unique solution to exist, and at least n+1 for degenerate solutions. This conclusion explicitly organizes previous results obtained by other modeling studies into a coherent framework, but is not surprising.
The authors speculate that the need for neurons to regulate several of their output features may have provided the evolutionary drive for the highly diverse sets of voltage-dependencies and kinetics observed in different ion channels in nature. This speculation is intriguing, but also raises the question whether and how the unrealistically simple (and not very diverse) set of model ion channel characteristics used in this work may have impacted the extent and shape of single-property and multi-property solution sets: none of the ion channels in the model appear to have inactivation variables, and the two primarily varied ionic currents, I_Na and I_K, are identical in their voltage-dependence and activation dynamics; they differ only in their reversal potentials. It is likely that channels with such similar characteristics are more able to compensate for each other, and therefore produce more extensive and differently shaped solution sets, than more dissimilar channels.
Further exploration of the influence of solution set dimensionality on the existence and tightness of linear correlations between pairs of maximal conductances demonstrates that higher-dimensional solution sets, and larger tolerances around output property target values, lead to fewer and weaker correlations. This again is an insight that puts an organizing perspective on previous studies.
Finally, the paper provides examples of conductance regulation schemes that rely on multiple error signals (deviations of output properties from their respective target values) to differentially regulate different membrane conductances. These schemes are shown to successfully regulate neuron models and allow them to "find" solution sets in many circumstances (if solutions exist). While providing a proof-of-principle that echoes previous work by others, the biological interpretability of these regulation schemes is somewhat limited, because they can not be tied back to how the molecular machinery in a neuron would implement them. For example, it is not clear how a neuron would measure its energy efficiency per spike.
We thank the reviewer for the detailed and insightful summary of our paper. We completely agree with the shortcomings identified by the author with respect to specific molecular machinery. We kept our model deliberately simple because many details are not yet experimentally established, and it was not the goal of this study to uncover such details (which would have required very different experiments/simulations).
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Evaluation Summary:
This paper examines model neuron solution sets - combinations of ionic membrane conductance parameters that allow the model to produce functional output properties - and how the extent and shape of these solutions sets depends on ion channel pleiotropy, i.e., that ion channels can influence multiple outputs simultaneously. The work provides an organizing framework for previous experimental and modeling studies related to degeneracy of solutions, ion channel correlations, and homeostatic regulation, and should therefore be of interest to researchers in this area.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 agreed to share their name with the authors.)
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Reviewer #1 (Public Review):
Yang. et al. use computational modeling to explore how neurons can co-regulate different properties (firing rate, excitability thresholds, energy consumption) by adjusting ion channel expression. To do so, they rely on the activity-dependent channel expression model introduced in O'Leary et al. 2014 and assume that any regulation loop can be boiled down to such a model. They thus propose a parallel feedback loop regulation model, in which each loop regulates a property by chasing its target value and regulating ion channel expression according to an integral control law.
The authors start by proving experimentally and computationally ion channel pleiotropy. This preliminary analysis is clearly developed and confirms/rediscovers in CA1 neurons known facts originally observed in invertebrate systems like the …
Reviewer #1 (Public Review):
Yang. et al. use computational modeling to explore how neurons can co-regulate different properties (firing rate, excitability thresholds, energy consumption) by adjusting ion channel expression. To do so, they rely on the activity-dependent channel expression model introduced in O'Leary et al. 2014 and assume that any regulation loop can be boiled down to such a model. They thus propose a parallel feedback loop regulation model, in which each loop regulates a property by chasing its target value and regulating ion channel expression according to an integral control law.
The authors start by proving experimentally and computationally ion channel pleiotropy. This preliminary analysis is clearly developed and confirms/rediscovers in CA1 neurons known facts originally observed in invertebrate systems like the STG. They subsequently use their regulation model to provide elegant geometric explanations for the emergence of ion channels correlations and for the success or failure of homeostatic regulation, as a function of the number of regulated properties and the number of ion channels.
The model they rely on exhibits two basic characteristics. Suppose the model possesses N different types of ion channels.
1. As in the model in O'Leary et al. 2014, when taken in isolation, the regulator of each neural property possesses an N-1 dimensional submanifold of steady-states in the N-dimensional maximal conductance space where the target for the regulated property is reached.
2. Regulation loops are independent of each other.
Given these two characteristics, in the presence of M regulated neural properties, the regulation target is reached simultaneously for each of them on an (N-M)-dimensional joint target submanifold of homeostatic steady states, i.e., the intersection of M (N-1)-dimensional target submanifolds. Depending on initial conditions, homeostatically regulated maximal conductances will spread out along this submanifold, thus creating N-M dimensional correlations. This (mathematically) elementary observation leads to the various general conclusions of the paper, the main of which are i) that increasing the number of regulated properties increase ion channel correlations (by reducing the dimension of the joint target submanifold) and ii) that increasing the number of regulated properties makes homeostatic regulation more likely to fail (when the intersection between the target submanifolds is empty). This simple geometric view on multi-property regulation is neat and, most importantly, experimentally verifiable/falsifiable.
The main drawback of this approach is that characteristics 1 and 2 (above) of the model used in the paper are non-generic and makes the model a useful but maybe oversimplified (and fragile) testbed. Let me develop.
Property 1 reflects one of the main limitations of the model in O'Leary et al. 2014, namely, that this model provides biologically meaningful results and predictions only when initial conditions are small and in the absence of any disturbance to its regulation dynamics (including the presence of multiple master regulators). This follows exactly from the fact that, in that model, the homeostatic regulator has zero effect once the state reaches the N-1 dimensional submanifold of target steady states. Thus, different initial conditions or exogenous disturbances will arbitrarily spread conductances on this submanifold, making it unrealistically unrobust to both types of perturbations. This limitation was overcome in a simple, biologically plausible way in Franci et al. 2020 by adding a molecular regulatory network between the homeostatic sensor and the regulated conductances. The revised model still exhibits the same correlated variability between maximal conductances as the original model. But only in the revised model the correlation ray exhibits biologically-meaningful levels of robustness both to disturbances and initial conditions. In dynamical systems terminology, the model in O'Leary et al. 2014 is structurally unstable (non generic) whereas its 2020 revision is structurally stable (generic). Crucially, in the 2020 model the homeostatic target is approximately reached at a unique steady state (i.e., the target submanifold is zero-dimensional) but the presence of a slow direction v_slow in the regulation space amplifies heterogeneities and disturbances, which leads to correlated variability (along v_slow).
Characteristic 2 of the model used in this paper is also non-generic. Consider the simple homeostatic parallel control scheme,
output = x+y
tau_x*dx/dt = tgt - (x+y)
tau_y*dy/dt = tgt - (x+y)
which (in line with the present paper) reaches the desired target output tgt on the 1-dimensional subspace of steady states x+y=tgt. Let's now introduce small coupling between the two variables as follows
output = x+y
tau_x*dx/dt = tgt - (x+y) - epsilon*y
tau_y*dy/dt = tgt - (x+y) - epsilon*x
where epsilon>0 is small. It is easy to verify that the new model has a unique exponentially stable steady state given by
x = y; x+y= 2/(2+epsilon)*tgt ~ tgt (for epsilon sufficiently small)
Thus, introducing an arbitrary small coupling between the two regulation loops is sufficient to change the dimension of the target subspace from 1 to zero (without introducing new regulated properties!) while only leading to a small (O(epsilon)) error in the regulated property. It is also easy to show that the uncoupled/parallel model is not structurally stable, while the weakly coupled model is.
These observations lead to the question of whether the mathematical/computational results of the paper are realistic or whether they are artifacts of the non-generic modeling assumptions used for the regulation loops.
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Reviewer #2 (Public Review):
Yang, Shakil, Ratté, and Prescott conducted a combined dynamic clamp and modeling exploration on the geometry and dimensionality of single-output and multi-output solution sets embedded in the cellular parameter spaces of single neurons, i.e. CA1 pyramidal neurons for the dynamic clamp studies, and a generic single-compartment model with several varieties of sodium and potassium channels for the computational studies. Both types of neurons were stimulated with a randomly fluctuating current, and output measures (termed 'properties') of the neurons were measured or calculated, including rheobase, firing rate, energy consumption, and energy efficiency per spike. Ion channel maximal conductances were then varied, and the dependence of the output properties on the conductance values and their combinations were …
Reviewer #2 (Public Review):
Yang, Shakil, Ratté, and Prescott conducted a combined dynamic clamp and modeling exploration on the geometry and dimensionality of single-output and multi-output solution sets embedded in the cellular parameter spaces of single neurons, i.e. CA1 pyramidal neurons for the dynamic clamp studies, and a generic single-compartment model with several varieties of sodium and potassium channels for the computational studies. Both types of neurons were stimulated with a randomly fluctuating current, and output measures (termed 'properties') of the neurons were measured or calculated, including rheobase, firing rate, energy consumption, and energy efficiency per spike. Ion channel maximal conductances were then varied, and the dependence of the output properties on the conductance values and their combinations were explored.
The authors define as a single-output solution set the subset of maximal conductance space containing conductance combinations that produce a value of one of the output properties within a tolerance range around a target value. These single-output solution sets can take the shape of points, curves, surfaces or volumes in parameter space. A multi-output solution set is then defined as the subset of parameter space that produces values within a tolerance range for multiple properties simultaneously, and thus lies at the intersection of several single-output solution sets (if such an intersection exists).
A major focus of the work is on the effect of channel pleiotropy - the impact of one ion channel type on more than one cellular output property - on the shape of solution sets, and whether homeostatic regulation schemes that adjust ionic membrane conductances to maintain and restore output properties in a target range can "find" the solution sets, and maintain the neuron within the solution set. The regulation schemes explored in this work directly use error signals of the output properties (the difference between an output property produced by a neuron and its target value) to increase or decrease maximal membrane constants, with pre-assigned regulation time constants.
The study is systematically executed, and results support the main conclusion, that successful regulation of n independent neuronal output properties requires that at least n ion channel densities be adjustable for a unique solution to exist, and at least n+1 for degenerate solutions. This conclusion explicitly organizes previous results obtained by other modeling studies into a coherent framework, but is not surprising.
The authors speculate that the need for neurons to regulate several of their output features may have provided the evolutionary drive for the highly diverse sets of voltage-dependencies and kinetics observed in different ion channels in nature. This speculation is intriguing, but also raises the question whether and how the unrealistically simple (and not very diverse) set of model ion channel characteristics used in this work may have impacted the extent and shape of single-property and multi-property solution sets: none of the ion channels in the model appear to have inactivation variables, and the two primarily varied ionic currents, I_Na and I_K, are identical in their voltage-dependence and activation dynamics; they differ only in their reversal potentials. It is likely that channels with such similar characteristics are more able to compensate for each other, and therefore produce more extensive and differently shaped solution sets, than more dissimilar channels.
Further exploration of the influence of solution set dimensionality on the existence and tightness of linear correlations between pairs of maximal conductances demonstrates that higher-dimensional solution sets, and larger tolerances around output property target values, lead to fewer and weaker correlations. This again is an insight that puts an organizing perspective on previous studies.
Finally, the paper provides examples of conductance regulation schemes that rely on multiple error signals (deviations of output properties from their respective target values) to differentially regulate different membrane conductances. These schemes are shown to successfully regulate neuron models and allow them to "find" solution sets in many circumstances (if solutions exist). While providing a proof-of-principle that echoes previous work by others, the biological interpretability of these regulation schemes is somewhat limited, because they can not be tied back to how the molecular machinery in a neuron would implement them. For example, it is not clear how a neuron would measure its energy efficiency per spike.
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