Ion Channel Degeneracy, Variability, and Covariation in Neuron and Circuit Resilience.

The large number of ion channels found in all nervous systems poses fundamental questions concerning how the characteristic intrinsic properties of single neurons are determined by the specific subsets of channels they express. All neurons display many different ion channels with overlapping voltage- and time-dependent properties. We speculate that these overlapping properties promote resilience in neuronal function. Individual neurons of the same cell type show variability in ion channel conductance densities even though they can generate reliable and similar behavior. This complicates a simple assignment of function to any conductance and is associated with variable responses of neurons of the same cell type to perturbations, deletions, and pharmacological manipulation. Ion channel genes often show strong positively correlated expression, which may result from the molecular and developmental rules that determine which ion channels are expressed in a given cell type.

A biophysical perspective on the resilience of neuronal excitability across timescales.

Neuronal membrane excitability must be resilient to perturbations that can take place over timescales from milliseconds to months (or even years in long-lived animals). A great deal of attention has been paid to classes of homeostatic mechanisms that contribute to long-term maintenance of neuronal excitability through processes that alter a key structural parameter: the number of ion channel proteins present at the neuronal membrane. However, less attention has been paid to the self-regulating 'automatic' mechanisms that contribute to neuronal resilience by virtue of the kinetic properties of ion channels themselves. Here, we propose that these two sets of mechanisms are complementary instantiations of feedback control, together enabling resilience on a wide range of temporal scales. We further point to several methodological and conceptual challenges entailed in studying these processes - both of which involve enmeshed feedback control loops - and consider the consequences of these mechanisms of resilience.

Neuronal morphology enhances robustness to perturbations of channel densities.

Biological neurons show significant cell-to-cell variability but have the striking ability to maintain their key firing properties in the face of unpredictable perturbations and stochastic noise. Using a population of multi-compartment models consisting of soma, neurites, and axon for the lateral pyloric neuron in the crab stomatogastric ganglion, we explore how rebound bursting is preserved when the 14 channel conductances in each model are all randomly varied. The coupling between the axon and other compartments is critical for the ability of the axon to spike during bursts and consequently determines the set of successful solutions. When the coupling deviates from a biologically realistic range, the neuronal tolerance of conductance variations is lessened. Thus, the gross morphological features of these neurons enhance their robustness to perturbations of channel densities and expand the space of individual variability that can maintain a desired output pattern.

Gating of homeostatic regulation of intrinsic excitability produces cryptic long-term storage of prior perturbations.

Neurons and neuronal circuits must maintain their function throughout the life of the organism despite changing environments. Previous theoretical and experimental work suggests that neurons monitor their activity using intracellular calcium concentrations to regulate their intrinsic excitability. Models with multiple sensors can distinguish among different patterns of activity, but previous work using models with multiple sensors produced instabilities that lead the models' conductances to oscillate and then to grow without bound and diverge. We now introduce a nonlinear degradation term that explicitly prevents the maximal conductances to grow beyond a bound. We combine the sensors' signals into a master feedback signal that can be used to modulate the timescale of conductance evolution. Effectively, this means that the negative feedback can be gated on and off according to how far the neuron is from its target. The modified model recovers from multiple perturbations. Interestingly, depolarizing the models to the same membrane potential with current injection or with simulated high extracellular K+ produces different changes in conductances, arguing that caution must be used in interpreting manipulations that serve as a proxy for increased neuronal activity. Finally, these models accrue traces of prior perturbations that are not visible in their control activity after perturbation but that shape their responses to subsequent perturbations. These cryptic or hidden changes may provide insight into disorders such as posttraumatic stress disorder that only become visible in response to specific perturbations.

Alterations in network robustness upon simultaneous temperature and pH perturbations.

Nervous systems have evolved to function consistently in the face of the normal environmental fluctuations experienced by animals. The stomatogastric nervous system (STNS) of the crab, Cancer borealis, produces a motor output that has been studied for its remarkable robustness in response to single global perturbations. Changes in environments, however, are often complex and multifactorial. Therefore, we studied the robustness of the pyloric network in the stomatogastric ganglion (STG) in response to simultaneous perturbations of temperature and pH. We compared the effects of elevated temperatures on the pyloric rhythm at control, acid, or base pHs. In each pH recordings were made at 11°C, and then the temperature was increased until the rhythms became disorganized ("crashed"). Pyloric burst frequencies and phase relationships showed minor differences between pH groups until reaching close to the crash temperatures. However, the temperatures at which the rhythms were disrupted were lower in the two extreme pH conditions. This indicates that one environmental stress can make an animal less resilient to a second stressor.NEW & NOTEWORTHY Resilience to environmental fluctuations is important for all animals. It is common that animals encounter multiple stressful events at the same time, the cumulative impacts of which are largely unknown. This study examines the effects of temperature and pH on the nervous system of crabs that live in the fluctuating environments of the Northern Atlantic Ocean. The ranges of tolerance to one perturbation, temperature, are reduced under the influence of a second, pH.

Interactions among diameter, myelination, and the Na/K pump affect axonal resilience to high-frequency spiking.

Axons reliably conduct action potentials between neurons and/or other targets. Axons have widely variable diameters and can be myelinated or unmyelinated. Although the effect of these factors on propagation speed is well studied, how they constrain axonal resilience to high-frequency spiking is incompletely understood. Maximal firing frequencies range from ∼1 Hz to >300 Hz across neurons, but the process by which Na/K pumps counteract Na+ influx is slow, and the extent to which slow Na+ removal is compatible with high-frequency spiking is unclear. Modeling the process of Na+ removal shows that large-diameter axons are more resilient to high-frequency spikes than are small-diameter axons, because of their slow Na+ accumulation. In myelinated axons, the myelinated compartments between nodes of Ranvier act as a "reservoir" to slow Na+ accumulation and increase the reliability of axonal propagation. We now find that slowing the activation of K+ current can increase the Na+ influx rate, and the effect of minimizing the overlap between Na+ and K+ currents on spike propagation resilience depends on complex interactions among diameter, myelination, and the Na/K pump density. Our results suggest that, in neurons with different channel gating kinetic parameters, different strategies may be required to improve the reliability of axonal propagation.

Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability.

Excitability-a threshold-governed transient in transmembrane voltage-is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin-Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model's sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in Xenopus oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin-Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.

From the Neuroscience of Individual Variability to Climate Change.

Years of basic neuroscience on the modulation of the small circuits found in the crustacean stomatogastric ganglion have led us to study the effects of temperature on the motor patterns produced by the stomatogastric ganglion. While the impetus for this work was the study of individual variability in the parameters determining intrinsic and synaptic conductances, we are confronting substantial fluctuations in the stability of the networks to extreme temperature; these may correlate with changes in ocean temperature. Interestingly, when studied under control conditions, these wild-caught animals appear to be unchanged, but it is only when challenged by extreme temperatures that we reveal the consequences of warming oceans.

Neuromodulation of circuits with variable parameters: single neurons and small circuits reveal principles of state-dependent and robust neuromodulation.

Neuromodulation underlies many behavioral states and has been extensively studied in small circuits. This has allowed the systematic exploration of how neuromodulatory substances and the neurons that release them can influence circuit function. The physiological state of a network and its level of activity can have profound effects on how the modulators act, a phenomenon known as state dependence. We provide insights from experiments and computational work that show how state dependence can arise and the consequences it can have for cellular and circuit function. These observations pose a general unsolved question that is relevant to all nervous systems: How is robust modulation achieved in spite of animal-to-animal variability and degenerate, nonlinear mechanisms for the production of neuronal and network activity?

Functional consequences of neuropeptide and small-molecule co-transmission.

Colocalization of small-molecule and neuropeptide transmitters is common throughout the nervous system of all animals. The resulting co-transmission, which provides conjoint ionotropic ('classical') and metabotropic ('modulatory') actions, includes neuropeptide- specific aspects that are qualitatively different from those that result from metabotropic actions of small-molecule transmitter release. Here, we focus on the flexibility afforded to microcircuits by such co-transmission, using examples from various nervous systems. Insights from such studies indicate that co-transmission mediated even by a single neuron can configure microcircuit activity via an array of contributing mechanisms, operating on multiple timescales, to enhance both behavioural flexibility and robustness.

Molecular profiling of single neurons of known identity in two ganglia from the crab Cancer borealis.

Understanding circuit organization depends on identification of cell types. Recent advances in transcriptional profiling methods have enabled classification of cell types by their gene expression. While exceptionally powerful and high throughput, the ground-truth validation of these methods is difficult: If cell type is unknown, how does one assess whether a given analysis accurately captures neuronal identity? To shed light on the capabilities and limitations of solely using transcriptional profiling for cell-type classification, we performed 2 forms of transcriptional profiling-RNA-seq and quantitative RT-PCR, in single, unambiguously identified neurons from 2 small crustacean neuronal networks: The stomatogastric and cardiac ganglia. We then combined our knowledge of cell type with unbiased clustering analyses and supervised machine learning to determine how accurately functionally defined neuron types can be classified by expression profile alone. The results demonstrate that expression profile is able to capture neuronal identity most accurately when combined with multimodal information that allows for post hoc grouping, so analysis can proceed from a supervised perspective. Solely unsupervised clustering can lead to misidentification and an inability to distinguish between 2 or more cell types. Therefore, this study supports the general utility of cell identification by transcriptional profiling, but adds a caution: It is difficult or impossible to know under what conditions transcriptional profiling alone is capable of assigning cell identity. Only by combining multiple modalities of information such as physiology, morphology, or innervation target can neuronal identity be unambiguously determined.

Neuronal oscillator robustness to multiple global perturbations.

Neuronal activity depends on ion channels and biophysical processes that are strongly and differentially sensitive to physical variables such as temperature and pH. Nonetheless, neuronal oscillators can be surprisingly resilient to perturbations in these variables. We study a three-neuron pacemaker ensemble that drives the pyloric rhythm of the crab, Cancer borealis. These crabs routinely experience a number of global perturbations, including changes in temperature and pH. Although pyloric oscillations are robust to such changes, for sufficiently large deviations the rhythm reversibly breaks down. As temperature increases beyond a tipping point, oscillators transition to silence. Acidic pH deviations also show tipping points, with a reliable transition first to tonic spiking, then to silence. Surprisingly, robustness to perturbations in pH only moderately affects temperature robustness. Consistent with high animal-to-animal variability in biophysical circuit parameters, tipping points in temperature and pH vary across animals. However, the ordering and discrete classes of transitions at critical points are conserved. This implies that qualitative oscillator dynamics are preserved across animals despite high quantitative parameter variability. A universal model of bursting dynamics predicts the existence of these transition types and the order in which they occur.

Cellular function given parametric variation in the Hodgkin and Huxley model of excitability.

How is reliable physiological function maintained in cells despite considerable variability in the values of key parameters of multiple interacting processes that govern that function? Here, we use the classic Hodgkin-Huxley formulation of the squid giant axon action potential to propose a possible approach to this problem. Although the full Hodgkin-Huxley model is very sensitive to fluctuations that independently occur in its many parameters, the outcome is in fact determined by simple combinations of these parameters along two physiological dimensions: structural and kinetic (denoted S and K, respectively). Structural parameters describe the properties of the cell, including its capacitance and the densities of its ion channels. Kinetic parameters are those that describe the opening and closing of the voltage-dependent conductances. The impacts of parametric fluctuations on the dynamics of the system-seemingly complex in the high-dimensional representation of the Hodgkin-Huxley model-are tractable when examined within the S-K plane. We demonstrate that slow inactivation, a ubiquitous activity-dependent feature of ionic channels, is a powerful local homeostatic control mechanism that stabilizes excitability amid changes in structural and kinetic parameters.

Rapid adaptation to elevated extracellular potassium in the pyloric circuit of the crab, Cancer borealis.

Elevated potassium concentration ([K+]) is often used to alter excitability in neurons and networks by shifting the potassium equilibrium potential (EK) and, consequently, the resting membrane potential. We studied the effects of increased extracellular [K+] on the well-described pyloric circuit of the crab Cancer borealis. A 2.5-fold increase in extracellular [K+] (2.5×[K+]) depolarized pyloric dilator (PD) neurons and resulted in short-term loss of their normal bursting activity. This period of silence was followed within 5-10 min by the recovery of spiking and/or bursting activity during continued superfusion of 2.5×[K+] saline. In contrast, when PD neurons were pharmacologically isolated from pyloric presynaptic inputs, they exhibited no transient loss of spiking activity in 2.5×[K+], suggesting the presence of an acute inhibitory effect mediated by circuit interactions. Action potential threshold in PD neurons hyperpolarized during an hour-long exposure to 2.5×[K+] concurrent with the recovery of spiking and/or bursting activity. Thus the initial loss of activity appears to be mediated by synaptic interactions within the network, but the secondary adaptation depends on changes in the intrinsic excitability of the pacemaker neurons. The complex sequence of events in the responses of pyloric neurons to elevated [K+] demonstrates that electrophysiological recordings are necessary to determine both the transient and longer term effects of even modest alterations of K+ concentrations on neuronal activity.NEW & NOTEWORTHY Solutions with elevated extracellular potassium are commonly used as a depolarizing stimulus. We studied the effects of high potassium concentration ([K+]) on the pyloric circuit of the crab stomatogastric ganglion. A 2.5-fold increase in extracellular [K+] caused a transient loss of activity that was not due to depolarization block, followed by a rapid increase in excitability and recovery of spiking within minutes. This suggests that changing extracellular potassium can have complex and nonstationary effects on neuronal circuits.

Graded Transmission without Action Potentials Sustains Rhythmic Activity in Some But Not All Modulators That Activate the Same Current.

Neurons in the central pattern-generating circuits in the crustacean stomatogastric ganglion (STG) release neurotransmitter both as a graded function of presynaptic membrane potential that persists in TTX and in response to action potentials. In the STG of the male crab Cancer borealis, the modulators oxotremorine, C. borealis tachykinin-related peptide Ia (CabTRP1a), red pigment concentrating hormone (RPCH), proctolin, TNRNFLRFamide, and crustacean cardioactive peptide (CCAP) produce and sustain robust pyloric rhythms by activating the same modulatory current (IMI), albeit on different subsets of pyloric network targets. The muscarinic agonist oxotremorine, and the peptides CabTRP1a and RPCH elicited rhythmic triphasic intracellular alternating fluctuations of activity in the presence of TTX. Intracellular waveforms of pyloric neurons in oxotremorine and CabTRP1a in TTX were similar to those in the intact rhythm, and phase relationships among neurons were conserved. Although cycle frequency was conserved in oxotremorine and TTX, it was altered in CabTRP1a in the presence of TTX. Both rhythms were primarily driven by the pacemaker kernel consisting of the Anterior Burster and Pyloric Dilator neurons. In contrast, in TTX the circuit remained silent in proctolin, TNRNFLRFamide, and CCAP. These experiments show that graded synaptic transmission in the absence of voltage-gated Na+ current is sufficient to sustain rhythmic motor activity in some, but not other, modulatory conditions, even when each modulator activates the same ionic current. This further demonstrates that similar rhythmic motor patterns can be produced by qualitatively different mechanisms, one that depends on the activity of voltage-gated Na+ channels, and one that can persist in their absence.SIGNIFICANCE STATEMENT The pyloric rhythm of the crab stomatogastric ganglion depends both on spike-mediated and graded synaptic transmission. We activate the pyloric rhythm with a wide variety of different neuromodulators, all of which converge on the same voltage-dependent inward current. Interestingly, when action potentials and spike-mediated transmission are blocked using TTX, we find that the muscarinic agonist oxotremorine and the neuropeptide CabTRP1a sustain rhythmic alternations and appropriate phases of activity in the absence of action potentials. In contrast, TTX blocks rhythmic activity in the presence of other modulators. This demonstrates fundamental differences in the burst-generation mechanisms in different modulators that would not be suspected on the basis of their cellular actions at the level of the targeted current.

In vivo effects of temperature on the heart and pyloric rhythms in the crab Cancer borealis.

The heart and pyloric rhythms of crustaceans have been studied separately and extensively over many years. Local and hormonal neuromodulation and sensory inputs into these central pattern generator circuits play a significant role in an animal's response to perturbations, but are usually lost or removed during in vitro studies. To examine simultaneously the in vivo motor output of the crustacean heart and pyloric rhythms, we used photoplethysmography. In the population measured (n=49), the heart rhythm frequency ranged from 0.3 to 2.3 Hz. The pyloric rhythm varied from 0.2 to 1.6 Hz. We observed a weak correlation between the frequencies of the heart and pyloric rhythms. During multiple hour-long recordings, many animals held at a controlled temperature showed strong inhibitory bouts in which the heart decreased in frequency or become quiescent and the pyloric rhythm decreased in frequency. We measured the simultaneous responses of the rhythms to temperature ramps by heating or cooling the saline bath while recording both the heart and pyloric muscle movements. Q10, critical temperature (temperature at which muscle function is compromised) and changes in frequency were calculated for each of the rhythms tested. The heart rhythm was more robust to high temperature than the pyloric rhythm.

Regulation of Eag by Ca2+/calmodulin controls presynaptic excitability in Drosophila.

Drosophila ether-à-go-go ( eag) is the founding member of a large family of voltage-gated K+ channels, the KCNH family, which includes Kv10, 11, and 12. Concurrent binding of calcium/calmodulin (Ca2+/CaM) to NH2- and COOH-terminal sites inhibits mammalian EAG1 channels at submicromolar Ca2+ concentrations, likely by causing pore constriction. Although the Drosophila EAG channel was believed to be Ca2+-insensitive (Schönherr R, Löber K, Heinemann SH. EMBO J 19: 3263-3271, 2000.), both the NH2- and COOH-terminal sites are conserved. In this study we show that Drosophila EAG is inhibited by high Ca2+ concentrations that are only present at plasma membrane Ca2+ channel microdomains. To test the role of this regulation in vivo, we engineered mutations that block CaM-binding to the major COOH-terminal site of the endogenous eag locus, disrupting Ca2+-dependent inhibition. eag CaMBD mutants have reduced evoked release from larval motor neuron presynaptic terminals and show decreased Ca2+ influx in stimulated adult projection neuron presynaptic terminals, consistent with an increase in K+ conductance. These results are predicted by a conductance-based multicompartment model of the presynaptic terminal in which some fraction of EAG is localized to the Ca2+ channel microdomains that control neurotransmitter release. The reduction of release in the larval neuromuscular junction drives a compensatory increase in motor neuron somatic excitability. This misregulation of synaptic and somatic excitability has consequences for systems-level processes and leads to defects in associative memory formation in adults. NEW & NOTEWORTHY Regulation of excitability is critical to tuning the nervous system for complex behaviors. We demonstrate in this article that the EAG family of voltage-gated K+ channels exhibit conserved gating by Ca2+/CaM. Disruption of this inhibition in Drosophila results in decreased evoked neurotransmitter release due to truncated Ca2+ influx in presynaptic terminals. In adults, disrupted Ca2+ dynamics cripples memory formation. These data demonstrate that the biophysical details of channels have important implications for cell function and behavior.

Understanding brains: details, intuition, and big data.

Understanding how the brain works requires a delicate balance between the appreciation of the importance of a multitude of biological details and the ability to see beyond those details to general principles. As technological innovations vastly increase the amount of data we collect, the importance of intuition into how to analyze and treat these data may, paradoxically, become more important.