Mathematical analysis of depolarization block mediated by slow inactivation of fast sodium channels in midbrain dopamine neurons

Membrane depolarization mediates both the inhibition of neural activity and cell-type-differences in response to high-frequency stimulation

Neuromodulation using high frequency (>1 kHz) electric stimulation (HFS) enables preferential activation or inhibition of individual neural types, offering the possibility of more effective treatments across a broad spectrum of neurological diseases. To improve effectiveness, it is important to better understand the mechanisms governing activation and inhibition with HFS so that selectivity can be optimized. In this study, we measure the membrane potential (Vm) and spiking responses of ON and OFF α-sustained retinal ganglion cells (RGCs) to a wide range of stimulus frequencies (100-2500 Hz) and amplitudes (10-100 µA). Our findings indicate that HFS induces shifts in Vm, with both the strength and polarity of the shifts dependent on the stimulus conditions. Spiking responses in each cell directly correlate with the shifts in Vm, where strong depolarization leads to spiking suppression. Comparisons between the two cell types reveal that ON cells are more depolarized by a given amplitude of HFS than OFF cells-this sensitivity difference enables the selective targeting. Computational modeling indicates that ion-channel dynamics largely account for the shifts in Vm, suggesting that a better understanding of the differences in ion-channel properties across cell types may improve the selectivity and ultimately, enhance HFS-based neurostimulation strategies.

Galvanic vs. pulsatile effects on decision-making networks: reshaping the neural activation landscape

Objective. Primarily due to safety concerns, biphasic pulsatile stimulation (PS) is the present standard for electrical excitation of neural tissue with a diverse set of applications. While pulses have been shown to be effective to achieve functional outcomes, they have well-known deficits. Due to recent technical advances, galvanic stimulation (GS), delivery of current for extended periods of time (>1 s), has re-emerged as an alternative to PS.Approach. In this paper, we use a winner-take-all decision-making cortical network model to investigate differences between pulsatile and GS in the context of a perceptual decision-making task.Main results. Based on previous work, we hypothesized that GS would produce more spatiotemporally distributed, network-sensitive neural responses, while PS would produce highly synchronized activation of a limited group of neurons. Our results in-silico support these hypotheses for low-amplitude GS but deviate when galvanic amplitudes are large enough to directly activate or block nearby neurons.Significance. We conclude that with careful parametrization, GS could overcome some limitations of PS to deliver more naturalistic firing patterns in the group of targeted neurons.

Electrophysiological Properties of Proprioception-Related Neurons in the Intermediate Thoracolumbar Spinal Cord

Proprioception, the sense of limb and body position, is required to produce accurate and precise movements. Proprioceptive sensory neurons transmit muscle length and tension information to the spinal cord. The function of excitatory neurons in the intermediate spinal cord, which receive this proprioceptive information, remains poorly understood. Using genetic labeling strategies and patch-clamp techniques in acute spinal cord preparations in mice, we set out to uncover how two sets of spinal neurons, Clarke's column (CC) and Atoh1-lineage neurons, respond to electrical activity and how their inputs are organized. Both sets of neurons are located in close proximity in laminae V-VII of the thoracolumbar spinal cord and have been described to receive proprioceptive signals. We find that a majority of CC neurons have a tonic-firing type and express a distinctive hyperpolarization-activated current (Ih). Atoh1-lineage neurons, which cluster into two spatially distinct populations, are mostly a fading-firing type and display similar electrophysiological properties to each other, possibly due to their common developmental lineage. Finally, we find that CC neurons respond to stimulation of lumbar dorsal roots, consistent with prior knowledge that CC neurons receive hindlimb proprioceptive information. In contrast, using a combination of electrical stimulation, optogenetic stimulation, and transsynaptic rabies virus tracing, we find that Atoh1-lineage neurons receive heterogeneous, predominantly local thoracic inputs that include parvalbumin-lineage sensory afferents and local interneuron presynaptic inputs. Altogether, we find that CC and Atoh1-lineage neurons have distinct membrane properties and sensory input organization, representing different subcircuit modes of proprioceptive information processing.

Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection

For single neuron models, reproducing characteristics of neuronal activity such as the firing rate, amplitude of spikes, and threshold potentials as functions of both synaptic current and conductance is a challenging task. In the present work, we measure these characteristics of regular spiking cortical neurons using the dynamic patch-clamp technique, compare the data with predictions from the standard Hodgkin-Huxley and Izhikevich models, and propose a relatively simple five-dimensional dynamical system model, based on threshold criteria. The model contains a single sodium channel with slow inactivation, fast activation and moderate deactivation, as well as, two fast repolarizing and slow shunting potassium channels. The model quantitatively reproduces characteristics of steady-state activity that are typical for a cortical pyramidal neuron, namely firing rate not exceeding 30 Hz; critical values of the stimulating current and conductance which induce the depolarization block not exceeding 80 mV and 3, respectively (both values are scaled by the resting input conductance); extremum of hyperpolarization close to the midpoint between spikes. The analysis of the model reveals that the spiking regime appears through a saddle-node-on-invariant-circle bifurcation, and the depolarization block is reached through a saddle-node bifurcation of cycles. The model can be used for realistic network simulations, and it can also be implemented within the so-called mean-field, refractory density framework.

Sodium channel slow inactivation normalizes firing in axons with uneven conductance distributions

The Na+ channels that are important for action potentials show rapid inactivation, a state in which they do not conduct, although the membrane potential remains depolarized.1,2 Rapid inactivation is a determinant of millisecond-scale phenomena, such as spike shape and refractory period. Na+ channels also inactivate orders of magnitude more slowly, and this slow inactivation has impacts on excitability over much longer timescales than those of a single spike or a single inter-spike interval.3,4,5,6,7,8,9,10 Here, we focus on the contribution of slow inactivation to the resilience of axonal excitability11,12 when ion channels are unevenly distributed along the axon. We study models in which the voltage-gated Na+ and K+ channels are unevenly distributed along axons with different variances, capturing the heterogeneity that biological axons display.13,14 In the absence of slow inactivation, many conductance distributions result in spontaneous tonic activity. Faithful axonal propagation is achieved with the introduction of Na+ channel slow inactivation. This "normalization" effect depends on relations between the kinetics of slow inactivation and the firing frequency. Consequently, neurons with characteristically different firing frequencies will need to implement different sets of channel properties to achieve resilience. The results of this study demonstrate the importance of the intrinsic biophysical properties of ion channels in normalizing axonal function.

Sex-Specific Timelines for Adaptations of Prefrontal Parvalbumin Neurons in Response to Stress and Changes in Anxiety- and Depressive-Like Behaviors

Women are twice as likely as men to experience emotional dysregulation after stress, resulting in substantially higher psychopathology for equivalent lifetime stress exposure, yet the mechanisms underlying this vulnerability remain unknown. Studies suggest changes in medial prefrontal cortex (mPFC) activity as a potential contributor. Whether maladaptive changes in inhibitory interneurons participate in this process, and whether adaptations in response to stress differ between men and women, producing sex-specific changes in emotional behaviors and mPFC activity, remained undetermined. This study examined whether unpredictable chronic mild stress (UCMS) in mice differentially alters behavior and mPFC parvalbumin (PV) interneuron activity by sex, and whether the activity of these neurons drives sex-specific behavioral changes. Four weeks of UCMS increased anxiety-like and depressive-like behaviors associated with FosB activation in mPFC PV neurons, particularly in females. After 8 weeks of UCMS, both sexes displayed these behavioral and neural changes. Chemogenetic activation of PV neurons in UCMS-exposed and nonstressed males induced significant changes in anxiety-like behaviors. Importantly, patch-clamp electrophysiology demonstrated altered excitability and basic neural properties on the same timeline as the emergence of behavioral effects: changes in females after 4 weeks and in males after 8 weeks of UCMS. These findings show, for the first time, that sex-specific changes in the excitability of prefrontal PV neurons parallel the emergence of anxiety-like behavior, revealing a potential novel mechanism underlying the enhanced vulnerability of females to stress-induced psychopathology and supporting further investigation of this neuronal population to identify new therapeutic targets for stress disorders.

Olfaction: The smell stops here

A recent study has shown that, in the fly Drosophila, olfactory neurons stop signaling when smells get too strong. This changes the way we think about odor encoding across concentrations.

Depolarization block in olfactory sensory neurons expands the dimensionality of odor encoding

Upon strong and prolonged excitation, neurons can undergo a silent state called depolarization block that is often associated with disorders such as epileptic seizures. Here, we show that neurons in the peripheral olfactory system undergo depolarization block as part of their normal physiological function. Typically, olfactory sensory neurons enter depolarization block at odor concentrations three orders of magnitude above their detection threshold, thereby defining receptive fields over concentration bands. The silencing of high-affinity olfactory sensory neurons produces sparser peripheral odor representations at high-odor concentrations, which might facilitate perceptual discrimination. Using a conductance-based model of the olfactory transduction cascade paired with spike generation, we provide numerical and experimental evidence that depolarization block arises from the slow inactivation of sodium channels-a process that could affect a variety of sensory neurons. The existence of ethologically relevant depolarization block in olfactory sensory neurons creates an additional dimension that expands the peripheral encoding of odors.

Disrupting Epileptiform Activity by Preventing Parvalbumin Interneuron Depolarization Block

Inhibitory synaptic mechanisms oppose epileptic network activity in the brain. The breakdown in this inhibitory restraint and propagation of seizure activity has been linked to the overwhelming of feedforward inhibition, which is provided in large part by parvalbumin-expressing (PV) interneurons in the cortex. The underlying cellular processes therefore represent potential targets for understanding and preventing the propagation of seizure activity. Here we use an optogenetic strategy to test the hypothesis that depolarization block in PV interneurons is a significant factor during the loss of inhibitory restraint. Depolarization block results from the inactivation of voltage-gated sodium channels and leads to impaired action potential firing. We used focal NMDA stimulation to elicit reproducible epileptiform discharges in hippocampal organotypic brain slices from male and female mice and combined this with targeted recordings from defined neuronal populations. Simultaneous patch-clamp recordings from PV interneurons and pyramidal neurons revealed epileptiform activity that was associated with an overwhelming of inhibitory synaptic mechanisms and the emergence of a partial, and then complete, depolarization block in PV interneurons. To counteract this depolarization block, we developed protocols for eliciting pulsed membrane hyperpolarization via the inhibitory opsin, archaerhodopsin. This optical approach was effective in counteracting cumulative inactivation of voltage-gated channels, maintaining PV interneuron action potential firing properties during the inhibitory restraint period, and reducing the probability of initiating epileptiform activity. These experiments support the idea that depolarization block is a point of weakness in feedforward inhibitory synaptic mechanisms and represents a target for preventing the initiation and spread of seizure activity.SIGNIFICANCE STATEMENT GABA_A receptor-mediated synaptic transmission opposes seizure activity by establishing an inhibitory restraint against spreading excitation. Parvalbumin-expressing (PV) interneurons contribute significantly to this inhibitory restraint, but it has been suggested that these cells are overwhelmed as they enter a state of "depolarization block." Here we test the importance of this process by devising an optogenetic strategy to selectively relieve depolarization block in PV interneurons. By inducing brief membrane hyperpolarization, we show that it is possible to reduce depolarization block in PV interneurons, maintain their action potential firing in the face of strong excitation, and disrupt epileptiform activity in an in vitro model. This represents a proof of principle that targeting rate-limiting processes can strengthen the inhibitory restraint of epileptiform activity.

Reduced Dopamine Signaling Impacts Pyramidal Neuron Excitability in Mouse Motor Cortex

Dopaminergic modulation is essential for the control of voluntary movement; however, the role of dopamine in regulating the neural excitability of the primary motor cortex (M1) is not well understood. Here, we investigated two modes by which dopamine influences the input/output function of M1 neurons. To test the direct regulation of M1 neurons by dopamine, we performed whole-cell recordings of excitatory neurons and measured excitability before and after local, acute dopamine receptor blockade. We then determined whether chronic depletion of dopaminergic input to the entire motor circuit, via a mouse model of Parkinson's disease, was sufficient to shift M1 neuron excitability. We show that D1 receptor (D1R) and D2R antagonism altered subthreshold and suprathreshold properties of M1 pyramidal neurons in a layer-specific fashion. The effects of D1R antagonism were primarily driven by changes to intrinsic properties, while the excitability shifts following D2R antagonism relied on synaptic transmission. In contrast, chronic depletion of dopamine to the motor circuit with 6-hydroxydopamine induced layer-specific synaptic transmission-dependent shifts in M1 neuron excitability that only partially overlapped with the effects of acute D1R antagonism. These results suggest that while acute and chronic changes in dopamine modulate the input/output function of M1 neurons, the mechanisms engaged are distinct depending on the duration and origin of the manipulation. Our study highlights the broad influence of dopamine on M1 excitability by demonstrating the consequences of local and global dopamine depletion on neuronal input/output function.

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons

Two subpopulations of midbrain dopamine (DA) neurons are known to have different dynamic firing ranges in vitro that correspond to distinct projection targets: the originally identified conventional DA neurons project to the dorsal striatum and the lateral shell of the nucleus accumbens, whereas an atypical DA population with higher maximum firing frequencies projects to prefrontal regions and other limbic regions including the medial shell of nucleus accumbens. Using a computational model, we show that previously identified differences in biophysical properties do not fully account for the larger dynamic range of the atypical population and predict that the major difference is that originally identified conventional cells have larger occupancy of voltage-gated sodium channels in a long-term inactivated state that recovers slowly; stronger sodium and potassium conductances during action potential firing are also predicted for the conventional compared to the atypical DA population. These differences in sodium channel gating imply that longer intervals between spikes are required in the conventional population for full recovery from long-term inactivation induced by the preceding spike, hence the lower maximum frequency. These same differences can also change the bifurcation structure to account for distinct modes of entry into depolarization block: abrupt versus gradual. The model predicted that in cells that have entered depolarization block, it is much more likely that an additional depolarization can evoke an action potential in conventional DA population. New experiments comparing lateral to medial shell projecting neurons confirmed this model prediction, with implications for differential synaptic integration in the two populations.

We developed a theoretical and mathematical framework that could explain the major electrophysiological differences between the conventional midbrain dopamine (DA) neurons with a low maximum firing rate, and the more recently identified atypical DA neurons. Testable predictions from this framework were then verified with in vitro patch-clamp recordings from DA neurons with identified phenotypes and projection targets. Since different subpopulations of DA neurons participate in different circuits, and these circuits are likely differentially dysregulated in diseases such as addiction, Parkinson disease, and schizophrenia, it is important to identify the differences of their intrinsic electrophysiological properties as a prelude to developing more precisely targeted therapies.

Dimensions of control for subthreshold oscillations and spontaneous firing in dopamine neurons

Dopaminergic neurons (DAs) of the rodent substantia nigra pars compacta (SNc) display varied electrophysiological properties in vitro. Despite this, projection patterns and functional inputs from DAs to other structures are conserved, so in vivo delivery of consistent, well-timed dopamine modulation to downstream circuits must be coordinated. Here we show robust coordination by linear parameter controllers, discovered through powerful mathematical analyses of data and models, and from which consistent control of DA subthreshold oscillations (STOs) and spontaneous firing emerges. These units of control represent coordinated intracellular variables, sufficient to regulate complex cellular properties with radical simplicity. Using an evolutionary algorithm and dimensionality reduction, we discovered metaparameters, which when regressed against STO features, revealed a 2-dimensional control plane for the neuron’s 22-dimensional parameter space that fully maps the natural range of DA subthreshold electrophysiology. This plane provided a basis for spiking currents to reproduce a large range of the naturally occurring spontaneous firing characteristics of SNc DAs. From it we easily produced a unique population of models, derived using unbiased parameter search, that show good generalization to channel blockade and compensatory intracellular mechanisms. From this population of models, we then discovered low-dimensional controllers for regulating spontaneous firing properties, and gain insight into how currents active in different voltage regimes interact to produce the emergent activity of SNc DAs. Our methods therefore reveal simple regulators of neuronal function lurking in the complexity of combined ion channel dynamics.

Electrophysiological activity of the neuronal membrane and concomitant ion channel properties are highly variable within groups of neurons of the same type from the same brain region. Reconciliation of the mechanisms generating neuronal activity is challenging due to the complexity of the interactions between the channel currents involved. Here we present a set of mathematical analyses that uncover the low-dimensional intracellular parameter combinations capable of regulating features of subthreshold oscillations and spontaneous firing in empirically constrained models of nigral dopaminergic neurons. This method generates, from a naive starting point, linear combinations of ion channel properties that are surprisingly capable of reliably controlling a wide variety of emergent electrophysiological activity, thereby predicting drug effects and shedding light on unsuspected compensatory mechanisms that contribute to neuronal function.

Simulation Study of Intermittent Axonal Block and Desynchronization Effect Induced by High-Frequency Stimulation of Electrical Pulses

Deep brain stimulation (DBS) has been successfully used in treating neural disorders in brain, such as Parkinson’s disease and epilepsy. However, the precise mechanisms of DBS remain unclear. Regular DBS therapy utilizes high-frequency stimulation (HFS) of electrical pulses. Among all of neuronal elements, axons are mostly inclined to be activated by electrical pulses. Therefore, the response of axons may play an important role in DBS treatment. To study the axonal responses during HFS, we developed a computational model of myelinated axon to simulate sequences of action potentials generated in single and multiple axons (an axon bundle) by stimulations. The stimulations are applied extracellularly by a point source of current pulses with a frequency of 50–200 Hz. Additionally, our model takes into account the accumulation of potassium ions in the peri-axonal spaces. Results show that the increase of extracellular potassium generates intermittent depolarization block in the axons during HFS. Under the state of alternate block and recovery, axons fire action potentials at a rate far lower than the frequency of stimulation pulses. In addition, the degree of axonal block is highly related to the distance between the axons and the stimulation point. The differences in the degree of block for individual axons in a bundle result in desynchronized firing among the axons. Stimulations with higher frequency and/or greater intensity can induce axonal block faster and increase the desynchronization effect on axonal firing. Presumably, the desynchronized axonal activity induced by HFS could generate asynchronous activity in the population of target neurons downstream thereby suppressing over-synchronized firing of neurons in pathological conditions. The desynchronization effect generated by intermittent activation of axons may be crucial for DBS therapy. The present study provides new insights into the mechanisms of DBS, which is significant for advancing the application of DBS.

Parameter Optimization Using Covariance Matrix Adaptation-Evolutionary Strategy (CMA-ES), an Approach to Investigate Differences in Channel Properties Between Neuron Subtypes

Computational models in neuroscience can be used to predict causal relationships between biological mechanisms in neurons and networks, such as the effect of blocking an ion channel or synaptic connection on neuron activity. Since developing a biophysically realistic, single neuron model is exceedingly difficult, software has been developed for automatically adjusting parameters of computational neuronal models. The ideal optimization software should work with commonly used neural simulation software; thus, we present software which works with models specified in declarative format for the MOOSE simulator. Experimental data can be specified using one of two different file formats. The fitness function is customizable as a weighted combination of feature differences. The optimization itself uses the covariance matrix adaptation-evolutionary strategy, because it is robust in the face of local fluctuations of the fitness function, and deals well with a high-dimensional and discontinuous fitness landscape. We demonstrate the versatility of the software by creating several model examples of each of four types of neurons (two subtypes of spiny projection neurons and two subtypes of globus pallidus neurons) by tuning to current clamp data. Optimizations reached convergence within 1,600-4,000 model evaluations (200-500 generations × population size of 8). Analysis of the parameters of the best fitting models revealed differences between neuron subtypes, which are consistent with prior experimental results. Overall our results suggest that this easy-to-use, automatic approach for finding neuron channel parameters may be applied to current clamp recordings from neurons exhibiting different biochemical markers to help characterize ionic differences between other neuron subtypes.

Dopamine Neurons Change the Type of Excitability in Response to Stimuli

The dynamics of neuronal excitability determine the neuron's response to stimuli, its synchronization and resonance properties and, ultimately, the computations it performs in the brain. We investigated the dynamical mechanisms underlying the excitability type of dopamine (DA) neurons, using a conductance-based biophysical model, and its regulation by intrinsic and synaptic currents. Calibrating the model to reproduce low frequency tonic firing results in N-methyl-D-aspartate (NMDA) excitation balanced by γ-Aminobutyric acid (GABA)-mediated inhibition and leads to type I excitable behavior characterized by a continuous decrease in firing frequency in response to hyperpolarizing currents. Furthermore, we analyzed how excitability type of the DA neuron model is influenced by changes in the intrinsic current composition. A subthreshold sodium current is necessary for a continuous frequency decrease during application of a negative current, and the low-frequency "balanced" state during simultaneous activation of NMDA and GABA receptors. Blocking this current switches the neuron to type II characterized by the abrupt onset of repetitive firing. Enhancing the anomalous rectifier Ih current also switches the excitability to type II. Key characteristics of synaptic conductances that may be observed in vivo also change the type of excitability: a depolarized γ-Aminobutyric acid receptor (GABAR) reversal potential or co-activation of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs) leads to an abrupt frequency drop to zero, which is typical for type II excitability. Coactivation of N-methyl-D-aspartate receptors (NMDARs) together with AMPARs and GABARs shifts the type I/II boundary toward more hyperpolarized GABAR reversal potentials. To better understand how altering each of the aforementioned currents leads to changes in excitability profile of DA neuron, we provide a thorough dynamical analysis. Collectively, these results imply that type I excitability in dopamine neurons might be important for low firing rates and fine-tuning basal dopamine levels, while switching excitability to type II during NMDAR and AMPAR activation may facilitate a transient increase in dopamine concentration, as type II neurons are more amenable to synchronization by mutual excitation.

Implications of cellular models of dopamine neurons for disease

This review addresses the present state of single-cell models of the firing pattern of midbrain dopamine neurons and the insights that can be gained from these models into the underlying mechanisms for diseases such as Parkinson's, addiction, and schizophrenia. We will explain the analytical technique of separation of time scales and show how it can produce insights into mechanisms using simplified single-compartment models. We also use morphologically realistic multicompartmental models to address spatially heterogeneous aspects of neural signaling and neural metabolism. Separation of time scale analyses are applied to pacemaking, bursting, and depolarization block in dopamine neurons. Differences in subpopulations with respect to metabolic load are addressed using multicompartmental models.

Synergy of AMPA and NMDA Receptor Currents in Dopaminergic Neurons: A Modeling Study

Dopaminergic (DA) neurons display two modes of firing: low-frequency tonic and high-frequency bursts. The high frequency firing within the bursts is attributed to NMDA, but not AMPA receptor activation. In our models of the DA neuron, both biophysical and abstract, the NMDA receptor current can significantly increase their firing frequency, whereas the AMPA receptor current is not able to evoke high-frequency activity and usually suppresses firing. However, both currents are produced by glutamate receptors and, consequently, are often co-activated. Here we consider combined influence of AMPA and NMDA synaptic input in the models of the DA neuron. Different types of neuronal activity (resting state, low frequency, or high frequency firing) are observed depending on the conductance of the AMPAR and NMDAR currents. In two models, biophysical and reduced, we show that the firing frequency increases more effectively if both receptors are co-activated for certain parameter values. In particular, in the more quantitative biophysical model, the maximal frequency is 40% greater than that with NMDAR alone. The dynamical mechanism of such frequency growth is explained in the framework of phase space evolution using the reduced model. In short, both the AMPAR and NMDAR currents flatten the voltage nullcline, providing the frequency increase, whereas only NMDA prevents complete unfolding of the nullcline, providing robust firing. Thus, we confirm a major role of the NMDAR in generating high-frequency firing and conclude that AMPAR activation further significantly increases the frequency.

Differential Regulation of Action Potential Shape and Burst-Frequency Firing by BK and Kv2 Channels in Substantia Nigra Dopaminergic Neurons

Little is known about the voltage-dependent potassium currents underlying spike repolarization in midbrain dopaminergic neurons. Studying mouse substantia nigra pars compacta dopaminergic neurons both in brain slice and after acute dissociation, we found that BK calcium-activated potassium channels and Kv2 channels both make major contributions to the depolarization-activated potassium current. Inhibiting Kv2 or BK channels had very different effects on spike shape and evoked firing. Inhibiting Kv2 channels increased spike width and decreased the afterhyperpolarization, as expected for loss of an action potential-activated potassium conductance. BK inhibition also increased spike width but paradoxically increased the afterhyperpolarization. Kv2 channel inhibition steeply increased the slope of the frequency-current (f-I) relationship, whereas BK channel inhibition had little effect on the f-I slope or decreased it, sometimes resulting in slowed firing. Action potential clamp experiments showed that both BK and Kv2 current flow during spike repolarization but with very different kinetics, with Kv2 current activating later and deactivating more slowly. Further experiments revealed that inhibiting either BK or Kv2 alone leads to recruitment of additional current through the other channel type during the action potential as a consequence of changes in spike shape. Enhancement of slowly deactivating Kv2 current can account for the increased afterhyperpolarization produced by BK inhibition and likely underlies the very different effects on the f-I relationship. The cross-regulation of BK and Kv2 activation illustrates that the functional role of a channel cannot be defined in isolation but depends critically on the context of the other conductances in the cell.

SIGNIFICANCE STATEMENT

This work shows that BK calcium-activated potassium channels and Kv2 voltage-activated potassium channels both regulate action potentials in dopamine neurons of the substantia nigra pars compacta. Although both channel types participate in action potential repolarization about equally, they have contrasting and partially opposite effects in regulating neuronal firing at frequencies typical of bursting. Our analysis shows that this results from their different kinetic properties, with fast-activating BK channels serving to short-circuit activation of Kv2 channels, which tend to slow firing by producing a deep afterhyperpolarization. The cross-regulation of BK and Kv2 activation illustrates that the functional role of a channel cannot be defined in isolation but depends critically on the context of the other conductances in the cell.

A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block

Midbrain dopamine neurons exhibit a novel type of bursting that we call "inverted square wave bursting" when exposed to Ca(2+)-activated small conductance (SK) K(+) channel blockers in vitro. This type of bursting has three phases: hyperpolarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca(2+) channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.