A small fraction of strongly cooperative sodium channels boosts neuronal encoding of high frequencies

Investigating the role of axonal ion channel cooperativity in action potential dynamics: Studies on Hodgkin-Huxley's model

Voltage-gated ion channels play an important role in generating action potential in neurons. These ion channels are found to be in localized cluster form on the axonal membrane surface and behave cooperatively. However, in Hodgkin & Huxley's model of action potential the ion channels are considered to function independently. According to some recent reports, the activity of an ion channel is influenced by the neighboring ion channels' activities. We have modified the Hodgkin-Huxley's model based on our previous studies on cooperativity among ion channels. Computational analysis of the proposed model shows that the initiation of the action potential, amplitude and hyperpolarization are affected significantly by the cooperative interactions among the voltage-gated ion channels present on the axonal membrane surface. These results are qualitatively supported by the existing experimental facts.

Voltage sensors of a Na+ channel dissociate from the pore domain and form inter-channel dimers in the resting state

Understanding voltage-gated sodium (Nav) channels is significant since they generate action potential. Nav channels consist of a pore domain (PD) and a voltage sensor domain (VSD). All resolved Nav structures in different gating states have VSDs that tightly interact with PDs; however, it is unclear whether VSDs attach to PDs during gating under physiological conditions. Here, we reconstituted three different voltage-dependent NavAb, which is cloned from Arcobacter butzleri, into a lipid membrane and observed their structural dynamics by high-speed atomic force microscopy on a sub-second timescale in the steady state. Surprisingly, VSDs dissociated from PDs in the mutant in the resting state and further dimerized to form cross-links between channels. This dimerization would occur at a realistic channel density, offering a potential explanation for the facilitation of positive cooperativity of channel activity in the rising phase of the action potential.

High-resolution volumetric imaging constrains compartmental models to explore synaptic integration and temporal processing by cochlear nucleus globular bushy cells

Globular bushy cells (GBCs) of the cochlear nucleus play central roles in the temporal processing of sound. Despite investigation over many decades, fundamental questions remain about their dendrite structure, afferent innervation, and integration of synaptic inputs. Here, we use volume electron microscopy (EM) of the mouse cochlear nucleus to construct synaptic maps that precisely specify convergence ratios and synaptic weights for auditory nerve innervation and accurate surface areas of all postsynaptic compartments. Detailed biophysically based compartmental models can help develop hypotheses regarding how GBCs integrate inputs to yield their recorded responses to sound. We established a pipeline to export a precise reconstruction of auditory nerve axons and their endbulb terminals together with high-resolution dendrite, soma, and axon reconstructions into biophysically detailed compartmental models that could be activated by a standard cochlear transduction model. With these constraints, the models predict auditory nerve input profiles whereby all endbulbs onto a GBC are subthreshold (coincidence detection mode), or one or two inputs are suprathreshold (mixed mode). The models also predict the relative importance of dendrite geometry, soma size, and axon initial segment length in setting action potential threshold and generating heterogeneity in sound-evoked responses, and thereby propose mechanisms by which GBCs may homeostatically adjust their excitability. Volume EM also reveals new dendritic structures and dendrites that lack innervation. This framework defines a pathway from subcellular morphology to synaptic connectivity, and facilitates investigation into the roles of specific cellular features in sound encoding. We also clarify the need for new experimental measurements to provide missing cellular parameters, and predict responses to sound for further in vivo studies, thereby serving as a template for investigation of other neuron classes.

Structure and dynamics that specialize neurons for high-frequency coincidence detection in the barn owl nucleus laminaris

A principal cue for sound source localization is the difference in arrival times of sounds at an animal's two ears (interaural time difference, ITD). Neurons that process ITDs are specialized to compare the timing of inputs with submillisecond precision. In the barn owl, ITD processing begins in the nucleus laminaris (NL) region of the auditory brain stem. Remarkably, NL neurons are sensitive to ITDs in high-frequency sounds (kilohertz-range). This contrasts with ITD-based sound localization in analogous regions in mammals where ITD sensitivity is typically restricted to lower-frequency sounds. Guided by previous experiments and modeling studies of tone-evoked responses of NL neurons, we propose NL neurons achieve high-frequency ITD sensitivity if they respond selectively to the small-amplitude, high-frequency oscillations in their inputs, and remain relatively non-responsive to mean input level. We use a biophysically based model to study the effects of soma-axon coupling on dynamics and function in NL neurons. First, we show that electrical separation of the soma from the axon region in the neuron enhances high-frequency ITD sensitivity. This soma-axon coupling configuration promotes linear subthreshold dynamics and rapid spike initiation, making the model more responsive to input oscillations, rather than mean input level. Second, we provide new evidence for the essential role of phasic dynamics for high-frequency neural coincidence detection. Transforming our model to the phasic firing mode further tunes the model to respond selectively to the oscillating inputs that carry ITD information. Similar structural and dynamical mechanisms specialize mammalian auditory brain stem neurons for ITD sensitivity, and thus, our work identifies common principles of ITD processing and neural coincidence detection across species and for sounds at widely different frequencies.

Ultrafast population coding and axo-somatic compartmentalization

Populations of cortical neurons respond to common input within a millisecond. Morphological features and active ion channel properties were suggested to contribute to this astonishing processing speed. Here we report an exhaustive study of ultrafast population coding for varying axon initial segment (AIS) location, soma size, and axonal current properties. In particular, we studied their impact on two experimentally observed features 1) precise action potential timing, manifested in a wide-bandwidth dynamic gain, and 2) high-frequency boost under slowly fluctuating correlated input. While the density of axonal channels and their distance from the soma had a very small impact on bandwidth, it could be moderately improved by increasing soma size. When the voltage sensitivity of axonal currents was increased we observed ultrafast coding and high-frequency boost. We conclude that these computationally relevant features are strongly dependent on axonal ion channels’ voltage sensitivity, but not their number or exact location. We point out that ion channel properties, unlike dendrite size, can undergo rapid physiological modification, suggesting that the temporal accuracy of neuronal population encoding could be dynamically regulated. Our results are in line with recent experimental findings in AIS pathologies and establish a framework to study structure-function relations in AIS molecular design.

In large nervous systems, a signal often diverges to hundreds or thousands of neurons. This population’s spike rate can track changes in this common input for frequencies up to several hundred Hertz. This ultrafast population response is experimentally well established and critically impacts cortical information processing. Its underlying biophysical determinants, however, are not understood. Experiments suggest that the ion channels at the axon initial segment strongly contribute to the ultrafast response, but recent theoretical studies emphasize the importance of neuron morphology and the resulting resistive coupling between axon and somato-dendritic compartments. We provide an exhaustive analysis of the population response of a simplified multi-compartment model. We vary the axo-somatic interaction and also active axonal properties and compare models at equivalent working points, avoiding bias. This approach provides a guideline for future experimental and theoretical studies. In this framework, the population response is closely associated with the AP generation speed at the AP initiation site, which is mostly determined by axonal ion channel voltage sensitivity. The resistive axo-somatic coupling has an additional modulatory influence. These insights are expected to hold for encoding mechanisms of more sophisticated models, suggesting that physiological changes to axonal ion channels could modulate the population response rapidly.

Mechanisms and Physiological Implications of Cooperative Gating of Ion Channels Clusters

Ion channels play a central role in the regulation of nearly every cellular process. Dating back to the classic 1952 Hodgkin-Huxley model of the generation of the action potential, ion channels have always been thought of as independent agents. A myriad of recent experimental findings exploiting advances in electrophysiology, structural biology, and imaging techniques, however, have posed a serious challenge to this long-held axiom as several classes of ion channels appear to open and close in a coordinated, cooperative manner. Ion channel cooperativity ranges from variable-sized oligomeric cooperative gating in voltage-gated, dihydropyridine-sensitive Ca_v1.2 and Ca_v1.3 channels to obligatory dimeric assembly and gating of voltage-gated Na_v1.5 channels. Potassium channels, transient receptor potential channels, hyperpolarization cyclic nucleotide-activated channels, ryanodine receptors (RyRs), and inositol trisphosphate receptors (IP₃Rs) have also been shown to gate cooperatively. The implications of cooperative gating of these ion channels range from fine tuning excitation-contraction coupling in muscle cells to regulating cardiac function and vascular tone, to modulation of action potential and conduction velocity in neurons and cardiac cells, and to control of pace-making activity in the heart. In this review, we discuss the mechanisms leading to cooperative gating of ion channels, their physiological consequences and how alterations in cooperative gating of ion channels may induce a range of clinically significant pathologies.

New methods for quantifying rapidity of action potential onset differentiate neuron types

Two new methods for quantifying the rapidity of action potential onset have lower relative standard deviations and better distinguish neuron cell types than current methods. Action potentials (APs) in most central mammalian neurons exhibit sharp onset dynamics. The main views explaining such an abrupt onset differ. Some studies suggest sharp onsets reflect cooperative sodium channels activation, while others suggest they reflect AP backpropagation from the axon initial segment. However, AP onset rapidity is defined subjectively in these studies, often using the slope at an arbitrary value on the phase plot. Thus, we proposed more systematic methods using the membrane potential's second-time derivative ([Formula: see text]) peak width. Here, the AP rapidity was measured for four different cortical and hippocampal neuron types using four quantification methods: the inverse of full-width at the half maximum of the [Formula: see text] peak (IFWd2), the inverse of half-width at the half maximum of the [Formula: see text] peak (IHWd2), the phase plot slope, and the error ratio method. The IFWd2 and IHWd2 methods show the smallest variation among neurons of the same type. Furthermore, the AP rapidity, using the [Formula: see text] peak width methods, significantly differentiates between different types of neurons, indicating that AP rapidity can be used to classify neuron types. The AP rapidity measured using the IFWd2 method was able to differentiate between all four neuron types analyzed. Therefore, the [Formula: see text] peak width methods provide another sensitive tool to investigate the mechanisms impacting the AP onset dynamics.

Collective Dynamics of Ion Channels on Bilayer Lipid Membranes

Ion channels self-organize on cellular and organelle membranes as clusters and mutually modulate their gating behavior. It has been reported that the efficient information transfer is achieved by cooperative clustering of ion channels. To address the origin and nature of collective dynamics in ion channel clusters, a statistical mechanical model, namely, the Zimm-Bragg-type model in two dimensions with unequal weight distribution in channel-channel interactions, has been proposed. Nearest neighbor interaction along with next-nearest neighbor interaction has been considered, assuming symmetric spatial organization. The multichannel bilayer electrophysiology recordings of the voltage-dependent anion channel (VDAC) from rat brain mitochondria have been analyzed in order to test and further extend the model. The model successfully describes the multichannel gating behavior and self-organization of the VDAC cluster.

Clusters of cooperative ion channels enable a membrane-potential-based mechanism for short-term memory

Across biological systems, cooperativity between proteins enables fast actions, supra-linear responses, and long-lasting molecular switches. In the nervous system, however, the function of cooperative interactions between voltage-dependent ionic channels remains largely unknown. Based on mathematical modeling, we here demonstrate that clusters of strongly cooperative ion channels can plausibly form bistable conductances. Consequently, clusters are permanently switched on by neuronal spiking, switched off by strong hyperpolarization, and remain in their state for seconds after stimulation. The resulting short-term memory of the membrane potential allows to generate persistent firing when clusters of cooperative channels are present together with non-cooperative spike-generating conductances. Dynamic clamp experiments in rodent cortical neurons confirm that channel cooperativity can robustly induce graded persistent activity - a single-cell based, multistable mnemonic firing mode experimentally observed in several brain regions. We therefore propose that ion channel cooperativity constitutes an efficient cell-intrinsic implementation for short-term memories at the voltage level.

Axonal Computations

Axons functionally link the somato-dendritic compartment to synaptic terminals. Structurally and functionally diverse, they accomplish a central role in determining the delays and reliability with which neuronal ensembles communicate. By combining their active and passive biophysical properties, they ensure a plethora of physiological computations. In this review, we revisit the biophysics of generation and propagation of electrical signals in the axon and their dynamics. We further place the computational abilities of axons in the context of intracellular and intercellular coupling. We discuss how, by means of sophisticated biophysical mechanisms, axons expand the repertoire of axonal computation, and thereby, of neural computation.

An axon initial segment is required for temporal precision in action potential encoding by neuronal populations

Central neurons initiate action potentials (APs) in the axon initial segment (AIS), a compartment characterized by a high concentration of voltage-dependent ion channels and specialized cytoskeletal anchoring proteins arranged in a regular nanoscale pattern. Although the AIS was a key evolutionary innovation in neurons, the functional benefits it confers are not clear. Using a mutation of the AIS cytoskeletal protein βIV-spectrin, we here establish an in vitro model of neurons with a perturbed AIS architecture that retains nanoscale order but loses the ability to maintain a high Na_V density. Combining experiments and simulations, we show that a high Na_V density in the AIS is not required for axonal AP initiation; it is, however, crucial for a high bandwidth of information encoding and AP timing precision. Our results provide the first experimental demonstration of axonal AP initiation without high axonal channel density and suggest that increasing the bandwidth of the neuronal code and, hence, the computational efficiency of network function, was a major benefit of the evolution of the AIS.

Reduced cooperativity of voltage-gated sodium channels in the hippocampal interneurons of an aged mouse model of Alzheimer's disease

Beta amyloid (A[Formula: see text] ) associated with Alzheimer's disease (AD) leads to abnormal behavior in inhibitory neurons, resulting in hyperactive neuronal networks, epileptiform behavior, disrupted gamma rhythms, and aberrant synaptic plasticity. Previously, we used a dual modeling-experimental approach to explain several observations, including failure to reliably produce action potentials (APs), smaller AP amplitudes, higher resting membrane potential, and higher membrane depolarization in response to a range of stimuli in hippocampal inhibitory neurons from 12- to 16-month-old female AP Pswe/PSEN1DeltaE9 (APdE9) AD mice as compared to age-matched non-transgenic (NTG) mice. Our experimental results also showed that AP initiation in interneurons from APdE9 mice are significantly different from that of NTG mice. APs in interneurons from NTG mice are characterized by abrupt onset and an upstroke that is much steeper and occurs with larger variability as compared to cells from APdE9 mice. The phase plot (the rate of change of membrane potential versus the instantaneous membrane potential) of APs produced by interneurons from APdE9 mice shows a biphasic behavior, whereas that from NTG mice shows a monophasic behavior. Here we show that using the classic Hodgkin-Huxley (HH) formalism for the gating of voltage-gated sodium channels (VGSCs) in a single-compartment neuron, we cannot reproduce these features, and a model that takes into account a cooperative activation of VGSCs is needed. We also argue that considering a realistic multi-compartment neuron where the kinetics of VGSC is modeled by HH formalism, as done in the past, would not explain our observations when APs from both NTG and APdE9 mice are considered simultaneously. We further show that VGSCs in interneurons from APdE9 mice exhibit significantly lower cooperativity in their activation as compared to those from NTG mice.

A biophysical modelling platform of the cochlear nucleus and other auditory circuits: From channels to networks

Models of the auditory brainstem have been an invaluable tool for testing hypotheses about auditory information processing and for highlighting the most important gaps in the experimental literature. Due to the complexity of the auditory brainstem, and indeed most brain circuits, the dynamic behavior of the system may be difficult to predict without a detailed, biologically realistic computational model. Despite the sensitivity of models to their exact construction and parameters, most prior models of the cochlear nucleus have incorporated only a small subset of the known biological properties. This confounds the interpretation of modelling results and also limits the potential future uses of these models, which require a large effort to develop. To address these issues, we have developed a general purpose, biophysically detailed model of the cochlear nucleus for use both in testing hypotheses about cochlear nucleus function and also as an input to models of downstream auditory nuclei. The model implements conductance-based Hodgkin-Huxley representations of cells using a Python-based interface to the NEURON simulator. Our model incorporates most of the quantitatively characterized intrinsic cell properties, synaptic properties, and connectivity available in the literature, and also aims to reproduce the known response properties of the canonical cochlear nucleus cell types. Although we currently lack the empirical data to completely constrain this model, our intent is for the model to continue to incorporate new experimental results as they become available.

The basis of sharp spike onset in standard biophysical models

In most vertebrate neurons, spikes initiate in the axonal initial segment (AIS). When recorded in the soma, they have a surprisingly sharp onset, as if sodium (Na) channels opened abruptly. The main view stipulates that spikes initiate in a conventional manner at the distal end of the AIS, then progressively sharpen as they backpropagate to the soma. We examined the biophysical models used to substantiate this view, and we found that spikes do not initiate through a local axonal current loop that propagates along the axon, but through a global current loop encompassing the AIS and soma, which forms an electrical dipole. Therefore, the phenomenon is not adequately modeled as the backpropagation of an electrical wave along the axon, since the wavelength would be as large as the entire system. Instead, in these models, we found that spike initiation rather follows the critical resistive coupling model proposed recently, where the Na current entering the AIS is matched by the axial resistive current flowing to the soma. Besides demonstrating it by examining the balance of currents at spike initiation, we show that the observed increase in spike sharpness along the axon is artifactual and disappears when an appropriate measure of rapidness is used; instead, somatic onset rapidness can be predicted from spike shape at initiation site. Finally, we reproduce the phenomenon in a two-compartment model, showing that it does not rely on propagation. In these models, the sharp onset of somatic spikes is therefore not an artifact of observing spikes at the incorrect location, but rather the signature that spikes are initiated through a global soma-AIS current loop forming an electrical dipole.

Encoding of High Frequencies Improves with Maturation of Action Potential Generation in Cultured Neocortical Neurons

The ability of neocortical neurons to detect and encode rapid changes at their inputs is crucial for basic neuronal computations, such as coincidence detection, precise synchronization of activity and spike-timing dependent plasticity. Indeed, populations of cortical neurons can respond to subtle changes of the input very fast, on a millisecond time scale. Theoretical studies and model simulations linked the encoding abilities of neuronal populations to the fast onset dynamics of action potentials (APs). Experimental results support this idea, however mechanisms of fast onset of APs in cortical neurons remain elusive. Studies in neuronal cultures, that are allowing for accurate control over conditions of growth and microenvironment during the development of neurons and provide better access to the spike initiation zone, may help to shed light on mechanisms of AP generation and encoding. Here we characterize properties of AP encoding in neocortical neurons grown for 11-25 days in culture. We show that encoding of high frequencies improves upon culture maturation, which is accompanied by the development of passive electrophysiological properties and AP generation. The onset of APs becomes faster with culture maturation. Statistical analysis using correlations and linear model approaches identified the onset dynamics of APs as a major predictor of age-dependent changes of encoding. Encoding of high frequencies strongly correlated also with the input resistance of neurons. Finally, we show that maturation of encoding properties of neurons in cultures is similar to the maturation of encoding in neurons studied in slices. These results show that maturation of AP generators and encoding is, to a large extent, determined genetically and takes place even without normal micro-environment and activity of the whole brain in vivo. This establishes neuronal cultures as a valid experimental model for studying mechanisms of AP generation and encoding, and their maturation.

Analyzing and Modeling the Dysfunction of Inhibitory Neurons in Alzheimer's Disease

Alzheimer's disease (AD) is characterized by the abnormal proteolytic processing of amyloid precursor protein, resulting in increased production of a self-aggregating form of beta amyloid (Aβ). Several lines of work on AD patients and transgenic mice with high Aβ levels exhibit altered rhythmicity, aberrant neuronal network activity and hyperexcitability reflected in clusters of hyperactive neurons, and spontaneous epileptic activity. Recent studies highlight that abnormal accumulation of Aβ changes intrinsic properties of inhibitory neurons, which is one of the main reasons underlying the impaired network activity. However, specific cellular mechanisms leading to interneuronal dysfunction are not completely understood. Using extended Hodgkin-Huxley (HH) formalism in conjunction with patch-clamp experiments, we investigate the mechanisms leading to the impaired activity of interneurons. Our detailed analysis indicates that increased Na+ leak explains several observations in inhibitory neurons, including their failure to reliably produce action potentials, smaller action potential amplitude, increased resting membrane potential, and higher membrane depolarization in response to a range of stimuli in a model of APPSWE/PSEN1DeltaE9 (APdE9) AD mice as compared to age-matched control mice. While increasing the conductance of hyperpolarization activated cyclic nucleotide-gated (HCN) ion channel could account for most of the observations, the extent of increase required to reproduce these observations render such changes unrealistic. Furthermore, increasing the conductance of HCN does not account for the observed changes in depolarizability of interneurons from APdE9 mice as compared to those from NTG mice. None of the other pathways tested could lead to all observations about interneuronal dysfunction. Thus we conclude that upregulated sodium leak is the most likely source of impaired interneuronal function.

Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics

The response of a neuronal population over a space of inputs depends on the intrinsic properties of its constituent neurons. Two main modes of single neuron dynamics–integration and resonance–have been distinguished. While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects. To understand better how a resonator’s frequency preference emerges from its intrinsic dynamics and contributes to its local area’s population firing rate dynamics, we analyze the dynamic gain of an analytically solvable two-degree of freedom neuron model. In the Fokker-Planck approach, the dynamic gain is intractable. The alternative Gauss-Rice approach lifts the resetting of the voltage after a spike. This allows us to derive a complete expression for the dynamic gain of a resonator neuron model in terms of a cascade of filters on the input. We find six distinct response types and use them to fully characterize the routes to resonance across all values of the relevant timescales. We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak. We determine the parameter regions for the existence of an intrinsic frequency and for subthreshold and spiking resonance, finding all possible intersections of the three. The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

Dynamic gain, the amount by which features at specific frequencies in the input to a neuron are amplified or attenuated in its output spiking, is fundamental for the encoding of information by neural populations. Most studies of dynamic gain have focused on neurons without intrinsic degrees of freedom exhibiting integrator-type subthreshold dynamics. Many neuron types in the brain, however, exhibit complex subthreshold dynamics such as resonance, found for instance in cortical interneurons, stellate cells, and mitral cells. A resonator neuron has at least two degrees of freedom for which the classical Fokker-Planck approach to calculating the dynamic gain is largely intractable. Here, we lift the voltage-reset rule after a spike, allowing us to derive a complete expression of the dynamic gain of a resonator neuron model. We find the gain can exhibit only six shapes. The resonant ones have peaks that become large due to intrinsic adaptation and become sharp due to an intrinsic frequency. A resonance can nevertheless result from either property. The analysis presented here helps explain how intrinsic neuron dynamics shape population-level response properties and provides a powerful tool for developing theories of inter-neuron correlations and dynamic responses of neural populations.

The interplay between cooperativity and diversity in model threshold ensembles

The interplay between cooperativity and diversity is crucial for biological ensembles because single molecule experiments show a significant degree of heterogeneity and also for artificial nanostructures because of the high individual variability characteristic of nanoscale units. We study the cross-effects between cooperativity and diversity in model threshold ensembles composed of individually different units that show a cooperative behaviour. The units are modelled as statistical distributions of parameters (the individual threshold potentials here) characterized by central and width distribution values. The simulations show that the interplay between cooperativity and diversity results in ensemble-averaged responses of interest for the understanding of electrical transduction in cell membranes, the experimental characterization of heterogeneous groups of biomolecules and the development of biologically inspired engineering designs with individually different building blocks.

Neuronal morphology generates high-frequency firing resonance

The attenuation of neuronal voltage responses to high-frequency current inputs by the membrane capacitance is believed to limit single-cell bandwidth. However, neuronal populations subject to stochastic fluctuations can follow inputs beyond this limit. We investigated this apparent paradox theoretically and experimentally using Purkinje cells in the cerebellum, a motor structure that benefits from rapid information transfer. We analyzed the modulation of firing in response to the somatic injection of sinusoidal currents. Computational modeling suggested that, instead of decreasing with frequency, modulation amplitude can increase up to high frequencies because of cellular morphology. Electrophysiological measurements in adult rat slices confirmed this prediction and displayed a marked resonance at 200 Hz. We elucidated the underlying mechanism, showing that the two-compartment morphology of the Purkinje cell, interacting with a simple spiking mechanism and dendritic fluctuations, is sufficient to create high-frequency signal amplification. This mechanism, which we term morphology-induced resonance, is selective for somatic inputs, which in the Purkinje cell are exclusively inhibitory. The resonance sensitizes Purkinje cells in the frequency range of population oscillations observed in vivo.

Action potential initiation in a multi-compartmental model with cooperatively gating Na channels in the axon initial segment

Somatic action potentials (AP) of cortical pyramidal neurons have characteristically high onset-rapidness. The onset of the AP waveform is an indirect measure for the ability of a neuron to respond to temporally fast-changing stimuli. Theoretical studies on the pyramidal neuron response usually involves a canonical Hodgkin-Huxley (HH) type ion channel gating model, which assumes statistically independent gating of each individual channel. However, cooperative activity of ion channels are observed for various cell types, meaning that the activity (e.g. opening) of one channel triggers the activity (e.g. opening) of a certain fraction of its neighbors and hence, these groups of channels behave as a unit. In this study, we describe a multi-compartmental conductance-based model with cooperatively gating voltage-gated Na channels in the axon initial segment. Our model successfully reproduced the somatic sharp AP onsets of cortical pyramidal neurons. The onset latencies from the initiation site to the soma and the conduction velocities were also in agreement with the previous experimental studies.

What is the most realistic single-compartment model of spike initiation?

A large variety of neuron models are used in theoretical and computational neuroscience, and among these, single-compartment models are a popular kind. These models do not explicitly include the dendrites or the axon, and range from the Hodgkin-Huxley (HH) model to various flavors of integrate-and-fire (IF) models. The main classes of models differ in the way spikes are initiated. Which one is the most realistic? Starting with some general epistemological considerations, I show that the notion of realism comes in two dimensions: empirical content (the sort of predictions that a model can produce) and empirical accuracy (whether these predictions are correct). I then examine the realism of the main classes of single-compartment models along these two dimensions, in light of recent experimental evidence.

Cortical Specializations Underlying Fast Computations

The time course of behaviorally relevant environmental events sets temporal constraints on neuronal processing. How does the mammalian brain make use of the increasingly complex networks of the neocortex, while making decisions and executing behavioral reactions within a reasonable time? The key parameter determining the speed of computations in neuronal networks is a time interval that neuronal ensembles need to process changes at their input and communicate results of this processing to downstream neurons. Theoretical analysis identified basic requirements for fast processing: use of neuronal populations for encoding, background activity, and fast onset dynamics of action potentials in neurons. Experimental evidence shows that populations of neocortical neurons fulfil these requirements. Indeed, they can change firing rate in response to input perturbations very quickly, within 1 to 3 ms, and encode high-frequency components of the input by phase-locking their spiking to frequencies up to 300 to 1000 Hz. This implies that time unit of computations by cortical ensembles is only few, 1 to 3 ms, which is considerably faster than the membrane time constant of individual neurons. The ability of cortical neuronal ensembles to communicate on a millisecond time scale allows for complex, multiple-step processing and precise coordination of neuronal activity in parallel processing streams, while keeping the speed of behavioral reactions within environmentally set temporal constraints.

Utrophin regulates modal gating of mechanosensitive ion channels in dystrophic skeletal muscle

Dystrophin is a large, submembrane cytoskeletal protein, absence of which causes Duchenne muscular dystrophy. Utrophin is a dystrophin homologue found in both muscle and brain whose physiological function is unknown. Recordings of single-channel activity were made from membrane patches on skeletal muscle from mdx, mdx/utrn(+/-) heterozygotes and mdx/utrn(-/-) double knockout mice to investigate the role of these cytoskeletal proteins in mechanosensitive (MS) channel gating. We find complex, gene dose-dependent effects of utrophin depletion in dystrophin-deficient mdx muscle: (1) increased MS channel open probability, (2) a shift of MS channel gating to larger pressures, (3) appearance of modal gating of MS channels and small conductance channels and (4) expression of large conductance MS channels. We suggest a physical model in which utrophin acts as a scaffolding protein that stabilizes lipid microdomains and clusters MS channel subunits. Depletion of utrophin disrupts domain composition in a manner that favours open channel area expansion, as well as allowing diffusion and aggregation of additional MS channel subunits.

Dynamical models of cortical circuits

Cortical neurons operate within recurrent neuronal circuits. Dissecting their operation is key to understanding information processing in the cortex and requires transparent and adequate dynamical models of circuit function. Convergent evidence from experimental and theoretical studies indicates that strong feedback inhibition shapes the operating regime of cortical circuits. For circuits operating in inhibition-dominated regimes, mathematical and computational studies over the past several years achieved substantial advances in understanding response modulation and heterogeneity, emergent stimulus selectivity, inter-neuron correlations, and microstate dynamics. The latter indicate a surprisingly strong dependence of the collective circuit dynamics on the features of single neuron action potential generation. New approaches are needed to definitely characterize the cortical operating regime.

A simple Markov model of sodium channels with a dynamic threshold

Characteristics of action potential generation are important to understanding brain functioning and, thus, must be understood and modeled. It is still an open question what model can describe concurrently the phenomena of sharp spike shape, the spike threshold variability, and the divisive effect of shunting on the gain of frequency-current dependence. We reproduced these three effects experimentally by patch-clamp recordings in cortical slices, but we failed to simulate them by any of 11 known neuron models, including one- and multi-compartment, with Hodgkin-Huxley and Markov equation-based sodium channel approximations, and those taking into account sodium channel subtype heterogeneity. Basing on our voltage-clamp data characterizing the dependence of sodium channel activation threshold on history of depolarization, we propose a 3-state Markov model with a closed-to-open state transition threshold dependent on slow inactivation. This model reproduces the all three phenomena. As a reduction of this model, a leaky integrate-and-fire model with a dynamic threshold also shows the effect of gain reduction by shunt. These results argue for the mechanism of gain reduction through threshold dynamics determined by the slow inactivation of sodium channels.

Sharpness of spike initiation in neurons explained by compartmentalization

In cortical neurons, spikes are initiated in the axon initial segment. Seen at the soma, they appear surprisingly sharp. A standard explanation is that the current coming from the axon becomes sharp as the spike is actively backpropagated to the soma. However, sharp initiation of spikes is also seen in the input–output properties of neurons, and not only in the somatic shape of spikes; for example, cortical neurons can transmit high frequency signals. An alternative hypothesis is that Na channels cooperate, but it is not currently supported by direct experimental evidence. I propose a simple explanation based on the compartmentalization of spike initiation. When Na channels are placed in the axon, the soma acts as a current sink for the Na current. I show that there is a critical distance to the soma above which an instability occurs, so that Na channels open abruptly rather than gradually as a function of somatic voltage.

Spike initiation determines how the combined inputs to a neuron are converted to an output. Since the pioneering work of Hodgkin and Huxley, it is known that spikes are generated by the opening of sodium channels with depolarization. According to this standard theory, these channels should open gradually when the membrane potential increases, but spikes measured at the soma appear to suddenly rise from rest. This apparent contradiction has triggered a controversy about the origin of spike “sharpness.” This study shows with biophysical modelling that if sodium channels are placed in the axon rather than in the soma, they open all at once when the somatic membrane potential exceeds a critical value. This work explains the sharpness of spike initiation and provides another demonstration that morphology plays a critical role in neural function.

S6 peptide derived from KvAP channel shows cooperativity in gating on bilayer lipid membrane

Collective behavior of S6 peptide channels derived from KvAP (a bacterial potassium channel) incorporated in lipid bilayer membrane, has been investigated at various applied potentials through multi-channel electrophysiological experiments. The current versus time traces at any particular membrane potential show clear steps for sequential opening of the multi-channels. The minimum current (representing one-channel current) was found out from the amplitude histograms. Accordingly, the number of open channels corresponding to a particular open state was calculated. It was observed that the above-mentioned one channel current is higher than the corresponding single-channel current at most of the applied membrane potentials. Moreover, the difference between the single and one channel conductances is a nonlinear function of the membrane potential. We conclude that the S6 multi-channels show co-operative gating. Voltage relaxation studies support the above-mentioned conclusion.

Fast computations in cortical ensembles require rapid initiation of action potentials

The abilities of neuronal populations to encode rapidly varying stimuli and respond quickly to abrupt input changes are crucial for basic neuronal computations, such as coincidence detection, grouping by synchrony, and spike-timing-dependent plasticity, as well as for the processing speed of neuronal networks. Theoretical analyses have linked these abilities to the fast-onset dynamics of action potentials (APs). Using a combination of whole-cell recordings from rat neocortical neurons and computer simulations, we provide the first experimental evidence for this conjecture and prove its validity for the case of distal AP initiation in the axon initial segment (AIS), typical for cortical neurons. Neocortical neurons with fast-onset APs in the soma can phase-lock their population firing to signal frequencies up to ∼300-400 Hz and respond within 1-2 ms to subtle changes of input current. The ability to encode high frequencies and response speed were dramatically reduced when AP onset was slowed by experimental manipulations or was intrinsically slow due to immature AP generation mechanisms. Multicompartment conductance-based models reproducing the initiation of spikes in the AIS could encode high frequencies only if AP onset was fast at the initiation site (e.g., attributable to cooperative gating of a fraction of sodium channels) but not when fast onset of somatic AP was produced solely by backpropagation. We conclude that fast-onset dynamics is a genuine property of cortical AP generators. It enables fast computations in cortical circuits that are rich in recurrent connections both within each region and across the hierarchy of areas.