Sharpness of spike initiation in neurons explained by compartmentalization

Impact on backpropagation of the spatial heterogeneity of sodium channel kinetics in the axon initial segment

In a variety of neurons, action potentials (APs) initiate at the proximal axon, within a region called the axon initial segment (AIS), which has a high density of voltage-gated sodium channels (NaVs) on its membrane. In pyramidal neurons, the proximal AIS has been reported to exhibit a higher proportion of NaVs with gating properties that are "right-shifted" to more depolarized voltages, compared to the distal AIS. Further, recent experiments have revealed that as neurons develop, the spatial distribution of NaV subtypes along the AIS can change substantially, suggesting that neurons tune their excitability by modifying said distribution. When neurons are stimulated axonally, computational modelling has shown that this spatial separation of gating properties in the AIS enhances the backpropagation of APs into the dendrites. In contrast, in the more natural scenario of somatic stimulation, our simulations show that the same distribution can impede backpropagation, suggesting that the choice of orthodromic versus antidromic stimulation can bias or even invert experimental findings regarding the role of NaV subtypes in the AIS. We implemented a range of hypothetical NaV distributions in the AIS of three multicompartmental pyramidal cell models and investigated the precise kinetic mechanisms underlying such effects, as the spatial distribution of NaV subtypes is varied. With axonal stimulation, proximal NaV availability dominates, such that concentrating right-shifted NaVs in the proximal AIS promotes backpropagation. However, with somatic stimulation, the models are insensitive to availability kinetics. Instead, the higher activation threshold of right-shifted NaVs in the AIS impedes backpropagation. Therefore, recently observed developmental changes to the spatial separation and relative proportions of NaV1.2 and NaV1.6 in the AIS differentially impact activation and availability. The observed effects on backpropagation, and potentially learning via its putative role in synaptic plasticity (e.g. through spike-timing-dependent plasticity), are opposite for orthodromic versus antidromic stimulation, which should inform hypotheses about the impact of the developmentally regulated subcellular localization of these NaV subtypes.

Cell anatomy and network input explain differences within but not between leech touch cells at two different locations

Mechanosensory cells in the leech share several common features with mechanoreceptors in the human glabrous skin. Previous studies showed that the six T (touch) cells in each body segment of the leech are highly variable in their responses to somatic current injection and change their excitability over time. Here, we investigate three potential reasons for this variability in excitability by comparing the responses of T cells at two soma locations (T2 and T3): (1) Differential effects of time-dependent changes in excitability, (2) divergent synaptic input from the network, and (3) different anatomical structures. These hypotheses were explored with a combination of electrophysiological double recordings, 3D reconstruction of neurobiotin-filled cells, and compartmental model simulations. Current injection triggered significantly more spikes with shorter latency and larger amplitudes in cells at soma location T2 than at T3. During longer recordings, cells at both locations increased their excitability over time in the same way. T2 and T3 cells received the same amount of synaptic input from the unstimulated network, and the polysynaptic connections between both T cells were mutually symmetric. However, we found a striking anatomical difference: While in our data set all T2 cells innervated two roots connecting the ganglion with the skin, 50% of the T3 cells had only one root process. The sub-sample of T3 cells with one root process was significantly less excitable than the T3 cells with two root processes and the T2 cells. To test if the additional root process causes higher excitability, we simulated the responses of 3D reconstructed cells of both anatomies with detailed multi-compartment models. The anatomical subtypes do not differ in excitability when identical biophysical parameters and a homogeneous channel distribution are assumed. Hence, all three hypotheses may contribute to the highly variable T cell responses, but none of them is the only factor accounting for the observed systematic difference in excitability between cells at T2 vs. T3 soma location. Therefore, future patch clamp and modeling studies are needed to analyze how biophysical properties and spatial distribution of ion channels on the cell surface contribute to the variability and systematic differences of electrophysiological phenotypes.

Compartment-specific dendritic information processing in striatal cholinergic interneurons is reconfigured by peptide neuromodulation

Cholinergic interneurons are central hubs of the striatal neuronal network, controlling information processing in a behavioral-state-dependent manner. It remains unknown, however, how such state transitions influence the integrative properties of these neurons. To address this, we made simultaneous somato-dendritic recordings from identified rodent cholinergic interneurons, revealing that action potentials are initiated at dendritic sites because of a dendritic axonal origin. Functionally, this anatomical arrangement ensured that the action potential initiation threshold was lowest at axon-bearing dendritic sites, a privilege efficacy powerfully accentuated at the hyperpolarized membrane potentials achieved in cholinergic interneurons following salient behavioral stimuli. Experimental analysis revealed the voltage-dependent attenuation of the efficacy of non-axon-bearing dendritic excitatory input was mediated by the recruitment of dendritic potassium channels, a regulatory mechanism that, in turn, was controlled by the pharmacological activation of neurokinin receptors. Together, these results indicate that the neuropeptide microenvironment dynamically controls state- and compartment-dependent dendritic information processing in striatal cholinergic interneurons.

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.

Simulations of cortical networks using spatially extended conductance-based neuronal models

The Hodgkin-Huxley model of action potential generation and propagation, published in the Journal of Physiology in 1952, initiated the field of biophysically detailed computational modelling in neuroscience, which has expanded to encompass a variety of species and components of the nervous system. Here we review the developments in this area with a focus on efforts in the community towards modelling the mammalian neocortex using spatially extended conductance-based neuronal models. The Hodgkin-Huxley formalism and related foundational contributions, such as Rall's cable theory, remain widely used in these efforts to the current day. We argue that at present the field is undergoing a qualitative change due to new very rich datasets describing the composition, connectivity and functional activity of cortical circuits, which are being integrated systematically into large-scale network models. This trend, combined with the accelerating development of convenient software tools supporting such complex modelling projects, is giving rise to highly detailed models of the cortex that are extensively constrained by the data, enabling computational investigation of a multitude of questions about cortical structure and function.

Alterations of the axon initial segment in multiple sclerosis grey matter

Grey matter damage has been established as a key contributor to disability progression in multiple sclerosis. Aside from neuronal loss and axonal transections, which predominate in cortical demyelinated lesions, synaptic alterations have been detected in both demyelinated plaques and normal-appearing grey matter, resulting in functional neuronal damage. The axon initial segment is a key element of neuronal function, responsible for action potential initiation and maintenance of neuronal polarity. Despite several reports of profound axon initial segment alterations in different pathological models, among which experimental auto-immune encephalomyelitis, whether the axon initial segment is affected in multiple sclerosis is still unknown. Using immunohistochemistry, we analysed axon initial segments from control and multiple sclerosis tissue, focusing on layer 5/6 pyramidal neurons in the neocortex and Purkinje cells in the cerebellum and performed analysis on the parameters known to control neuronal excitability, i.e. axon initial segment length and position. We found that the axon initial segment length was increased only in pyramidal neurons of inactive demyelinated lesions, compared with normal appearing grey matter tissue. In contrast, in both cell types, the axon initial segment position was altered, with an increased soma-axon initial segment gap, in both active and inactive demyelinated lesions. In addition, using a computational model, we show that this increased gap between soma and axon initial segment might increase neuronal excitability. Taken together, these results show, for the first time, changes of axon initial segments in multiple sclerosis, in active as well as inactive grey matter lesions in both neocortex and cerebellum, which might alter neuronal function.

Morphological Determinants of Cell-to-Cell Variations in Action Potential Dynamics in Substantia Nigra Dopaminergic Neurons

Action potential (AP) shape is a critical electrophysiological parameter, in particular because it strongly modulates neurotransmitter release. As it greatly varies between neuronal types, AP shape is also used to distinguish neuronal populations. For instance, AP duration ranges from hundreds of microseconds in cerebellar granule cells to 2-3 ms in SNc dopaminergic (DA) neurons. While most of this variation across cell types seems to arise from differences in the voltage- and calcium-gated ion channels expressed, a few studies suggested that dendritic morphology also affects AP shape. AP duration also displays significant variability in a same neuronal type, although the determinants of these variations are poorly known. Using electrophysiological recordings, morphological reconstructions, and realistic Hodgkin-Huxley modeling, we investigated the relationships between dendritic morphology and AP shape in rat SNc DA neurons from both sexes. In this neuronal type where the axon arises from an axon-bearing dendrite (ABD), the duration of the somatic AP could be predicted from a linear combination of the ABD and non-ABDs' complexities. Dendrotomy experiments and simulation showed that these correlations arise from the causal influence of dendritic topology on AP duration, due in particular to a high density of sodium channels in the somatodendritic compartment. Surprisingly, computational modeling suggested that this effect arises from the influence of sodium currents on the decaying phase of the AP. Consistent with previous findings, these results demonstrate that dendritic morphology plays a major role in defining the electrophysiological properties of SNc DA neurons and their cell-to-cell variations.SIGNIFICANCE STATEMENT Action potential (AP) shape is a critical electrophysiological parameter, in particular because it strongly modulates neurotransmitter release. AP shape (e.g., duration) greatly varies between neuronal types but also within a same neuronal type. While differences in ion channel expression seem to explain most of AP shape variation across cell types, the determinants of cell-to-cell variations in a same neuronal type are mostly unknown. We used electrophysiological recordings, neuronal reconstruction, and modeling to show that, because of the presence of sodium channels in the somatodendritic compartment, a large part of cell-to-cell variations in somatic AP duration in substantia nigra pars compacta dopaminergic neurons is explained by variations in dendritic topology.

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.

Neural excitability increases with axonal resistance between soma and axon initial segment

The position of the axon initial segment (AIS) is thought to play a critical role in neuronal excitability. Previous experimental studies have found that a distal shift in AIS position correlates with a reduction in excitability. Yet theoretical work has suggested the opposite, because of increased electrical isolation. A distal shift in AIS position corresponds to an elevation of axial resistance R _a We therefore examined how changes in R _a at the axon hillock impact the voltage threshold (V_th) of the somatic action potential in L5 pyramidal neurons. Increasing R _a by mechanically pinching the axon between the soma and the AIS was found to lower V_th by ∼6 mV. Conversely, decreasing R _a by substituting internal ions with higher mobility elevated V_th All R _a -dependent changes in V_th could be reproduced in a Hodgkin-Huxley compartmental model. We conclude that in L5 pyramidal neurons, excitability increases with axial resistance and therefore with a distal shift of the AIS.

Electrical match between initial segment and somatodendritic compartment for action potential backpropagation in retinal ganglion cells

The action potential of most vertebrate neurons initiates in the axon initial segment (AIS) and is then transmitted to the soma where it is regenerated by somatodendritic sodium channels. For successful transmission, the AIS must produce a strong axial current, so as to depolarize the soma to the threshold for somatic regeneration. Theoretically, this axial current depends on AIS geometry and Na⁺ conductance density. We measured the axial current of mouse retinal ganglion cells using whole cell recordings with post hoc AIS labeling. We found that this current is large, implying high Na⁺ conductance density, and carries a charge that covaries with capacitance so as to depolarize the soma by ∼30 mV. Additionally, we observed that the axial current attenuates strongly with depolarization, consistent with sodium channel inactivation, but temporally broadens so as to preserve the transmitted charge. Thus, the AIS appears to be organized so as to reliably backpropagate the axonal action potential.NEW & NOTEWORTHY We measured the axial current produced at spike initiation by the axon initial segment of mouse retinal ganglion cells. We found that it is a large current, requiring high sodium channel conductance density, which covaries with cell capacitance so as to ensure a ∼30 mV depolarization. During sustained depolarization the current attenuated, but it broadened to preserve somatic depolarization. Thus, properties of the initial segment are adjusted to ensure backpropagation of the axonal action potential.

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.

Channel current fluctuations conclusively explain neuronal encoding of internal potential into spike trains

Hodgkin and Huxley's seminal neuron model describes the propagation of voltage spikes in axons, but it cannot explain certain full-neuron features crucial for understanding the neural code. We consider channel current fluctuations in a trisection of the Hodgkin-Huxley model, allowing an analytic-mechanistic explanation of these features and yielding consistently excellent matches with in vivo recordings of cerebellar Purkinje neurons, which we use as model systems. This shows that the neuronal encoding is described conclusively by a soft-thresholding function having just three parameters.

The location of the axon initial segment affects the bandwidth of spike initiation dynamics

The dynamics and the sharp onset of action potential (AP) generation have recently been the subject of intense experimental and theoretical investigations. According to the resistive coupling theory, an electrotonic interplay between the site of AP initiation in the axon and the somato-dendritic load determines the AP waveform. This phenomenon not only alters the shape of APs recorded at the soma, but also determines the dynamics of excitability across a variety of time scales. Supporting this statement, here we generalize a previous numerical study and extend it to the quantification of the input-output gain of the neuronal dynamical response. We consider three classes of multicompartmental mathematical models, ranging from ball-and-stick simplified descriptions of neuronal excitability to 3D-reconstructed biophysical models of excitatory neurons of rodent and human cortical tissue. For each model, we demonstrate that increasing the distance between the axonal site of AP initiation and the soma markedly increases the bandwidth of neuronal response properties. We finally consider the Liquid State Machine paradigm, exploring the impact of altering the site of AP initiation at the level of a neuronal population, and demonstrate that an optimal distance exists to boost the computational performance of the network in a simple classification task.

The neurons in the brain encode information through electrical impulses. The performance of a cell in terms of its ability to process and transfer information downstream thus depends heavily on the machinery of initiation of these impulses. In this work, we consider both the cell morphology and the biophysical properties of impulse initiation as the primary parameters that influence information processing in single neurons, as well as in networks. We specifically analyze the location of nerve impulse initiation along the cell’s axon and the way the neuron transfers incoming information. By using single-cell models of various complexity as well as network models, we conclude that information processing is sensitive to the geometrical details of impulse initiation.

Theoretical relation between axon initial segment geometry and excitability

In most vertebrate neurons, action potentials are triggered at the distal end of the axon initial segment (AIS). Both position and length of the AIS vary across and within neuron types, with activity, development and pathology. What is the impact of AIS geometry on excitability? Direct empirical assessment has proven difficult because of the many potential confounding factors. Here, we carried a principled theoretical analysis to answer this question. We provide a simple formula relating AIS geometry and sodium conductance density to the somatic voltage threshold. A distal shift of the AIS normally produces a (modest) increase in excitability, but we explain how this pattern can reverse if a hyperpolarizing current is present at the AIS, due to resistive coupling with the soma. This work provides a theoretical tool to assess the significance of structural AIS plasticity for electrical function.

Diversity of Axonal and Dendritic Contributions to Neuronal Output

Our general understanding of neuronal function is that dendrites receive information that is transmitted to the axon, where action potentials (APs) are initiated and propagated to eventually trigger neurotransmitter release at synaptic terminals. Even though this canonical division of labor is true for a number of neuronal types in the mammalian brain (including neocortical and hippocampal pyramidal neurons or cerebellar Purkinje neurons), many neuronal types do not comply with this classical polarity scheme. In fact, dendrites can be the site of AP initiation and propagation, and even neurotransmitter release. In several interneuron types, all functions are carried out by dendrites as these neurons are devoid of a canonical axon. In this article, we present a few examples of "misbehaving" neurons (with a non-canonical polarity scheme) to highlight the diversity of solutions that are used by mammalian neurons to transmit information. Moreover, we discuss how the contribution of dendrites and axons to neuronal excitability may impose constraints on the morphology of these compartments in specific functional contexts.

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.

Loss of Neurofascin-186 Disrupts Alignment of AnkyrinG Relative to Its Binding Partners in the Axon Initial Segment

The axon initial segment (AIS) is a specialized region within the proximal portion of the axon that initiates action potentials thanks in large part to an enrichment of sodium channels. The scaffolding protein ankyrinG (AnkG) is essential for the recruitment of sodium channels as well as several other intracellular and extracellular proteins to the AIS. In the present study, we explore the role of the cell adhesion molecule (CAM) neurofascin-186 (NF-186) in arranging the individual molecular components of the AIS in cultured rat hippocampal neurons. Using a CRISPR depletion strategy to ablate NF expression, we found that the loss of NF selectively perturbed AnkG accumulation and its relative proximal distribution within the AIS. We found that the overexpression of sodium channels could restore AnkG accumulation, but not its altered distribution within the AIS without NF present. We go on to show that although the loss of NF altered AnkG distribution, sodium channel function within the AIS remained normal. Taken together, these results demonstrate that the regulation of AnkG and sodium channel accumulation within the AIS can occur independently of one another, potentially mediated by other binding partners such as NF.

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.

Extracellular waveforms reveal an axonal origin of spikelets in pyramidal neurons

Spikelets are small spike-like membrane depolarizations measured at the soma whose origin in pyramidal neurons is still unresolved. We investigated the mechanism of spikelet generation using detailed models of pyramidal neurons. We simulated extracellular waveforms associated with action potentials and spikelets and compared these with experimental data obtained by Chorev and Brecht ( J Neurophysiol 108: 1584-1593, 2012) from hippocampal pyramidal neurons in vivo. We considered spikelets originating in the axon of a single cell as well as spikelets generated in two cells coupled with gap junctions. We found that in both cases the experimental data can be explained by an axonal origin of spikelets: in the single-cell case, action potentials are generated in the axon but fail to activate the soma. Such spikelets can be evoked by dendritic input. Alternatively, spikelets resulting from axoaxonal gap junction coupling with a large (greater than several hundred μm) distance between the somata of the coupled cells are also consistent with the data. Our results demonstrate that a cell firing a somatic spikelet generates a detectable extracellular potential that is different from the action potential-correlated extracellular waveform generated by the same cell and recorded at the same location. This, together with the absence of a refractory period between action potentials and spikelets, implies that spikelets and action potentials generated in one cell may easily get misclassified in extracellular recordings as two different cells, albeit they both constitute the output of a single pyramidal neuron. NEW & NOTEWORTHY We addressed the origin of spikelets, using compartmental models of pyramidal neurons. Comparing our simulation results with published extracellular spikelet recordings revealed an axonal origin of spikelets. Our results imply that action potential- and spikelet-associated extracellular waveforms may easily get misclassified as two different cells, albeit they both constitute the output of a single pyramidal cell.

A thermodynamic description for physiological transmembrane transport

A general formulation for both passive and active transmembrane transport is derived from basic thermodynamical principles. The derivation takes into account the energy required for the motion of molecules across membranes and includes the possibility of modeling asymmetric flow. Transmembrane currents can then be described by the general model in the case of electrogenic flow. As it is desirable in new models, it is possible to derive other well-known expressions for transmembrane currents as particular cases of the general formulation. For instance, the conductance-based formulation for current turns out to be a linear approximation of the general formula for current. Also, under suitable assumptions, other formulas for current based on electrodiffusion, like the constant field approximation by Goldman, can be recovered from the general formulation. The applicability of the general formulations is illustrated first with fits to existing data, and after, with models of transmembrane potential dynamics for pacemaking cardiocytes and neurons. The general formulations presented here provide a common ground for the biophysical study of physiological phenomena that depend on transmembrane transport.

Morphological and Biophysical Determinants of the Intracellular and Extracellular Waveforms in Nigral Dopaminergic Neurons: A Computational Study

Action potentials (APs) in nigral dopaminergic neurons often exhibit two separate components: the first reflecting spike initiation in the dendritically located axon initial segment (AIS) and the second the subsequent dendro-somatic spike. These components are separated by a notch in the ascending phase of the somatic extracellular waveform and in the temporal derivative of the somatic intracellular waveform. Still, considerable variability exists in the presence and magnitude of the notch across neurons. To systematically address the contribution of AIS, dendritic and somatic compartments to shaping the two-component APs, we modeled APs of previously in vivo electrophysiologically characterized and 3D-reconstructed male mouse and rat dopaminergic neurons. A parsimonious two-domain model, with high (AIS) and lower (dendro-somatic) Na⁺ conductance, reproduced the notch in the temporal derivatives, but not in the extracellular APs, regardless of morphology. The notch was only revealed when somatic active currents were reduced, constraining the model to three domains. Thus, an initial AIS spike is followed by an actively generated spike by the axon-bearing dendrite (ABD), in turn followed mostly passively by the soma. The transition from being a source compartment for the AIS spike to a source compartment for the ABD spike satisfactorily explains the extracellular somatic notch. Larger AISs and thinner ABD (but not soma-to-AIS distance) accentuate the AIS component. We conclude that variability in AIS size and ABD caliber explains variability in AP extracellular waveform and separation of AIS and dendro-somatic components, given the presence of at least three functional domains with distinct excitability characteristics.SIGNIFICANCE STATEMENT Midbrain dopamine neurons make an important contribution to circuits mediating motivation and movement. Understanding the basic rules that govern the electrical activity of single dopaminergic neurons is therefore essential to reveal how they ultimately contribute to movement and motivation as well as what goes wrong in associated disorders. Our computational study focuses on the generation and propagation of action potentials and shows that different morphologies and excitability characteristics of the cell body, dendrites and proximal axon can explain the diversity of action potentials shapes in this population. These compartments likely make differential contributions both to normal dopaminergic signaling and could potentially underlie pathological dopaminergic signaling implicated in addiction, schizophrenia, Parkinson's disease, and other disorders.

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

Gain control with A-type potassium current: IA as a switch between divisive and subtractive inhibition

Neurons process and convey information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition typically suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron’s output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two “modes” of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. In this study, we use simulations and mathematical analysis of a neuron model to find the specific conditions (parameter sets) for which inhibitory inputs have subtractive or divisive effects. Significantly, we identify a novel role for the A-type Potassium current (I_A). In our model, this fast-activating, slowly-inactivating outward current acts as a switch between subtractive and divisive inhibition. In particular, if I_A is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if I_A is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods (plane analysis and fast-slow dissection) to define how a spike threshold condition depends on synaptic inputs and I_A. Our findings suggest that neurons can “self-regulate” the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for I_A would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract.

Neurons process information by generating spikes in response to two types of synaptic inputs. Excitatory inputs increase spike rates and inhibitory inputs decrease spike rates (typically). The interaction between these two input types and the transformation of these inputs into spike outputs is not, however, a simple matter of addition and subtraction. Inhibitory inputs can suppress outputs in a variety of ways. For instance, in some cases, inhibition adjusts the rate of spiking activity while preserving the range of inputs that evoke spiking activity; an important computational principle known as gain control. We use simulations and mathematical analysis of a neuron model to identify properties of a neuron that determine how inhibitory inputs affect spiking activity. Specifically, we demonstrate how the gain control effects of inhibition depend on the A-type Potassium current. This novel role for the A-type Potassium current provides a way for neurons to flexibly regulate how they process synaptic inputs and transmit signals to other areas of the brain.

Contribution of the Axon Initial Segment to Action Potentials Recorded Extracellularly

Action potentials (APs) are electric phenomena that are recorded both intracellularly and extracellularly. APs are usually initiated in the short segment of the axon called the axon initial segment (AIS). It was recently proposed that at the onset of an AP the soma and the AIS form a dipole. We study the extracellular signature [the extracellular AP (EAP)] generated by such a dipole. First, we demonstrate the formation of the dipole and its extracellular signature in detailed morphological models of a reconstructed pyramidal neuron. Then, we study the EAP waveform and its spatial dependence in models with axonal AP initiation and contrast it with the EAP obtained in models with somatic AP initiation. We show that in the models with axonal AP initiation the dipole forms between somatodendritic compartments and the AIS, and not between soma and dendrites as in the classical models. The soma-dendrites dipole is present only in models with somatic AP initiation. Our study has consequences for interpreting extracellular recordings of single-neuron activity and determining electrophysiological neuron types, but also for better understanding the origins of the high-frequency macroscopic extracellular potentials recorded in the brain.

Systematic generation of biophysically detailed models for diverse cortical neuron types

The cellular components of mammalian neocortical circuits are diverse, and capturing this diversity in computational models is challenging. Here we report an approach for generating biophysically detailed models of 170 individual neurons in the Allen Cell Types Database to link the systematic experimental characterization of cell types to the construction of cortical models. We build models from 3D morphologies and somatic electrophysiological responses measured in the same cells. Densities of active somatic conductances and additional parameters are optimized with a genetic algorithm to match electrophysiological features. We evaluate the models by applying additional stimuli and comparing model responses to experimental data. Applying this technique across a diverse set of neurons from adult mouse primary visual cortex, we verify that models preserve the distinctiveness of intrinsic properties between subsets of cells observed in experiments. The optimized models are accessible online alongside the experimental data. Code for optimization and simulation is also openly distributed.

Nanodiamonds-induced effects on neuronal firing of mouse hippocampal microcircuits

Fluorescent nanodiamonds (FND) are carbon-based nanomaterials that can efficiently incorporate optically active photoluminescent centers such as the nitrogen-vacancy complex, thus making them promising candidates as optical biolabels and drug-delivery agents. FNDs exhibit bright fluorescence without photobleaching combined with high uptake rate and low cytotoxicity. Focusing on FNDs interference with neuronal function, here we examined their effect on cultured hippocampal neurons, monitoring the whole network development as well as the electrophysiological properties of single neurons. We observed that FNDs drastically decreased the frequency of inhibitory (from 1.81 Hz to 0.86 Hz) and excitatory (from 1.61 to 0.68 Hz) miniature postsynaptic currents, and consistently reduced action potential (AP) firing frequency (by 36%), as measured by microelectrode arrays. On the contrary, bursts synchronization was preserved, as well as the amplitude of spontaneous inhibitory and excitatory events. Current-clamp recordings revealed that the ratio of neurons responding with AP trains of high-frequency (fast-spiking) versus neurons responding with trains of low-frequency (slow-spiking) was unaltered, suggesting that FNDs exerted a comparable action on neuronal subpopulations. At the single cell level, rapid onset of the somatic AP (“kink”) was drastically reduced in FND-treated neurons, suggesting a reduced contribution of axonal and dendritic components while preserving neuronal excitability.

The Axon Initial Segment: An Updated Viewpoint

At the base of axons sits a unique compartment called the axon initial segment (AIS). The AIS generates and shapes the action potential before it is propagated along the axon. Neuronal excitability thus depends crucially on the AIS composition and position, and these adapt to developmental and physiological conditions. The AIS also demarcates the boundary between the somatodendritic and axonal compartments. Recent studies have brought insights into the molecular architecture of the AIS and how it regulates protein trafficking. This Viewpoints article summarizes current knowledge about the AIS and highlights future challenges in understanding this key actor of neuronal physiology.

Dynamical response properties of neocortical neurons to conductance-driven time-varying inputs

Ensembles of cortical neurons can track fast-varying inputs and relay them in their spike trains, far beyond the cut-off imposed by membrane passive electrical properties and mean firing rates. Initially explored in silico and later demonstrated experimentally, investigating how neurons respond to sinusoidally modulated stimuli provides a deeper insight into spike initiation mechanisms and information processing than conventional F-I curve methodologies. Besides net membrane currents, physiological synaptic inputs can also induce a stimulus-dependent modulation of the total membrane conductance, which is not reproduced by standard current-clamp protocols. Here, we investigated whether rat cortical neurons can track fast temporal modulations over a noisy conductance background. We also determined input-output transfer properties over a range of conditions, including: distinct presynaptic activation rates, postsynaptic firing rates and variability and type of temporal modulations. We found a very broad signal transfer bandwidth across all conditions, similar large cut-off frequencies and power-law attenuations of fast-varying inputs. At slow and intermediate input modulations, the response gain decreased for increasing output mean firing rates. The gain also decreased significantly for increasing intensities of background synaptic activity, thus generalising earlier studies on F-I curves. We also found a direct correlation between the action potentials' onset rapidness and the neuronal bandwidth. Our novel results extend previous investigations of dynamical response properties to non-stationary and conductance-driven conditions, and provide computational neuroscientists with a novel set of observations that models must capture when aiming to replicate cortical cellular excitability.

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.

The Segregated Expression of Voltage-Gated Potassium and Sodium Channels in Neuronal Membranes: Functional Implications and Regulatory Mechanisms

Neurons are highly polarized cells with apparent functional and morphological differences between dendrites and axon. A critical determinant for the molecular and functional identity of axonal and dendritic segments is the restricted expression of voltage-gated ion channels (VGCs). Several studies show an uneven distribution of ion channels and their differential regulation within dendrites and axons, which is a prerequisite for an appropriate integration of synaptic inputs and the generation of adequate action potential (AP) firing patterns. This review article will focus on the signaling pathways leading to segmented expression of voltage-gated potassium and sodium ion channels at the neuronal plasma membrane and the regulatory mechanisms ensuring segregated functions. We will also discuss the relevance of proper ion channel targeting for neuronal physiology and how alterations in polarized distribution contribute to neuronal pathology.

Spikelets in Pyramidal Neurons: Action Potentials Initiated in the Axon Initial Segment That Do Not Activate the Soma

Spikelets are small spike-like depolarizations that can be measured in somatic intracellular recordings. Their origin in pyramidal neurons remains controversial. To explain spikelet generation, we propose a novel single-cell mechanism: somato-dendritic input generates action potentials at the axon initial segment that may fail to activate the soma and manifest as somatic spikelets. Using mathematical analysis and numerical simulations of compartmental neuron models, we identified four key factors controlling spikelet generation: (1) difference in firing threshold, (2) impedance mismatch, and (3) electrotonic separation between the soma and the axon initial segment, as well as (4) input amplitude. Because spikelets involve forward propagation of action potentials along the axon while they avoid full depolarization of the somato-dendritic compartments, we conjecture that this mode of operation saves energy and regulates dendritic plasticity while still allowing for a read-out of results of neuronal computations.

Action potentials (APs) are digital, all-or-none signals by which neurons communicate with each other. Therefore, APs are the basis of neural function, yet some of their fundamental features are still not well understood. Here we focus on pyramidal cells, which are the principal neurons in neocortex and hippocampus. According to textbook knowledge, an AP in pyramidal neurons is initiated at the axon initial segment and propagates along the axon to the next cell. Concurrently, the AP also propagates back to the soma and into the dendrites where it might trigger synaptic plasticity, which is the basis of learning and memory. However, besides APs, pyramidal cells sometimes also show somatic spikelets—small depolarizations with an AP-like shape—whose origin remains unclear. Here, we propose that spikelets occur when an AP initiated at the axon initial segment only propagates down the axon, but fails to activate sodium currents in the soma and dendrites. As a result, spikelet firing saves energy, and moreover, might be a means to control synaptic plasticity and thereby control learning and memory.

Covariation of axon initial segment location and dendritic tree normalizes the somatic action potential

In mammalian neurons, the axon initial segment (AIS) electrically connects the somatodendritic compartment with the axon and converts the incoming synaptic voltage changes into a temporally precise action potential (AP) output code. Although axons often emanate directly from the soma, they may also originate more distally from a dendrite, the implications of which are not well-understood. Here, we show that one-third of the thick-tufted layer 5 pyramidal neurons have an axon originating from a dendrite and are characterized by a reduced dendritic complexity and thinner main apical dendrite. Unexpectedly, the rising phase of somatic APs is electrically indistinguishable between neurons with a somatic or a dendritic axon origin. Cable analysis of the neurons indicated that the axonal axial current is inversely proportional to the AIS distance, denoting the path length between the soma and the start of the AIS, and to produce invariant somatic APs, it must scale with the local somatodendritic capacitance. In agreement, AIS distance inversely correlates with the apical dendrite diameter, and model simulations confirmed that the covariation suffices to normalize the somatic AP waveform. Therefore, in pyramidal neurons, the AIS location is finely tuned with the somatodendritic capacitive load, serving as a homeostatic regulation of the somatic AP in the face of diverse neuronal morphologies.

Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons

The way in which single neurons transform input into output spike trains has fundamental consequences for network coding. Theories and modeling studies based on standard Integrate-and-Fire models implicitly assume that, in response to increasingly strong inputs, neurons modify their coding strategy by progressively reducing their selective sensitivity to rapid input fluctuations. Combining mathematical modeling with in vitro experiments, we demonstrate that, in L5 pyramidal neurons, the firing threshold dynamics adaptively adjust the effective timescale of somatic integration in order to preserve sensitivity to rapid signals over a broad range of input statistics. For that, a new Generalized Integrate-and-Fire model featuring nonlinear firing threshold dynamics and conductance-based adaptation is introduced that outperforms state-of-the-art neuron models in predicting the spiking activity of neurons responding to a variety of in vivo-like fluctuating currents. Our model allows for efficient parameter extraction and can be analytically mapped to a Generalized Linear Model in which both the input filter—describing somatic integration—and the spike-history filter—accounting for spike-frequency adaptation—dynamically adapt to the input statistics, as experimentally observed. Overall, our results provide new insights on the computational role of different biophysical processes known to underlie adaptive coding in single neurons and support previous theoretical findings indicating that the nonlinear dynamics of the firing threshold due to Na⁺-channel inactivation regulate the sensitivity to rapid input fluctuations.

Over the last decades, a variety of simplified spiking models have been shown to achieve a surprisingly high performance in predicting the neuronal responses to in vitro somatic current injections. Because of the complex adaptive behavior featured by cortical neurons, this success is however restricted to limited stimulus ranges: model parameters optimized for a specific input regime are often inappropriate to describe the response to input currents with different statistical properties. In the present study, a new spiking neuron model is introduced that captures single-neuron computation over a wide range of input statistics and explains different aspects of the neuronal dynamics within a single framework. Our results indicate that complex forms of single neuron adaptation are mediated by the nonlinear dynamics of the firing threshold and that the input-output transformation performed by cortical pyramidal neurons can be intuitively understood in terms of an enhanced Generalized Linear Model in which both the input filter and the spike-history filter adapt to the input statistics.

Functional implications of axon initial segment cytoskeletal disruption in stroke

Axon initial segment (AIS) is the proximal part of the axon, which is not covered with a myelin sheath and possesses a distinctive, specialized assembly of voltage-gated ion channels and associated proteins. AIS plays critical roles in synaptic integration and action potential generation in central neurons. Recent evidence shows that stroke causes rapid, irreversible calpain-mediated proteolysis of the AIS cytoskeleton of neurons surrounding the ischemic necrotic core. A better understanding of the molecular mechanisms underlying this "non-lethal" neuronal damage might provide new therapeutic strategies for improving stroke outcome. Here, we present a brief overview of the structure and function of the AIS. We then discuss possible mechanisms underlying stroke-induced AIS damage, including the roles of calpains and possible sources of Ca(2+) ions, which are necessary for the activation of calpains. Finally, we discuss the potential functional implications of the loss of the AIS cytoskeleton and ion channel clusters for neuronal excitability.

Modeling neural activity with cumulative damage distributions

Neurons transmit information as action potentials or spikes. Due to the inherent randomness of the inter-spike intervals (ISIs), probabilistic models are often used for their description. Cumulative damage (CD) distributions are a family of probabilistic models that has been widely considered for describing time-related cumulative processes. This family allows us to consider certain deterministic principles for modeling ISIs from a probabilistic viewpoint and to link its parameters to values with biological interpretation. The CD family includes the Birnbaum-Saunders and inverse Gaussian distributions, which possess distinctive properties and theoretical arguments useful for ISI description. We expand the use of CD distributions to the modeling of neural spiking behavior, mainly by testing the suitability of the Birnbaum-Saunders distribution, which has not been studied in the setting of neural activity. We validate this expansion with original experimental and simulated electrophysiological data.

Dendritic sodium spikes are required for long-term potentiation at distal synapses on hippocampal pyramidal neurons

Dendritic integration of synaptic inputs mediates rapid neural computation as well as longer-lasting plasticity. Several channel types can mediate dendritically initiated spikes (dSpikes), which may impact information processing and storage across multiple timescales; however, the roles of different channels in the rapid vs long-term effects of dSpikes are unknown. We show here that dSpikes mediated by Nav channels (blocked by a low concentration of TTX) are required for long-term potentiation (LTP) in the distal apical dendrites of hippocampal pyramidal neurons. Furthermore, imaging, simulations, and buffering experiments all support a model whereby fast Nav channel-mediated dSpikes (Na-dSpikes) contribute to LTP induction by promoting large, transient, localized increases in intracellular calcium concentration near the calcium-conducting pores of NMDAR and L-type Cav channels. Thus, in addition to contributing to rapid neural processing, Na-dSpikes are likely to contribute to memory formation via their role in long-lasting synaptic plasticity.

Biophysical Insights into How Spike Threshold Depends on the Rate of Membrane Potential Depolarization in Type I and Type II Neurons

Dynamic spike threshold plays a critical role in neuronal input-output relations. In many neurons, the threshold potential depends on the rate of membrane potential depolarization (dV/dt) preceding a spike. There are two basic classes of neural excitability, i.e., Type I and Type II, according to input-output properties. Although the dynamical and biophysical basis of their spike initiation has been established, the spike threshold dynamic for each cell type has not been well described. Here, we use a biophysical model to investigate how spike threshold depends on dV/dt in two types of neuron. It is observed that Type II spike threshold is more depolarized and more sensitive to dV/dt than Type I. With phase plane analysis, we show that each threshold dynamic arises from the different separatrix and K+ current kinetics. By analyzing subthreshold properties of membrane currents, we find the activation of hyperpolarizing current prior to spike initiation is a major factor that regulates the threshold dynamics. The outward K+ current in Type I neuron does not activate at the perithresholds, which makes its spike threshold insensitive to dV/dt. The Type II K+ current activates prior to spike initiation and there is a large net hyperpolarizing current at the perithresholds, which results in a depolarized threshold as well as a pronounced threshold dynamic. These predictions are further attested in several other functionally equivalent cases of neural excitability. Our study provides a fundamental description about how intrinsic biophysical properties contribute to the threshold dynamics in Type I and Type II neurons, which could decipher their significant functions in neural coding.

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.

Subcellular localization of K+ channels in mammalian brain neurons: remarkable precision in the midst of extraordinary complexity

Potassium channels (KChs) are the most diverse ion channels, in part due to extensive combinatorial assembly of a large number of principal and auxiliary subunits into an assortment of KCh complexes. Their structural and functional diversity allows KChs to play diverse roles in neuronal function. Localization of KChs within specialized neuronal compartments defines their physiological role and also fundamentally impacts their activity, due to localized exposure to diverse cellular determinants of channel function. Recent studies in mammalian brain reveal an exquisite refinement of KCh subcellular localization. This includes axonal KChs at the initial segment, and near/within nodes of Ranvier and presynaptic terminals, dendritic KChs found at sites reflecting specific synaptic input, and KChs defining novel neuronal compartments. Painting the remarkable diversity of KChs onto the complex architecture of mammalian neurons creates an elegant picture of electrical signal processing underlying the sophisticated function of individual neuronal compartments, and ultimately neurotransmission and behavior.

Subthreshold membrane currents confer distinct tuning properties that enable neurons to encode the integral or derivative of their input

Neurons rely on action potentials, or spikes, to encode information. But spikes can encode different stimulus features in different neurons. We show here through simulations and experiments how neurons encode the integral or derivative of their input based on the distinct tuning properties conferred upon them by subthreshold currents. Slow-activating subthreshold inward (depolarizing) current mediates positive feedback control of subthreshold voltage, sustaining depolarization and allowing the neuron to spike on the basis of its integrated stimulus waveform. Slow-activating subthreshold outward (hyperpolarizing) current mediates negative feedback control of subthreshold voltage, truncating depolarization and forcing the neuron to spike on the basis of its differentiated stimulus waveform. Depending on its direction, slow-activating subthreshold current cooperates or competes with fast-activating inward current during spike initiation. This explanation predicts that sensitivity to the rate of change of stimulus intensity differs qualitatively between integrators and differentiators. This was confirmed experimentally in spinal sensory neurons that naturally behave as specialized integrators or differentiators. Predicted sensitivity to different stimulus features was confirmed by covariance analysis. Integration and differentiation, which are themselves inverse operations, are thus shown to be implemented by the slow feedback mediated by oppositely directed subthreshold currents expressed in different neurons.

Dendrites impact the encoding capabilities of the axon

This study highlights a new and powerful direct impact of the dendritic tree (the input region of neurons) on the encoding capability of the axon (the output region). We show that the size of the dendritic arbors (its impedance load) strongly modulates the shape of the action potential (AP) onset at the axon initial segment; it is accelerated in neurons with larger dendritic surface area. AP onset rapidness is key in determining the capability of the axonal spikes to encode (phase lock to) rapid changes in synaptic inputs. Hence, our findings imply that neurons with larger dendritic arbors have improved encoding capabilities. This "dendritic size effect" was explored both analytically as well as numerically, in simplified and detailed models of 3D reconstructed layer 2/3 cortical pyramidal cells of rats and humans. The cutoff frequency of spikes phase locking to modulated inputs increased from 100 to 200 Hz in pyramidal cells of young rats to 400-600 Hz in human cells. In the latter case, phase locking reached close to 1 KHz in in vivo-like conditions. This work highlights new and functionally profound cross talk between the dendritic tree and the axon initial segment, providing new understanding of neurons as sophisticated nonlinear input/output devices.

Spike-threshold adaptation predicted by membrane potential dynamics in vivo

Neurons encode information in sequences of spikes, which are triggered when their membrane potential crosses a threshold. In vivo, the spiking threshold displays large variability suggesting that threshold dynamics have a profound influence on how the combined input of a neuron is encoded in the spiking. Threshold variability could be explained by adaptation to the membrane potential. However, it could also be the case that most threshold variability reflects noise and processes other than threshold adaptation. Here, we investigated threshold variation in auditory neurons responses recorded in vivo in barn owls. We found that spike threshold is quantitatively predicted by a model in which the threshold adapts, tracking the membrane potential at a short timescale. As a result, in these neurons, slow voltage fluctuations do not contribute to spiking because they are filtered by threshold adaptation. More importantly, these neurons can only respond to input spikes arriving together on a millisecond timescale. These results demonstrate that fast adaptation to the membrane potential captures spike threshold variability in vivo.

Neurons spike when their membrane potential exceeds a threshold value, but this value has been shown to be variable in the same neuron recorded in vivo. This variability could reflect noise, or deterministic processes that make the threshold vary with the membrane potential. The second alternative would have important functional consequences. Here, we show that threshold variability is a genuine feature of neurons, which reflects adaptation to the membrane potential at a short timescale, with little contribution from noise. This demonstrates that a deterministic model can predict spikes based only on the membrane potential.

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.