The Na+/K+-ATPase generically enables deterministic bursting in class I neurons by shearing the spike-onset bifurcation structure.

Slow brain rhythms, for example during slow-wave sleep or pathological conditions like seizures and spreading depolarization, can be accompanied by oscillations in extracellular potassium concentration. Such slow brain rhythms typically have a lower frequency than tonic action-potential firing. They are assumed to arise from network-level mechanisms, involving synaptic interactions and delays, or from intrinsically bursting neurons. Neuronal burst generation is commonly attributed to ion channels with slow kinetics. Here, we explore an alternative mechanism generically available to all neurons with class I excitability. It is based on the interplay of fast-spiking voltage dynamics with a one-dimensional slow dynamics of the extracellular potassium concentration, mediated by the activity of the Na+/K+-ATPase. We use bifurcation analysis of the complete system as well as the slow-fast method to reveal that this coupling suffices to generate a hysteresis loop organized around a bistable region that emerges from a saddle-node loop bifurcation-a common feature of class I excitable neurons. Depending on the strength of the Na+/K+-ATPase, bursts are generated from pump-induced shearing the bifurcation structure, spiking is tonic, or cells are silenced via depolarization block. We suggest that transitions between these dynamics can result from disturbances in extracellular potassium regulation, such as glial malfunction or hypoxia affecting the Na+/K+-ATPase activity. The identified minimal mechanistic model outlining the sodium-potassium pump's generic contribution to burst dynamics can, therefore, contribute to a better mechanistic understanding of pathologies such as epilepsy syndromes and, potentially, inform therapeutic strategies.

How neuronal morphology impacts the synchronisation state of neuronal networks.

The biophysical properties of neurons not only affect how information is processed within cells, they can also impact the dynamical states of the network. Specifically, the cellular dynamics of action-potential generation have shown relevance for setting the (de)synchronisation state of the network. The dynamics of tonically spiking neurons typically fall into one of three qualitatively distinct types that arise from distinct mathematical bifurcations of voltage dynamics at the onset of spiking. Accordingly, changes in ion channel composition or even external factors, like temperature, have been demonstrated to switch network behaviour via changes in the spike onset bifurcation and hence its associated dynamical type. A thus far less addressed modulator of neuronal dynamics is cellular morphology. Based on simplified and anatomically realistic mathematical neuron models, we show here that the extent of dendritic arborisation has an influence on the neuronal dynamical spiking type and therefore on the (de)synchronisation state of the network. Specifically, larger dendritic trees prime neuronal dynamics for in-phase-synchronised or splayed-out activity in weakly coupled networks, in contrast to cells with otherwise identical properties yet smaller dendrites. Our biophysical insights hold for generic multicompartmental classes of spiking neuron models (from ball-and-stick-type to anatomically reconstructed models) and establish a connection between neuronal morphology and the susceptibility of neural tissue to synchronisation in health and disease.

KCC2 reverse mode helps to clear postsynaptically released potassium at glutamatergic synapses.

Extracellular potassium [K+]o elevation during synaptic activity retrogradely modifies presynaptic release and astrocytic uptake of glutamate. Hence, local K+ clearance and replenishment mechanisms are crucial regulators of glutamatergic transmission and plasticity. Based on recordings of astrocytic inward rectifier potassium current IKir and K+-sensitive electrodes as sensors of [K+]o as well as on in silico modeling, we demonstrate that the neuronal K+-Cl- co-transporter KCC2 clears local perisynaptic [K+]o during synaptic excitation by operating in an activity-dependent reversed mode. In reverse mode, KCC2 replenishes K+ in dendritic spines and complements clearance of [K+]o, therewith attenuating presynaptic glutamate release and shortening LTP. We thus demonstrate a physiological role of KCC2 in neuron-glial interactions and regulation of synaptic signaling and plasticity through the uptake of postsynaptically released K+.

Gap junctions desynchronize a neural circuit to stabilize insect flight.

Insect asynchronous flight is one of the most prevalent forms of animal locomotion used by more than 600,000 species. Despite profound insights into the motor patterns1, biomechanics2,3 and aerodynamics underlying asynchronous flight4,5, the architecture and function of the central-pattern-generating (CPG) neural network remain unclear. Here, on the basis of an experiment-theory approach including electrophysiology, optophysiology, Drosophila genetics and mathematical modelling, we identify a miniaturized circuit solution with unexpected properties. The CPG network consists of motoneurons interconnected by electrical synapses that, in contrast to doctrine, produce network activity splayed out in time instead of synchronized across neurons. Experimental and mathematical evidence support a generic mechanism for network desynchronization that relies on weak electrical synapses and specific excitability dynamics of the coupled neurons. In small networks, electrical synapses can synchronize or desynchronize network activity, depending on the neuron-intrinsic dynamics and ion channel composition. In the asynchronous flight CPG, this mechanism translates unpatterned premotor input into stereotyped neuronal firing with fixed sequences of cell activation that ensure stable wingbeat power and, as we show, is conserved across multiple species. Our findings prove a wider functional versatility of electrical synapses in the dynamic control of neural circuits and highlight the relevance of detecting electrical synapses in connectomics.

Temperature elevations can induce switches to homoclinic action potentials that alter neural encoding and synchronization.

Almost seventy years after the discovery of the mechanisms of action potential generation, some aspects of their computational consequences are still not fully understood. Based on mathematical modeling, we here explore a type of action potential dynamics - arising from a saddle-node homoclinic orbit bifurcation - that so far has received little attention. We show that this type of dynamics is to be expected by specific changes in common physiological parameters, like an elevation of temperature. Moreover, we demonstrate that it favours synchronization patterns in networks - a feature that becomes particularly prominent when system parameters change such that homoclinic spiking is induced. Supported by in-vitro hallmarks for homoclinic spikes in the rodent brain, we hypothesize that the prevalence of homoclinic spikes in the brain may be underestimated and provide a missing link between the impact of biophysical parameters on abrupt transitions between asynchronous and synchronous states of electrical activity in the brain.

A feedforward inhibitory premotor circuit for auditory-vocal interactions in zebra finches.

During vocal exchanges, hearing specific auditory signals can provoke vocal responses or suppress vocalizations to avoid interference. These abilities result in the widespread phenomenon of vocal turn taking, yet little is known about the neural circuitry that regulates the input-dependent timing of vocal replies. Previous work in vocally interacting zebra finches has highlighted the importance of premotor inhibition for precisely timed vocal output. By developing physiologically constrained mathematical models, we derived circuit mechanisms based on feedforward inhibition that enable both the temporal modulation of vocal premotor drive as well as auditory suppression of vocalization during listening. Extracellular recordings in HVC during the listening phase confirmed the presence of auditory-evoked response patterns in putative inhibitory interneurons, along with corresponding signatures of auditory-evoked activity suppression. Further, intracellular recordings of identified neurons projecting to HVC from the upstream sensorimotor nucleus, nucleus interfacialis (NIf), shed light on the timing of auditory inputs to this network. The analysis of incrementally time-lagged interactions between auditory and premotor activity in the model resulted in the prediction of a window of auditory suppression, which could be, in turn, verified in behavioral data. A phasic feedforward inhibition model consistently explained the experimental results. This mechanism highlights a parsimonious and generalizable principle for how different driving inputs (vocal and auditory related) can be integrated in a single sensorimotor circuit to regulate two opposing vocal behavioral outcomes: the controlled timing of vocal output or the suppression of overlapping vocalizations.

A dynamic clamp protocol to artificially modify cell capacitance.

Dynamics of excitable cells and networks depend on the membrane time constant, set by membrane resistance and capacitance. Whereas pharmacological and genetic manipulations of ionic conductances of excitable membranes are routine in electrophysiology, experimental control over capacitance remains a challenge. Here, we present capacitance clamp, an approach that allows electrophysiologists to mimic a modified capacitance in biological neurons via an unconventional application of the dynamic clamp technique. We first demonstrate the feasibility to quantitatively modulate capacitance in a mathematical neuron model and then confirm the functionality of capacitance clamp in in vitro experiments in granule cells of rodent dentate gyrus with up to threefold virtual capacitance changes. Clamping of capacitance thus constitutes a novel technique to probe and decipher mechanisms of neuronal signaling in ways that were so far inaccessible to experimental electrophysiology.

Hebbian plasticity in parallel synaptic pathways: A circuit mechanism for systems memory consolidation.

Systems memory consolidation involves the transfer of memories across brain regions and the transformation of memory content. For example, declarative memories that transiently depend on the hippocampal formation are transformed into long-term memory traces in neocortical networks, and procedural memories are transformed within cortico-striatal networks. These consolidation processes are thought to rely on replay and repetition of recently acquired memories, but the cellular and network mechanisms that mediate the changes of memories are poorly understood. Here, we suggest that systems memory consolidation could arise from Hebbian plasticity in networks with parallel synaptic pathways-two ubiquitous features of neural circuits in the brain. We explore this hypothesis in the context of hippocampus-dependent memories. Using computational models and mathematical analyses, we illustrate how memories are transferred across circuits and discuss why their representations could change. The analyses suggest that Hebbian plasticity mediates consolidation by transferring a linear approximation of a previously acquired memory into a parallel pathway. Our modelling results are further in quantitative agreement with lesion studies in rodents. Moreover, a hierarchical iteration of the mechanism yields power-law forgetting-as observed in psychophysical studies in humans. The predicted circuit mechanism thus bridges spatial scales from single cells to cortical areas and time scales from milliseconds to years.

Neural optimization: Understanding trade-offs with Pareto theory.

Nervous systems, like any organismal structure, have been shaped by evolutionary processes to increase fitness. The resulting neural 'bauplan' has to account for multiple objectives simultaneously, including computational function, as well as additional factors such as robustness to environmental changes and energetic limitations. Oftentimes these objectives compete, and quantification of the relative impact of individual optimization targets is non-trivial. Pareto optimality offers a theoretical framework to decipher objectives and trade-offs between them. We, therefore, highlight Pareto theory as a useful tool for the analysis of neurobiological systems from biophysically detailed cells to large-scale network structures and behavior. The Pareto approach can help to assess optimality, identify relevant objectives and their respective impact, and formulate testable hypotheses.

Biophysical models of intrinsic homeostasis: Firing rates and beyond.

In view of ever-changing conditions both in the external world and in intrinsic brain states, maintaining the robustness of computations poses a challenge, adequate solutions to which we are only beginning to understand. At the level of cell-intrinsic properties, biophysical models of neurons permit one to identify relevant physiological substrates that can serve as regulators of neuronal excitability and to test how feedback loops can stabilize crucial variables such as long-term calcium levels and firing rates. Mathematical theory has also revealed a rich set of complementary computational properties arising from distinct cellular dynamics and even shaping processing at the network level. Here, we provide an overview over recently explored homeostatic mechanisms derived from biophysical models and hypothesize how multiple dynamical characteristics of cells, including their intrinsic neuronal excitability classes, can be stably controlled.

Spatially structured inhibition defined by polarized parvalbumin interneuron axons promotes head direction tuning.

In cortical microcircuits, it is generally assumed that fast-spiking parvalbumin interneurons mediate dense and nonselective inhibition. Some reports indicate sparse and structured inhibitory connectivity, but the computational relevance and the underlying spatial organization remain unresolved. In the rat superficial presubiculum, we find that inhibition by fast-spiking interneurons is organized in the form of a dominant super-reciprocal microcircuit motif where multiple pyramidal cells recurrently inhibit each other via a single interneuron. Multineuron recordings and subsequent 3D reconstructions and analysis further show that this nonrandom connectivity arises from an asymmetric, polarized morphology of fast-spiking interneuron axons, which individually cover different directions in the same volume. Network simulations assuming topographically organized input demonstrate that such polarized inhibition can improve head direction tuning of pyramidal cells in comparison to a "blanket of inhibition." We propose that structured inhibition based on asymmetrical axons is an overarching spatial connectivity principle for tailored computation across brain regions.

Activity-mediated accumulation of potassium induces a switch in firing pattern and neuronal excitability type.

During normal neuronal activity, ionic concentration gradients across a neuron's membrane are often assumed to be stable. Prolonged spiking activity, however, can reduce transmembrane gradients and affect voltage dynamics. Based on mathematical modeling, we investigated the impact of neuronal activity on ionic concentrations and, consequently, the dynamics of action potential generation. We find that intense spiking activity on the order of a second suffices to induce changes in ionic reversal potentials and to consistently induce a switch from a regular to an intermittent firing mode. This transition is caused by a qualitative alteration in the system's voltage dynamics, mathematically corresponding to a co-dimension-two bifurcation from a saddle-node on invariant cycle (SNIC) to a homoclinic orbit bifurcation (HOM). Our electrophysiological recordings in mouse cortical pyramidal neurons confirm the changes in action potential dynamics predicted by the models: (i) activity-dependent increases in intracellular sodium concentration directly reduce action potential amplitudes, an effect typically attributed solely to sodium channel inactivation; (ii) extracellular potassium accumulation switches action potential generation from tonic firing to intermittently interrupted output. Thus, individual neurons may respond very differently to the same input stimuli, depending on their recent patterns of activity and/or the current brain-state.

Firing statistics in the bistable regime of neurons with homoclinic spike generation.

Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixed point and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increase the variability of the spike train compared to neurons with only one stable attractor. Qualitative features of the resulting spike statistics depend on the spike onset bifurcations. This paper focuses on the creation of the spiking limit cycle via the saddle-homoclinic orbit (HOM) bifurcation and derives interspike interval (ISI) densities for a conductance-based neuron model in the bistable regime. The ISI densities of bistable homoclinic neurons are found to be unimodal yet distinct from the inverse Gaussian distribution associated with the saddle-node-on-invariant-cycle bifurcation. It is demonstrated that for the HOM bifurcation the transition between rest and spiking is mainly determined along the downstroke of the action potential-a dynamical feature that is not captured by the commonly used reset neuron models. The deduced spike statistics can help to identify HOM dynamics in experimental data.

Modifications of sodium channel voltage dependence induce arrhythmia-favouring dynamics of cardiac action potentials.

Heart arrhythmia is a pathological condition where the sequence of electrical impulses in the heart deviates from the normal rhythm. It is often associated with specific channelopathies in cardiac tissue, yet how precisely the changes in ionic channels affect the electrical activity of cardiac cells is still an open question. Even though sodium channel mutations that underlie cardiac syndromes like the Long-Q-T and the Brugada-syndrome are known to affect a number of channel parameters simultaneously, previous studies have predominantly focused on the persistent late component of the sodium current as the causal explanation for an increased risk of heart arrhythmias in these cardiac syndromes. A systematic analysis of the impact of other important sodium channel parameters is currently lacking. Here, we investigate the reduced ten-Tusscher-model for single human epicardium ventricle cells and use mathematical bifurcation analysis to predict the dependence of the cardiac action potential on sodium channel activation and inactivation time-constants and voltage dependence. We show that, specifically, shifts of the voltage dependence of activation and inactivation curve can lead to drastic changes in the action potential dynamics, inducing oscillations of the membrane potential as well as bistability. Our results not only demonstrate a new way to induce multiple co-existing states of excitability (biexcitability) but also emphasize the critical role of the voltage dependence of sodium channel activation and inactivation curves for the induction of heart-arrhythmias.

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.

Spikelets in pyramidal neurons: generating mechanisms, distinguishing properties, and functional implications.

Spikelets are small spike-like depolarizations that are found in somatic recordings of many neuron types. Spikelets have been assigned important functions, ranging from neuronal synchronization to the regulation of synaptic plasticity, which are specific to the particular mechanism of spikelet generation. As spikelets reflect spiking activity in neuronal compartments that are electrotonically distinct from the soma, four modes of spikelet generation can be envisaged: (1) dendritic spikes or (2) axonal action potentials occurring in a single cell as well as action potentials transmitted via (3) gap junctions or (4) ephaptic coupling in pairs of neurons. In one of the best studied neuron type, cortical pyramidal neurons, the origins and functions of spikelets are still unresolved; all four potential mechanisms have been proposed, but the experimental evidence remains ambiguous. Here we attempt to reconcile the scattered experimental findings in a coherent theoretical framework. We review in detail the various mechanisms that can give rise to spikelets. For each mechanism, we present the biophysical underpinnings as well as the resulting properties of spikelets and compare these predictions to experimental data from pyramidal neurons. We also discuss the functional implications of each mechanism. On the example of pyramidal neurons, we illustrate that several independent spikelet-generating mechanisms fulfilling vastly different functions might be operating in a single cell.

How to correctly quantify neuronal phase-response curves from noisy recordings.

At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.

Function and energy consumption constrain neuronal biophysics in a canonical computation: Coincidence detection.

Neural morphology and membrane properties vary greatly between cell types in the nervous system. The computations and local circuit connectivity that neurons support are thought to be the key factors constraining the cells' biophysical properties. Nevertheless, additional constraints can be expected to further shape neuronal design. Here, we focus on a particularly energy-intense system (as indicated by metabolic markers): principal neurons in the medial superior olive (MSO) nucleus of the auditory brainstem. Based on a modeling approach, we show that a trade-off between the level of performance of a functionally relevant computation and energy consumption predicts optimal ranges for cell morphology and membrane properties. The biophysical parameters appear most strongly constrained by functional needs, while energy use is minimized as long as function can be maintained. The key factors that determine model performance and energy consumption are 1) the saturation of the synaptic conductance input and 2) the temporal resolution of the postsynaptic signals as they reach the soma, which is largely determined by active membrane properties. MSO cells seem to operate close to pareto optimality, i.e., the trade-off boundary between performance and energy consumption that is formed by the set of optimal models. Good performance for drastically lower costs could in theory be achieved by small neurons without dendrites, as seen in the avian auditory system, pointing to additional constraints for mammalian MSO cells, including their circuit connectivity.

Dendrite-Specific Amplification of Weak Synaptic Input during Network Activity In Vivo.

Excitatory synaptic input reaches the soma of a cortical excitatory pyramidal neuron via anatomically segregated apical and basal dendrites. In vivo, dendritic inputs are integrated during depolarized network activity, but how network activity affects apical and basal inputs is not understood. Using subcellular two-photon stimulation of Channelrhodopsin2-expressing layer 2/3 pyramidal neurons in somatosensory cortex, nucleus-specific thalamic optogenetic stimulation, and paired recordings, we show that slow, depolarized network activity amplifies small-amplitude synaptic inputs targeted to basal dendrites but reduces the amplitude of all inputs from apical dendrites and the cell soma. Intracellular pharmacology and mathematical modeling suggests that the amplification of weak basal inputs is mediated by postsynaptic voltage-gated channels. Thus, network activity dynamically reconfigures the relative somatic contribution of apical and basal inputs and could act to enhance the detectability of weak synaptic inputs.

Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation.

Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.