Classification of bursting patterns: A tale of two ducks.

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

Cross-scale excitability in networks of quadratic integrate-and-fire neurons.

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. Here, we show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.

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

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

Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by NaV1.1/SCN1A mutations.

NaV1.1 (SCN1A) is a voltage-gated sodium channel mainly expressed in GABAergic neurons. Loss of function mutations of NaV1.1 lead to epileptic disorders, while gain of function mutations cause a migraine in which cortical spreading depolarizations (CSDs) are involved. It is still debated how these opposite effects initiate two different manifestations of neuronal hyperactivity: epileptic seizures and CSD. To investigate this question, we previously built a conductance-based model of two neurons (GABAergic and pyramidal), with dynamic ion concentrations (Lemaire et al. in PLoS Comput Biol 17(7):e1009239, 2021. https://doi.org/10.1371/journal.pcbi.1009239 ). When implementing either NaV1.1 migraine or epileptogenic mutations, ion concentration modifications acted as slow processes driving the system to the corresponding pathological firing regime. However, the large dimensionality of the model complicated the exploitation of its implicit multi-timescale structure. Here, we substantially simplify our biophysical model to a minimal version more suitable for bifurcation analysis. The explicit timescale separation allows us to apply slow-fast theory, where slow variables are treated as parameters in the fast singular limit. In this setting, we reproduce both pathological transitions as dynamic bifurcations in the full system. In the epilepsy condition, we shift the spike-terminating bifurcation to lower inputs for the GABAergic neuron, to model an increased susceptibility to depolarization block. The resulting failure of synaptic inhibition triggers hyperactivity of the pyramidal neuron. In the migraine scenario, spiking-induced release of potassium leads to the abrupt increase of the extracellular potassium concentration. This causes a dynamic spike-terminating bifurcation of both neurons, which we interpret as CSD initiation.

Synchronization in STDP-driven memristive neural networks with time-varying topology.

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing P (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing F (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay [Formula: see text], the average degree [Formula: see text], and the rewiring probability [Formula: see text] have some appropriate values. When [Formula: see text], [Formula: see text], and [Formula: see text] are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when F increases, depending on the network topology. It is also found that the time delay [Formula: see text] can induce intermittent CS and PS whose occurrence is independent F. Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena.

Modeling NaV1.1/SCN1A sodium channel mutations in a microcircuit with realistic ion concentration dynamics suggests differential GABAergic mechanisms leading to hyperexcitability in epilepsy and hemiplegic migraine.

Loss of function mutations of SCN1A, the gene coding for the voltage-gated sodium channel NaV1.1, cause different types of epilepsy, whereas gain of function mutations cause sporadic and familial hemiplegic migraine type 3 (FHM-3). However, it is not clear yet how these opposite effects can induce paroxysmal pathological activities involving neuronal networks' hyperexcitability that are specific of epilepsy (seizures) or migraine (cortical spreading depolarization, CSD). To better understand differential mechanisms leading to the initiation of these pathological activities, we used a two-neuron conductance-based model of interconnected GABAergic and pyramidal glutamatergic neurons, in which we incorporated ionic concentration dynamics in both neurons. We modeled FHM-3 mutations by increasing the persistent sodium current in the interneuron and epileptogenic mutations by decreasing the sodium conductance in the interneuron. Therefore, we studied both FHM-3 and epileptogenic mutations within the same framework, modifying only two parameters. In our model, the key effect of gain of function FHM-3 mutations is ion fluxes modification at each action potential (in particular the larger activation of voltage-gated potassium channels induced by the NaV1.1 gain of function), and the resulting CSD-triggering extracellular potassium accumulation, which is not caused only by modifications of firing frequency. Loss of function epileptogenic mutations, on the other hand, increase GABAergic neurons' susceptibility to depolarization block, without major modifications of firing frequency before it. Our modeling results connect qualitatively to experimental data: potassium accumulation in the case of FHM-3 mutations and facilitated depolarization block of the GABAergic neuron in the case of epileptogenic mutations. Both these effects can lead to pyramidal neuron hyperexcitability, inducing in the migraine condition depolarization block of both the GABAergic and the pyramidal neuron. Overall, our findings suggest different mechanisms of network hyperexcitability for migraine and epileptogenic NaV1.1 mutations, implying that the modifications of firing frequency may not be the only relevant pathological mechanism.

Hyperactivation of monocytes and macrophages in MCI patients contributes to the progression of Alzheimer's disease.

BACKGROUND: Alzheimer's disease (AD) is the most common neurodegenerative disease ultimately manifesting as clinical dementia. Despite considerable effort and ample experimental data, the role of neuroinflammation related to systemic inflammation is still unsettled. While the implication of microglia is well recognized, the exact contribution of peripheral monocytes/macrophages is still largely unknown, especially concerning their role in the various stages of AD. OBJECTIVES: AD develops over decades and its clinical manifestation is preceded by subjective memory complaints (SMC) and mild cognitive impairment (MCI); thus, the question arises how the peripheral innate immune response changes with the progression of the disease. Therefore, to further investigate the roles of monocytes/macrophages in the progression of AD we assessed their phenotypes and functions in patients at SMC, MCI and AD stages and compared them with cognitively healthy controls. We also conceptualised an idealised mathematical model to explain the functionality of monocytes/macrophages along the progression of the disease. RESULTS: We show that there are distinct phenotypic and functional changes in monocyte and macrophage populations as the disease progresses. Higher free radical production upon stimulation could already be observed for the monocytes of SMC patients. The most striking results show that activation of peripheral monocytes (hyperactivation) is the strongest in the MCI group, at the prodromal stage of the disease. Monocytes exhibit significantly increased chemotaxis, free radical production, and cytokine production in response to TLR2 and TLR4 stimulation. CONCLUSION: Our data suggest that the peripheral innate immune system is activated during the progression from SMC through MCI to AD, with the highest levels of activation being in MCI subjects and the lowest in AD patients. Some of these parameters may be used as biomarkers, but more holistic immune studies are needed to find the best period of the disease for clinical intervention.

Canard-induced complex oscillations in an excitatory network.

In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitable responses to an input current, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes in multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by demonstrating canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and give rise to mixed-mode bursting oscillations (MMBOs). We unveil complicated isolas along which MMBOs exist in parameter space. The profile of solutions along each isola undergoes canard-mediated transitions between the sub-threshold regime and the bursting regime; these explosive transitions change the number of oscillations in each regime. Finally, we relate the MMBO dynamics to experimental recordings and discuss their effects on the silent phases of bursting patterns as well as their potential role in creating subthreshold fluctuations that are often interpreted as noise. The mathematical framework used in this paper is relevant for modelling multiple timescale dynamics in excitable systems.

Excitable dynamics in neural and cardiac systems.

The application of mathematics, physics and engineering to medical research is continuously growing; interactions among these disciplines have become increasingly important and have contributed to an improved understanding of clinical and biological phenomena, with implications for disease prevention, diagnosis and treatment. This special issue presents examples of this synergy, with a particular focus on the investigation of cardiac and neural excitability. This issue includes 24 original research papers and covers a broad range of topics related to the physiological and pathophysiological function of the brain and the heart. Studies span scales from isolated neurons and small networks of neurons to whole-organ dynamics for the brain and from cardiac subcellular domains and cardiomyocytes to one-dimensional tissues for the heart. This preface is part of the Special Issue on "Excitable Dynamics in Neural and Cardiac Systems".

Modeling cortical spreading depression induced by the hyperactivity of interneurons.

Cortical spreading depression (CSD) is a wave of transient intense neuronal firing leading to a long lasting depolarizing block of neuronal activity. It is a proposed pathological mechanism of migraine with aura. Some forms of migraine are associated with a genetic mutation of the Nav1.1 channel, resulting in its gain of function and implying hyperexcitability of interneurons. This leads to the counterintuitive hypothesis that intense firing of interneurons can cause CSD ignition. To test this hypothesis in silico, we developed a computational model of an E-I pair (a pyramidal cell and an interneuron), in which the coupling between the cells in not just synaptic, but takes into account also the effects of the accumulation of extracellular potassium caused by the activity of the neurons and of the synapses. In the context of this model, we show that the intense firing of the interneuron can lead to CSD. We have investigated the effect of various biophysical parameters on the transition to CSD, including the levels of glutamate or GABA, frequency of the interneuron firing and the efficacy of the KCC2 co-transporter. The key element for CSD ignition in our model was the frequency of interneuron firing and the related accumulation of extracellular potassium, which induced a depolarizing block of the pyramidal cell. This constitutes a new mechanism of CSD ignition.

Conductance-Based Refractory Density Approach for a Population of Bursting Neurons.

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modelling interacting populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study, we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting population model makes use of slow-fast analysis, which leads to a novel methodology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explorations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).

Time-coded neurotransmitter release at excitatory and inhibitory synapses.

Communication between neurons at chemical synapses is regulated by hundreds of different proteins that control the release of neurotransmitter that is packaged in vesicles, transported to an active zone, and released when an input spike occurs. Neurotransmitter can also be released asynchronously, that is, after a delay following the spike, or spontaneously in the absence of a stimulus. The mechanisms underlying asynchronous and spontaneous neurotransmitter release remain elusive. Here, we describe a model of the exocytotic cycle of vesicles at excitatory and inhibitory synapses that accounts for all modes of vesicle release as well as short-term synaptic plasticity (STSP). For asynchronous release, the model predicts a delayed inertial protein unbinding associated with the SNARE complex assembly immediately after vesicle priming. Experiments are proposed to test the model's molecular predictions for differential exocytosis. The simplicity of the model will also facilitate large-scale simulations of neural circuits.

Anticipation via canards in excitable systems.

Neurons can anticipate incoming signals by exploiting a physiological mechanism that is not well understood. This article offers a novel explanation on how a receiver neuron can predict the sender's dynamics in a unidirectionally-coupled configuration, in which both sender and receiver follow the evolution of a multi-scale excitable system. We present a novel theoretical viewpoint based on a mathematical object, called canard, to explain anticipation in excitable systems. We provide a numerical approach, which allows to determine the transient effects of canards. To demonstrate the general validity of canard-mediated anticipation in the context of excitable systems, we illustrate our framework in two examples, a multi-scale radio-wave circuit (the van der Pol model) that inspired a caricature neuronal model (the FitzHugh-Nagumo model) and a biophysical neuronal model (a 2-dimensional reduction of the Hodgkin-Huxley model), where canards act as messengers to the senders' prediction. We also propose an experimental paradigm that would enable experimental neuroscientists to validate our predictions. We conclude with an outlook to possible fascinating research avenues to further unfold the mechanisms underpinning anticipation. We envisage that our approach can be employed by a wider class of excitable systems with appropriate theoretical extensions.

Short-term synaptic plasticity in the deterministic Tsodyks-Markram model leads to unpredictable network dynamics.

Short-term synaptic plasticity strongly affects the neural dynamics of cortical networks. The Tsodyks and Markram (TM) model for short-term synaptic plasticity accurately accounts for a wide range of physiological responses at different types of cortical synapses. Here, we report a route to chaotic behavior via a Shilnikov homoclinic bifurcation that dynamically organizes some of the responses in the TM model. In particular, the presence of such a homoclinic bifurcation strongly affects the shape of the trajectories in the phase space and induces highly irregular transient dynamics; indeed, in the vicinity of the Shilnikov homoclinic bifurcation, the number of population spikes and their precise timing are unpredictable and highly sensitive to the initial conditions. Such an irregular deterministic dynamics has its counterpart in stochastic/network versions of the TM model: The existence of the Shilnikov homoclinic bifurcation generates complex and irregular spiking patterns and--acting as a sort of springboard--facilitates transitions between the down-state and unstable periodic orbits. The interplay between the (deterministic) homoclinic bifurcation and stochastic effects may give rise to some of the complex dynamics observed in neural systems.

Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster.

This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which consist of alternating SAOs and LAOs, without the LAOs being organized into burst events. In this article, we show how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike-adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that bifurcation. Canards are central to the dynamics of MMBOs, and their role in shaping the MMBOs is two-fold: saddle-type canards are involved in the spike-adding mechanism of the underlying burster and permit one to understand the number of LAOs in each burst event, and folded-node canards arise due to the slow passage effect and control the number of SAOs. The analysis is carried out for a prototypical fourth-order system of this type, which consists of the third-order Hindmarsh-Rose system, known to have the spike-adding mechanism, and in which one of the key bifurcation parameters also varies slowly. We also include a discussion of the MMBO phenomenon for the Morris-Lecar-Terman system. Finally, we discuss the role of the MMBOs to a biological modeling of secreting neurons.

A method for the reconstruction of unknown non-monotonic growth functions in the chemostat.

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.

Viruses - a major cause of amyloid deposition in the brain.

INTRODUCTION: Clinically, Alzheimer's disease (AD) is a syndrome with a spectrum of various cognitive disorders. There is a complete dissociation between the pathology and the clinical presentation. Therefore, we need a disruptive new approach to be able to prevent and treat AD. AREAS COVERED: In this review, the authors extensively discuss the evidence why the amyloid beta is not the pathological cause of AD which makes therefore the amyloid hypothesis not sustainable anymore. They review the experimental evidence underlying the role of microbes, especially that of viruses, as a trigger/cause for the production of amyloid beta leading to the establishment of a chronic neuroinflammation as the mediator manifesting decades later by AD as a clinical spectrum. In this context, the emergence and consequences of the infection/antimicrobial protection hypothesis are described. The epidemiological and clinical data supporting this hypothesis are also analyzed. EXPERT OPINION: For decades, we have known that viruses are involved in the pathogenesis of AD. This discovery was ignored and discarded for a long time. Now we should accept this fact, which is not a hypothesis anymore, and stimulate the research community to come up with new ideas, new treatments, and new concepts.

Geometry of spiking patterns in early visual cortex: a topological data analytic approach.

In the brain, spiking patterns live in a high-dimensional space of neurons and time. Thus, determining the intrinsic structure of this space presents a theoretical and experimental challenge. To address this challenge, we introduce a new framework for applying topological data analysis (TDA) to spike train data and use it to determine the geometry of spiking patterns in the visual cortex. Key to our approach is a parametrized family of distances based on the timing of spikes that quantifies the dissimilarity between neuronal responses. We applied TDA to visually driven single-unit and multiple single-unit spiking activity in macaque V1 and V2. TDA across timescales reveals a common geometry for spiking patterns in V1 and V2 which, among simple models, is most similar to that of a low-dimensional space endowed with Euclidean or hyperbolic geometry with modest curvature. Remarkably, the inferred geometry depends on timescale and is clearest for the timescales that are important for encoding contrast, orientation and spatial correlations.

Interaction Mechanism Between the HSV-1 Glycoprotein B and the Antimicrobial Peptide Amyloid-ß.

Background: Unravelling the mystery of Alzheimer's disease (AD) requires urgent resolution given the worldwide increase of the aging population. There is a growing concern that the current leading AD hypothesis, the amyloid cascade hypothesis, does not stand up to validation with respect to emerging new data. Indeed, several paradoxes are being discussed in the literature, for instance, both the deposition of the amyloid-ß peptide (Aß) and the intracellular neurofibrillary tangles could occur within the brain without any cognitive pathology. Thus, these paradoxes suggest that something more fundamental is at play in the onset of the disease and other key and related pathomechanisms must be investigated. Objective: The present study follows our previous investigations on the infectious hypothesis, which posits that some pathogens are linked to late onset AD. Our studies also build upon the finding that Aß is a powerful antimicrobial agent, produced by neurons in response to viral infection, capable of inhibiting pathogens as observed in in vitro experiments. Herein, we ask what are the molecular mechanisms in play when Aß neutralizes infectious pathogens? Methods: To answer this question, we probed at nanoscale lengths with FRET (Förster Resonance Energy Transfer), the interaction between Aß peptides and glycoprotein B (responsible of virus-cell binding) within the HSV-1 virion. Results: The experiments show an energy transfer between Aß peptides and glycoprotein B when membrane is intact. No energy transfer occurs after membrane disruption or treatment with blocking antibody. Conclusion: We concluded that Aß insert into viral membrane, close to glycoprotein B, and participate in virus neutralization.

Immunosenescence and Altered Vaccine Efficiency in Older Subjects: A Myth Difficult to Change.

Organismal ageing is associated with many physiological changes, including differences in the immune system of most animals. These differences are often considered to be a key cause of age-associated diseases as well as decreased vaccine responses in humans. The most often cited vaccine failure is seasonal influenza, but, while it is usually the case that the efficiency of this vaccine is lower in older than younger adults, this is not always true, and the reasons for the differential responses are manifold. Undoubtedly, changes in the innate and adaptive immune response with ageing are associated with failure to respond to the influenza vaccine, but the cause is unclear. Moreover, recent advances in vaccine formulations and adjuvants, as well as in our understanding of immune changes with ageing, have contributed to the development of vaccines, such as those against herpes zoster and SARS-CoV-2, that can protect against serious disease in older adults just as well as in younger people. In the present article, we discuss the reasons why it is a myth that vaccines inevitably protect less well in older individuals, and that vaccines represent one of the most powerful means to protect the health and ensure the quality of life of older adults.