A Novel Bursting Mechanism of Type A Neurons in Injured Dorsal Root Ganglia

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.

Bursting is ubiquitous in cellular excitable rhythms and comes in a plethora of patterns, both experimentally recorded and reproduced through models. As these different patterns may reflect different coding or information properties, it is therefore crucial to develop modeling frameworks that can both capture them and understand their characteristics. In this review, we propose a comprehensive account of the main bursting classification systems that have been developed over the past 40 years, together with recent developments allowing us to extend these classifications. Based upon bifurcation theory and heavily reliant on timescale separation, these schemes take full advantage of the fast subsystem analysis, obtained when slow variables are frozen and considered as bifurcation parameters. We complement this classical view by showing that nontrivial slow subsystem may also encode key informations important to classify bursting rhythms, due to the presence of so-called folded-node singularities. We provide minimal idealized models as well as one generic conductance-based example displaying bursting oscillations that require our extended classification in order to be fully characterized. We also highlight examples of biological data that could be suitably revisited with the lenses of this extended classifications and could lead to new models of complex cellular activity.

Canonical models for torus canards in elliptic bursters

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards may appear, the difference between the two being the fast subsystem bifurcation that they approach: saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan-type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich [SIAM J. Appl. Math. 60, 503-535 (2000)] and adapted to elliptic bursting by Ju et al. [Chaos 28, 106317 (2018)] and we show that it does not produce mixed-type torus canards due to a nongeneric transition at one end of the bursting regime. We, therefore, introduce a perturbative term in the slow equation, which extends this canonical form to a new one that we call Leidenator and which supports the right transitions to and from elliptic bursting via classical and mixed-type torus canards, respectively. Throughout the study, we use singular flows ( ε=0) to predict the full system's dynamics ( ε>0 small enough). We consider three singular flows, slow, fast, and average slow, so as to appropriately construct singular orbits corresponding to all relevant dynamics pertaining to elliptic bursting and torus canards. Finally, we comment on possible links with mixed-type torus canards and folded-saddle-node singularities in non-canonical elliptic bursters that possess a natural three-timescale structure.

Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker

The transition from chaotic bursting to chaotic spiking has been simulated and analyzed in theoretical neuronal models. In the present study, we report experimental observations in a neural pacemaker of a transition from chaotic bursting to chaotic spiking within a bifurcation scenario from period-1 bursting to period-1 spiking. This was induced by adjusting extracellular calcium or potassium concentrations. The bifurcation scenario began from period-doubling bifurcations or period-adding sequences of bursting pattern. This chaotic bursting is characterized by alternations between multiple continuous spikes and a long duration of quiescence, whereas chaotic spiking is comprised of fast, continuous spikes without periods of quiescence. Chaotic bursting changed to chaotic spiking as long interspike intervals (ISIs) of quiescence disappeared within bursting patterns, drastically decreasing both ISIs and the magnitude of the chaotic attractors. Deterministic structures of the chaotic bursting and spiking patterns are also identified by a short-term prediction. The experimental observations, which agree with published findings in theoretical neuronal models, demonstrate the existence and reveal the dynamics of a neuronal transition from chaotic bursting to chaotic spiking in the nervous system.

Dorsal root ganglion compression as an animal model of sciatica and low back pain

As sciatica and low back pain are among the most common medical complaints, many studies have duplicated these conditions in animals. Chronic compression of the dorsal root ganglion (CCD) is one of these models. The surgery is simple: after exposing the L4/L5 intervertebral foramina, stainless steel rods are implanted unilaterally, one rod for each vertebra, to chronically compress the lumbar dorsal root ganglion (DRG). Then, CCD can be used to simulate the clinical conditions caused by stenosis, such as a laterally herniated disc or foraminal stenosis. As the intraforaminal implantation of a rod results in neuronal somal hyperexcitability and spontaneous action potentials associated with hyperalgesia, spontaneous pain, and mechanical allodynia, CCD provides an animal model that mimics radicular pain in humans. This review concerns the mechanisms of neuronal hyperexcitability, focusing on various patterns of spontaneous discharge including one possible pain signal for mechanical allodynia - evoked bursting. Also, new data regarding its significant property of maintaining peripheral input are also discussed. Investigations using this animal model will enhance our understanding of the neural mechanisms for low back pain and sciatica. Furthermore, the peripheral location of the DRG facilitates its use as a locus for controlling pain with minimal central effects, in the hope of ultimately uncovering analgesics that block neuropathic pain without influencing physiological pain.

Identification of molecular pathologies sufficient to cause neuropathic excitability in primary somatosensory afferents using dynamical systems theory

Pain caused by nerve injury (i.e. neuropathic pain) is associated with development of neuronal hyperexcitability at several points along the pain pathway. Within primary afferents, numerous injury-induced changes have been identified but it remains unclear which molecular changes are necessary and sufficient to explain cellular hyperexcitability. To investigate this, we built computational models that reproduce the switch from a normal spiking pattern characterized by a single spike at the onset of depolarization to a neuropathic one characterized by repetitive spiking throughout depolarization. Parameter changes that were sufficient to switch the spiking pattern also enabled membrane potential oscillations and bursting, suggesting that all three pathological changes are mechanistically linked. Dynamical analysis confirmed this prediction by showing that excitability changes co-develop when the nonlinear mechanism responsible for spike initiation switches from a quasi-separatrix-crossing to a subcritical Hopf bifurcation. This switch stems from biophysical changes that bias competition between oppositely directed fast- and slow-activating conductances operating at subthreshold potentials. Competition between activation and inactivation of a single conductance can be similarly biased with equivalent consequences for excitability. “Bias” can arise from a multitude of molecular changes occurring alone or in combination; in the latter case, changes can add or offset one another. Thus, our results identify pathological change in the nonlinear interaction between processes affecting spike initiation as the critical determinant of how simple injury-induced changes at the molecular level manifest complex excitability changes at the cellular level. We demonstrate that multiple distinct molecular changes are sufficient to produce neuropathic changes in excitability; however, given that nerve injury elicits numerous molecular changes that may be individually sufficient to alter spike initiation, our results argue that no single molecular change is necessary to produce neuropathic excitability. This deeper understanding of degenerate causal relationships has important implications for how we understand and treat neuropathic pain.

Neuropathic pain results from damage to the nervous system. Much is known about the multitude of molecular and cellular changes that are triggered by nerve injury (and which correlate with development of neuropathic pain), but little is understood about how those changes cause neuropathic pain. Rather than identifying what changes occur after nerve injury (which has already been the focus of countless studies), our study focuses on identifying which changes are functionally important. Specifically, we explain how certain molecular changes, acting alone or in combination, cause a triad of neuropathic changes in primary afferent excitability. Through computational modeling and nonlinear dynamical analysis, we demonstrate that the entire triad of excitability changes arises from a single switch in the nonlinear mechanism responsible for spike initiation. Going further, we demonstrate that many distinct molecular changes are sufficient to produce that switch but that no single molecular change is necessary if more than one sufficient change co-occurs after nerve injury, which appears to be the case. The issue becomes whether molecular changes combine to reach some tipping point whereupon cellular excitability is qualitatively altered. This highlights the importance of nonlinearities for neuropathic pain and the need for more computational pain research.

Deterministic Dynamics in Neuronal Discharge from Pallidotomy Targets

The non-linear dynamic specificity of the firing pattern discharged from neurons of the internal globus pallidus (GPi) was investigated by recording their spontaneous firing using a microelectrode during posteroventral pallidotomy in eight patients with Parkinson's disease. Raw data from the cells were processed to extract spiking events (discharges above a selected threshold) and the interspike interval was measured. Using the unstable periodic orbits extraction method, significant period-1, −2 and −3 orbits were identified in burst firing discharged from the GPi cells in all eight patients, suggesting that deterministic dynamics exist in the timing of the discharges. As well as providing a useful peri-operative technique for locating posteroventral pallidotomy targets in Parkinson's disease, this method also provides a promising basis for investigating characteristic neuronal discharges in other regions of the brain and for various other neurological disorders.

Synaptic transmission of chaotic spike trains between primary afferent fiber and spinal dorsal horn neuron in the rat

Primary sensory neurons can generate irregular burst firings in which the existence of significant deterministic behaviors of chaotic dynamics has been proved with nonlinear time series analysis. But how well the deterministic characteristics and neural information of presynaptic chaotic spike trains were transmitted into postsynaptic spike trains is still an open question. Here we investigated the synaptic transmission of chaotic spike trains between primary Adelta afferent fiber and spinal dorsal horn neuron. Two kinds of basic stimulus unit, brief burst and single pulse, were employed by us to comprise chaotic stimulus trains. For time series analysis, we defined "events" as the longest sequences of spikes with all interspike intervals less than or equal to a certain threshold and extracted the interevent intervals (IEIs) from spike trains. Return map analysis of the IEI series showed that the main temporal structure of chaotic input trains could be detected in postsynaptic output trains, especially under brief-burst stimulation. Using correlation dimension and nonlinear prediction methods, we found that synaptic transmission could influence the nonlinear characteristics of chaotic trains, such as fractal dimension and short-term predictability, with greater influence made under single-pulse stimulation. By calculating the mutual information between input and output trains, we found the information carried by presynaptic spike trains could not be completely transmitted at primary afferent synapses, and that brief bursts could more reliably transmit the information carried by chaotic input trains across synapses. These results indicate that although unreliability exists during synaptic transmission, the main deterministic characteristics of chaotic burst trains can be transmitted across primary afferent synapses. Moreover, brief bursts that come from the periphery can more reliably transmit neural information between primary afferent fibers and spinal dorsal horn neurons.