William H. Barnett

Bistability of silence and seizure-like bursting

Neuronal circuits exhibiting seizure episodes have been shown to be prone to multistability. The coexistence of normal and pathological regimes could explain why seizures suddenly start and stop. Methods developed in dynamical systems theory are powerful tools for determining the cellular mechanisms that underlie multistable seizure dynamics. Here, we present two different approaches to assess multistability in a model neuron. In this model, we identified a bursting regime and a silent regime. First, we investigated properties of a square pulse of injected current which produced a switch from seizure-like bursting into silence. By systematically varying the phase and amplitude of the pulse, we found contiguous pulse parameter sets, so-called windows, that satisfied this criterion, and we determined the dependence of these windows on the parameter gleak. As gleak increased, the size of each window scaled according to the same law as the amplitude of the saddle orbit. Second, we examined the role of each current in supporting bistability of bursting and silence. We defined the index of propensity for multistability as the range of gleak for which bursting and silence coexisted. We computed this quantity while iteratively varying the maximal conductance of each voltage-gated current one at a time. Increasing the maximal conductance of the slow potassium current or the hyperpolarization-activated current increased the range of bistability. In contrast, decreasing the maximal conductance of the persistent sodium current increased the range of bistability.

Chemoreception and neuroplasticity in respiratory circuits

ABSTRACT The respiratory central pattern generator must respond to chemosensory cues to maintain oxygen (O2) and carbon dioxide (CO2) homeostasis in the blood and tissues. To do this, sensorial cells located in the periphery and central nervous system monitor the arterial partial pressure of O2 and CO2 and initiate respiratory and autonomic reflex adjustments in conditions of hypoxia and hypercapnia. In conditions of chronic intermittent hypoxia (CIH), repeated peripheral chemoreceptor input mediated by the nucleus of the solitary tract induces plastic changes in respiratory circuits that alter baseline respiratory and sympathetic motor outputs and result in chemoreflex sensitization, active expiration, and arterial hypertension. Herein, we explored the hypothesis that the CIH‐induced neuroplasticity primarily consists of increased excitability of pre‐inspiratory/inspiratory neurons in the pre‐Bötzinger complex. To evaluate this hypothesis and elucidate neural mechanisms for the emergence of active expiration and sympathetic overactivity in CIH‐treated animals, we extended a previously developed computational model of the brainstem respiratory‐sympathetic network to reproduce experimental data on peripheral and central chemoreflexes post‐CIH. The model incorporated neuronal connections between the 2nd‐order NTS neurons and peripheral chemoreceptors afferents, the respiratory pattern generator, and sympathetic neurons in the rostral ventrolateral medulla in order to capture key features of sympathetic and respiratory responses to peripheral chemoreflex stimulation. Our model identifies the potential neuronal groups recruited during peripheral chemoreflex stimulation that may be required for the development of inspiratory, expiratory and sympathetic reflex responses. Moreover, our model predicts that pre‐inspiratory neurons in the pre‐Bötzinger complex experience plasticity of channel expression due to excessive excitation during peripheral chemoreflex. Simulations also show that, due to positive interactions between pre‐inspiratory neurons in the pre‐Bötzinger complex and expiratory neurons in the retrotrapezoid nucleus, increased excitability of the former may lead to the emergence of the active expiratory pattern at normal CO2 levels found after CIH exposure. We conclude that neuronal type specific neuroplasticity in the pre‐Bötzinger complex induced by repetitive episodes of peripheral chemoreceptor activation by hypoxia may contribute to the development of sympathetic over‐activity and hypertension.

High prevalence of multistability of rest states and bursting in a database of a model neuron.

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.

Na + /K + pump interacts with the h -current to control bursting activity in central pattern generator neurons of leeches

The dynamics of different ionic currents shape the bursting activity of neurons and networks that control motor output. Despite being ubiquitous in all animal cells, the contribution of the Na+/K+ pump current to such bursting activity has not been well studied. We used monensin, a Na+/H+ antiporter, to examine the role of the pump on the bursting activity of oscillator heart interneurons in leeches. When we stimulated the pump with monensin, the period of these neurons decreased significantly, an effect that was prevented or reversed when the h-current was blocked by Cs+. The decreased period could also occur if the pump was inhibited with strophanthidin or K+-free saline. Our monensin results were reproduced in model, which explains the pump’s contributions to bursting activity based on Na+ dynamics. Our results indicate that a dynamically oscillating pump current that interacts with the h-current can regulate the bursting activity of neurons and networks. DOI: http://dx.doi.org/10.7554/eLife.19322.001

A Codimension-2 Bifurcation Controlling Endogenous Bursting Activity and Pulse-Triggered Responses of a Neuron Model

The dynamics of individual neurons are crucial for producing functional activity in neuronal networks. An open question is how temporal characteristics can be controlled in bursting activity and in transient neuronal responses to synaptic input. Bifurcation theory provides a framework to discover generic mechanisms addressing this question. We present a family of mechanisms organized around a global codimension-2 bifurcation. The cornerstone bifurcation is located at the intersection of the border between bursting and spiking and the border between bursting and silence. These borders correspond to the blue sky catastrophe bifurcation and the saddle-node bifurcation on an invariant circle (SNIC) curves, respectively. The cornerstone bifurcation satisfies the conditions for both the blue sky catastrophe and SNIC. The burst duration and interburst interval increase as the inverse of the square root of the difference between the corresponding bifurcation parameter and its bifurcation value. For a given set of burst duration and interburst interval, one can find the parameter values supporting these temporal characteristics. The cornerstone bifurcation also determines the responses of silent and spiking neurons. In a silent neuron with parameters close to the SNIC, a pulse of current triggers a single burst. In a spiking neuron with parameters close to the blue sky catastrophe, a pulse of current temporarily silences the neuron. These responses are stereotypical: the durations of the transient intervals–the duration of the burst and the duration of latency to spiking–are governed by the inverse-square-root laws. The mechanisms described here could be used to coordinate neuromuscular control in central pattern generators. As proof of principle, we construct small networks that control metachronal-wave motor pattern exhibited in locomotion. This pattern is determined by the phase relations of bursting neurons in a simple central pattern generator modeled by a chain of oscillators.

Coregulation of ionic currents maintaining the duty cycle of bursting

Central pattern generators (CPGs) are the oscillatory neuronal networks which control rhythmic movements of animals. Some CPGs keep the phase relationships between the neurons’ oscillatory activities over a wide range of the cycle periods. Maintenance of the duty cycle of the bursting activity could be a key feature for a variety of dynamical mechanisms supporting phase constancy in oscillatory neuronal networks. It is a form of cellular homeostasis of neuronal activity. Among other currents, hyperpolarization-activated currents and potassium currents have been shown to be a ubiquitous target for modulation and homeostasis [1,2]. Here we present a novel mechanism of coregulation of currents which preserves duty cycle of bursting activity over a range of cycle periods. We develop a generic lowdimensional Hodgkin-Huxley type model stemming from a model of the leech heart interneuron under certain pharmacological conditions [3]. Application of Co 2+ and 4-AP blocks Ca 2+ currents, the synaptic currents and most of the K + currents. The model contains the slow potassium current (IK2), the fast sodium current (INa). Our new model also includes the hyperpolarization activated current (Ih). Bifurcation theory allows us to make predictions concerning the temporal characteristics of the dynamics of bursting nearby the critical transitions between activities. Shilnikov & Cymblayuk sho wed that the transition from bursting into tonic spiking (blue sky catastrophe) determines the dependence of the burst duration on the voltage of half-activation of IK2 (θ K2 )a s one over square root of the parameter value [4]. Here we show that the half-activation potential of Ih (θ h) controls the interburst interval as one over square root of the parameter value. We investigate the activity of the model to identify mechanisms of coregulation of IK2 and Ih maintaining the duty cycle. Bifurcation analysis of the model was performed using θ K2 and θ h, as controlling parameters. We investigated the temporal characteristics of bursting activity. We identified a saddle node bifurcation for periodic orbits determining the blue sky catastrophe [4] and a saddle node bifurcation for stationary states (SNIC) [5]. We showed the temporal characteristics of bursting depend on the location of the bifurcation curves. By coordinating the two parameters, we were able to increase the period such that the burst duration and interburst interval maintained constant proportion. The coregulation consists of a negative correlation of θ K2 and θ h, which is steeper for the higher duty cycles.

The Kölliker-Fuse orchestrates the timing of expiratory abdominal nerve bursting

Coordination of respiratory pump and valve muscle activity is essential for normal breathing. A hallmark respiratory response to hypercapnia and hypoxia is the emergence of active exhalation, characterized by abdominal muscle pumping during the late one-third of expiration (late-E phase). Late-E abdominal activity during hypercapnia has been attributed to the activation of expiratory neurons located within the parafacial respiratory group (pFRG). However, the mechanisms that control emergence of active exhalation, and its silencing in restful breathing, are not completely understood. We hypothesized that inputs from the Kölliker-Fuse nucleus (KF) control the emergence of late-E activity during hypercapnia. Previously, we reported that reversible inhibition of the KF reduced postinspiratory (post-I) motor output to laryngeal adductor muscles and brought forward the onset of hypercapnia-induced late-E abdominal activity. Here we explored the contribution of the KF for late-E abdominal recruitment during hypercapnia by pharmacologically disinhibiting the KF in in situ decerebrate arterially perfused rat preparations. These data were combined with previous results and incorporated into a computational model of the respiratory central pattern generator. Disinhibition of the KF through local parenchymal microinjections of gabazine (GABAA receptor antagonist) prolonged vagal post-I activity and inhibited late-E abdominal output during hypercapnia. In silico, we reproduced this behavior and predicted a mechanism in which the KF provides excitatory drive to post-I inhibitory neurons, which in turn inhibit late-E neurons of the pFRG. Although the exact mechanism proposed by the model requires testing, our data confirm that the KF modulates the formation of late-E abdominal activity during hypercapnia. NEW & NOTEWORTHY The pons is essential for the formation of the three-phase respiratory pattern, controlling the inspiratory-expiratory phase transition. We provide functional evidence of a novel role for the Kölliker-Fuse nucleus (KF) controlling the emergence of abdominal expiratory bursts during active expiration. A computational model of the respiratory central pattern generator predicts a possible mechanism by which the KF interacts indirectly with the parafacial respiratory group and exerts an inhibitory effect on the expiratory conditional oscillator.

Reward Based Motor Adaptation Mediated by Basal Ganglia

It is widely accepted that the basal ganglia (BG) play a key role in action selection and reinforcement learning. However, despite considerable number of studies, the BG architecture and function are not completely understood. Action selection and reinforcement learning are facilitated by the activity of dopaminergic neurons, which encode reward prediction errors when reward outcomes are higher or lower than expected. The BG are thought to select proper motor responses by gating appropriate actions, and suppressing inappropriate ones. The direct striato-nigral (GO) and the indirect striato-pallidal (NOGO) pathways have been suggested to provide the functions of BG in the two-pathway concept. Previous models confirmed the idea that these two pathways can mediate the behavioral choice, but only for a relatively small number of potential behaviors. Recent studies have provided new evidence of BG involvement in motor adaptation tasks, in which adaptation occurs in a non-error-based manner. In such tasks, there is a continuum of possible actions, each represented by a complex neuronal activity pattern. We extended the classical concept of the two-pathway BG by creating a model of BG interacting with a movement execution system, which allows for an arbitrary number of possible actions. The model includes sensory and premotor cortices, BG, a spinal cord network, and a virtual mechanical arm performing 2D reaching movements. The arm is composed of 2 joints (shoulder and elbow) controlled by 6 muscles (4 mono-articular and 2 bi-articular). The spinal cord network contains motoneurons, controlling the muscles, and sensory interneurons that receive afferent feedback and mediate basic reflexes. Given a specific goal-oriented motor task, the BG network through reinforcement learning constructs a behavior from an arbitrary number of basic actions represented by cortical activity patterns. Our study confirms that, with slight modifications, the classical two-pathway BG concept is consistent with results of previous studies, including non-error based motor adaptation experiments, pharmacological manipulations with BG nuclei, and functional deficits observed in BG-related motor disorders.

Bifurcation control of gait transition in insect locomotion

Insect locomotion presents a type of pattern adaptation in which the phase relations change with the speed of walking. In forward walking, the protraction of legs progresses sequentially one after the other from posterior to anterior: a metachronal wave gait. The duration of protraction is roughly invariant to the speed of locomotion, and the duration of retraction is linearly dependent on the step period [1]. These constraints determine phase relations in a gait over a range of speeds. We described a model of insect locomotion employing the cornerstone Shilnikov bifurcation [2,3]. This bifurcation generates mechanisms that control burst duration and interburst interval in endogenous bursting and the duration of pulse-triggered bursts in endogenously silent neurons [3,4]. We suggest that the mechanism describing the stereotypical burst responses of silent neurons explains smooth transitions between gaits. The burst duration is controlled by the half-activation voltage of a potassium current (-θK2). In the model, each leg was controlled by one oscillator consisting of two mutually inhibitory interneurons: protraction and retraction interneurons. The model central pattern generator (CPG) contains three coupled oscillators: PP-PR, MsP-MsR, MtP-MtR labeling protractor and retractor interneurons each for the prothoracic, mesothoracic, and metathoracic segments, respectively. The bifurcation-generated mechanisms make quantitative predictions on the CPG activity. The duration of the burst was governed by the inverse-square-root law (Figure 1). The burst duration grows linear with the number of spikes per burst in retractor interneurons. The burst duration of the retractor interneuron determined the period of the network. The retractor burst duration determined what type of gait was exhibited by network activity. As such, we were able to control the smooth transition from metachronal wave to tripod gait. While the duty cycle of retractor interneurons was greater than 50%, we observed a gait comprised of metachronal progression of bursts from posterior to anterior. When the duty cycle became 50%, we observed the tripod gait, where the activity in the prothoracic and metathoracic protractor interneurons was synchronous. In conclusion, we constructed a locomotor CPG model using a mechanism generated by the cornerstone bifurcation. This mechanism controls the duration of pulse-triggered bursts in endogenously silent neurons and governs a smooth transition from a metachronal gait to a tripod gait.

Cellular mechanisms generating bursting activity in neuronal networks

An open question in neuroscience is how the temporal characteristics of bursting activity are controlled by intrinsic biophysical characteristics. We present two mechanisms organized around the cornerstone bifurcation in a 3D Hodgkin-Huxley style neuronal model. This bifurcation satisfies the criteria for both the Shilnikov blue sky catastrophe and the saddle-node bifurcation on an invariant circle (SNIC) [1-3]. The burst duration (BD) and interburst interval (IBI) increase as the inverse of the square root of the difference between the corresponding parameter and its bifurcation value. The cornerstone bifurcation also determines the stereotypical transient responses of silent and spiking neurons [3]. The mechanisms presented here are based on these transient responses. The first mechanism described half-center oscillator consisting of two intrinsically silent, mutually inhibitory neurons (Figure 1AB). These two neurons were in the silent parameter regime but close to the cornerstone bifurcation. The two coupled neurons showed anti-phase bursting activity without the release and escape mechanisms. We found that if the half-activation voltage of a non-inactivating potassium current (V1/2,mk2) was systematically shifted towards the bifurcation value for the saddle-node bifurcation of periodic orbits, the BDs of both neurons increased in accordance with the inverse-square-root law and exhibited linear dependence on the spike number per burst. Figure 1 The cornerstone bifurcation generates mechanisms controlling bursting in neural networks. (A). Two endogenously silent, mutually inhibitory coupled neurons produce alternating bursting activity. V1/2,mk2 controls BDs. (B). The BDs are depicted as blue ... The second mechanism described the bursting activity of two intrinsically spiking, mutually excitatory neurons. The parameters of the neurons were in vicinity of the cornerstone bifurcation. This network exhibited synchronized bursting (Figure 1CD). Remarkably, excitatory interaction between endogenously spiking neurons essentially led to reduction of excitability of the network. When the half-activation voltage of hyperpolarization-activated current (V1/2,mh) was systematically shifted to the bifurcation value, IBIs of both neurons increased in accordance with the inverse-square-root law and the BDs and the number of spikes per burst stayed constant (6 spikes per burst). This study reveals new mechanisms controlling bursting activity in small neuronal networks based on cellular properties determining transient responses of endogenously spiking and silent neurons. These mechanisms are generic and could govern the bursting regimes in rhythmic neuronal network such as central pattern generators.