Neurons as oscillators

Mechanisms of pacemaking in mammalian neurons

Many neurons in the mammalian brain show pacemaking activity: rhythmic generation of action potentials in the absence of sensory or synaptic input. Slow pacemaking of neurons releasing modulatory transmitters is easy to rationalize. More surprisingly, many neurons in the motor system also show pacemaking activity, often rapid, including cerebellar Purkinje neurons that fire spontaneously at 20-100 Hz, as well as key neurons in the basal ganglia, including subthalamic nucleus neurons and globus pallidus neurons. Although the spontaneous rhythmic firing of pacemaking neurons is phenomenologically similar to cardiac pacemaking, the underlying ionic mechanism in most neurons is quite different than for cardiac pacemaking. Few spontaneously active neurons rely on HCN 'pacemaker' channels for their activity. Most commonly, a central element is 'persistent' sodium current, steady-state subthreshold current carried by the same voltage-dependent sodium channels that underlie fast action potentials. Persistent sodium current is a steeply voltage-dependent current with a midpoint near -60 mV, which results in regenerative spontaneous depolarization once it produces a net inward current when summed with all other background currents, often at voltages as negative as -70 mV. This 'engine' of pacemaking is present in almost all neurons and must be held in check in non-pacemaking neurons by sufficiently large competing outward currents from background potassium channels. The intrinsic propensity of neurons to fire spontaneously underlies key normal functions such as respiration and generates the complex background oscillatory circuits revealed in EEGs, but can also produce out-of-control oscillations of overall brain function in epilepsy, ataxia and tremor.

Impact of Unitary Synaptic Inhibition on Spike Timing in Ventral Tegmental Area Dopamine Neurons

Midbrain dopamine neurons receive convergent synaptic input from multiple brain areas, which perturbs rhythmic pacemaking to produce the complex firing patterns observed in vivo. This study investigated the impact of single and multiple inhibitory inputs on ventral tegmental area (VTA) dopamine neuron firing in mice of both sexes using novel experimental measurements and modeling. We first measured unitary inhibitory postsynaptic currents produced by single axons using both minimal electrical stimulation and minimal optical stimulation of rostromedial tegmental nucleus and ventral pallidum afferents. We next determined the phase resetting curve, the reversal potential for GABAA receptor-mediated inhibitory postsynaptic currents (IPSCs), and the average interspike membrane potential trajectory during pacemaking. We combined these data in a phase oscillator model of a VTA dopamine neuron, simulating the effects of unitary inhibitory postsynaptic conductances (uIPSGs) on spike timing and rate. The effect of a uIPSG on spike timing was predicted to vary according to its timing within the interspike interval or phase. Simulations were performed to predict the pause duration resulting from the synchronous arrival of multiple uIPSGs and the changes in firing rate and regularity produced by asynchronous uIPSGs. The model data suggest that asynchronous inhibition is more effective than synchronous inhibition, because it tends to hold the neuron at membrane potentials well positive to the IPSC reversal potential. Our results indicate that small fluctuations in the inhibitory synaptic input arriving from the many afferents to each dopamine neuron are sufficient to produce highly variable firing patterns, including pauses that have been implicated in reinforcement.

Resilience of the slow component in timescale-separated synchronized oscillators

Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations.

Synchronization of delayed coupled neurons with multiple synaptic connections

Synchronization is a key feature of the brain dynamics and is necessary for information transmission across brain regions and in higher brain functions like cognition, learning and memory. Experimental findings demonstrated that in cortical microcircuits there are multiple synapses between pairs of connected neurons. Synchronization of neurons in the presence of multiple synaptic connections may be relevant for optimal learning and memory, however, its effect on the dynamics of the neurons is not adequately studied. Here, we address the question that how changes in the strength of the synaptic connections and transmission delays between neurons impact synchronization in a two-neuron system with multiple synapses. To this end, we analytically and computationally investigated synchronization dynamics by considering both phase oscillator model and conductance-based Hodgkin-Huxley (HH) model. Our results show that symmetry/asymmetry of feedforward and feedback connections crucially determines stability of the phase locking of the system based on the strength of connections and delays. In both models, the two-neuron system with multiple synapses achieves in-phase synchrony in the presence of small and large delays, whereas an anti-phase synchronization state is favored for median delays. Our findings can expand the understanding of the functional role of multisynaptic contacts in neuronal synchronization and may shed light on the dynamical consequences of pathological multisynaptic connectivity in a number of brain disorders.

Coalescence of limit cycles in the presence of noise

Complex dynamical systems may exhibit multiple steady states, including time-periodic limit cycles, where the final trajectory depends on initial conditions. With tuning of parameters, limit cycles can proliferate or merge at an exceptional point. Here we ask how dynamics in the vicinity of such a bifurcation are influenced by noise. A pitchfork bifurcation can be used to induce bifurcation behavior. We model a limit cycle with the normal form of the Hopf oscillator, couple it to the pitchfork, and investigate the resulting dynamical system in the presence of noise. We show that the generating functional for the averages of the dynamical variables factorizes between the pitchfork and the oscillator. The statistical properties of the pitchfork in the presence of noise in its various regimes are investigated and a scaling theory is developed for the correlation and response functions, including a possible symmetry-breaking field. The analysis is done by perturbative calculations as well as numerical means. Finally, observables illustrating the coupling of a system with a limit cycle to a pitchfork are discussed and the phase-phase correlations are shown to exhibit nondiffusive behavior with universal scaling.

Modeling of cerebellar transcranial electrical stimulation effects on hand tremor in Parkinson's disease

Introduction

Parkinson's disease (PD) is a neurodegenerative disorder with different motor and neurocognitive symptoms. Tremor is a well-known symptom of this disease. Increasing evidence suggested that the cerebellum may substantially contribute to tremors as a clinical symptom of PD. However, the theoretical foundations behind these observations are not yet fully understood.

Methods

In this study, a computational model is proposed to consider the role of the cerebellum and to show the effectiveness of cerebellar transcranial alternating current stimulation (tACS) on the rest tremor in participants with PD. The proposed model consists of the cortex, cerebellum, spinal circuit-muscular system (SC-MS), and basal ganglia blocks as the most critical parts of the brain, which are involved in generating rest tremors. The cortex, cerebellum, and SC-MS blocks were modeled using Van der Pol oscillators that interacted through synchronization procedures. Basal ganglia are considered as a regulator of the coupling weights defined between oscillators. In order to evaluate the global behavior of the model, we applied tACS on the cerebellum of fifteen PD patients for 15 min at each patient's peak frequency of their rest tremors. A tri-axial accelerometer recorded rest tremors before, during, and after the tACS.

Results and Discussion

The simulation of the model provides a suggestion for the possible role of the cerebellum on rest tremors and how cerebellar tACS can affect these tremors. Results of human experiments also showed that the online and offline effects of cerebellar tACS could lead to the reduction of rest tremors significantly by about %76 and %68, respectively. Our findings suggest that the cerebellar tACS could serve as a reliable, therapeutic technique to suppress the PD tremor.

Active oscillations in microscale navigation

Living organisms routinely navigate their surroundings in search of better conditions, more food, or to avoid predators. Typically, animals do so by integrating sensory cues from the environment with their locomotor apparatuses. For single cells or small organisms that possess motility, fundamental physical constraints imposed by their small size have led to alternative navigation strategies that are specific to the microscopic world. Intriguingly, underlying these myriad exploratory behaviours or sensory functions is the onset of periodic activity at multiple scales, such as the undulations of cilia and flagella, the vibrations of hair cells, or the oscillatory shape modes of migrating neutrophils. Here, I explore oscillatory dynamics in basal microeukaryotes and hypothesize that these active oscillations play a critical role in enhancing the fidelity of adaptive sensorimotor integration.

Propagation of Oscillations in the Indirect Pathway of the Basal Ganglia

Oscillatory signals propagate in the basal ganglia from prototypic neurons in the external globus pallidus (GPe) to their target neurons in the substantia nigra pars reticulata (SNr), internal pallidal segment, and subthalamic nucleus. Neurons in the GPe fire spontaneously, so oscillatory input signals can be encoded as changes in timing of action potentials within an ongoing spike train. When GPe neurons were driven by an oscillatory current in male and female mice, these spike-timing changes produced spike-oscillation coherence over a range of frequencies extending at least to 100 Hz. Using the known kinetics of the GPe→SNr synapse, we calculated the postsynaptic currents that would be generated in SNr neurons from the recorded GPe spike trains. The ongoing synaptic barrage from spontaneous firing, frequency-dependent short-term depression, and stochastic fluctuations at the synapse embed the input oscillation into a noisy sequence of synaptic currents in the SNr. The oscillatory component of the resulting synaptic current must compete with the noisy spontaneous synaptic barrage for control of postsynaptic SNr neurons, which have their own frequency-dependent sensitivities. Despite this, SNr neurons subjected to synaptic conductance changes generated from recorded GPe neuron firing patterns also became coherent with oscillations over a broad range of frequencies. The presynaptic, synaptic, and postsynaptic frequency sensitivities were all dependent on the firing rates of presynaptic and postsynaptic neurons. Firing rate changes, often assumed to be the propagating signal in these circuits, do not encode most oscillation frequencies, but instead determine which signal frequencies propagate effectively and which are suppressed.SIGNIFICANCE STATEMENT Oscillations are present in all the basal ganglia nuclei, include a range of frequencies, and change over the course of learning and behavior. Exaggerated oscillations are a hallmark of basal ganglia pathologies, and each has a specific frequency range. Because of its position as a hub in the basal ganglia circuitry, the globus pallidus is a candidate origin for oscillations propagating between nuclei. We imposed low-amplitude oscillations on individual globus pallidus neurons at specific frequencies and measured the coherence between the oscillation and firing as a function of frequency. We then used these responses to measure the effectiveness of oscillatory propagation to other basal ganglia nuclei. Propagation was effective for oscillation frequencies as high as 100 Hz.

Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles

The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.

Optimal synchronization in pulse-coupled oscillator networks using reinforcement learning

Spontaneous synchronization is ubiquitous in natural and man-made systems. It underlies emergent behaviors such as neuronal response modulation and is fundamental to the coordination of robot swarms and autonomous vehicle fleets. Due to its simplicity and physical interpretability, pulse-coupled oscillators has emerged as one of the standard models for synchronization. However, existing analytical results for this model assume ideal conditions, including homogeneous oscillator frequencies and negligible coupling delays, as well as strict requirements on the initial phase distribution and the network topology. Using reinforcement learning, we obtain an optimal pulse-interaction mechanism (encoded in phase response function) that optimizes the probability of synchronization even in the presence of nonideal conditions. For small oscillator heterogeneities and propagation delays, we propose a heuristic formula for highly effective phase response functions that can be applied to general networks and unrestricted initial phase distributions. This allows us to bypass the need to relearn the phase response function for every new network.

[Frequency-Specific Alterations of Spontaneous Brain Activity in First-Episode Drug-Naïve Schizophrenia]

Objective

To investigate frequency-specific alterations of spontaneous brain activity in first-episode drug-naïve schizophrenia (SZ) patients and the associations with clinical symptoms.

Methods

We collected the resting-state functional MRI (rs-fMRI) data from 84 first-episode drug-naïve SZ patients and 94 healthy controls (HCs) and calculated the amplitude of low-frequency fluctuations (ALFF) and regional homogeneity (ReHo) of four frequency bands, including slow-2, slow-3, slow-4, and slow-5. Two-sample t-tests were used to evaluate the intergroup differences in ALFF and ReHo, while partial correlation analyses were conducted to explore the associations between abnormal ALFF and ReHo and the severity of clinical symptoms in the SZ group.

Results

Compared with HCs, the SZ group showed reduced ALFF in superior cerebellum and cerebellar vermis across slow-2, slow-3, and slow-4 bands, while increased ALFF was found in left superior temporal gyrus, middle temporal gyrus, and superior temporal pole at slow-4 band. Moreover, reduced ReHo was observed in the right precentral and postcentral gyri at slow-3 band in the SZ group. Additionally, the ALFF of left superior temporal gyrus, middle temporal gyrus, and superior temporal pole in slow-4 band showed a trend of positive correlation with the excited factor score of Positive and Negative Syndrome Scale (PANSS) in the SZ group.

Conclusion

Our results suggest that local alterations of spontaneous brain activity were frequency-specific in first-episode drug-naïve SZ patients.

Computational modeling predicts regulation of central pattern generator oscillations by size and density of the underlying heterogenous network

Central pattern generators are characterized by a heterogeneous cellular composition, with different cell types playing distinct roles in the production and transmission of rhythmic signals. However, little is known about the functional implications of individual variation in the relative distributions of cells and their connectivity patterns. Here, we addressed this question through a combination of morphological data analysis and computational modeling, using the pacemaker nucleus of the weakly electric fish Apteronotus leptorhynchus as case study. A neural network comprised of 60-110 interconnected pacemaker cells and 15-30 relay cells conveying its output to electromotoneurons in the spinal cord, this nucleus continuously generates neural signals at frequencies of up to 1 kHz with high temporal precision. We systematically explored the impact of network size and density on oscillation frequencies and their variation within and across cells. To accurately determine effect sizes, we minimized the likelihood of complex dynamics using a simplified setup precluding differential delays. To identify natural constraints, parameter ranges were extended beyond experimentally recorded numbers of cells and connections. Simulations revealed that pacemaker cells have higher frequencies and lower within-population variability than relay cells. Within-cell precision and between-cells frequency synchronization increased with the number of pacemaker cells and of connections of either type, and decreased with relay cell count in both populations. Network-level frequency-synchronized oscillations occurred in roughly half of simulations, with maximized likelihood and firing precision within biologically observed parameter ranges. These findings suggest the structure of the biological pacemaker nucleus is optimized for generating synchronized sustained oscillations.

Dynamic Instability and Time Domain Response of a Model Halide Perovskite Memristor for Artificial Neurons

Memristors are candidate devices for constructing artificial neurons, synapses, and computational networks for brainlike information processing and sensory-motor autonomous systems. However, the dynamics of natural neurons and synapses are challenging and cannot be well reproduced with standard electronic components. Halide perovskite memristors operate by mixed ionic-electronic properties that may lead to replicate the live computation elements. Here we explore the dynamical behavior of a halide perovskite memristor model to evaluate the response to a step perturbation and the self-sustained oscillations that produce analog neuron spiking. As the system contains a capacitor and a voltage-dependent chemical inductor, it can mimic an action potential in response to a square current pulse. Furthermore, we discover a property that cannot occur in the standard two-dimensional model systems: a three-dimensional model shows a dynamical instability that produces a spiking regime without the need for an intrinsic negative resistance. These results open a new pathway to create spiking neurons without the support of electronic circuits.

Resonance as a Design Strategy for AI and Social Robots

Resonance, a powerful and pervasive phenomenon, appears to play a major role in human interactions. This article investigates the relationship between the physical mechanism of resonance and the human experience of resonance, and considers possibilities for enhancing the experience of resonance within human-robot interactions. We first introduce resonance as a widespread cultural and scientific metaphor. Then, we review the nature of "sympathetic resonance" as a physical mechanism. Following this introduction, the remainder of the article is organized in two parts. In part one, we review the role of resonance (including synchronization and rhythmic entrainment) in human cognition and social interactions. Then, in part two, we review resonance-related phenomena in robotics and artificial intelligence (AI). These two reviews serve as ground for the introduction of a design strategy and combinatorial design space for shaping resonant interactions with robots and AI. We conclude by posing hypotheses and research questions for future empirical studies and discuss a range of ethical and aesthetic issues associated with resonance in human-robot interactions.

Role of Interaction Delays in the Synchronization of Inhibitory Networks

Neural oscillations provide a means for efficient and flexible communication among different brain areas. Understanding the mechanisms of the generation of brain oscillations is crucial to determine principles of communication and information transfer in the brain circuits. It is well known that the inhibitory neurons play a major role in the generation of oscillations in the gamma range, in pure inhibitory networks, or in the networks composed of excitatory and inhibitory neurons. In this study, we explore the impact of different parameters and, in particular, the delay in the transmission of the signals between the neurons, on the dynamics of inhibitory networks. We show that increasing delay in a reasonable range increases the synchrony and stabilizes the oscillations. Unstable gamma oscillations characterized by a highly variable amplitude of oscillations can be observed in an intermediate range of delays. We show that in this range of delays, other experimentally observed phenomena such as sparse firing, variable amplitude and period, and the correlation between the instantaneous amplitude and period could be observed. The results broaden our understanding of the mechanism of the generation of the gamma oscillations in the inhibitory networks, known as the ING (interneuron-gamma) mechanism.

Dynamical robustness of complex networks subject to long-range connectivity

In spite of a few attempts in understanding the dynamical robustness of complex networks, this extremely important subject of research is still in its dawn as compared to the other dynamical processes on networks. We hereby consider the concept of long-range interactions among the dynamical units of complex networks and demonstrate for the first time that such a characteristic can have noteworthy impacts on the dynamical robustness of networked systems, regardless of the underlying network topology. We present a comprehensive analysis of this phenomenon on top of diverse network architectures. Such dynamical damages being able to substantially affect the network performance, determining mechanisms that boost the robustness of networks becomes a fundamental question. In this work, we put forward a prescription based upon self-feedback that can efficiently resurrect global rhythmicity of complex networks composed of active and inactive dynamical units, and thus can enhance the network robustness. We have been able to delineate the whole proposition analytically while dealing with all d -path adjacency matrices, having an excellent agreement with the numerical results. For the numerical computations, we examine scale-free networks, Watts–Strogatz small-world model and also Erdös–Rényi random network, along with Landau–Stuart oscillators for casting the local dynamics.

Chemical Inductor

A multitude of chemical, biological, and material systems present an inductive behavior that is not electromagnetic in origin. Here, it is termed a chemical inductor. We show that the structure of the chemical inductor consists of a two-dimensional system that couples a fast conduction mode and a slowing down element. Therefore, it is generally defined in dynamical terms rather than by a specific physicochemical mechanism. The chemical inductor produces many familiar features in electrochemical reactions, including catalytic, electrodeposition, and corrosion reactions in batteries and fuel cells, and in solid-state semiconductor devices such as solar cells, organic light-emitting diodes, and memristors. It generates the widespread phenomenon of negative capacitance, it causes negative spikes in voltage transient measurements, and it creates inverted hysteresis effects in current-voltage curves and cyclic voltammetry. Furthermore, it determines stability, bifurcations, and chaotic properties associated to self-sustained oscillations in biological neurons and electrochemical systems. As these properties emerge in different types of measurement techniques such as impedance spectroscopy and time-transient decays, the chemical inductor becomes a useful framework for the interpretation of the electrical, optoelectronic, and electrochemical responses in a wide variety of systems. In the paper, we describe the general dynamical structure of the chemical inductor and we comment on a broad range of examples from different research areas.

Satiety Associated with Calorie Restriction and Time-Restricted Feeding: Central Neuroendocrine Integration

This review focuses on summarizing current knowledge on how time-restricted feeding (TRF) and continuous caloric restriction (CR) affect central neuroendocrine systems involved in regulating satiety. Several interconnected regions of the hypothalamus, brainstem, and cortical areas of the brain are involved in the regulation of satiety. Following CR and TRF, the increase in hunger and reduction in satiety signals of the melanocortin system [neuropeptide Y (NPY), proopiomelanocortin (POMC), and agouti-related peptide (AgRP)] appear similar between CR and TRF protocols, as do the dopaminergic responses in the mesocorticolimbic circuit. However, ghrelin and leptin signaling via the melanocortin system appears to improve energy balance signals and reduce hyperphagia following TRF, which has not been reported in CR. In addition to satiety systems, CR and TRF also influence circadian rhythms. CR influences the suprachiasmatic nucleus (SCN) or the primary circadian clock as seen by increased clock gene expression. In contrast, TRF appears to affect both the SCN and the peripheral clocks, as seen by phasic changes in the non-SCN (potentially the elusive food entrainable oscillator) and metabolic clocks. The peripheral clocks are influenced by the primary circadian clock but are also entrained by food timing, sleep timing, and other lifestyle parameters, which can supersede the metabolic processes that are regulated by the primary circadian clock. Taken together, TRF influences hunger/satiety, energy balance systems, and circadian rhythms, suggesting a role for adherence to CR in the long run if implemented using the TRF approach. However, these suggestions are based on only a few studies, and future investigations that use standardized protocols for the evaluation of the effect of these diet patterns (time, duration, meal composition, sufficiently powered) are necessary to verify these preliminary observations.

Heterogeneity in GnRH and kisspeptin neurons and their significance in vertebrate reproductive biology

Vertebrate reproduction is essentially controlled by the hypothalamus-pituitary-gonadal (HPG) axis, which is a central dogma of reproductive biology. Two major hypothalamic neuroendocrine cell groups containing gonadotropin-releasing hormone (GnRH) and kisspeptin are crucial for control of the HPG axis in vertebrates. GnRH and kisspeptin neurons exhibit high levels of heterogeneity including their cellular morphology, biochemistry, neurophysiology and functions. However, the molecular foundation underlying heterogeneities in GnRH and kisspeptin neurons remains unknown. More importantly, the biological and physiological significance of their heterogeneity in reproductive biology is poorly understood. In this review, we first describe the recent advances in the neuroendocrine functions of kisspeptin-GnRH pathways. We then view the recent emerging progress in the heterogeneity of GnRH and kisspeptin neurons using morphological and single-cell transcriptomic analyses. Finally, we discuss our views on the significance of functional heterogeneity of reproductive endocrine cells and their potential relevance to reproductive health.

A Frequency Domain Analysis of the Excitability and Bifurcations of the FitzHugh-Nagumo Neuron Model

The dynamics of neurons consist of oscillating patterns of a membrane potential that underpin the operation of biological intelligence. The FitzHugh-Nagumo (FHN) model for neuron excitability generates rich dynamical regimes with a simpler mathematical structure than the Hodgkin-Huxley model. Because neurons can be understood in terms of electrical and electrochemical methods, here we apply the analysis of the impedance response to obtain the characteristic spectra and their evolution as a function of applied voltage. We convert the two nonlinear differential equations of FHN into an equivalent circuit model, classify the different impedance spectra, and calculate the corresponding trajectories in the phase plane of the variables. In analogy to the field of electrochemical oscillators, impedance spectroscopy detects the Hopf bifurcations and the spiking regimes. We show that a neuron element needs three essential internal components: capacitor, inductor, and negative differential resistance. The method supports the fabrication of memristor-based artificial neural networks.

Cell and circuit origins of fast network oscillations in the mammalian main olfactory bulb

Neural synchrony generates fast network oscillations throughout the brain, including the main olfactory bulb (MOB), the first processing station of the olfactory system. Identifying the mechanisms synchronizing neurons in the MOB will be key to understanding how network oscillations support the coding of a high-dimensional sensory space. Here, using paired recordings and optogenetic activation of glomerular sensory inputs in MOB slices, we uncovered profound differences in principal mitral cell (MC) vs. tufted cell (TC) spike-time synchrony: TCs robustly synchronized across fast- and slow-gamma frequencies, while MC synchrony was weaker and concentrated in slow-gamma frequencies. Synchrony among both cell types was enhanced by shared glomerular input but was independent of intraglomerular lateral excitation. Cell-type differences in synchrony could also not be traced to any difference in the synchronization of synaptic inhibition. Instead, greater TC than MC synchrony paralleled the more periodic firing among resonant TCs than MCs and emerged in patterns consistent with densely synchronous network oscillations. Collectively, our results thus reveal a mechanism for parallel processing of sensory information in the MOB via differential TC vs. MC synchrony, and further contrast mechanisms driving fast network oscillations in the MOB from those driving the sparse synchronization of irregularly firing principal cells throughout cortex.

Deciphering how interneuron specific 3 cells control oriens lacunosum-moleculare cells to contribute to circuit function

The wide diversity of inhibitory cells across the brain makes them suitable to contribute to network dynamics in specialized fashions. However, the contributions of a particular inhibitory cell type in a behaving animal are challenging to untangle as one needs to both record cellular activities and identify the cell type being recorded. Thus, using computational modeling and theory to predict and hypothesize cell-specific contributions is desirable. Here, we examine potential contributions of interneuron-specific 3 (I-S3) cells - an inhibitory interneuron found in CA1 hippocampus that only targets other inhibitory interneurons - during simulated theta rhythms. We use previously developed multi-compartment models of oriens lacunosum-moleculare (OLM) cells, the main target of I-S3 cells, and explore how I-S3 cell inputs during in vitro and in vivo scenarios contribute to theta. We find that I-S3 cells suppress OLM cell spiking, rather than engender its spiking via post-inhibitory rebound mechanisms, and contribute to theta frequency spike resonance during simulated in vivo scenarios. To elicit recruitment similar to in vitro experiments, inclusion of disinhibited pyramidal cell inputs is necessary, implying that I-S3 cell firing broadens the window for pyramidal cell disinhibition. Using in vivo virtual networks, we show that I-S3 cells contribute to a sharpening of OLM cell recruitment at theta frequencies. Further, shifting the timing of I-S3 cell spiking due to external modulation shifts the timing of the OLM cell firing and thus disinhibitory windows. We propose a specialized contribution of I-S3 cells to create temporally precise coordination of modulation pathways.

Evidence of two modes of spiking evoked in human firing motoneurones by Ia afferent electrical stimulation

Neurone firing behaviour is a result of complex interaction between synaptic inputs and cellular intrinsic properties. Intriguing firing behaviour, delayed spiking, was shown in some neurones, in particular, in cat neocortical neurones and rat pyramidal hippocampal neurones. In contrast, the similar spiking mode was not reported for animal spinal motoneurones. In the present study, an attempt was made to look for possible evidence of delayed spiking in human motoneurones firing within the low-frequency, sub-primary range, characteristic for voluntary muscle contractions and postural tasks. Forty-seven firing motor units (MUs) were analyzed in ten experiments on three muscles (the flexor carpi ulnaris, the tibialis anterior, and the abductor pollicis brevis) in four healthy humans. Single MUs were activated by gentle voluntary muscle contractions. MU peri-stimulus time histograms, durations of inter-spike intervals, and motoneurone excitability changes within a target interspike interval were analyzed. It was found that during testing the firing motoneurone excitability by small, transient excitatory Ia afferent volley, depending firstly on volley timing within a target interspike interval and excitatory volley strength, the same motoneurone displayed either the direct short-latency response (the H-reflex) or the delayed response (with prolonged and variable latency). Thus, the findings, for the first time, provide evidence for a possibility of two modes of spiking in firing motoneurones. Methods of the estimation of delayed responses and their possible functional significance are discussed. It is emphasized that, for understanding of this issue, the integration of data from studies on experimental animals and humans is desirable.

Basin of attraction for chimera states in a network of Rössler oscillators

Chimera states are spatiotemporal patterns in which coherent and incoherent dynamics coexist simultaneously. These patterns were observed in both locally and nonlocally coupled oscillators. We study the existence of chimera states in networks of coupled Rössler oscillators. The Rössler oscillator can exhibit periodic or chaotic behavior depending on the control parameters. In this work, we show that the existence of coherent, incoherent, and chimera states depends not only on the coupling strength, but also on the initial state of the network. The initial states can belong to complex basins of attraction that are not homogeneously distributed. Due to this fact, we characterize the basins by means of the uncertainty exponent and basin stability. In our simulations, we find basin boundaries with smooth, fractal, and riddled structures.

Indirect pathway control of firing rate and pattern in the substantia nigra pars reticulata

Unitary pallido-nigral synaptic currents were measured using optogenetic stimulation, which activated up to three unitary synaptic inputs to each substantia nigra pars reticulata (SNr) cell. Episodic barrages of synaptic conductances were generated based on in vivo firing patterns of globus pallidus pars externa (GPe) cells and applied to SNr cells using conductance clamp. Barrage inputs were compared to continuous step conductances with the same mean. Barrage inputs and steps both slowed SNr neuron firing and produced disinhibition responses seen in peristimulus histograms. Barrages were less effective than steps at producing inhibition and disinhibition responses. Barrages, but not steps, produced irregular firing during the inhibitory response. Phase models of SNr neurons were constructed from their phase-resetting curves. The phase models reproduced the inhibition and disinhibition responses to the same inputs applied to the neurons. The disinhibition response did not require rebound currents but arose from reset of the cells' oscillation. The differences in firing rate and irregularity in response to barrage and step inhibition resulted from the high sensitivity of SNr neurons to inhibition at late phases in their intrinsic oscillation. During step inhibition, cells continued rhythmic firing at a reduced rate. During barrages, brief bouts of intense inhibition stalled the cells' phase evolution late in their cycle, close to firing, and even very brief respites from inhibition rapidly released single action potentials. The SNr cell firing pattern reflected the fine structure of the synaptic barrage from GPe, as well as its onset and offset.NEW & NOTEWORTHY The pallido-nigral pathway connects the striatum to spontaneously active basal ganglia output neurons in the substantia nigra. Each substantia nigra neuron receives powerful inhibitory synaptic connections from a small group of globus pallidus cells and may fire during pauses in pallidal activity. Despite lacking any hyperpolarization-activated rebound currents, they fire quickly to even brief pauses in the pallido-nigral inhibition. The mechanism of their rapid disinhibitory response is explained by features of their phase-resetting curves.

Frequency-dependent entrainment of striatal fast-spiking interneurons

Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20-200 spikes/s separated by pauses. In vivo, or with fluctuating applied current, both runs and pauses become briefer and more variable. During runs, spikes are entrained specifically to gamma-frequency components of the input fluctuations. We stimulated parvalbumin-expressing striatal FSIs in mouse brain slices with broadband noise currents added to direct current steps and measured spike entrainment across all frequencies. As the constant current level was increased, FSIs produced longer runs and showed sharper frequency tuning, with best entrainment at the stimulus frequency matching their intrarun firing rate. We separated the contributions of previous spikes from that of the fluctuating stimulus, revealing a strong contribution of previous action potentials to gamma-frequency entrainment. In contrast, after subtraction of the effect inherited from the previous spike, the remaining stimulus contribution to spike generation was less sharply tuned, showing a larger contribution of lower frequencies. The frequency specificity of entrainment within a run was reproduced with a phase resetting model based on experimentally measured phase resetting curves of the same FSIs. In the model, broadly tuned phase entrainment for the first spike in a run evolved into sharply tuned gamma entrainment over the next few spikes. The data and modeling results indicate that for FSIs firing in brief runs and pauses firing within runs is entrained by gamma-frequency components of the input, whereas the onset timing of runs may be sensitive to a wider range of stimulus frequency components.NEW & NOTEWORTHY Specific types of neurons entrain their spikes to particular oscillation frequencies in their synaptic input. This entrainment is commonly understood in terms of the subthreshold voltage response, but how this translates to spiking is not clear. We show that in striatal fast-spiking interneurons, entrainment to gamma-frequency input depends on rhythmic spike runs and is explained by the phase resetting curve, whereas run initiation can be triggered by a broad range of input frequencies.

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.

Inferring the dynamics of oscillatory systems using recurrent neural networks

We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.

Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits

Macroscopic oscillations of different brain regions show multiple phase relationships that are persistent across time and have been implicated in routing information. While multiple cellular mechanisms influence the network oscillatory dynamics and structure the macroscopic firing motifs, one of the key questions is to identify the biophysical neuronal and synaptic properties that permit such motifs to arise. A second important issue is how the different neural activity coherence states determine the communication between the neural circuits. Here we analyse the emergence of phase-locking within bidirectionally delayed-coupled spiking circuits in which global gamma band oscillations arise from synaptic coupling among largely excitable neurons. We consider both the interneuronal (ING) and the pyramidal-interneuronal (PING) population gamma rhythms and the inter coupling targeting the pyramidal or the inhibitory neurons. Using a mean-field approach together with an exact reduction method, we reduce each spiking network to a low dimensional nonlinear system and derive the macroscopic phase resetting-curves (mPRCs) that determine how the phase of the global oscillation responds to incoming perturbations. This is made possible by the use of the quadratic integrate-and-fire model together with a Lorentzian distribution of the bias current. Depending on the type of gamma (PING vs. ING), we show that incoming excitatory inputs can either speed up the macroscopic oscillation (phase advance; type I PRC) or induce both a phase advance and a delay (type II PRC). From there we determine the structure of macroscopic coherence states (phase-locking) of two weakly synaptically-coupled networks. To do so we derive a phase equation for the coupled system which links the synaptic mechanisms to the coherence states of the system. We show that a synaptic transmission delay is a necessary condition for symmetry breaking, i.e. a non-symmetric phase lag between the macroscopic oscillations. This potentially provides an explanation to the experimentally observed variety of gamma phase-locking modes. Our analysis further shows that symmetry-broken coherence states can lead to a preferred direction of signal transfer between the oscillatory networks where this directionality also depends on the timing of the signal. Hence we suggest a causal theory for oscillatory modulation of functional connectivity between cortical circuits.

Large scale brain oscillations emerge from synaptic interactions within neuronal circuits. Over the past years, such macroscopic rhythms have been suggested to play a crucial role in routing the flow of information across cortical regions, resulting in a functional connectome. The underlying mechanism is cortical oscillations that bind together following a well-known motif called phase-locking. While there is significant experimental support for multiple phase-locking modes in the brain, it is still unclear what is the underlying mechanism that permits macroscopic rhythms to phase lock. In the present paper we take up with this issue, and to show that, one can study the emergent macroscopic phase-locking within the mathematical framework of weakly coupled oscillators. We find that under synaptic delays, fully symmetrically coupled networks can display symmetry-broken states of activity, where one network starts to lead in phase the second (also sometimes known as stuttering states). When we analyse how incoming transient signals affect the coupled system, we find that in the symmetry-broken state, the effect depends strongly on which network is targeted (the leader or the follower) as well as the timing of the input. Hence we show how the dynamics of the emergent phase-locked activity imposes a functional directionality on how signals are processed. We thus offer clarification on the synaptic and circuit properties responsible for the emergence of multiple phase-locking patterns and provide support for its functional implication in information transfer.

Predicting responses to inhibitory synaptic input in substantia nigra pars reticulata neurons

The changes in firing probability produced by a synaptic input are usually visualized using the poststimulus time histogram (PSTH). It would be useful if postsynaptic firing patterns could be predicted from patterns of afferent synaptic activation, but attempts to predict the PSTH from synaptic potential waveforms using reasoning based on voltage trajectory and spike threshold have not been successful, especially for inhibitory inputs. We measured PSTHs for substantia nigra pars reticulata (SNr) neurons inhibited by optogenetic stimulation of striato-nigral inputs or by matching artificial inhibitory conductances applied by dynamic clamp. The PSTH was predicted by a model based on each SNr cell's phase-resetting curve (PRC). Optogenetic activation of striato-nigral input or artificial synaptic inhibition produced a PSTH consisting of an initial depression of firing followed by oscillatory increases and decreases repeating at the SNr cell's baseline firing rate. The phase resetting model produced PSTHs closely resembling the cell data, including the primary pause in firing and the oscillation. Key features of the PSTH, including the onset rate and duration of the initial inhibitory phase, and the subsequent increase in firing probability could be explained from the characteristic shape of the SNr cell's PRC. The rate of damping of the late oscillation was explained by the influence of asynchronous phase perturbations producing firing rate jitter and wander. Our results demonstrate the utility of phase-resetting models as a general method for predicting firing in spontaneously active neurons and their value in interpretation of the striato-nigral PSTH. NEW & NOTEWORTHY The coupling of patterned presynaptic input to sequences of postsynaptic firing is a Gordian knot, complicated by the multidimensionality of neuronal state and the diversity of potential initial states. Even so, it is fundamental for even the simplest understanding of network dynamics. We show that a simple phase-resetting model constructed from experimental measurements can explain and predict the sequence of spike rate changes following synaptic inhibition of an oscillating basal ganglia output neuron.

Systems approaches to optimizing deep brain stimulation therapies in Parkinson's disease

Over the last 30 years, deep brain stimulation (DBS) has been used to treat chronic neurological diseases like dystonia, obsessive-compulsive disorders, essential tremor, Parkinson's disease, and more recently, dementias, depression, cognitive disorders, and epilepsy. Despite its wide use, DBS presents numerous challenges for both clinicians and engineers. One challenge is the design of novel, more efficient DBS therapies, which are hampered by the lack of complete understanding about the cellular mechanisms of therapeutic DBS. Another challenge is the existence of redundancy in clinical outcomes, that is, different DBS programs can result in similar clinical benefits but very little information (e.g., predictive models, longitudinal data, metrics, etc.) is available to select one program over another. Finally, there is high variability in patients' responses to DBS, which forces clinicians to carefully adjust the stimulation settings to each patient via lengthy programming sessions. Researchers in neural engineering and systems biology have been tackling these challenges over the past few years with the specific goal of developing novel DBS therapies, design methodologies, and computational tools that optimize the therapeutic effects of DBS in each patient. Furthermore, efforts are being made to automatically adapt the DBS treatment to the fluctuations of disease symptoms. A review of the quantitative approaches currently available for the treatment of Parkinson's disease is presented here with an emphasis on the contributions that systems theoretical approaches have provided to understand the global dynamics of complex neuronal circuits in the brain under DBS. This article is categorized under: Translational, Genomic, and Systems Medicine > Therapeutic Methods Analytical and Computational Methods > Computational Methods Analytical and Computational Methods > Dynamical Methods Physiology > Mammalian Physiology in Health and Disease.

Macroscopic phase-resetting curves for spiking neural networks

The study of brain rhythms is an open-ended, and challenging, subject of interest in neuroscience. One of the best tools for the understanding of oscillations at the single neuron level is the phase-resetting curve (PRC). Synchronization in networks of neurons, effects of noise on the rhythms, effects of transient stimuli on the ongoing rhythmic activity, and many other features can be understood by the PRC. However, most macroscopic brain rhythms are generated by large populations of neurons, and so far it has been unclear how the PRC formulation can be extended to these more common rhythms. In this paper, we describe a framework to determine a macroscopic PRC (mPRC) for a network of spiking excitatory and inhibitory neurons that generate a macroscopic rhythm. We take advantage of a thermodynamic approach combined with a reduction method to simplify the network description to a small number of ordinary differential equations. From this simplified but exact reduction, we can compute the mPRC via the standard adjoint method. Our theoretical findings are illustrated with and supported by numerical simulations of the full spiking network. Notably our mPRC framework allows us to predict the difference between effects of transient inputs to the excitatory versus the inhibitory neurons in the network.

Measurement of phase resetting curves using optogenetic barrage stimuli

BACKGROUND

The phase resetting curve (PRC) is a primary measure of a rhythmically firing neuron's responses to synaptic input, quantifying the change in phase of the firing oscillation as a function of the input phase. PRCs provide information about whether neurons will synchronize due to synaptic coupling or shared input. However, PRC estimation has been limited to in vitro preparations where stable intracellular recordings can be obtained and background activity is minimal, and new methods are required for in vivo applications.

NEW METHOD

We estimated PRCs using dense optogenetic stimuli and extracellular spike recording. Autonomously firing neurons in substantia nigra pars reticulata (SNr) of Thy1-channelrhodopsin 2 (ChR2) transgenic mice were stimulated with random barrages of light pulses, and PRCs were determined using multiple linear regression.

RESULTS

The PRCs obtained were type-I, showing only phase advances in response to depolarizing input, and generally sloped upward from early to late phases. Secondary PRCs, indicating the effect on the subsequent ISI, showed phase delays primarily for stimuli arriving at late phases. Phase models constructed from the optogenetic PRCs accounted for a large fraction of the variance in ISI length and provided a good approximation of the spike-triggered average stimulus.

COMPARISON WITH EXISTING METHODS

Compared to methods based on intracellular current injection, the new method sacrifices some temporal resolution. However, it should be much more widely applicable in vivo, because only extracellular recording and optogenetic stimulation are required.

CONCLUSIONS

These results demonstrate PRC estimation using methods suitable for in vivo applications.

Predicting the response of striatal spiny neurons to sinusoidal input

Spike-timing effects of small-amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively. Spike-timing reliability varied with the stimulus frequency. For frequencies near the cell's firing rate, the cells altered firing rate to match the stimulus and became phase locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell's unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells' phase-resetting curve. Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cell's unperturbed rate, and interspike intervals were accurately predicted by the map. Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency. Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model. When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell's unperturbed firing rate. Their spike times encoded the frequency and phase of that single oscillatory input.NEW & NOTEWORTHY During repetitive firing, the timing of action potentials is determined by the interaction between the input and voltage-sensitive currents throughout the interspike interval. This interaction is encapsulated in the neuron's phase-resetting curve. The phase-resetting curve predicted spike timing to small sinusoidal currents over a wide range of stimulus frequencies. Firing patterns were most sensitive to oscillatory components near the cell's own firing rate, even in the presence of noise and other inputs.

Synchrony and so much more: Diverse roles for electrical synapses in neural circuits

Electrical synapses are neuronal gap junctions that are ubiquitous across brain regions and species. The biophysical properties of most electrical synapses are relatively simple-transcellular channels allow nearly ohmic, bidirectional flow of ionic current. Yet these connections can play remarkably diverse roles in different neural circuit contexts. Recent findings illustrate how electrical synapses may excite or inhibit, synchronize or desynchronize, augment or diminish rhythms, phase-shift, detect coincidences, enhance signals relative to noise, adapt, and interact with nonlinear membrane and transmitter-release mechanisms. Most of these functions are likely to be widespread in central nervous systems. © 2016 Wiley Periodicals, Inc. Develop Neurobiol 77: 610-624, 2017.