Electrical synapses and synchrony: the role of intrinsic currents

ExaFlexHH: an exascale-ready, flexible multi-FPGA library for biologically plausible brain simulations

Introduction

In-silico simulations are a powerful tool in modern neuroscience for enhancing our understanding of complex brain systems at various physiological levels. To model biologically realistic and detailed systems, an ideal simulation platform must possess: (1) high performance and performance scalability, (2) flexibility, and (3) ease of use for non-technical users. However, most existing platforms and libraries do not meet all three criteria, particularly for complex models such as the Hodgkin-Huxley (HH) model or for complex neuron-connectivity modeling such as gap junctions.

Methods

This work introduces ExaFlexHH, an exascale-ready, flexible library for simulating HH models on multi-FPGA platforms. Utilizing FPGA-based Data-Flow Engines (DFEs) and the dataflow programming paradigm, ExaFlexHH addresses all three requirements. The library is also parameterizable and compliant with NeuroML, a prominent brain-description language in computational neuroscience. We demonstrate the performance scalability of the platform by implementing a highly demanding extended-Hodgkin-Huxley (eHH) model of the Inferior Olive using ExaFlexHH.

Results

Model simulation results show linear scalability for unconnected networks and near-linear scalability for networks with complex synaptic plasticity, with a 1.99 × performance increase using two FPGAs compared to a single FPGA simulation, and 7.96 × when using eight FPGAs in a scalable ring topology. Notably, our results also reveal consistent performance efficiency in GFLOPS per watt, further facilitating exascale-ready computing speeds and pushing the boundaries of future brain-simulation platforms.

Discussion

The ExaFlexHH library shows superior resource efficiency, quantified in FLOPS per hardware resources, benchmarked against other competitive FPGA-based brain simulation implementations.

D-type K+ current rules the function of electrically coupled neurons in a species-specific fashion

Electrical synapses supported by gap junctions are known to form networks of electrically coupled neurons in many regions of the mammalian brain, where they play relevant functional roles. Yet, how electrical coupling supports sophisticated network operations and the contribution of the intrinsic electrophysiological properties of neurons to these operations remain incompletely understood. Here, a comparative analysis of electrically coupled mesencephalic trigeminal (MesV) neurons uncovered remarkable difference in the operation of these networks in highly related species. While spiking of MesV neurons might support the recruitment of coupled cells in rats, this rarely occurs in mice. Using whole-cell recordings, we determined that the higher efficacy in postsynaptic recruitment in rat's MesV neurons does not result from coupling strength of larger magnitude, but instead from the higher excitability of coupled neurons. Consistently, MesV neurons from rats present a lower rheobase, more hyperpolarized threshold, as well as a higher ability to generate repetitive discharges, in comparison to their counterparts from mice. This difference in neuronal excitability results from a significantly higher magnitude of the D-type K+ current (ID) in MesV neurons from mice, indicating that the magnitude of this current gates the recruitment of postsynaptic-coupled neurons. Since MesV neurons are primary afferents critically involved in the organization of orofacial behaviors, activation of a coupled partner could support lateral excitation, which by amplifying sensory inputs may significantly contribute to information processing and the organization of motor outputs.

Linking α-synuclein-induced synaptopathy and neural network dysfunction in early Parkinson’s disease

The prodromal phase of Parkinson’s disease is characterized by aggregation of the misfolded pathogenic protein α-synuclein in select neural centres, co-occurring with non-motor symptoms including sensory and cognitive loss, and emotional disturbances. It is unclear whether neuronal loss is significant during the prodrome. Underlying these symptoms are synaptic impairments and aberrant neural network activity. However, the relationships between synaptic defects and network-level perturbations are not established. In experimental models, pathological α-synuclein not only impacts neurotransmission at the synaptic level, but also leads to changes in brain network-level oscillatory dynamics—both of which likely contribute to non-motor deficits observed in Parkinson’s disease. Here we draw upon research from both human subjects and experimental models to propose a ‘synapse to network prodrome cascade’ wherein before overt cell death, pathological α-synuclein induces synaptic loss and contributes to aberrant network activity, which then gives rise to prodromal symptomology. As the disease progresses, abnormal patterns of neural activity ultimately lead to neuronal loss and clinical progression of disease. Finally, we outline goals and research needed to unravel the basis of functional impairments in Parkinson’s disease and other α-synucleinopathies.

On the Diverse Functions of Electrical Synapses

Electrical synapses are the neurophysiological product of gap junctional pores between neurons that allow bidirectional flow of current between neurons. They are expressed throughout the mammalian nervous system, including cortex, hippocampus, thalamus, retina, cerebellum, and inferior olive. Classically, the function of electrical synapses has been associated with synchrony, logically following that continuous conductance provided by gap junctions facilitates the reduction of voltage differences between coupled neurons. Indeed, electrical synapses promote synchrony at many anatomical and frequency ranges across the brain. However, a growing body of literature shows there is greater complexity to the computational function of electrical synapses. The paired membranes that embed electrical synapses act as low-pass filters, and as such, electrical synapses can preferentially transfer spike after hyperpolarizations, effectively providing spike-dependent inhibition. Other functions include driving asynchronous firing, improving signal to noise ratio, aiding in discrimination of dissimilar inputs, or dampening signals by shunting current. The diverse ways by which electrical synapses contribute to neuronal integration merits furthers study. Here we review how functions of electrical synapses vary across circuits and brain regions and depend critically on the context of the neurons and brain circuits involved. Computational modeling of electrical synapses embedded in multi-cellular models and experiments utilizing optical control and measurement of cellular activity will be essential in determining the specific roles performed by electrical synapses in varying contexts.

Cortical control of striatal fast-spiking interneuron synchrony

KEY POINTS

Electrical synapses between striatal fast-spiking interneurons in adult mice occur in ∼8% of assayed pairs. Coincident, convergent cortical input onto fast-spiking interneurons significantly contributes to fast-spiking interneuron synchrony Electrical synapses between fast-spiking interneurons provide only minor enhancement of fast-spiking interneuron synchrony. These results suggest a mechanism by which adult mouse fast-spiking interneurons of the striatum synchronize in the face of declining expression of the electrical synapse-forming connexin-36 protein.

ABSTRACT

Inhibitory fast-spiking interneurons in the dorsal striatum regulate actions and action strategies, including habits. Fast-spiking interneurons are widely believed to synchronize their firing due to the electrical synapses formed between these neurons. However, neuronal modeling data suggest convergent cortical input may also drive synchrony in fast-spiking interneuron networks. To better understand how fast-spiking interneuron synchrony arises, we performed dual whole-cell patch clamp electrophysiology experiments to inform a simple Bayesian network modeling cortico-fast-spiking interneuron circuitry. Dual whole-cell patch clamp electrophysiology revealed that while responsivity to corticostriatal input activation was high in fast-spiking interneurons, few of these neurons exhibited electrical coupling in adult mice. In simulations of a cortico-fast-spiking interneuron network informed by these data, the degree of glutamatergic cortical convergence onto fast-spiking interneurons significantly increased fast-spiking interneuron synchronization while manipulations of electrical coupling between these neurons exerted relatively little impact. These results suggest that the primary source of functional coordination of fast-spiking interneuron activity in adulthood arises from convergent corticostriatal input activation. Abstract figure legend Dual whole-cell patch clamp recordings of dorsal striatal fast-spiking interneurons (FSIs; red circles) rarely (8 percentage) form electrical synapses with other FSIs in adult mouse. In a two-layer in silico model of cortical pyramidal neuron (gray triangles) input to FSIs using empirically defined cortico-FSI synaptic weights, synchronous FSI-FSI activity (in the absence of abundant electrical synapses) is achievable by convergent cortical pyramidal excitation of FSIs. This article is protected by copyright. All rights reserved.

Function and Plasticity of Electrical Synapses in the Mammalian Brain: Role of Non-Junctional Mechanisms

Simple Summary

Relevant brain functions, such as perception, organization of behavior, and cognitive processes, are the outcome of information processing by neural circuits. Within these circuits, communication between neurons mainly relies on two modalities of synaptic transmission: chemical and electrical. Moreover, changes in the strength of these connections, aka synaptic plasticity, are believed to underlie processes of learning and memory, and its dysfunction has been suggested to underlie a variety of neurological disorders. While the relevance of chemical transmission and its plastic changes are known in great detail, analogous mechanisms and functional impact of their electrical counterparts were only recently acknowledged. In this article, we review the basic physical principles behind electrical transmission between neurons, the plethora of functional operations supported by this modality of neuron-to-neuron communication, as well as the basic principles of plasticity at these synapses.

Abstract

Electrical transmission between neurons is largely mediated by gap junctions. These junctions allow the direct flow of electric current between neurons, and in mammals, they are mostly composed of the protein connexin36. Circuits of electrically coupled neurons are widespread in these animals. Plus, experimental and theoretical evidence supports the notion that, beyond synchronicity, these circuits are able to perform sophisticated operations such as lateral excitation and inhibition, noise reduction, as well as the ability to selectively respond upon coincident excitatory inputs. Although once considered stereotyped and unmodifiable, we now know that electrical synapses are subject to modulation and, by reconfiguring neural circuits, these modulations can alter relevant operations. The strength of electrical synapses depends on the gap junction resistance, as well as on its functional interaction with the electrophysiological properties of coupled neurons. In particular, voltage and ligand gated channels of the non-synaptic membrane critically determine the efficacy of transmission at these contacts. Consistently, modulatory actions on these channels have been shown to represent relevant mechanisms of plasticity of electrical synaptic transmission. Here, we review recent evidence on the regulation of electrical synapses of mammals, the underlying molecular mechanisms, and the possible ways in which they affect circuit function.

Excitatory synapses and gap junctions cooperate to improve Pv neuronal burst firing and cortical social cognition in Shank2-mutant mice

NMDA receptor (NMDAR) and GABA neuronal dysfunctions are observed in animal models of autism spectrum disorders, but how these dysfunctions impair social cognition and behavior remains unclear. We report here that NMDARs in cortical parvalbumin (Pv)-positive interneurons cooperate with gap junctions to promote high-frequency (>80 Hz) Pv neuronal burst firing and social cognition. Shank2^–/– mice, displaying improved sociability upon NMDAR activation, show impaired cortical social representation and inhibitory neuronal burst firing. Cortical Shank2^–/– Pv neurons show decreased NMDAR activity, which suppresses the cooperation between NMDARs and gap junctions (GJs) for normal burst firing. Shank2^–/– Pv neurons show compensatory increases in GJ activity that are not sufficient for social rescue. However, optogenetic boosting of Pv neuronal bursts, requiring GJs, rescues cortical social cognition in Shank2^–/– mice, similar to the NMDAR-dependent social rescue. Therefore, NMDARs and gap junctions cooperate to promote cortical Pv neuronal bursts and social cognition.

How NMDAR and GABA neuronal dysfunctions result in impaired social behaviour is unclear. Here, the authors show that NMDARs and gap junctions in cortical PV interneurons modulate burst firing, affecting social behaviour.

Regulating synchronous oscillations of cerebellar granule cells by different types of inhibition

Synchronous oscillations in neural populations are considered being controlled by inhibitory neurons. In the granular layer of the cerebellum, two major types of cells are excitatory granular cells (GCs) and inhibitory Golgi cells (GoCs). GC spatiotemporal dynamics, as the output of the granular layer, is highly regulated by GoCs. However, there are various types of inhibition implemented by GoCs. With inputs from mossy fibers, GCs and GoCs are reciprocally connected to exhibit different network motifs of synaptic connections. From the view of GCs, feedforward inhibition is expressed as the direct input from GoCs excited by mossy fibers, whereas feedback inhibition is from GoCs via GCs themselves. In addition, there are abundant gap junctions between GoCs showing another form of inhibition. It remains unclear how these diverse copies of inhibition regulate neural population oscillation changes. Leveraging a computational model of the granular layer network, we addressed this question to examine the emergence and modulation of network oscillation using different types of inhibition. We show that at the network level, feedback inhibition is crucial to generate neural oscillation. When short-term plasticity was equipped on GoC-GC synapses, oscillations were largely diminished. Robust oscillations can only appear with additional gap junctions. Moreover, there was a substantial level of cross-frequency coupling in oscillation dynamics. Such a coupling was adjusted and strengthened by GoCs through feedback inhibition. Taken together, our results suggest that the cooperation of distinct types of GoC inhibition plays an essential role in regulating synchronous oscillations of the GC population. With GCs as the sole output of the granular network, their oscillation dynamics could potentially enhance the computational capability of downstream neurons.

Neural Interactions in Developing Rhythmogenic Spinal Networks: Insights From Computational Modeling

The mechanisms involved in generation of rhythmic locomotor activity in the mammalian spinal cord remain poorly understood. These mechanisms supposedly rely on both intrinsic properties of constituting neurons and interactions between them. A subset of Shox2 neurons was suggested to contribute to generation of spinal locomotor activity, but the possible cellular basis for rhythmic bursting in these neurons remains unknown. Ha and Dougherty (2018) recently revealed the presence of bidirectional electrical coupling between Shox2 neurons in neonatal spinal cords, which can be critically involved in neuronal synchronization and generation of populational bursting. Gap junctional connections found between functionally-related Shox2 interneurons decrease with age, possibly being replaced by increasing interactions through chemical synapses. Here, we developed a computational model of a heterogeneous population of neurons sparsely connected by electrical or/and chemical synapses and investigated the dependence of frequency of populational bursting on the type and strength of neuronal interconnections. The model proposes a mechanistic explanation that can account for the emergence of a synchronized rhythmic activity in the neuronal population and provides insights into the possible role of gap junctional coupling between Shox2 neurons in the spinal mechanisms for locomotor rhythm generation.

Exact Mean-Field Theory Explains the Dual Role of Electrical Synapses in Collective Synchronization

Electrical synapses play a major role in setting up neuronal synchronization, but the precise mechanisms whereby these synapses contribute to synchrony are subtle and remain elusive. To investigate these mechanisms mean-field theories for quadratic integrate-and-fire neurons with electrical synapses have been recently put forward. Still, the validity of these theories is controversial since they assume that the neurons produce unrealistic, symmetric spikes, ignoring the well-known impact of spike shape on synchronization. Here, we show that the assumption of symmetric spikes can be relaxed in such theories. The resulting mean-field equations reveal a dual role of electrical synapses: First, they equalize membrane potentials favoring the emergence of synchrony. Second, electrical synapses act as "virtual chemical synapses," which can be either excitatory or inhibitory depending upon the spike shape. Our results offer a precise mathematical explanation of the intricate effect of electrical synapses in collective synchronization. This reconciles previous theoretical and numerical works, and confirms the suitability of recent low-dimensional mean-field theories to investigate electrically coupled neuronal networks.

Efficient Communication in Distributed Simulations of Spiking Neuronal Networks With Gap Junctions

Investigating the dynamics and function of large-scale spiking neuronal networks with realistic numbers of synapses is made possible today by state-of-the-art simulation code that scales to the largest contemporary supercomputers. However, simulations that involve electrical interactions, also called gap junctions, besides chemical synapses scale only poorly due to a communication scheme that collects global data on each compute node. In comparison to chemical synapses, gap junctions are far less abundant. To improve scalability we exploit this sparsity by integrating an existing framework for continuous interactions with a recently proposed directed communication scheme for spikes. Using a reference implementation in the NEST simulator we demonstrate excellent scalability of the integrated framework, accelerating large-scale simulations with gap junctions by more than an order of magnitude. This allows, for the first time, the efficient exploration of the interactions of chemical and electrical coupling in large-scale neuronal networks models with natural synapse density distributed across thousands of compute nodes.

Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.

Interneuronal gap junctions increase synchrony and robustness of hippocampal ripple oscillations

Sharp wave-ripples (SWRs) are important for memory consolidation. Their signature in the hippocampal extracellular field potential can be decomposed into a ≈100 ms long sharp wave superimposed by ≈200 Hz ripple oscillations. How ripple oscillations are generated is currently not well understood. A promising model for the genesis of ripple oscillations is based on recurrent interneuronal networks (INT-INT). According to this hypothesis, the INT-INT network in CA1 receives a burst of excitation from CA3 that generates the sharp wave, and recurrent inhibition leads to an ultrafast synchronization of the CA1 network causing the ripple oscillations; fast-spiking parvalbumin-positive basket cells (PV⁺  BCs) may constitute the ripple-generating interneuronal network. PV⁺  BCs are also coupled by gap junctions (GJs) but the function of GJs for ripple oscillations has not been quantified. Using simulations of CA1 hippocampal networks of PV⁺  BCs, we show that GJs promote synchrony beyond a level that could be obtained by only inhibition. GJs also increase the neuronal firing rate of the interneuronal ensemble, while they affect the ripple frequency only mildly. The promoting effect of GJs on ripple oscillations depends on fast GJ transmission ( ≲ 0.5 ms), which requires proximal GJ coupling ( ≲ 100 μm from soma), but is robust to variability in the delay and the amplitude of GJ coupling.

Synchronization of Electrically Coupled Resonate-and-Fire Neurons

Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with electrical coupling show a diversity of post-spike subthreshold fluctuations, often linked to subthreshold resonance, which are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with damped subthreshold oscillations and a range of post-spike voltage dynamics. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuous post-spike reset rule. We find that both spikes and subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau potential. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs.

Neuronal Dynamics Underlying Communication Signals in a Weakly Electric Fish: Implications for Connectivity in a Pacemaker Network

Neuronal networks can produce stable oscillations and synchrony that are under tight control yet flexible enough to rapidly switch between dynamical states. The pacemaker nucleus in the weakly electric fish comprises a network of electrically coupled neurons that fire synchronously at high frequency. This activity sets the timing for an oscillating electric organ discharge with the lowest cycle-to-cycle variability of all known biological oscillators. Despite this high temporal precision, pacemaker activity is behaviorally modulated on millisecond time-scales for the generation of electrocommunication signals. The network mechanisms that allow for this combination of stability and flexibility are not well understood. In this study, we use an in vitro pacemaker preparation from Apteronotus leptorhynchus to characterize the neural responses elicited by the synaptic inputs underlying electrocommunication. These responses involve a variable increase in firing frequency and a prominent desynchronization of neurons that recovers within 5 oscillation cycles. Using a previously developed computational model of the pacemaker network, we show that the frequency changes and rapid resynchronization observed experimentally are most easily explained when model neurons are interconnected more densely and with higher coupling strengths than suggested by published data. We suggest that the pacemaker network achieves both stability and flexibility by balancing coupling strength with interconnectivity and that variation in these network features may provide a substrate for species-specific evolution of electrocommunication signals.

Nonautonomous driving induces stability in network of identical oscillators

Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilizing complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows as the amplitude of the frequency modulation is increased, through the birth of an intermittent synchronization regime. For repulsive couplings, we propose a control strategy to stabilize the dynamics by altering very slightly the network topology. We also show how, without changing the topology, time-variability in the driving frequency can itself stabilize the dynamics. As a byproduct of the analysis, we observe chimeralike states. We conclude that time-variability-induced stability phenomena are also present in networks, reinforcing the idea that this is a quite realistic scenario for living systems to use in maintaining their functioning in the face of ongoing perturbations.

Dual Mechanism for the Emergence of Synchronization in Inhibitory Neural Networks

During cognitive tasks cortical microcircuits synchronize to bind stimuli into unified perception. The emergence of coherent rhythmic activity is thought to be inhibition-driven and stimulation-dependent. However, the exact mechanisms of synchronization remain unknown. Recent optogenetic experiments have identified two neuron sub-types as the likely inhibitory vectors of synchronization. Here, we show that local networks mimicking the soma-targeting properties observed in fast-spiking interneurons and the dendrite-projecting properties observed in somatostatin interneurons synchronize through different mechanisms which may provide adaptive advantages by combining flexibility and robustness. We probed the synchronization phase diagrams of small all-to-all inhibitory networks in-silico as a function of inhibition delay, neurotransmitter kinetics, timings and intensity of stimulation. Inhibition delay is found to induce coherent oscillations over a broader range of experimental conditions than high-frequency entrainment. Inhibition delay boosts network capacity (ln2)^−N-fold by stabilizing locally coherent oscillations. This work may inform novel therapeutic strategies for moderating pathological cortical oscillations.

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

Gamma oscillations are thought to play an important role in brain function. Interneuron gamma (ING) and pyramidal interneuron gamma (PING) mechanisms have been proposed as generation mechanisms for these oscillations. However, the relation between the generation mechanisms and the dynamical properties of the gamma oscillation are still unclear. Among the dynamical properties of the gamma oscillation, the phase response function (PRF) is important because it encodes the response of the oscillation to inputs. Recently, the PRF for an inhibitory population of modified theta neurons that generate an ING rhythm was computed by the adjoint method applied to the associated Fokker-Planck equation (FPE) for the model. The modified theta model incorporates conductance-based synapses as well as the voltage and current dynamics. Here, we extended this previous work by creating an excitatory-inhibitory (E-I) network using the modified theta model and described the population dynamics with the corresponding FPE. We conducted a bifurcation analysis of the FPE to find parameter regions which generate gamma oscillations. In order to label the oscillatory parameter regions by their generation mechanisms, we defined ING- and PING-type gamma oscillation in a mathematically plausible way based on the driver of the inhibitory population. We labeled the oscillatory parameter regions by these generation mechanisms and derived PRFs via the adjoint method on the FPE in order to investigate the differences in the responses of each type of oscillation to inputs. PRFs for PING and ING mechanisms are derived and compared. We found the amplitude of the PRF for the excitatory population is larger in the PING case than in the ING case. Finally, the E-I population of the modified theta neuron enabled us to analyze the PRFs of PING-type gamma oscillation and the entrainment ability of E and I populations. We found a parameter region in which PRFs of E and I are both purely positive in the case of PING oscillations. The different entrainment abilities of E and I stimulation as governed by the respective PRFs was compared to direct simulations of finite populations of model neurons. We find that it is easier to entrain the gamma rhythm by stimulating the inhibitory population than by stimulating the excitatory population as has been found experimentally.

Gap junction plasticity as a mechanism to regulate network-wide oscillations

Cortical oscillations are thought to be involved in many cognitive functions and processes. Several mechanisms have been proposed to regulate oscillations. One prominent but understudied mechanism is gap junction coupling. Gap junctions are ubiquitous in cortex between GABAergic interneurons. Moreover, recent experiments indicate their strength can be modified in an activity-dependent manner, similar to chemical synapses. We hypothesized that activity-dependent gap junction plasticity acts as a mechanism to regulate oscillations in the cortex. We developed a computational model of gap junction plasticity in a recurrent cortical network based on recent experimental findings. We showed that gap junction plasticity can serve as a homeostatic mechanism for oscillations by maintaining a tight balance between two network states: asynchronous irregular activity and synchronized oscillations. This homeostatic mechanism allows for robust communication between neuronal assemblies through two different mechanisms: transient oscillations and frequency modulation. This implies a direct functional role for gap junction plasticity in information transmission in cortex.

Does layer 4 in the barrel cortex function as a balanced circuit when responding to whisker movements?

Neurons in one barrel in layer 4 (L4) in the mouse vibrissa somatosensory cortex are innervated mostly by neurons from the VPM nucleus and by other neurons within the same barrel. During quiet wakefulness or whisking in air, thalamic inputs vary slowly in time, and excitatory neurons rarely fire. A barrel in L4 contains a modest amount of neurons; the synaptic conductances are not very strong and connections are not sparse. Are the dynamical properties of the L4 circuit similar to those expected from fluctuation-dominated, balanced networks observed for large, strongly coupled and sparse cortical circuits? To resolve this question, we analyze a network of 150 inhibitory parvalbumin-expressing fast-spiking inhibitory interneurons innervated by the VPM thalamus with random connectivity, without or with 1600 low-firing excitatory neurons. Above threshold, the population-average firing rate of inhibitory cortical neurons increases linearly with the thalamic firing rate. The coefficient of variation CV is somewhat less than 1. Moderate levels of synchrony are induced by converging VPM inputs and by inhibitory interaction among neurons. The strengths of excitatory and inhibitory currents during whisking are about three times larger than threshold. We identify values of numbers of presynaptic neurons, synaptic delays between inhibitory neurons, and electrical coupling within the experimentally plausible ranges for which spike synchrony levels are low. Heterogeneity in in-degrees increases the width of the firing rate distribution to the experimentally observed value. We conclude that an L4 circuit in the low-synchrony regime exhibits qualitative dynamical properties similar to those of balanced networks.

Effect of Phase Response Curve Shape and Synaptic Driving Force on Synchronization of Coupled Neuronal Oscillators

The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

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.

Circadian Rhythms in Diet-Induced Obesity

The biological clocks of the circadian timing system coordinate cellular and physiological processes and synchronizes these with daily cycles, feeding patterns also regulates circadian clocks. The clock genes and adipocytokines show circadian rhythmicity. Dysfunction of these genes are involved in the alteration of these adipokines during the development of obesity. Food availability promotes the stimuli associated with food intake which is a circadian oscillator outside of the suprachiasmatic nucleus (SCN). Its circadian rhythm is arranged with the predictable daily mealtimes. Food anticipatory activity is mediated by a self-sustained circadian timing and its principal component is food entrained oscillator. However, the hypothalamus has a crucial role in the regulation of energy balance rather than food intake. Fatty acids or their metabolites can modulate neuronal activity by brain nutrient-sensing neurons involved in the regulation of energy and glucose homeostasis. The timing of three-meal schedules indicates close association with the plasma levels of insulin and preceding food availability. Desynchronization between the central and peripheral clocks by altered timing of food intake and diet composition can lead to uncoupling of peripheral clocks from the central pacemaker and to the development of metabolic disorders. Metabolic dysfunction is associated with circadian disturbances at both central and peripheral levels and, eventual disruption of circadian clock functioning can lead to obesity. While CLOCK expression levels are increased with high fat diet-induced obesity, peroxisome proliferator-activated receptor (PPAR) alpha increases the transcriptional level of brain and muscle ARNT-like 1 (BMAL1) in obese subjects. Consequently, disruption of clock genes results in dyslipidemia, insulin resistance and obesity. Modifying the time of feeding alone can greatly affect body weight. Changes in the circadian clock are associated with temporal alterations in feeding behavior and increased weight gain. Thus, shift work is associated with increased risk for obesity, diabetes and cardio-vascular diseases as a result of unusual eating time and disruption of circadian rhythm.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Synergistic plasticity of intrinsic conductance and electrical coupling restores synchrony in an intact motor network

Motor neurons of the crustacean cardiac ganglion generate virtually identical, synchronized output despite the fact that each neuron uses distinct conductance magnitudes. As a result of this variability, manipulations that target ionic conductances have distinct effects on neurons within the same ganglion, disrupting synchronized motor neuron output that is necessary for proper cardiac function. We hypothesized that robustness in network output is accomplished via plasticity that counters such destabilizing influences. By blocking high-threshold K⁺ conductances in motor neurons within the ongoing cardiac network, we discovered that compensation both resynchronized the network and helped restore excitability. Using model findings to guide experimentation, we determined that compensatory increases of both G_A and electrical coupling restored function in the network. This is one of the first direct demonstrations of the physiological regulation of coupling conductance in a compensatory context, and of synergistic plasticity across cell- and network-level mechanisms in the restoration of output.

DOI:http://dx.doi.org/10.7554/eLife.16879.001

Neurons can communicate with each other by releasing chemicals called neurotransmitters, or by forming direct connections with each other known as gap junctions. These direct connections allow electrical impulses to flow from one neuron to another via pores in the membranes between the cells. Unlike communication via neurotransmitters, gap junctions are usually thought to be hard-wired and unchanging over the life of the animal.

Lane et al. recorded electrical activity in a network of neurons that generates rhythmic heart contractions in the Jonah crab. Neurons in this network usually all fire an electrical impulse at the same time, which is crucial to make sure that the whole heart contracts at the same time. The experiments show that drugs that block potassium channel pores in the membrane cause the neurons to fire too much and at different times to each other.

However, the network of neurons soon adapted to the changes caused by the drugs and returned to working as normal. Mimicking these changes in a computer model of the neuron network, together with experimental data, showed that changes to the gap junctions play a major role in restoring normal activity to the network.

The next step following on from this research is to understand how a network of neurons ‘senses’ that it is not working normally and changes its electrical activity.

DOI:http://dx.doi.org/10.7554/eLife.16879.002

From Neuron Biophysics to Orientation Selectivity in Electrically Coupled Networks of Neocortical L2/3 Large Basket Cells

In the neocortex, inhibitory interneurons of the same subtype are electrically coupled with each other via dendritic gap junctions (GJs). The impact of multiple GJs on the biophysical properties of interneurons and thus on their input processing is unclear. The present experimentally based theoretical study examined GJs in L2/3 large basket cells (L2/3 LBCs) with 3 goals in mind: (1) To evaluate the errors due to GJs in estimating the cable properties of individual L2/3 LBCs and suggest ways to correct these errors when modeling these cells and the networks they form; (2) to bracket the GJ conductance value (0.05-0.25 nS) and membrane resistivity (10 000-40 000 Ω cm(2)) of L2/3 LBCs; these estimates are tightly constrained by in vitro input resistance (131 ± 18.5 MΩ) and the coupling coefficient (1-3.5%) of these cells; and (3) to explore the functional implications of GJs, and show that GJs: (i) dynamically modulate the effective time window for synaptic integration; (ii) improve the axon's capability to encode rapid changes in synaptic inputs; and (iii) reduce the orientation selectivity, linearity index, and phase difference of L2/3 LBCs. Our study provides new insights into the role of GJs and calls for caution when using in vitro measurements for modeling electrically coupled neuronal networks.

Conditional Spike Transmission Mediated by Electrical Coupling Ensures Millisecond Precision-Correlated Activity among Interneurons In Vivo

Many GABAergic interneurons are electrically coupled and in vitro can display correlated activity with millisecond precision. However, the mechanisms underlying correlated activity between interneurons in vivo are unknown. Using dual patch-clamp recordings in vivo, we reveal that in the presence of spontaneous background synaptic activity, electrically coupled cerebellar Golgi cells exhibit robust millisecond precision-correlated activity which is enhanced by sensory stimulation. This precisely correlated activity results from the cooperative action of two mechanisms. First, electrical coupling ensures slow subthreshold membrane potential correlations by equalizing membrane potential fluctuations, such that coupled neurons tend to approach action potential threshold together. Second, fast spike-triggered spikelets transmitted through gap junctions conditionally trigger postjunctional spikes, depending on both neurons being close to threshold. Electrical coupling therefore controls the temporal precision and degree of both spontaneous and sensory-evoked correlated activity between interneurons, by the cooperative effects of shared synaptic depolarization and spikelet transmission.

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Double patch-clamp recordings from Golgi cells reveal millisecond synchrony in vivo

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Millisecond synchrony requires gap junctions and is enhanced by sensory stimuli

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Gap junctions drive synchrony via slow V_m equalization and fast spikelet transmission

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Modeling shows these findings can be generalized to any electrically coupled neurons

Network burst dynamics under heterogeneous cholinergic modulation of neural firing properties and heterogeneous synaptic connectivity

The characteristics of neural network activity depend on intrinsic neural properties and synaptic connectivity in the network. In brain networks, both of these properties are critically affected by the type and levels of neuromodulators present. The expression of many of the most powerful neuromodulators, including acetylcholine (ACh), varies tonically and phasically with behavioural state, leading to dynamic, heterogeneous changes in intrinsic neural properties and synaptic connectivity properties. Namely, ACh significantly alters neural firing properties as measured by the phase response curve in a manner that has been shown to alter the propensity for network synchronization. The aim of this simulation study was to build an understanding of how heterogeneity in cholinergic modulation of neural firing properties and heterogeneity in synaptic connectivity affect the initiation and maintenance of synchronous network bursting in excitatory networks. We show that cells that display different levels of ACh modulation have differential roles in generating network activity: weakly modulated cells are necessary for burst initiation and provide synchronizing drive to the rest of the network, whereas strongly modulated cells provide the overall activity level necessary to sustain burst firing. By applying several quantitative measures of network activity, we further show that the existence of network bursting and its characteristics, such as burst duration and intraburst synchrony, are dependent on the fraction of cell types providing the synaptic connections in the network. These results suggest mechanisms underlying ACh modulation of brain oscillations and the modulation of seizure activity during sleep states.

Cooperation and competition of gamma oscillation mechanisms

Oscillations of neuronal activity in different frequency ranges are thought to reflect important aspects of cortical network dynamics. Here we investigate how various mechanisms that contribute to oscillations in neuronal networks may interact. We focus on networks with inhibitory, excitatory, and electrical synapses, where the subnetwork of inhibitory interneurons alone can generate interneuron gamma (ING) oscillations and the interactions between interneurons and pyramidal cells allow for pyramidal-interneuron gamma (PING) oscillations. What type of oscillation will such a network generate? We find that ING and PING oscillations compete: The mechanism generating the higher oscillation frequency "wins"; it determines the frequency of the network oscillation and suppresses the other mechanism. For type I interneurons, the network oscillation frequency is equal to or slightly above the higher of the ING and PING frequencies in corresponding reduced networks that can generate only either of them; if the interneurons belong to the type II class, it is in between. In contrast to ING and PING, oscillations mediated by gap junctions and oscillations mediated by inhibitory synapses may cooperate or compete, depending on the type (I or II) of interneurons and the strengths of the electrical and chemical synapses. We support our computer simulations by a theoretical model that allows a full theoretical analysis of the main results. Our study suggests experimental approaches to deciding to what extent oscillatory activity in networks of interacting excitatory and inhibitory neurons is dominated by ING or PING oscillations and of which class the participating interneurons are.

A unified framework for spiking and gap-junction interactions in distributed neuronal network simulations

Contemporary simulators for networks of point and few-compartment model neurons come with a plethora of ready-to-use neuron and synapse models and support complex network topologies. Recent technological advancements have broadened the spectrum of application further to the efficient simulation of brain-scale networks on supercomputers. In distributed network simulations the amount of spike data that accrues per millisecond and process is typically low, such that a common optimization strategy is to communicate spikes at relatively long intervals, where the upper limit is given by the shortest synaptic transmission delay in the network. This approach is well-suited for simulations that employ only chemical synapses but it has so far impeded the incorporation of gap-junction models, which require instantaneous neuronal interactions. Here, we present a numerical algorithm based on a waveform-relaxation technique which allows for network simulations with gap junctions in a way that is compatible with the delayed communication strategy. Using a reference implementation in the NEST simulator, we demonstrate that the algorithm and the required data structures can be smoothly integrated with existing code such that they complement the infrastructure for spiking connections. To show that the unified framework for gap-junction and spiking interactions achieves high performance and delivers high accuracy in the presence of gap junctions, we present benchmarks for workstations, clusters, and supercomputers. Finally, we discuss limitations of the novel technology.

Synchronization of Biological Clock Neurons by Light and Peripheral Feedback Systems Promotes Circadian Rhythms and Health

In mammals, the suprachiasmatic nucleus (SCN) functions as a circadian clock that drives 24-h rhythms in both physiology and behavior. The SCN is a multicellular oscillator in which individual neurons function as cell-autonomous oscillators. The production of a coherent output rhythm is dependent upon mutual synchronization among single cells and requires both synaptic communication and gap junctions. Changes in phase-synchronization between individual cells have consequences on the amplitude of the SCN’s electrical activity rhythm, and these changes play a major role in the ability to adapt to seasonal changes. Both aging and sleep deprivation negatively affect the circadian amplitude of the SCN, whereas behavioral activity (i.e., exercise) has a positive effect on amplitude. Given that the amplitude of the SCN’s electrical activity rhythm is essential for achieving robust rhythmicity in physiology and behavior, the mechanisms that underlie neuronal synchronization warrant further study. A growing body of evidence suggests that the functional integrity of the SCN contributes to health, well-being, cognitive performance, and alertness; in contrast, deterioration of the 24-h rhythm is a risk factor for neurodegenerative disease, cancer, depression, and sleep disorders.

Analysis of the role of the low threshold currents IT and Ih in intrinsic delta oscillations of thalamocortical neurons

Thalamocortical neurons are involved in the generation and maintenance of brain rhythms associated with global functional states. The repetitive burst firing of TC neurons at delta frequencies (1–4 Hz) has been linked to the oscillations recorded during deep sleep and during episodes of absence seizures. To get insight into the biophysical properties that are the basis for intrinsic delta oscillations in these neurons, we performed a bifurcation analysis of a minimal conductance-based thalamocortical neuron model including only the I_T channel and the sodium and potassium leak channels. This analysis unveils the dynamics of repetitive burst firing of TC neurons, and describes how the interplay between the amplifying variable m_T and the recovering variable h_T of the calcium channel I_T is sufficient to generate low threshold oscillations in the delta band. We also explored the role of the hyperpolarization activated cationic current I_h in this reduced model and determine that, albeit not required, I_h amplifies and stabilizes the oscillation.

Oscillations emerging from noise-driven steady state in networks with electrical synapses and subthreshold resonance

Oscillations play a critical role in cognitive phenomena and have been observed in many brain regions. Experimental evidence indicates that classes of neurons exhibit properties that could promote oscillations, such as subthreshold resonance and electrical gap junctions. Typically, these two properties are studied separately but it is not clear which is the dominant determinant of global network rhythms. Our aim is to provide an analytical understanding of how these two effects destabilize the fluctuation-driven state, in which neurons fire irregularly, and lead to an emergence of global synchronous oscillations. Here we show how the oscillation frequency is shaped by single neuron resonance, electrical and chemical synapses.The presence of both gap junctions and subthreshold resonance are necessary for the emergence of oscillations. Our results are in agreement with several experimental observations such as network responses to oscillatory inputs and offer a much-needed conceptual link connecting a collection of disparate effects observed in networks.

Using a computational model to analyze the effects of firing frequency on synchrony of a network of gap junction-coupled hypoglossal motoneurons

Hypoglossal motoneurons (HMs) are located in the brainstem and play an important role in the maintenance of upper airway patency. HMs are known to be coupled to one another via gap junctions and exhibit synchronous firing behavior when driven by premotor inputs. In the current study, we used a computational model to analyze the influence of firing frequency on synchronous firing behavior of a network of gap junction-coupled HMs. As there are many factors that can influence excitability and firing frequency of HMs, our simulations were focused on the effects of modulation of SK channel conductance and the magnitude of the input currents as a means of modifying firing frequency. Our simulations revealed that regardless of the mechanism by which firing frequency was modulated, increasing the firing frequency disrupted the synchrony in gap junction-coupled HM networks with low, medium, and high levels of gap junction coupling.

Electrical coupling and innexin expression in the stomatogastric ganglion of the crab Cancer borealis

Gap junctions are intercellular channels that allow for the movement of small molecules and ions between the cytoplasm of adjacent cells and form electrical synapses between neurons. In invertebrates, the gap junction proteins are coded for by the innexin family of genes. The stomatogastric ganglion (STG) in the crab Cancer borealis contains a small number of identified and electrically coupled neurons. We identified Innexin 1 (Inx1), Innexin 2 (Inx2), Innexin 3 (Inx3), Innexin 4 (Inx4), Innexin 5 (Inx5), and Innexin 6 (Inx6) members of the C. borealis innexin family. We also identified six members of the innexin family from the lobster Homarus americanus transcriptome. These innexins show significant sequence similarity to other arthropod innexins. Using in situ hybridization and reverse transcriptase-quantitative PCR (RT-qPCR), we determined that all the cells in the crab STG express multiple innexin genes. Electrophysiological recordings of coupling coefficients between identified pairs of pyloric dilator (PD) cells and PD-lateral posterior gastric (LPG) neurons show that the PD-PD electrical synapse is nonrectifying while the PD-LPG synapse is apparently strongly rectifying.

Predicting the responses of repetitively firing neurons to current noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.

Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.

Population dynamics of the modified theta model: macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation

Gamma oscillations of the local field potential are organized by collective dynamics of numerous neurons and have many functional roles in cognition and/or attention. To mathematically and physiologically analyse relationships between individual inhibitory neurons and macroscopic oscillations, we derive a modification of the theta model, which possesses voltage-dependent dynamics with appropriate synaptic interactions. Bifurcation analysis of the corresponding Fokker-Planck equation (FPE) enables us to consider how synaptic interactions organize collective oscillations. We also develop the adjoint method (infinitesimal phase resetting curve) for simultaneous equations consisting of ordinary differential equations representing synaptic dynamics and a partial differential equation for determining the probability distribution of the membrane potential. This method provides a macroscopic phase response function (PRF), which gives insights into how it is modulated by external perturbation or internal changes of parameters. We investigate the effects of synaptic time constants and shunting inhibition on these gamma oscillations. The sensitivity of rising and decaying time constants is analysed in the oscillatory parameter regions; we find that these sensitivities are not largely dependent on rate of synaptic coupling but, rather, on current and noise intensity. Analyses of shunting inhibition reveal that it can affect both promotion and elimination of gamma oscillations. When the macroscopic oscillation is far from the bifurcation, shunting promotes the gamma oscillations and the PRF becomes flatter as the reversal potential of the synapse increases, indicating the insensitivity of gamma oscillations to perturbations. By contrast, when the macroscopic oscillation is near the bifurcation, shunting eliminates gamma oscillations and a stable firing state appears. More interestingly, under appropriate balance of parameters, two branches of bifurcation are found in our analysis of the FPE. In this case, shunting inhibition can effect both promotion and elimination of the gamma oscillation depending only on the reversal potential.

The coupling features of electrical synapses modulate neuronal synchrony in hypothalamic superachiasmatic nucleus

Electrical synapses (gap junctions) exist in many types of neurons in the mammalian brain, especially during early development period; one of the most important roles of electrical synapses is to mediate the synchrony of the neuronal networks and to coordinate the neural circuits function precisely. Previous reports show that electrical coupling is involved in modulating synchronous activity among coupled neurons, but related dynamics and mechanisms are still poorly understood. Here, in order to investigate the correlation between gap junctions and synchrony we focus on the electrically coupled neurons in suprachiasmatic nucleus (SCN) by using calcium imaging with two-photon microscopy and electrophysiology. We observed that coupled neurons in SCN present a dynamic regulation on synchrony based on their coupling strengths and are modulated by vasoactive intestinal peptide (VIP) - a neuropeptide whose receptors are expressed throughout the SCN. Modification of coupling efficiency of electrical synapses changes the synchrony level of the neuronal networks in the SCN. Our results provide new insights into the causal relationship between gap junctions and synchrony in the SCN. We further demonstrate the importance of VIP in coordinating the gap junctions-mediated signal transmission and implicate that a homeostasis environment is important for SCN to modulate the rhythmic circadian activities.

Shifted encoding strategy in retinal luminance adaptation: from firing rate to neural correlation

Neuronal responses to prolonged stimulation attenuate over time. Here, we ask a fundamental question: is adaptation a simple process for the neural system during which sustained input is ignored, or is it actually part of a strategy for the neural system to adjust its encoding properties dynamically? After simultaneously recording the activities of a group of bullfrog's retinal ganglion cells (dimming detectors) in response to sustained dimming stimulation, we applied a combination of information analysis approaches to explore the time-dependent nature of information encoding during the adaptation. We found that at the early stage of the adaptation, the stimulus information was mainly encoded in firing rates, whereas at the late stage of the adaptation, it was more encoded in neural correlations. Such a transition in encoding properties is not a simple consequence of the attenuation of neuronal firing rates, but rather involves an active change in the neural correlation strengths, suggesting that it is a strategy adopted by the neural system for functional purposes. Our results reveal that in encoding a prolonged stimulation, the neural system may utilize concerted, but less active, firings of neurons to encode information.

Interaction function of oscillating coupled neurons

Large scale simulations of electrically coupled neuronal oscillators often employ the phase coupled oscillator paradigm to understand and predict network behavior. We study the nature of the interaction between such coupled oscillators using weakly coupled oscillator theory. By employing piecewise linear approximations for phase response curves and voltage time courses and parametrizing their shapes, we compute the interaction function for all such possible shapes and express it in terms of discrete Fourier modes. We find that reasonably good approximation is achieved with four Fourier modes that comprise of both sine and cosine terms.

Effect of phase response curve skewness on synchronization of electrically coupled neuronal oscillators

We investigate why electrically coupled neuronal oscillators synchronize or fail to synchronize using the theory of weakly coupled oscillators. Stability of synchrony and antisynchrony is predicted analytically and verified using numerical bifurcation diagrams. The shape of the phase response curve (PRC), the shape of the voltage time course, and the frequency of spiking are freely varied to map out regions of parameter spaces that hold stable solutions. We find that type 1 and type 2 PRCs can hold both synchronous and antisynchronous solutions, but the shape of the PRC and the voltage determine the extent of their stability. This is achieved by introducing a five-piecewise linear model to the PRC and a three-piecewise linear model to the voltage time course, and then analyzing the resultant eigenvalue equations that determine the stability of the phase-locked solutions. A single time parameter defines the skewness of the PRC, and another single time parameter defines the spike width and frequency. Our approach gives a comprehensive picture of the relation of the PRC shape, voltage time course, and stability of the resultant synchronous and antisynchronous solutions.

Spike width and frequency alter stability of phase-locking in electrically coupled neurons

The stability of phase-locked states of electrically coupled type-1 phase response curve neurons is studied using piecewise linear formulations for their voltage profile and phase response curves. We find that at low frequency and/or small spike width, synchrony is stable, and antisynchrony unstable. At high frequency and/or large spike width, these phase-locked states switch their stability. Increasing the ratio of spike width to spike height causes the antisynchronous state to transition into a stable synchronous state. We compute the interaction function and the boundaries of stability of both these phase-locked states, and present analytical expressions for them. We also study the effect of phase response curve skewness on the boundaries of synchrony and antisynchrony.

Gap junctions facilitate propagation of synchronous firing in the cortical neural population: a numerical simulation study

This study investigates the effect of gap junctions on firing propagation in a feedforward neural network by a numerical simulation with biologically plausible parameters. Gap junctions are electrical couplings between two cells connected by a binding protein, connexin. Recent electrophysiological studies have reported that a large number of inhibitory neurons in the mammalian cortex are mutually connected by gap junctions, and synchronization of gap junctions, spread over several hundred microns, suggests that these have a strong effect on the dynamics of the cortical network. However, the effect of gap junctions on firing propagation in cortical circuits has not been examined systematically. In this study, we perform numerical simulations using biologically plausible parameters to clarify this effect on population firing in a feedforward neural network. The results suggest that gap junctions switch the temporally uniform firing in a layer to temporally clustered firing in subsequent layers, resulting in an enhancement in the propagation of population firing in the feedforward network. Because gap junctions are often modulated in physiological conditions, we speculate that gap junctions could be related to a gating function of population firing in the brain.

Effect of sharp jumps at the edges of phase response curves on synchronization of electrically coupled neuronal oscillators

We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.

The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance (gc) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of gc are calculated from the experimentally measured coupling coefficient (CC) based on a single-compartment description of a neuron, and therefore may be severe underestimates of gc. With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of CC, which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.

Analyzing the effects of gap junction blockade on neural synchrony via a motoneuron network computational model

In specific regions of the central nervous system (CNS), gap junctions have been shown to participate in neuronal synchrony. Amongst the CNS regions identified, some populations of brainstem motoneurons are known to be coupled by gap junctions. The application of various gap junction blockers to these motoneuron populations, however, has led to mixed results regarding their synchronous firing behavior, with some studies reporting a decrease in synchrony while others surprisingly find an increase in synchrony. To address this discrepancy, we employ a neuronal network model of Hodgkin-Huxley-style motoneurons connected by gap junctions. Using this model, we implement a series of simulations and rigorously analyze their outcome, including the calculation of a measure of neuronal synchrony. Our simulations demonstrate that under specific conditions, uncoupling of gap junctions is capable of producing either a decrease or an increase in neuronal synchrony. Subsequently, these simulations provide mechanistic insight into these different outcomes.

Stimulus features, resetting curves, and the dependence on adaptation

We derive a formula that relates the spike-triggered covariance (STC) to the phase resetting curve (PRC) of a neural oscillator. We use this to show how changes in the shape of the PRC alter the sensitivity of the neuron to different stimulus features, which are the eigenvectors of the STC. We compute the PRC and STC for some biophysical models. We compare the STCs and their spectral properties for a two-parameter family of PRCs. Surprisingly, the skew of the PRC has a larger effect on the spectrum and shape of the STC than does the bimodality of the PRC (which plays a large role in synchronization properties). Finally, we relate the STC directly to the spike-triggered average and apply this theory to an olfactory bulb mitral cell recording.

The shape of phase-resetting curves in oscillators with a saddle node on an invariant circle bifurcation

We introduce a simple two-dimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.

Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization

Synchronous neural oscillations are found throughout the brain and are thought to contribute to neural coding and the propagation of activity. Several proposed mechanisms of synchronization have gained support through combined theoretical and experimental investigation, including mechanisms based on coupling and correlated input. Here, we ask how correlation-induced synchrony is affected by physiological heterogeneity across neurons. To address this question, we examined cell-to-cell differences in phase-response curves (PRCs), which characterize the response of periodically firing neurons to weak perturbations. Using acute slice electrophysiology, we measured PRCs across a single class of principal neurons capable of sensory-evoked oscillations in vivo: the olfactory bulb mitral cells (MCs). Periodically firing MCs displayed a broad range of PRCs, each of which was well fit by a simple three-parameter model. MCs also displayed differences in firing rate-current relationships and in preferred firing rate ranges. Both the observed PRC heterogeneity and moderate firing rate differences (∼10 Hz) separately reduced the maximum correlation-induced synchrony between MCs by up to 25-30%. Simulations further demonstrated that these components of heterogeneity alone were sufficient to account for the difference in synchronization among heterogeneous vs. homogeneous populations in vitro. Within this simulation framework, independent modulation of specific PRC features additionally revealed which aspects of PRC heterogeneity most strongly impact correlation-induced synchronization. Finally, we demonstrated good agreement of novel mathematical theory with our experimental and simulation results, providing a theoretical basis for the influence of heterogeneity on correlation-induced neural synchronization.