Short duty cycle destabilizes a half-center oscillator, but gap junctions can restabilize the anti-phase pattern

In-phase and anti-phase bursting dynamics and synchronisation scenario in neural network by varying coupling phase

We have analysed the synchronisation scenario and the rich spatiotemporal patterns in the network of Hindmarsh-Rose neurons under the influence of self, mixed and cross coupling of state variables which are realised by varying coupling phase. We have introduced a coupling matrix in the model to vary coupling phase. The excitatory and inhibitory couplings in the membrane potential induce in-phase and anti-phase bursting dynamics, respectively, in the two coupled system. When the off-diagonal elements of the matrix are zero, the system shows self coupling of the three variables, which helps to attain synchrony. The off-diagonal elements give cross interactions between the variables, which reduces synchrony. The stability of the synchrony attained is analysed using Lyapunov function approach. In our study, we found that self coupling in three variables is sufficient to induce chimera states in non-local coupling. The strength of incoherence and discontinuity measure validates the existence of chimera and multichimera states. The inhibitor self coupling in local interaction induces interesting patterns like Mixed Oscillatory State and clusters. The results may help in understanding the spatiotemporal communications of the brain, within the limitations of the size of the network analysed in this study.

Distinct Mechanisms Underlie Electrical Coupling Resonance and Its Interaction with Membrane Potential Resonance

Neurons in oscillatory networks often exhibit membrane potential resonance, a peak impedance at a non-zero input frequency. In electrically coupled oscillatory networks, the coupling coefficient (the ratio of post- and prejunctional voltage responses) could also show resonance. Such coupling resonance may emerge from the interaction between the coupling current and resonance properties of the coupled neurons, but this relationship has not been clearly described. Additionally, it is unknown if the gap-junction mediated electrical coupling conductance may have frequency dependence. We examined these questions by recording a pair of electrically coupled neurons in the oscillatory pyloric network of the crab Cancer borealis. We performed dual current- and voltage-clamp recordings and quantified the frequency preference of the coupled neurons, the coupling coefficient, the electrical conductance, and the postjunctional neuronal response. We found that all components exhibit frequency selectivity, but with distinct preferred frequencies. Mathematical and computational analysis showed that membrane potential resonance of the postjunctional neuron was sufficient to give rise to resonance properties of the coupling coefficient, but not the coupling conductance. A distinct coupling conductance resonance frequency therefore emerges either from other circuit components or from the gating properties of the gap junctions. Finally, to explore the functional effect of the resonance of the coupling conductance, we examined its role in synchronizing neuronal the activities of electrically coupled bursting model neurons. Together, our findings elucidate factors that produce electrical coupling resonance and the function of this resonance in oscillatory networks.

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.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Generalized half-center oscillators with short-term synaptic plasticity

How can we develop simple yet realistic models of the small neural circuits known as central pattern generators (CPGs), which contribute to generate complex multiphase locomotion in living animals? In this paper we introduce a new model (with design criteria) of a generalized half-center oscillator, (pools of) neurons reciprocally coupled by fast/slow inhibitory and excitatory synapses, to produce either alternating bursting or other rhythmic patterns, characterized by different phase lags, depending on the sensory or other external input. We also show how to calibrate its parameters, based on both physiological and functional criteria and on bifurcation analysis. This model accounts for short-term neuromodulation in a biophysically plausible way and is a building block to develop more realistic and functionally accurate CPG models. Examples and counterexamples are used to point out the generality and effectiveness of our design approach.

Computational Modeling of Spinal Locomotor Circuitry in the Age of Molecular Genetics

Neuronal circuits in the spinal cord are essential for the control of locomotion. They integrate supraspinal commands and afferent feedback signals to produce coordinated rhythmic muscle activations necessary for stable locomotion. For several decades, computational modeling has complemented experimental studies by providing a mechanistic rationale for experimental observations and by deriving experimentally testable predictions. This symbiotic relationship between experimental and computational approaches has resulted in numerous fundamental insights. With recent advances in molecular and genetic methods, it has become possible to manipulate specific constituent elements of the spinal circuitry and relate them to locomotor behavior. This has led to computational modeling studies investigating mechanisms at the level of genetically defined neuronal populations and their interactions. We review literature on the spinal locomotor circuitry from a computational perspective. By reviewing examples leading up to and in the age of molecular genetics, we demonstrate the importance of computational modeling and its interactions with experiments. Moving forward, neuromechanical models with neuronal circuitry modeled at the level of genetically defined neuronal populations will be required to further unravel the mechanisms by which neuronal interactions lead to locomotor behavior.

A biophysical model of striatal microcircuits suggests gamma and beta oscillations interleaved at delta/theta frequencies mediate periodicity in motor control

Striatal oscillatory activity is associated with movement, reward, and decision-making, and observed in several interacting frequency bands. Local field potential recordings in rodent striatum show dopamine- and reward-dependent transitions between two states: a "spontaneous" state involving β (∼15-30 Hz) and low γ (∼40-60 Hz), and a state involving θ (∼4-8 Hz) and high γ (∼60-100 Hz) in response to dopaminergic agonism and reward. The mechanisms underlying these rhythmic dynamics, their interactions, and their functional consequences are not well understood. In this paper, we propose a biophysical model of striatal microcircuits that comprehensively describes the generation and interaction of these rhythms, as well as their modulation by dopamine. Building on previous modeling and experimental work suggesting that striatal projection neurons (SPNs) are capable of generating β oscillations, we show that networks of striatal fast-spiking interneurons (FSIs) are capable of generating δ/θ (ie, 2 to 6 Hz) and γ rhythms. Under simulated low dopaminergic tone our model FSI network produces low γ band oscillations, while under high dopaminergic tone the FSI network produces high γ band activity nested within a δ/θ oscillation. SPN networks produce β rhythms in both conditions, but under high dopaminergic tone, this β oscillation is interrupted by δ/θ-periodic bursts of γ-frequency FSI inhibition. Thus, in the high dopamine state, packets of FSI γ and SPN β alternate at a δ/θ timescale. In addition to a mechanistic explanation for previously observed rhythmic interactions and transitions, our model suggests a hypothesis as to how the relationship between dopamine and rhythmicity impacts motor function. We hypothesize that high dopamine-induced periodic FSI γ-rhythmic inhibition enables switching between β-rhythmic SPN cell assemblies representing the currently active motor program, and thus that dopamine facilitates movement in part by allowing for rapid, periodic shifts in motor program execution.

The Potential Role of Gap Junctional Plasticity in the Regulation of State

Electrical synapses are the functional correlate of gap junctions and allow transmission of small molecules and electrical current between coupled neurons. Instead of static pores, electrical synapses are actually plastic, similar to chemical synapses. In the thalamocortical system, gap junctions couple inhibitory neurons that are similar in their biochemical profile, morphology, and electrophysiological properties. We postulate that electrical synaptic plasticity among inhibitory neurons directly interacts with the switching between different firing patterns in a state-dependent and type-dependent manner. In neuronal networks, electrical synapses may function as a modifiable resonance feedback system that enables stable oscillations. Furthermore, the plasticity of electrical synapses may play an important role in regulation of state, synchrony, and rhythmogenesis in the mammalian thalamocortical system, similar to chemical synaptic plasticity. Based on their plasticity, rich diversity, and specificity, electrical synapses are thus likely to participate in the control of consciousness and attention.

When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly non-trivial, combined effect of electrical and inhibitory connections. We report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism, which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

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.

Asymmetry Factors Shaping Regular and Irregular Bursting Rhythms in Central Pattern Generators

Central Pattern Generator (CPG) circuits are neural networks that generate rhythmic motor patterns. These circuits are typically built of half-center oscillator subcircuits with reciprocally inhibitory connections. Another common property in many CPGs is the remarkable rich spiking-bursting dynamics of their constituent cells, which balance robustness and flexibility to generate their joint coordinated rhythms. In this paper, we use conductance-based models and realistic connection topologies inspired by the crustacean pyloric CPG to address the study of asymmetry factors shaping CPG bursting rhythms. In particular, we assess the role of asymmetric maximal synaptic conductances, time constants and gap-junction connectivity to establish the regularity of half-center oscillator based CPGs. We map and characterize the synaptic parameter space that lead to regular and irregular bursting activity in these networks. The analysis indicates that asymmetric configurations display robust regular rhythms and that large regions of both regular and irregular but coordinated rhythms exist as a function of the asymmetry in the circuit. Our results show that asymmetry both in the maximal conductances and in the temporal dynamics of mutually inhibitory neurons can synergistically contribute to shape wide regimes of regular spiking-bursting activity in CPGs. Finally, we discuss how a closed-loop protocol driven by a regularity goal can be used to find and characterize regular regimes when there is not time to perform an exhaustive search, as in most experimental studies.

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.

Variety of alternative stable phase-locking in networks of electrically coupled relaxation oscillators

We studied the dynamics of a large-scale model network comprised of oscillating electrically coupled neurons. Cells are modeled as relaxation oscillators with short duty cycle, so they can be considered either as models of pacemaker cells, spiking cells with fast regenerative and slow recovery variables or firing rate models of excitatory cells with synaptic depression or cellular adaptation.

It was already shown that electrically coupled relaxation oscillators exhibit not only synchrony but also anti-phase behavior if electrical coupling is weak. We show that a much wider spectrum of spatiotemporal patterns of activity can emerge in a network of electrically coupled cells as a result of switching from synchrony, produced by short external signals of different spatial profiles. The variety of patterns increases with decreasing rate of neuronal firing (or duty cycle) and with decreasing strength of electrical coupling. We study also the effect of network topology - from all-to-all – to pure ring connectivity, where only the closest neighbors are coupled.

We show that the ring topology promotes anti-phase behavior as compared to all-to-all coupling. It also gives rise to a hierarchical organization of activity: during each of the main phases of a given pattern cells fire in a particular sequence determined by the local connectivity. We have analyzed the behavior of the network using geometric phase plane methods and we give heuristic explanations of our findings.

Our results show that complex spatiotemporal activity patterns can emerge due to the action of stochastic or sensory stimuli in neural networks without chemical synapses, where each cell is equally coupled to others via gap junctions. This suggests that in developing nervous systems where only electrical coupling is present such a mechanism can lead to the establishment of proto-networks generating premature multiphase oscillations whereas the subsequent emergence of chemical synapses would later stabilize generated patterns.

Rectifying electrical synapses can affect the influence of synaptic modulation on output pattern robustness

Rectifying electrical synapses are commonplace, but surprisingly little is known about how rectification alters the dynamics of neuronal networks. In this study, we use computational models to investigate how rectifying electrical synapses change the behavior of a small neuronal network that exhibits complex rhythmic output patterns. We begin with an electrically coupled circuit of three oscillatory neurons with different starting frequencies, and subsequently add two additional neurons and inhibitory chemical synapses. The five-cell model represents a pattern-generating neuronal network with two simultaneous rhythms competing for the recruitment of a hub neuron. We compare four different configurations of rectifying synapse placement and polarity, and we investigate how rectification changes the functional output of this network. Rectification can have a striking effect on the network's sensitivity to alterations of the strengths of the chemical synapses in the network. For some configurations, the rectification makes the circuit dynamics remarkably robust against changes in synaptic strength compared with the nonrectifying case. Based on our findings, we predict that modulation of rectifying electrical synapses could have functional consequences for the neuronal circuits that express them.

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.

Multiple mechanisms switch an electrically coupled, synaptically inhibited neuron between competing rhythmic oscillators

Rhythmic oscillations are common features of nervous systems. One of the fundamental questions posed by these rhythms is how individual neurons or groups of neurons are recruited into different network oscillations. We modeled competing fast and slow oscillators connected to a hub neuron with electrical and inhibitory synapses. We explore the patterns of coordination shown in the network as a function of the electrical coupling and inhibitory synapse strengths with the help of a novel visualization method that we call the "parameterscape." The hub neuron can be switched between the fast and slow oscillators by multiple network mechanisms, indicating that a given change in network state can be achieved by degenerate cellular mechanisms. These results have importance for interpreting experiments employing optogenetic, genetic, and pharmacological manipulations to understand circuit dynamics.

Adaptive and phase selective spike timing dependent plasticity in synaptically coupled neuronal oscillators

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits.

Spikes matter for phase-locked bursting in inhibitory neurons

We show that inhibitory networks composed of two endogenously bursting neurons can robustly display several coexistent phase-locked states in addition to stable antiphase and in-phase bursting. This work complements and enhances our recent result [Jalil, Belykh, and Shilnikov, Phys. Rev. E 81, 045201(R) (2010)] that fast reciprocal inhibition can synchronize bursting neurons due to spike interactions. We reveal the role of spikes in generating multiple phase-locked states and demonstrate that this multistability is generic by analyzing diverse models of bursting networks with various fast inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, and the Purkinje neuron model.

Co-existent activity patterns in inhibitory neuronal networks with short-term synaptic depression

A network of two neurons mutually coupled through inhibitory synapses that display short-term synaptic depression is considered. We show that synaptic depression expands the number of possible activity patterns that the network can display and allows for co-existence of different patterns. Specifically, the network supports different types of n-m anti-phase firing patterns, where one neuron fires n spikes followed by the other neuron firing m spikes. When maximal synaptic conductances are identical, n-n anti-phase firing patterns are obtained and there are conductance intervals over which different pairs of these solutions co-exist. The multitude of n-m anti-phase patterns and their co-existence are not found when the synapses are non-depressing. Geometric singular perturbation methods for dynamical systems are applied to the original eight-dimensional model system to derive a set of one-dimensional conditions for the existence and co-existence of different anti-phase solutions. The generality and validity of these conditions are demonstrated through numerical simulations utilizing the Hodgkin-Huxley and Morris-Lecar neuronal models.

Multi-stability and pattern-selection in oscillatory networks with fast inhibition and electrical synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives.

Multifunctional pattern-generating circuits

The ability of distinct anatomical circuits to generate multiple behavioral patterns is widespread among vertebrate and invertebrate species. These multifunctional neuronal circuits are the result of multistable neural dynamics and modular organization. The evidence suggests multifunctional circuits can be classified by distinct architectures, yet the activity patterns of individual neurons involved in more than one behavior can vary dramatically. Several mechanisms, including sensory input, the parallel activity of projection neurons, neuromodulation, and biomechanics, are responsible for the switching between patterns. Recent advances in both analytical and experimental tools have aided the study of these complex circuits.

Inhibition potentiates the synchronizing action of electrical synapses

In vivo and in vitro experimental studies have found that blocking electrical interactions connecting GABAergic interneurons reduces oscillatory activity in the gamma range in cortex. However, recent theoretical works have shown that the ability of electrical synapses to promote or impede synchrony, when alone, depends on their location on the dendritic tree of the neurons, the intrinsic properties of the neurons and the connectivity of the network. The goal of the present paper is to show that this versatility in the synchronizing ability of electrical synapses is greatly reduced when the neurons also interact via inhibition. To this end, we study a model network comprising two-compartment conductance-based neurons interacting with both types of synapses. We investigate the effect of electrical synapses on the dynamical state of the network as a function of the strength of the inhibition. We find that for weak inhibition, electrical synapses reinforce inhibition-generated synchrony only if they promote synchrony when they are alone. In contrast, when inhibition is sufficiently strong, electrical synapses improve synchrony even if when acting alone they would stabilize asynchronous firing. We clarify the mechanism underlying this cooperative interplay between electrical and inhibitory synapses. We show that it is relevant in two physiologically observed regimes: spike-to-spike synchrony, where neurons fire at almost every cycle of the population oscillations, and stochastic synchrony, where neurons fire irregularly and at a rate which is substantially lower than the frequency of the global population rhythm.

Sustained rhythmic activity in gap-junctionally coupled networks of model neurons depends on the diameter of coupled dendrites

Gap junctions are known to be important for many network functions such as synchronization of activity and the generation of waves and oscillations. Gap junctions have also been proposed to be essential for the generation of early embryonic activity. We have previously shown that the amplitude of electrical signals propagating across gap-junctionally coupled passive cables is maximized at a unique diameter. This suggests that threshold-dependent signals may propagate through gap junctions for a finite range of diameters around this optimal value. Here we examine the diameter dependence of action potential propagation across model networks of dendro-dendritically coupled neurons. The neurons in these models have passive soma and dendrites and an action potential-generating axon. We show that propagation of action potentials across gap junctions occurs only over a finite range of dendritic diameters and that propagation delay depends on this diameter. Additionally, in networks of gap-junctionally coupled neurons, rhythmic activity can emerge when closed loops (re-entrant paths) occur but again only for a finite range of dendrite diameters. The frequency of such rhythmic activity depends on the length of the path and the dendrite diameter. For large networks of randomly coupled neurons, we find that the re-entrant paths that underlie rhythmic activity also depend on dendrite diameter. These results underline the potential importance of dendrite diameter as a determinant of network activity in gap-junctionally coupled networks, such as network rhythms that are observed during early nervous system development.

Forced phase-locked states and information retrieval in a two-layer network of oscillatory neurons with directional connectivity

We propose two-layer architecture of associative memory oscillatory network with directional interlayer connectivity. The network is capable to store information in the form of phase-locked (in-phase and antiphase) oscillatory patterns. The first (input) layer takes an input pattern to be recognized and their units are unidirectionally connected with all units of the second (control) layer. The connection strengths are weighted using the Hebbian rule. The output (retrieved) patterns appear as forced-phase locked states of the control layer. The conditions are found and analytically expressed for pattern retrieval in response on incoming stimulus. It is shown that the system is capable to recover patterns with a certain level of distortions or noises in their profiles. The architecture is implemented with the Kuramoto phase model and using synaptically coupled neural oscillators with spikes. It is found that the spiking model is capable to retrieve patterns using the spiking phase that translates memorized patterns into the spiking phase shifts at different time scales.

Innexins in the lobster stomatogastric nervous system: cloning, phylogenetic analysis, developmental changes and expression within adult identified dye and electrically coupled neurons

Gap junctions play a key role in the operation of neuronal networks by enabling direct electrical and metabolic communication between neurons. Suitable models to investigate their role in network operation and plasticity are invertebrate motor networks, which are built of comparatively few identified neurons, and can be examined throughout development; an excellent example is the lobster stomatogastric nervous system. In invertebrates, gap junctions are formed by proteins that belong to the innexin family. Here, we report the first molecular characterization of two crustacean innexins: the lobster Homarus gammarus innexin 1 (Hg-inx1) and 2 (Hg-inx2). Phylogenetic analysis reveals that innexin gene duplication occurred within the arthropod clade before the separation of insect and crustacean lineages. Using in situ hybridization, we find that each innexin is expressed within the adult and developing lobster stomatogastric nervous system and undergoes a marked down-regulation throughout development within the stomatogastric ganglion (STG). The number of innexin expressing neurons is significantly higher in the embryo than in the adult. By combining in situ hybridization, dye and electrical coupling experiments on identified neurons, we demonstrate that adult neurons that express at least one innexin are dye and electrically coupled with at least one other STG neuron. Finally, two STG neurons display no detectable amount of either innexin mRNAs but may express weak electrical coupling with other STG neurons, suggesting the existence of other forms of innexins. Altogether, we provide evidence that innexins are expressed within small neuronal networks built of dye and electrically coupled neurons and may be developmentally regulated.

The synchronization properties of a network of inhibitory interneurons depend on the biophysical model

The synchronization properties of a pair of coupled fast spiking interneurons are studied by using the theory of weakly coupled oscillators. Four different biophysical models of the single fast spiking interneuron are used and the corresponding results are compared. It is shown that for a pair of identical coupled cells, the synchronization properties are model-dependent. In particular, the firing coherence of the network is strongly affected by the reversal potential, the kinetics of the inhibitory postsynaptic current and the electrical coupling; the activation properties of the sodium and potassium currents play a significant role too.

Just a phase they're going through: the complex interaction of intrinsic high-threshold bursting and gap junctions in the generation of thalamic alpha and theta rhythms

Rhythms in the alpha frequency band (8-13 Hz) are a defining feature of the human EEG during relaxed wakefulness and are known to be influenced by the thalamus. In the early stages of sleep and in several neurological and psychiatric conditions alpha rhythms are replaced by slower activity in the theta (3-7 Hz) band. Of particular interest is how these alpha and theta rhythms are generated at the cellular level. Recently we identified a subset of thalamocortical (TC) neurons in the lateral geniculate nucleus (LGN) which exhibit rhythmic high-threshold (>-55 mV) bursting at approximately 2-13 Hz and which are interconnected by gap junctions (GJs). These cells combine to generate a locally synchronized continuum of alpha and theta oscillations, thus providing direct evidence that the thalamus can act as an independent pacemaker of alpha and theta rhythms. Interestingly, GJ coupled pairs of TC neurons can exhibit both in-phase and anti-phase synchrony and will often spontaneously alternate between these two states. This dictates that the local field oscillation amplitude is not simply linked to the extent of cell recruitment into a single synchronized neuronal assembly but also to the degree of destructive interference between dynamic, spatially overlapping, competing anti-phase groups of continuously bursting neurons. Thus, the waxing and waning of thalamic alpha/theta rhythms should not be assumed to reflect a wholesale increase and reduction, respectively, in underlying neuronal synchrony. We argue that these network dynamics might have important consequences for relating changes in the amplitude of EEG alpha and theta rhythms to the activity of thalamic networks.

On the dynamics of electrically-coupled neurons with inhibitory synapses

We study the dynamics and bifurcations of noise-free neurons coupled by gap junctions and inhibitory synapses, using both delayed delta functions and alpha functions to model the latter. We focus on the case of two cells, as in the studies of Chow and Kopell (2000) and Lewis and Rinzel (2003), but also show that stable asynchronous splay states exist for globally coupled networks of N cells dominated by subthreshold electrical coupling. Our results agree with those of Lewis and Rinzel (2003) in the weak coupling range, but our Poincaré map analysis yields more information about global behavior and domains of attraction, and we show that the explicit discontinuous maps derived using delayed delta functions compare well with the continuous history-dependent, implicitly-defined maps derived from alpha functions. We find that increased bias currents, super-threshold electrical coupling and synaptic delays promote synchrony, while sub-threshold electrical coupling and fast synapses promote asynchrony. We compare our analytical results with simulations of an ionic current model of spiking cells, and briefly discuss implications for stimulus response modes of locus coeruleus and for central pattern generators.

Signal transmission between gap-junctionally coupled passive cables is most effective at an optimal diameter

We analyze simple morphological configurations that represent gap-junctional coupling between neuronal processes or between muscle fibers. Specifically, we use cable theory and simulations to examine the consequences of current flow from one cable to other gap-junctionally coupled passive cables. When the proximal end of the first cable is voltage clamped, the amplitude of the electrical signal in distal portions of the second cable depends on the cable diameter. However, this amplitude does not simply increase if cable diameter is increased, as expected from the larger length constant; instead, an optimal diameter exists. The optimal diameter arises because the dependency of voltage attenuation along the second cable on cable diameter follows two opposing rules. As cable diameter increases, the attenuation decreases because of a larger length constant yet increases because of a reduction in current density arising from the limiting effect of the gap junction on current flow into the second cable. The optimal diameter depends on the gap junction resistance and cable parameters. In branched cables, dependency on diameter is local and thus may serve to functionally compartmentalize branches that are coupled to other cells. Such compartmentalization may be important when periodic signals or action potentials cause the current flow across gap junctions.

Synchronization in networks with random interactions: theory and applications

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to systems with time delayed coupling and then nonlinear systems where the synchronized dynamics can be periodic or chaotic. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.

Electrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons

Information processing in higher brain structures is thought to rely on the synchronization of spiking neurons. Increasing evidence indicates that, within these structures, inhibitory neurons are linked by both chemical and electrical synapses. However, how synchronized states may emerge from such circuits is not fully understood. Using snail neurons interconnected through a dynamic-clamp system, we show that networks of spiking neurons linked by both reciprocal inhibition and electrical coupling can express two coexisting coordination patterns of different rhythms. One of these patterns consists of antiphase firing of the network partners whereas, in the other, neurons fire synchronously. Switching between patterns may be evoked immediately by transient stimuli, demonstrating bistability of the network. Thus electrical coupling can provide a potent way for instantaneous reconfiguration of activity patterns in inhibitory spiking networks without alteration of intrinsic network properties by modulatory processes.

Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling

We study the synchronization dynamics for a system of two Hodgkin-Huxley (HH) neurons coupled diffusively or through pulselike interactions. By calculating the maximum transverse Lyapunov exponent, we found that, with diffusive coupling, there are three regions in the parameter space, corresponding to qualitatively distinct behaviors of the coupled dynamics. In particular, the two neurons can synchronize in two regions and desynchronize in the third. When excitatory and inhibitory pulse coupling is considered, we found that synchronized dynamics becomes more difficult to achieve in the sense that the parameter regions where the synchronous state is stable are smaller. Numerical simulations of the coupled system are presented to validate these results. The stability of a network of coupled HH neurons is then analyzed and the stability regions in the parameter space are exactly obtained.

The combined effects of inhibitory and electrical synapses in synchrony

Recent experimental results have shown that GABAergic interneurons in the central nervous system are frequently connected via electrical synapses. Hence, depending on the area or the subpopulation, interneurons interact via inhibitory synapses or electrical synapses alone or via both types of interactions. The theoretical work presented here addresses the significance of these different modes of interactions for the interneuron networks dynamics. We consider the simplest system in which this issue can be investigated in models or in experiments: a pair of neurons, interacting via electrical synapses, inhibitory synapses, or both, and activated by the injection of a noisy external current. Assuming that the couplings and the noise are weak, we derive an analytical expression relating the cross-correlation (CC) of the activity of the two neurons to the phase response function of the neurons. When electrical and inhibitory interactions are not too strong, they combine their effect in a linear manner. In this regime, the effect of electrical and inhibitory interactions when combined can be deduced knowing the effects of each of the interactions separately. As a consequence, depending on intrinsic neuronal properties, electrical and inhibitory synapses may cooperate, both promoting synchrony, or may compete, with one promoting synchrony while the other impedes it. In contrast, for sufficiently strong couplings, the two types of synapses combine in a nonlinear fashion. Remarkably, we find that in this regime, combining electrical synapses with inhibition amplifies synchrony, whereas electrical synapses alone would desynchronize the activity of the neurons. We apply our theory to predict how the shape of the CC of two neurons changes as a function of ionic channel conductances, focusing on the effect of persistent sodium conductance, of the firing rate of the neurons and the nature and the strength of their interactions. These predictions may be tested using dynamic clamp techniques.

Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks

Electrical and chemical synapses exist within the same networks of inhibitory cells, and each kind of synapse is known to be able to foster synchrony among oscillating neurons. Using numerical and analytical techniques, we show here that the electrical and inhibitory coupling play different roles in the synchronization of rhythms in inhibitory networks. The parameter range chosen is motivated by gamma rhythms, in which the gamma-aminobutyric acid type A (GABAA)-mediated inhibition is relatively strong. Under this condition, addition of a small electrical conductance can increase the degree of synchronization far more than a much larger increase in inhibitory conductance. The inhibitory synapses act to eliminate the effects of different initial conditions, whereas the electrical synapses mitigate suppression of firing due to heterogeneity in the network. Analytical techniques include tracking trajectories of coupled cells between spikes; the analysis shows that, in networks in which the degree of excitability is heterogeneous, inhibition can increase the dispersion of the voltages between spikes, whereas electrical coupling reduces such dispersion.