Inhibitory synchronization of bursting in biological neurons: dependence on synaptic time constant

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

The neuronal and synaptic dynamics underlying post-inhibitory rebound burst related to major depressive disorder in the lateral habenula neuron model

A burst behavior observed in the lateral habenula (LHb) neuron related to major depressive disorder has attracted much attention. The burst is induced from silence by the excitatory N-methyl-D-aspartate (NMDA) synapse or by the inhibitory stimulation, i.e., a post-inhibitory rebound (PIR) burst, which has not been explained clearly. In the present paper, the neuronal and synaptic dynamics for the PIR burst are acquired in a theoretical neuron model. At first, dynamic cooperations between the fast rise of inhibitory γ-aminobutyric acid (GABA) synapse, slow rise of NMDA synapse, and T-type calcium current to evoke the PIR burst are obtained. Similar to the inhibitory pulse stimulation, fast rising GABA current can reduce the membrane potential to a level low enough to de-inactivate the low threshold T-type calcium current to evoke a PIR spike, which can enhance the slow rising NMDA current activated at a time before or after the PIR spike. The NMDA current following the PIR spike exhibits slow decay to induce multiple spikes to form the PIR burst. Such results present a theoretical explanation and a candidate for the PIR burst in real LHb neurons. Then, the dynamical mechanism for the PIR spike mediated by the T-type calcium channel is obtained. At large conductance of T-type calcium channel, the resting state corresponds to a stable focus near Hopf bifurcation and exhibits an "uncommon" threshold curve with membrane potential much lower than the resting membrane potential. Inhibitory modulation induces membrane potential decreased to run across the threshold curve to evoke the PIR spike. At small conductance of the T-type calcium channel, a stable node appears and manifests a common threshold curve with higher membrane potential, resulting in non-PIR phenomenon. The results present the dynamic cooperations between neuronal dynamics and fast/slow dynamics of different synapses for the PIR burst observed in the LHb neuron, which is helpful for the modulations to major depressive disorder.

Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling

Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.

Influence of inhibitory autapses on synchronization of inhibitory network gamma oscillations

A recent experimental study showed that inhibitory autapses favor firing synchronization of parvalbumin interneurons in the neocortex during gamma oscillations. In the present paper, to provide a comprehensive and deep understanding to the experimental observation, the influence of inhibitory autapses on synchronization of interneuronal network gamma oscillations is theoretically investigated. Weak, middle, and strong synchronizations of a globally inhibitory coupled network composed of Wang-Buzsáki model without autapses appear at the bottom-left, middle, and top-right of the parameter plane with the conductance (gsyn) and the decay constant (τsyn) of inhibitory synapses taken as the x-axis and y-axis, respectively. After introducing inhibitory autapses, the border between the strong and middle synchronizations in the (gsyn, τsyn) plane moves to the top-right with increasing the conductance (gaut) and the decay constant (τaut) of autapses, due to that interspike interval of the single neuron becomes longer, leading to that larger τsyn is needed to ensure the strong synchronization. Then, the synchronization degree of middle and strong synchronizations around the border in the (gsyn, τsyn) plane decreases, while of strong synchronization in the remaining region remains unchanged. The synchronization degree of weak synchronization increases with increasing τaut and gaut, due to that the inhibitory autaptic current becomes strong and long to facilitate synchronization. The enhancement of weak synchronization modulated by inhibitory autapses is also simulated in the random, small-world, and scale-free networks, which may provide explanations to the experimental observation. These results present complex dynamics of synchronization modulated by inhibitory autapses, which needs future experimental demonstrations.

Nonlinear mechanism for the enhanced bursting activities induced by fast inhibitory autapse and reduced activities by fast excitatory autapse

The paradoxical phenomena that excitatory modulation does not enhance but reduces or inhibitory modulation not suppresses but promotes neural firing activities have attracted increasing attention. In the present study, paradoxical phenomena induced by both fast excitatory and inhibitory autapses in a "Fold/Big Homoclinic" bursting are simulated, and the corresponding nonlinear and biophysical mechanisms are presented. Firstly, the enhanced conductance of excitatory autapse induces the number of spikes per burst and firing rate reduced, while the enhanced inhibitory autapse cause both indicators increased. Secondly, with fast-slow variable dissection, the burst of bursting is identified to locate between a fold bifurcation and a big saddle-homoclinic orbit bifurcation of the fast subsystem. Enhanced excitatory or inhibitory autapses cannot induce changes of both bifurcation points, i.e., burst width. However, width of slow variable between two successive spikes within a burst becomes wider for the excitatory autapse and narrower for the inhibitory autapse, resulting in the less and more spikes per burst, respectively. Last, the autaptic current of fast autapse mainly plays a role during the peak of action potential, differing from the slow autaptic current with exponential decay, which can play roles following the peak of action potential. The fast excitatory autaptic current enhances the amplitude of the action potential and reduces the repolarization of the action potential to lengthen the interspike interval (ISI) of the spiking of the fast subsystem, resulting in the wide width of slow variable between successive spikes. The fast inhibitory autaptic current reduces the amplitude of action potential and ISI of spiking, resulting in narrow width of slow variable. The novel example of the paradoxical responses for both fast modulations and nonlinear mechanism extend the contents of neurodynamics, which presents potential functions of the fast autapse.

Bio-Inspired Design of Superconducting Spiking Neuron and Synapse

The imitative modelling of processes in the brain of living beings is an ambitious task. However, advances in the complexity of existing hardware brain models are limited by their low speed and high energy consumption. A superconducting circuit with Josephson junctions closely mimics the neuronal membrane with channels involved in the operation of the sodium-potassium pump. The dynamic processes in such a system are characterised by a duration of picoseconds and an energy level of attojoules. In this work, two superconducting models of a biological neuron are studied. New modes of their operation are identified, including the so-called bursting mode, which plays an important role in biological neural networks. The possibility of switching between different modes in situ is shown, providing the possibility of dynamic control of the system. A synaptic connection that mimics the short-term potentiation of a biological synapse is developed and demonstrated. Finally, the simplest two-neuron chain comprising the proposed bio-inspired components is simulated, and the prospects of superconducting hardware biosimilars are briefly discussed.

Neurobiological reduction: From cellular explanations of behavior to interventions

Scientific reductionism, the view that higher level functions can be explained by properties at some lower-level or levels, has been an assumption of nervous system analyses since the acceptance of the neuron doctrine in the late 19th century, and became a dominant experimental approach with the development of intracellular recording techniques in the mid-20th century. Subsequent refinements of electrophysiological approaches and the continual development of molecular and genetic techniques have promoted a focus on molecular and cellular mechanisms in experimental analyses and explanations of sensory, motor, and cognitive functions. Reductionist assumptions have also influenced our views of the etiology and treatment of psychopathologies, and have more recently led to claims that we can, or even should, pharmacologically enhance the normal brain. Reductionism remains an area of active debate in the philosophy of science. In neuroscience and psychology, the debate typically focuses on the mind-brain question and the mechanisms of cognition, and how or if they can be explained in neurobiological terms. However, these debates are affected by the complexity of the phenomena being considered and the difficulty of obtaining the necessary neurobiological detail. We can instead ask whether features identified in neurobiological analyses of simpler aspects in simpler nervous systems support current molecular and cellular approaches to explaining systems or behaviors. While my view is that they do not, this does not invite the opposing view prevalent in dichotomous thinking that molecular and cellular detail is irrelevant and we should focus on computations or representations. We instead need to consider how to address the long-standing dilemma of how a nervous system that ostensibly functions through discrete cell to cell communication can generate population effects across multiple spatial and temporal scales to generate behavior.

Noise-activated barrier crossing in multiattractor dissipative neural networks

Noise-activated transitions between coexisting attractors are investigated in a chaotic spiking network. At low noise level, attractor hopping consists of discrete bifurcation events that conserve the memory of initial conditions. When the escape probability becomes comparable to the intrabasin hopping probability, the lifetime of attractors is given by a detailed balance where the less coherent attractors act as a sink for the more coherent ones. In this regime, the escape probability follows an activation law allowing us to assign pseudoactivation energies to limit cycle attractors. These pseudoenergies introduce a useful metric for evaluating the resilience of biological rhythms to perturbations.

Spiral waves in a hybrid discrete excitable media with electromagnetic flux coupling

Though there are many neuron models based on differential equations, the complexity in realizing them into digital circuits is still a challenge. Hence, many new discrete neuron models have been recently proposed, which can be easily implemented in digital circuits. We consider the well-known FitzHugh-Nagumo model and derive the discrete version of the model considering the sigmoid type of recovery variable and electromagnetic flux coupling. We show the various time series plots confirming the existence of periodic and chaotic bursting as in differential equation type neuron models. Also, we have used the bifurcation plots, Lyapunov exponents, and frequency bifurcations to investigate the dynamics of the proposed discrete neuron model. Different topologies of networks like single, two, and three layers are considered to analyze the wave propagation phenomenon in the network. We introduce the concept of using energy levels of nodes to study the spiral wave existence and compare them with the spatiotemporal snapshots. Interestingly, the energy plots clearly show that when the energy level of nodes is different and distributed, the occurrence of the spiral waves is identified in the network.

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.

Influence of Autapses on Synchronization in Neural Networks With Chemical Synapses

A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However, only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neural synchronization. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemical synapses, equipped with autapses, to study the effect of the latter on synchronous behavior. We consider time delay in the conductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting of both excitatory and inhibitory neurons, we uncover that synchronous behavior depends on their synapse type. Our results provide evidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory and excitatory synapses where neurons are equipped with autapses.

Neural mechanisms for sigh generation during prenatal development

The respiratory network of the preBötzinger complex (preBötC), which controls inspiratory behavior, can in normal conditions simultaneously produce two types of inspiration-related rhythmic activities: the eupneic rhythm composed of monophasic, low-amplitude, and relatively high-frequency bursts, interspersed with sigh rhythmic activity, composed of biphasic, high-amplitude, and lower frequency bursts. By combining electrophysiological recordings from transverse brainstem slices with computational modeling, new advances in the mechanisms underlying sigh production have been obtained during prenatal development. The present review summarizes recent findings that establish when sigh rhythmogenesis starts to be produced during embryonic development as well as the cellular, membrane, and synaptic properties required for its expression. Together, the results demonstrate that although generated by the same network, the eupnea and sigh rhythms have different developmental onset times and rely on distinct network properties. Because sighs (also known as augmented breaths) are important in maintaining lung function (by reopening collapsed alveoli), gaining insight into their underlying neural mechanisms at early developmental stages is likely to help in the treatment of prematurely born babies often suffering from breathing deficiencies.

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.

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'.

Developmental metaplasticity in neural circuit codes of firing and structure

Firing-rate dynamics have been hypothesized to mediate inter-neural information transfer in the brain. While the Hebbian paradigm, relating learning and memory to firing activity, has put synaptic efficacy variation at the center of cortical plasticity, we suggest that the external expression of plasticity by changes in the firing-rate dynamics represents a more general notion of plasticity. Hypothesizing that time constants of plasticity and firing dynamics increase with age, and employing the filtering property of the neuron, we obtain the elementary code of global attractors associated with the firing-rate dynamics in each developmental stage. We define a neural circuit connectivity code as an indivisible set of circuit structures generated by membrane and synapse activation and silencing. Synchronous firing patterns under parameter uniformity, and asynchronous circuit firing are shown to be driven, respectively, by membrane and synapse silencing and reactivation, and maintained by the neuronal filtering property. Analytic, graphical and simulation representation of the discrete iteration maps and of the global attractor codes of neural firing rate are found to be consistent with previous empirical neurobiological findings, which have lacked, however, a specific correspondence between firing modes, time constants, circuit connectivity and cortical developmental stages.

Noise effects on robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design

In this paper, we report on the synchronization of a pacemaker neuronal ensemble constituted of an AB neuron electrically coupled to two PD neurons. By the virtue of this electrical coupling, they can fire synchronous bursts of action potential. An external master neuron is used to induce to the whole system the desired dynamics, via a nonlinear controller. Such controller is obtained by a combination of sliding mode and feedback control. The proposed controller is able to offset uncertainties in the synchronized systems. We show how noise affects the synchronization of the pacemaker neuronal ensemble, and briefly discuss its potential benefits in our synchronization scheme. An extended Hindmarsh-Rose neuronal model is used to represent a single cell dynamic of the network. Numerical simulations and Pspice implementation of the synchronization scheme are presented. We found that, the proposed controller reduces the stochastic resonance of the network when its gain increases.

Dynamics of Time Delay-Induced Multiple Synchronous Behaviors in Inhibitory Coupled Neurons

The inhibitory synapse can induce synchronous behaviors different from the anti-phase synchronous behaviors, which have been reported in recent studies. In the present paper, synchronous behaviors are investigated in the motif model composed of reciprocal inhibitory coupled neurons with endogenous bursting and time delay. When coupling strength is weak, synchronous behavior appears at a single interval of time delay within a bursting period. When coupling strength is strong, multiple synchronous behaviors appear at different intervals of time delay within a bursting period. The different bursting patterns of synchronous behaviors, and time delays and coupling strengths that can induce the synchronous bursting patterns can be well interpreted by the dynamics of the endogenous bursting pattern of isolated neuron, which is acquired by the fast-slow dissection method, combined with the inhibitory coupling current. For an isolated neuron, when a negative impulsive current with suitable strength is applied at different phases of the bursting, multiple different bursting patterns can be induced. For a neuron in the motif, the inhibitory coupling current, of which the application time and strength is modulated by time delay and coupling strength, can cause single or multiple synchronous firing patterns like the negative impulsive current when time delay and coupling strength is suitable. The difference compared to the previously reported multiple synchronous behaviors that appear at time delays wider than a period of the endogenous firing is discussed. The results present novel examples of synchronous behaviors in the neuronal network with inhibitory synapses and provide a reasonable explanation.

Synergistic effect of repulsive inhibition in synchronization of excitatory networks

We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one's expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrate that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. This effect is generic and observed in different models of bursting neurons and fast synaptic interactions. We also find a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. This general law is in sharp contrast with linearly coupled networks with positive (attractive) and negative (repulsive) coupling where the placement and structure of negative connections heavily affect synchronization.

Switching mechanisms and bout times in a pair of reciprocally inhibitory neurons

Within the appropriate parameter regime, a deterministic model of a pair of mutually inhibitory neurons receiving excitatory driving currents exhibits bistability-each of the two stable states corresponds to one neuron being active and the other being quiescent. The presence of noise in the driving currents results in a system that randomly switches back and forth between these two states, causing alternating bouts of spiking activity. In this work, we examine the random bout durations of the two neurons and dependence on system parameters. We find that bout durations of each neuron are exponentially distributed, with changes in system parameters altering only the mean of the distribution. Synaptic inhibition independently controls the bout durations of the two neurons-the mean bout time of a neuron is a function of efferent (or outgoing) inhibition, and is independent of afferent (or incoming) inhibition. Furthermore, we find that the mean bout time of a neuron exhibits a critical dependence on the time course (rather than amplitude) of efferent inhibition-mean bout time of a neuron grows exponentially with the time course of efferent inhibition, and the growth rate of this exponential function depends only on the excitatory driving current to that neuron (and not on any other system parameters). We discuss the relevance of our results to the regulation of sleep-wake cycling by medullary and pontine structures within the brain.

Emergence of sigh rhythmogenesis in the embryonic mouse

In mammals, eupnoeic breathing is periodically interrupted by spontaneous augmented breaths (sighs) that include a larger-amplitude inspiratory effort, typically followed by a post-sigh apnoea. Previous in vitro studies in newborn rodents have demonstrated that the respiratory oscillator of the pre-Bötzinger complex (preBötC) can generate the distinct inspiratory motor patterns for both eupnoea- and sigh-related behaviour. During mouse embryonic development, the preBötC begins to generate eupnoeic rhythmicity at embryonic day (E) 15.5, but the network's ability to also generate sigh-like activity remains unexplored at prenatal stages. Using transverse brainstem slice preparations we monitored the neuronal population activity of the preBötC at different embryonic ages. Spontaneous sigh-like rhythmicity was found to emerge progressively, being expressed in 0/32 slices at E15.5, 7/30 at E16.5, 9/22 at E17.5 and 23/26 at E18.5. Calcium imaging showed that the preBötC cell population that participates in eupnoeic-like discharge was also active during fictive sighs. However, patch-clamp recordings revealed the existence of an additional small subset of neurons that fired exclusively during sigh activity. Changes in glycinergic inhibitory synaptic signalling, either by pharmacological blockade, functional perturbation or natural maturation of the chloride co-transporters KCC2 or NKCC1 selectively, and in an age-dependent manner, altered the bi-phasic nature of sigh bursts and their coordination with eupnoeic bursting, leading to the generation of an atypical monophasic sigh-related event. Together our results demonstrate that the developmental emergence of a sigh-generating capability occurs after the onset of eupnoeic rhythmogenesis and requires the proper maturation of chloride-mediated glycinergic synaptic transmission.

Global attractor alphabet of neural firing modes

The elementary set, or alphabet, of neural firing modes is derived from the widely accepted conductance-based rectified firing-rate model. The firing dynamics of interacting neurons are shown to be governed by a multidimensional bilinear threshold discrete iteration map. The parameter-dependent global attractors of the map morph into 12 attractor types. Consistent with the dynamic modes observed in biological neuronal firing, the global attractor alphabet is highly visual and intuitive in the scalar, single-neuron case. As synapse permeability varies from high depression to high potentiation, the global attractor type varies from chaotic to multiplexed, oscillatory, fixed, and saturated. As membrane permeability decreases, the global attractor transforms from active to passive state. Under the same activation, learning and retrieval end at the same global attractor. The bilinear threshold structure of the multidimensional map associated with interacting neurons generalizes the global attractor alphabet of neuronal firing modes to multineuron systems. Selective positive or negative activation and neural interaction yield combinatorial revelation and concealment of stored neuronal global attractors.

Dynamic systems approaches and levels of analysis in the nervous system

Various analyses are applied to physiological signals. While epistemological diversity is necessary to address effects at different levels, there is often a sense of competition between analyses rather than integration. This is evidenced by the differences in the criteria needed to claim understanding in different approaches. In the nervous system, neuronal analyses that attempt to explain network outputs in cellular and synaptic terms are rightly criticized as being insufficient to explain global effects, emergent or otherwise, while higher-level statistical and mathematical analyses can provide quantitative descriptions of outputs but can only hypothesize on their underlying mechanisms. The major gap in neuroscience is arguably our inability to translate what should be seen as complementary effects between levels. We thus ultimately need approaches that allow us to bridge between different spatial and temporal levels. Analytical approaches derived from critical phenomena in the physical sciences are increasingly being applied to physiological systems, including the nervous system, and claim to provide novel insight into physiological mechanisms and opportunities for their control. Analyses of criticality have suggested several important insights that should be considered in cellular analyses. However, there is a mismatch between lower-level neurophysiological approaches and statistical phenomenological analyses that assume that lower-level effects can be abstracted away, which means that these effects are unknown or inaccessible to experimentalists. As a result experimental designs often generate data that is insufficient for analyses of criticality. This review considers the relevance of insights from analyses of criticality to neuronal network analyses, and highlights that to move the analyses forward and close the gap between the theoretical and neurobiological levels, it is necessary to consider that effects at each level are complementary rather than in competition.

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.

Noninvertibility, chaotic coding, and chaotic multiplexity of synaptically modulated neural firing

Widely accepted neural firing and synaptic potentiation rules specify a cross-dependence of the two processes, which, evolving on different timescales, have been separated for analytic purposes, concealing essential dynamics. Here, the morphology of the firing rates process, modulated by synaptic potentiation, is shown to be described by a discrete iteration map in the form of a thresholded polynomial. Given initial synaptic weights, a firing activity is triggered by conductance. Elementary dynamic modes are defined by fixed points, cycles, and saddles of the map, building blocks of the underlying firing code. Showing parameter-dependent multiplicity of real polynomial roots, the map is proved to be noninvertible. The incidence of chaos is then implied by the parameter-dependent existence of snap-back repellers. The highly patterned geometric and statistical structures of the associated chaotic attractors suggest that these attractors are an integral part of the neural code. It further suggests the chaotic attractor as a natural mechanism for statistical encoding and temporal multiplexing of neural information. The analytic findings are supported by simulation.

Dynamic clamp with StdpC software

Dynamic clamp is a powerful method that allows the introduction of artificial electrical components into target cells to simulate ionic conductances and synaptic inputs. This method is based on a fast cycle of measuring the membrane potential of a cell, calculating the current of a desired simulated component using an appropriate model and injecting this current into the cell. Here we present a dynamic clamp protocol using free, fully integrated, open-source software (StdpC, for spike timing-dependent plasticity clamp). Use of this protocol does not require specialist hardware, costly commercial software, experience in real-time operating systems or a strong programming background. The software enables the configuration and operation of a wide range of complex and fully automated dynamic clamp experiments through an intuitive and powerful interface with a minimal initial lead time of a few hours. After initial configuration, experimental results can be generated within minutes of establishing cell recording.

Delay-induced intermittent transition of synchronization in neuronal networks with hybrid synapses

We study the dependence of synchronization transitions in scale-free networks of bursting neurons with hybrid synapses on the information transmission delay and the probability of inhibitory synapses. It is shown that, irrespective of the probability of inhibitory synapses, the delay always plays a subtle role during synchronization transition of the scale-free neuronal networks. In particular, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. In addition, it is found that, for smaller and larger probability of inhibitory synapses, intermittent synchronization transition is relatively profound, while for the moderate probability of inhibitory synapses, synchronization transition seems less profound. More interestingly, it is found that as the probability of inhibitory synapses is large, regions of synchronization are upscattering.

Neuronal network analyses: premises, promises and uncertainties

Neuronal networks assemble the cellular components needed for sensory, motor and cognitive functions. Any rational intervention in the nervous system will thus require an understanding of network function. Obtaining this understanding is widely considered to be one of the major tasks facing neuroscience today. Network analyses have been performed for some years in relatively simple systems. In addition to the direct insights these systems have provided, they also illustrate some of the difficulties of understanding network function. Nevertheless, in more complex systems (including human), claims are made that the cellular bases of behaviour are, or will shortly be, understood. While the discussion is necessarily limited, this issue will examine these claims and highlight some traditional and novel aspects of network analyses and their difficulties. This introduction discusses the criteria that need to be satisfied for network understanding, and how they relate to traditional and novel approaches being applied to addressing network function.

A population level computational model of the basal ganglia that generates parkinsonian Local Field Potential activity

Recordings from the basal ganglia's subthalamic nucleus are acquired via microelectrodes immediately prior to the application of Deep Brain Stimulation (DBS) treatment for Parkinson's Disease (PD) to assist in the selection of the final point for the implantation of the DBS electrode. The acquired recordings reveal a persistent characteristic beta band peak in the power spectral density function of the Local Field Potential (LFP) signals. This peak is considered to lie at the core of the causality-effect relationships of the parkinsonian pathophysiology. Based on LFPs acquired from human subjects during DBS for PD, we constructed a computational model of the basal ganglia on the population level that generates LFPs to identify the critical pathophysiological alterations that lead to the expression of the beta band peak. To this end, we used experimental data reporting that the strengths of the synaptic connections are modified under dopamine depletion. The hypothesis that the altered dopaminergic modulation may affect both the amplitude and the time course of the postsynaptic potentials is validated by the model. The results suggest a pivotal role of both of these parameters to the pathophysiology of PD.

Variability of bursting patterns in a neuron model in the presence of noise

Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.

Polyrhythmic synchronization in bursting networking motifs

We study the emergence of polyrhythmic dynamics of motifs which are the building block for small inhibitory-excitatory networks, such as central pattern generators controlling various locomotive behaviors of animals. We discover that the pacemaker determining the specific rhythm of such a network composed of realistic Hodgkin-Huxley-type neurons is identified through the order parameter, which is the ratio of the neurons' burst durations or of duty cycles. We analyze different configurations of the motifs and describe the universal mechanisms for synergetics of the bursting patterns. We discuss also the multistability of inhibitory networks that results in polyrhythmicity of its emergent synchronous behaviors.

When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons

We show that weak common inhibition applied to a network of bursting neurons with strong desynchronizing connections can induce burst and complete synchronization. We demonstrate that the weak synchronizing inhibition from the same pacemaker neuron can win out over much stronger desynchronizing connections within the network, provided that the neuron's duty cycle is sufficiently long. We also gain insight into how the changes in burst duty cycles can trigger unexpected clusters of synchrony in bursting networks.

Inhibition of kainate receptors reduces the frequency of hippocampal theta oscillations

We investigated the role of kainate receptors in the generation of theta oscillations using (S)-1-(2-amino-2-carboxyethyl)-3-(2-carboxythiophene-3-yl-methyl)pyrimidine-2,4-dione (UBP304), a novel, potent and highly selective antagonist of GLU(K5)-containing kainate receptors. EEG and single-unit recordings were made from the dorsal hippocampus of awake, freely moving rats trained to forage for food. Bilateral intracerebroventricular injections of UBP304 (2.0 microl, two times; 2.08 mM) caused a clear (approximately 25%) reduction in theta frequency that was dissociable from behavioral effects of the drug. The locations of firing fields of principal cells in the hippocampal formation were generally preserved, but both field firing rates and the precision of field organization decreased. UBP304 lowered the frequency of the theta modulation of hippocampal interneuron discharge, accurately matching the reduced frequency of the theta field oscillation. UBP308 [(R)-1-(2-amino-2-carboxyethyl)-3-(2-carboxythiophene-3-yl-methyl)pyrimidine-2,4-dione], the inactive enantiomer of UBP304, caused none of these effects. Our results suggest that GLU(K5) receptors have an important role in modulating theta activity. In addition, the effects on cellular responses provide both insight into the mechanisms of theta pacing, and useful information for models of temporal coding.

Dynamic clamp analyses of cardiac, endocrine, and neural function

The dynamic clamp introduces artificial conductances into cells to simulate electrical coupling, votage-dependent, leak, and synaptic conductances. This review describes how the dynamic clamp has been used to address various questions in the cardiac, endocrine, and nervous systems.

Oscillations and oscillatory behavior in small neural circuits

In order to determine the dynamical properties of central pattern generators (CPGs), we have examined the lobster stomatogastric ganglion using the tools of nonlinear dynamics. The lobster pyloric and gastric mill central pattern generators can be analyzed at both the cellular and network levels because they are small, i.e., contain only 25 neurons between them and each neuron and synapse are repeatedly identifiable from animal to animal. We discuss how the biophysical properties of each neuron and synapse in the two circuits act cooperatively to generate two different patterns of sequential activity, how these patterns are altered by neuromodulators and perturbed by noise and sensory inputs. Finally, we show how simplified Hindmarsh-Rose models can be made into analog electronic neurons that mimic the lobster neurons and in addition be incorporated into artificial CPGs with robotic applications.

Synchronization and propagation of bursts in networks of coupled map neurons

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Complexities and uncertainties of neuronal network function

The nervous system generates behaviours through the activity in groups of neurons assembled into networks. Understanding these networks is thus essential to our understanding of nervous system function. Understanding a network requires information on its component cells, their interactions and their functional properties. Few networks come close to providing complete information on these aspects. However, even if complete information were available it would still only provide limited insight into network function. This is because the functional and structural properties of a network are not fixed but are plastic and can change over time. The number of interacting network components, their (variable) functional properties, and various plasticity mechanisms endows networks with considerable flexibility, but these features inevitably complicate network analyses. This review will initially discuss the general approaches and problems of network analyses. It will then examine the success of these analyses in a model spinal cord locomotor network in the lamprey, to determine to what extent in this relatively simple vertebrate system it is possible to claim detailed understanding of network function and plasticity.

Bifurcation of synchronous oscillations into torus in a system of two reciprocally inhibitory silicon neurons: experimental observation and modeling

Oscillatory activity in the central nervous system is associated with various functions, like motor control, memory formation, binding, and attention. Quasiperiodic oscillations are rarely discussed in the neurophysiological literature yet they may play a role in the nervous system both during normal function and disease. Here we use a physical system and a model to explore scenarios for how quasiperiodic oscillations might arise in neuronal networks. An oscillatory system of two mutually inhibitory neuronal units is a ubiquitous network module found in nervous systems and is called a half-center oscillator. Previously we created a half-center oscillator of two identical oscillatory silicon (analog Very Large Scale Integration) neurons and developed a mathematical model describing its dynamics. In the mathematical model, we have shown that an in-phase limit cycle becomes unstable through a subcritical torus bifurcation. However, the existence of this torus bifurcation in experimental silicon two-neuron system was not rigorously demonstrated or investigated. Here we demonstrate the torus predicted by the model for the silicon implementation of a half-center oscillator using complex time series analysis, including bifurcation diagrams, mapping techniques, correlation functions, amplitude spectra, and correlation dimensions, and we investigate how the properties of the quasiperiodic oscillations depend on the strengths of coupling between the silicon neurons. The potential advantages and disadvantages of quasiperiodic oscillations (torus) for biological neural systems and artificial neural networks are discussed.

Long-lasting nicotinic modulation of GABAergic synaptic transmission in the rat nucleus accumbens associated with behavioural sensitization to amphetamine

A robust increase in dopaminergic transmission in the nucleus accumbens (NAc) shell has been reported to be consistently associated with the long-term expression of behavioural sensitization to drugs of abuse. However, little is known about how this affects the neuronal network of the NAc. We made cellular recordings in NAc slices of saline- and amphetamine-pretreated adult rats and found that expression of behavioural sensitization was associated with long-lasting changes in the basal firing pattern of cholinergic interneurons up to 3 weeks after the last drug injection. Consequently, upon amphetamine sensitization, an inhibiting effect of the nicotinic receptor blocker mecamylamine on the amplitudes of spontaneous GABAergic synaptic currents as well as on the failure rate of electrically evoked GABAergic currents was found that was not present under control conditions. Thus, behavioural sensitization to amphetamine is associated with an up-regulation of the endogenous activation of nicotinic receptors that, in turn, stimulate the GABAergic synaptic transmission within the NAc shell. This is a new mechanism by which drugs of abuse may induce alterations in the processing and integration of NAc inputs involved in psychomotor sensitization.

The dynamic clamp comes of age

The dynamic clamp uses computer simulation to introduce artificial membrane or synaptic conductances into biological neurons and to create hybrid circuits of real and model neurons. In the ten years since it was first developed, the dynamic clamp has become a widely used tool for the study of neural systems at the cellular and circuit levels. This review describes recent state-of-the-art implementations of the dynamic clamp and summarizes insights gained through its use, ranging from the role of voltage-dependent conductances in shaping neuronal activity to the effects of synaptic dynamics on network behavior and the impact of in vivo-like input on neuronal information processing.

Electrical coupling and neuronal synchronization in the Mammalian brain

Certain neurons in the mammalian brain have long been known to be joined by gap junctions, which are the most common type of electrical synapse. More recently, cloning of neuron-specific connexins, increased capability of visualizing cells within brain tissue, labeling of cell types by transgenic methods, and generation of connexin knockouts have spurred a rapid increase in our knowledge of the role of gap junctions in neural activity. This article reviews the many subtleties of transmission mediated by gap junctions and the mechanisms whereby these junctions contribute to synchronous firing.

The functional consequences of changes in the strength and duration of synaptic inputs to oscillatory neurons

We studied the effect of synaptic inputs of different amplitude and duration on neural oscillators by simulating synaptic conductance pulses in a bursting conductance-based pacemaker model and by injecting artificial synaptic conductance pulses into pyloric pacemaker neurons of the lobster stomatogastric ganglion using the dynamic clamp. In the model and the biological neuron, the change in burst period caused by inhibitory and excitatory inputs of increasing strength saturated, such that synaptic inputs above a certain strength all had the same effect on the firing pattern of the oscillatory neuron. In contrast, increasing the duration of the synaptic conductance pulses always led to changes in the burst period, indicating that neural oscillators are sensitive to changes in the duration of synaptic input but are not sensitive to changes in the strength of synaptic inputs above a certain conductance. This saturation of the response to progressively stronger synaptic inputs occurs not only in bursting neurons but also in tonically spiking neurons. We identified inward currents at hyperpolarized potentials as the cause of the saturation in the model neuron. Our findings imply that activity-dependent or modulator-induced changes in synaptic strength are not necessarily accompanied by changes in the functional impact of a synapse on the timing of postsynaptic spikes or bursts.