The number of synaptic inputs and the synchrony of large, sparse neuronal networks

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Temporal and spatial characteristics of vibrissa responses to motor commands

A mechanistic description of the generation of whisker movements is essential for understanding the control of whisking and vibrissal active touch. We explore how facial-motoneuron spikes are translated, via an intrinsic muscle, to whisker movements. This is achieved by constructing, simulating, and analyzing a computational, biomechanical model of the motor plant, and by measuring spiking to movement transformations at small and large angles using high-precision whisker tracking in vivo. Our measurements revealed a supralinear summation of whisker protraction angles in response to consecutive motoneuron spikes with moderate interspike intervals (5 ms < Deltat < 30 ms). This behavior is explained by a nonlinear transformation from intracellular changes in Ca(2+) concentration to muscle force. Our model predicts the following spatial constraints: (1) Contraction of a single intrinsic muscle results in movement of its two attached whiskers with different amplitudes; the relative amplitudes depend on the resting angles and on the attachment location of the intrinsic muscle on the anterior whisker. Counterintuitively, for a certain range of resting angles, activation of a single intrinsic muscle can lead to a retraction of one of its two attached whiskers. (2) When a whisker is pulled by its two adjacent muscles with similar forces, the protraction amplitude depends only weakly on the resting angle. (3) Contractions of two adjacent muscles sums up linearly for small amplitudes and supralinearly for larger amplitudes. The model provides a direct translation from motoneuron spikes to whisker movements and can serve as a building block in closed-loop motor-sensory models of active touch.

Neural cytoskeleton capabilities for learning and memory

This paper proposes a physical model involving the key structures within the neural cytoskeleton as major players in molecular-level processing of information required for learning and memory storage. In particular, actin filaments and microtubules are macromolecules having highly charged surfaces that enable them to conduct electric signals. The biophysical properties of these filaments relevant to the conduction of ionic current include a condensation of counterions on the filament surface and a nonlinear complex physical structure conducive to the generation of modulated waves. Cytoskeletal filaments are often directly connected with both ionotropic and metabotropic types of membrane-embedded receptors, thereby linking synaptic inputs to intracellular functions. Possible roles for cable-like, conductive filaments in neurons include intracellular information processing, regulating developmental plasticity, and mediating transport. The cytoskeletal proteins form a complex network capable of emergent information processing, and they stand to intervene between inputs to and outputs from neurons. In this manner, the cytoskeletal matrix is proposed to work with neuronal membrane and its intrinsic components (e.g., ion channels, scaffolding proteins, and adaptor proteins), especially at sites of synaptic contacts and spines. An information processing model based on cytoskeletal networks is proposed that may underlie certain types of learning and memory.

Sequentially switching cell assemblies in random inhibitory networks of spiking neurons in the striatum

The striatum is composed of GABAergic medium spiny neurons with inhibitory collaterals forming a sparse random asymmetric network and receiving an excitatory glutamatergic cortical projection. Because the inhibitory collaterals are sparse and weak, their role in striatal network dynamics is puzzling. However, here we show by simulation of a striatal inhibitory network model composed of spiking neurons that cells form assemblies that fire in sequential coherent episodes and display complex identity-temporal spiking patterns even when cortical excitation is simply constant or fluctuating noisily. Strongly correlated large-scale firing rate fluctuations on slow behaviorally relevant timescales of hundreds of milliseconds are shown by members of the same assembly whereas members of different assemblies show strong negative correlation, and we show how randomly connected spiking networks can generate this activity. Cells display highly irregular spiking with high coefficients of variation, broadly distributed low firing rates, and interspike interval distributions that are consistent with exponentially tailed power laws. Although firing rates vary coherently on slow timescales, precise spiking synchronization is absent in general. Our model only requires the minimal but striatally realistic assumptions of sparse to intermediate random connectivity, weak inhibitory synapses, and sufficient cortical excitation so that some cells are depolarized above the firing threshold during up states. Our results are in good qualitative agreement with experimental studies, consistent with recently determined striatal anatomy and physiology, and support a new view of endogenously generated metastable state switching dynamics of the striatal network underlying its information processing operations.

Systematic fluctuation expansion for neural network activity equations

Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.

Clustering predicted by an electrophysiological model of the suprachiasmatic nucleus

Despite the wealth of experimental data on the electrophysiology of individual neurons in the suprachiasmatic nuclei (SCN), the neural code of the SCN remains largely unknown. To predict the electrical activity of the SCN, the authors simulated networks of 10,000 GABAergic SCN neurons using a detailed model of the ionic currents within SCN neurons. Their goal was to understand how neuronal firing, which occurs on a time scale faster than a second, can encode a set phase of the circadian (24-h) cycle. The authors studied the effects of key network properties including: 1) the synaptic density within the SCN, 2) the magnitude of postsynaptic currents, 3) the heterogeneity of circadian phase in the neuronal population, 4) the degree of synaptic noise, and 5) the balance between excitation and inhibition. Their main result was that under a wide variety of conditions, the SCN network spontaneously organized into (typically 3) groups of synchronously firing neurons. They showed that this type of clustering can lead to the silencing of neurons whose intracellular clocks are out of circadian phase with the rest of the population. Their results provide clues to how the SCN may generate a coherent electrical output signal at the tissue level to time rhythms throughout the body.

Spatial attention in area V4 is mediated by circuits in primary visual cortex

The ability to covertly select visual stimuli in our environment based on their behavioral relevance is an important skill. Stimulus selection has been studied experimentally, at the single neuron as well as at the population level, by recording from the visual cortex of subjects performing attention-demanding tasks, but studies at the local circuit level are lacking. We conducted simulations of a primary visual cortex (V1) model to provide insight into the local circuit computation underlying stimulus selection in V4. Two small oriented rectangular bars were placed at different locations in the 4 by 4 degree visual field represented by the V1 model, such that they activated different V1 neurons but such that they were both inside the classical receptive field (CRF) of the same V4 neuron. The biased competition framework [Desimone, R., & Duncan, J. (1995). Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18, 193-222] makes predictions for the response of V4 neurons and the modulation thereof by spatial and feature attention. In our simulation of the V1 network, we obtained results consistent with these predictions for V4 when the model had long-range excitatory projections targeting inhibitory neurons and when spatial attention was mediated by a spatially restricted projection that either inhibited the inhibitory neurons or excited the excitatory neurons. Although it is not clear whether attention effects measured in V4 neurons are generated mostly by local circuits within V4, our simulations suggest that spatial attention at a resolution less than the size of the CRF of a V4 neuron is inherited from upstream areas like V1 and relies on circuits mediating surround suppression at the single neuron level. Furthermore, the model displayed global oscillations in the alpha frequency range (around 10 Hz), whose coherence was highest in the absence of visual stimulation, which is consistent with electroencephalograms recorded in humans. By contrast, when a stimulus was presented the alpha oscillation sped up and became less coherent, whereas at the single column level (40-480 cells) transient beta/gamma oscillations were observed with a frequency between 25 and 50 Hz.

Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons

We present results of an extensive numerical study of the dynamics of networks of integrate-and-fire neurons connected randomly through inhibitory interactions. We first consider delayed interactions with infinitely fast rise and decay. Depending on the parameters, the network displays transients which are short or exponentially long in the network size. At the end of these transients, the dynamics settle on a periodic attractor. If the number of connections per neuron is large ( approximately 1000) , this attractor is a cluster state with a short period. In contrast, if the number of connections per neuron is small ( approximately 100) , the attractor has complex dynamics and very long period. During the long transients the neurons fire in a highly irregular manner. They can be viewed as quasistationary states in which, depending on the coupling strength, the pattern of activity is asynchronous or displays population oscillations. In the first case, the average firing rates and the variability of the single-neuron activity are well described by a mean-field theory valid in the thermodynamic limit. Bifurcations of the long transient dynamics from asynchronous to synchronous activity are also well predicted by this theory. The transient dynamics display features reminiscent of stable chaos. In particular, despite being linearly stable, the trajectories of the transient dynamics are destabilized by finite perturbations as small as O(1/N) . We further show that stable chaos is also observed for postsynaptic currents with finite decay time. However, we report in this type of network that chaotic dynamics characterized by positive Lyapunov exponents can also be observed. We show in fact that chaos occurs when the decay time of the synaptic currents is long compared to the synaptic delay, provided that the network is sufficiently large.

Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrate-and-fire neurons

We investigate numerically the collective dynamical behavior of pulse-coupled nonleaky integrate-and-fire neurons that are arranged on a two-dimensional small-world network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Besides a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulentlike patterns, which-depending on network parameters-interfere with the global collective firing of the neurons. We attribute the various network dynamics to distinct regimes in the parameter space. For the same network parameters different network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spatiotemporal dynamics of biological networks.

Asynchronous response of coupled pacemaker neurons

We study a network model of two conductance-based pacemaker neurons of differing natural frequency, coupled with either mutual excitation or inhibition, and receiving shared random inhibitory synaptic input. The networks may phase lock spike to spike for strong mutual coupling. But the shared input can desynchronize the locked spike pairs by selectively eliminating the lagging spike or modulating its timing with respect to the leading spike depending on their separation time window. Such loss of synchrony is also found in a large network of sparsely coupled heterogeneous spiking neurons receiving shared input.

Persistent fluctuations of activity in undriven continuum neural field models with power-law connections

We study the effect of random inhomogeneous connections on a continuous field description of neural tissue. We focus on a regime in which persistent random fluctuations in activity arise spontaneously in the absence of either time-varying or spatially inhomogeneous input. While present in real tissue and network models of discrete neurons, such behavior has not been reported in continuum models of this type. The activity contains frequencies similar to those seen experimentally. We consider a power-law envelope r(-alpha) for the inhomogeneity and present evidence that the statistical coherence (a measure of two-point correlation) rapidly percolates across the system as alpha is reduced below alphac approximately 1,2 in one and two dimensions, respectively.

Bistability between synchrony and incoherence in limit-cycle oscillators with coupling strength inhomogeneity

The effect of coupling strength inhomogeneity on the synchronization of identical oscillators is investigated. Through simulations and analysis of phase-reduced models, it is shown that the mean value of coupling function and the degree of inhomogeneity in the total of coupling strength to the each oscillator cooperate to stabilize incoherent states. Under some circumstances, there can be bistability between coherent and incoherent states. Various cases of coupled Morris-Lecar oscillators are studied as examples of our results.

Colloid-guided assembly of oriented 3D neuronal networks

A central challenge in neuroscience is to understand the formation and function of three-dimensional (3D) neuronal networks. In vitro studies have been mainly limited to measurements of small numbers of neurons connected in two dimensions. Here we demonstrate the use of colloids as moveable supports for neuronal growth, maturation, transfection and manipulation, where the colloids serve as guides for the assembly of controlled 3D, millimeter-sized neuronal networks. Process growth can be guided into layered connectivity with a density similar to what is found in vivo. The colloidal superstructures are optically transparent, enabling remote stimulation and recording of neuronal activity using layer-specific expression of light-activated channels and indicator dyes. The modular approach toward in vitro circuit construction provides a stepping stone for applications ranging from basic neuroscience to neuron-based screening of targeted drugs.

Partially locked states in coupled oscillators due to inhomogeneous coupling

We investigate coupled identical phase oscillators with scale-free distribution of coupling strength. It is shown that partially locked states can occur due to the inhomogeneity in coupling and some properties of the coupling function. Various quantities of the partially locked states are computed through a self-consistency argument and the values show good agreement with simulation results.

Class-II neurons display a higher degree of stochastic synchronization than class-I neurons

We describe the relationship between the shape of the phase-resetting curve (PRC) and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated Poisson inputs in addition to inputs from independent sources. We use perturbation methods to derive an expression relating the shape of the PRC to the probability density function (PDF) of the phase difference between the oscillators. We compute various measures of the degree of synchrony and cross correlation from the PDF's and use the same to compare and contrast differently shaped PRCs, with respect to their ability to undergo stochastic synchronization. Since the shape of the PRC depends on underlying dynamical details of the oscillator system, we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators. It is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the PRC (type I or type II). It is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation.

The high-conductance state of cortical networks

We studied the dynamics of large networks of spiking neurons with conductance-based (nonlinear) synapses and compared them to networks with current-based (linear) synapses. For systems with sparse and inhibition-dominated recurrent connectivity, weak external inputs induced asynchronous irregular firing at low rates. Membrane potentials fluctuated a few millivolts below threshold, and membrane conductances were increased by a factor 2 to 5 with respect to the resting state. This combination of parameters characterizes the ongoing spiking activity typically recorded in the cortex in vivo. Many aspects of the asynchronous irregular state in conductance-based networks could be sufficiently well characterized with a simple numerical mean field approach. In particular, it correctly predicted an intriguing property of conductance-based networks that does not appear to be shared by current-based models: they exhibit states of low-rate asynchronous irregular activity that persist for some period of time even in the absence of external inputs and without cortical pacemakers. Simulations of larger networks (up to 350,000 neurons) demonstrated that the survival time of self-sustained activity increases exponentially with network size.

Stochastic dynamics of a finite-size spiking neural network

We present a simple Markov model of spiking neural dynamics that can be analytically solved to characterize the stochastic dynamics of a finite-size spiking neural network. We give closed-form estimates for the equilibrium distribution, mean rate, variance, and autocorrelation function of the network activity. The model is applicable to any network where the probability of firing of a neuron in the network depends on only the number of neurons that fired in a previous temporal epoch. Networks with statistically homogeneous connectivity and membrane and synaptic time constants that are not excessively long could satisfy these conditions. Our model completely accounts for the size of the network and correlations in the firing activity. It also allows us to examine how the network dynamics can deviate from mean field theory. We show that the model and solutions are applicable to spiking neural networks in biophysically plausible parameter regimes.

Temporal decorrelation of collective oscillations in neural networks with local inhibition and long-range excitation

We consider two neuronal networks coupled by long-range excitatory interactions. Oscillations in the gamma frequency band are generated within each network by local inhibition. When long-range excitation is weak, these oscillations phase lock with a phase shift dependent on the strength of local inhibition. Increasing the strength of long-range excitation induces a transition to chaos via period doubling or quasiperiodic scenarios. In the chaotic regime, oscillatory activity undergoes fast temporal decorrelation. The generality of these dynamical properties is assessed in firing-rate models as well as in large networks of conductance-based neurons.

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.

Correlations, fluctuations, and stability of a finite-size network of coupled oscillators

The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e., when the population is not synchronous. This demonstration is facilitated by the construction of a nonequilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory is consistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finite size corrections, which can be calculated in an expansion in 1/N, where N is the number of oscillators. The N-->infinity limit of this theory is what is traditionally called mean field theory for the Kuramoto model.

Modulation of synchrony without changes in firing rates

It was often reported and suggested that the synchronization of spikes can occur without changes in the firing rate. However, few theoretical studies have tested its mechanistic validity. In the present study, we investigate whether changes in synaptic weights can induce an independent modulation of synchrony while the firing rate remains constant. We study this question at the level of both single neurons and neuronal populations using network simulations of conductance based integrate-and-fire neurons. The network consists of a single layer that includes local excitatory and inhibitory recurrent connections, as well as long-range excitatory projections targeting both classes of neurons. Each neuron in the network receives external input consisting of uncorrelated Poisson spike trains. We find that increasing this external input leads to a linear increase of activity in the network, as well as an increase in the peak frequency of oscillation. In contrast, balanced changes of the synaptic weight of excitatory long-range projections for both classes of postsynaptic neurons modulate the degree of synchronization without altering the firing rate. These results demonstrate that, in a simple network, synchronization and firing rate can be modulated independently, and thus, may be used as independent coding dimensions.

Asymptotic behavior and synchronizability characteristics of a class of recurrent neural networks

We propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model. The model we introduce is a variation of the Zhang model. We investigate the time-asymptotic behavior of the corresponding dynamical system (in particular, the conditions for the existence of an invariant compact asymptotic set) and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with Poisson connectivity distribution.

Singular unlocking transition in the Winfree model of coupled oscillators

The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the unlocking transition, in which the frequency-locked state disappears abruptly as the spread of natural frequencies exceeds a critical width. Viewed as a function of the coupling strength, this critical width defines a bifurcation curve in parameter space. For the special case where the frequency distribution is uniform, earlier work had uncovered a puzzling singularity in this bifurcation curve. Here we seek to understand what causes the singularity. Using the Poincaré-Lindstedt method of perturbation theory, we analyze the locked state and its associated unlocking transition, first for an arbitrary distribution of natural frequencies, and then for discrete systems of N oscillators. We confirm that the bifurcation curve becomes singular for a continuum uniform distribution, yet find that it remains well behaved for any finite N , suggesting that the continuum limit is responsible for the singularity.

Kinetic theory of coupled oscillators

We present an approach for the description of fluctuations that are due to finite system size induced correlations in the Kuramoto model of coupled oscillators. We construct a hierarchy for the moments of the density of oscillators that is analogous to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the kinetic theory of plasmas and gases. To calculate the lowest order system size effect, we truncate this hierarchy at second order and solve the resulting closed equations for the two-oscillator correlation function around the incoherent state. We use this correlation function to compute the fluctuations of the order parameter, including the effect of transients, and compare this computation with numerical simulations.

How Noise Affects the Synchronization Properties of Recurrent Networks of Inhibitory Neurons

GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.

Attentional modulation of firing rate and synchrony in a model cortical network

The response of a neuron in the visual cortex to stimuli of different contrast placed in its receptive field is commonly characterized using the contrast response curve. When attention is directed into the receptive field of a V4 neuron, its contrast response curve is shifted to lower contrast values (Reynolds et al., 2000). The neuron will thus be able to respond to weaker stimuli than it responded to without attention. Attention also increases the coherence between neurons responding to the same stimulus (Fries et al., 2001). We studied how the firing rate and synchrony of a densely interconnected cortical network varied with contrast and how they were modulated by attention. The changes in contrast and attention were modeled as changes in driving current to the network neurons. We found that an increased driving current to the excitatory neurons increased the overall firing rate of the network, whereas variation of the driving current to inhibitory neurons modulated the synchrony of the network. We explain the synchrony modulation in terms of a locking phenomenon during which the ratio of excitatory to inhibitory firing rates is approximately constant for a range of driving current values. We explored the hypothesis that contrast is represented primarily as a drive to the excitatory neurons, whereas attention corresponds to a reduction in driving current to the inhibitory neurons. Using this hypothesis, the model reproduces the following experimental observations: (1) the firing rate of the excitatory neurons increases with contrast; (2) for high contrast stimuli, the firing rate saturates and the network synchronizes; (3) attention shifts the contrast response curve to lower contrast values; (4) attention leads to stronger synchronization that starts at a lower value of the contrast compared with the attend-away condition. In addition, it predicts that attention increases the delay between the inhibitory and excitatory synchronous volleys produced by the network, allowing the stimulus to recruit more downstream neurons.

Competition between feedback loops underlies normal and pathological dynamics in the basal ganglia

Experiments performed in normal animals suggest that the basal ganglia (BG) are crucial in motor program selection. BG are also involved in movement disorders. In particular, BG neuronal activity in parkinsonian animals and patients is more oscillatory and more synchronous than in normal individuals. We propose a new model for the function and dysfunction of the motor part of BG. We hypothesize that the striatum, the subthalamic nucleus, the internal pallidum (GPi), the thalamus, and the cortex are involved in closed feedback loops. The direct (cortex-striatum-GPi-thalamus-cortex) and the hyperdirect loops (cortex-subthalamic nucleus-GPi-thalamus-cortex), which have different polarities, play a key role in the model. We show that the competition between these two loops provides the BG-cortex system with the ability to perform motor program selection. Under the assumption that dopamine potentiates corticostriatal synaptic transmission, we demonstrate that, in our model, moderate dopamine depletion leads to a complete loss of action selection ability. High depletion can lead to synchronous oscillations. These modifications of the network dynamical state stem from an imbalance between the feedback in the direct and hyperdirect loops when dopamine is depleted. Our model predicts that the loss of selection ability occurs before the appearance of oscillations, suggesting that Parkinson's disease motor impairments are not directly related to abnormal oscillatory activity. Another major prediction of our model is that synchronous oscillations driven by the hyperdirect loop appear in BG after inactivation of the striatum.

Persistent Synchronized Bursting Activity in Cortical Tissues With Low Magnesium Concentration: A Modeling Study

We explore the mechanism of synchronized bursting activity with frequency of ∼10 Hz that appears in cortical tissues at low extracellular magnesium concentration [Mg2+]o. We hypothesize that this activity is persistent, namely coexists with the quiescent state and depends on slow N-methyl-d-aspartate (NMDA) conductances. To explore this hypothesis, we construct and investigate a conductance-based model of excitatory cortical networks. Population bursting activity can persist for physiological values of the NMDA decay time constant (∼100 ms). Neurons are synchronized at the time scale of bursts but not of single spikes. A reduced model of a cell coupled to itself can encompass most of this highly synchronized network behavior and is analyzed using the fast-slow method. Synchronized bursts appear for intermediate values of the NMDA conductance gNMDA if NMDA conductances are not too fast. Regular spiking activity appears for larger gNMDA. If the single cell is a conditional burster, persistent synchronized bursts become more robust. Weakly synchronized states appear for zero AMPA conductance gAMPA. Enhancing gAMPA increases both synchrony and the number of spikes within bursts and decreases the bursting frequency. Too strong gAMPA, however, prevents the activity because it enhances neuronal intrinsic adaptation. When [Mg2+]o is increased, higher gNMDA values are needed to maintain bursting activity. Bursting frequency decreases with [Mg2+]o, and the network is silent with physiological [Mg2+]o. Inhibition weakly decreases the bursting frequency if inhibitory cells receive enough NMDA-mediated excitation. This study explains the importance of conditional bursters in layer V in supporting epileptiform activity at low [Mg2+]o.

A model of the effects of applied electric fields on neuronal synchronization

We examine the effects of applied electric fields on neuronal synchronization. Two-compartment model neurons were synaptically coupled and embedded within a resistive array, thus allowing the neurons to interact both chemically and electrically. In addition, an external electric field was imposed on the array. The effects of this field were found to be nontrivial, giving rise to domains of synchrony and asynchrony as a function of the heterogeneity among the neurons. A simple phase oscillator reduction was successful in qualitatively reproducing these domains. The findings form several readily testable experimental predictions, and the model can be extended to a larger scale in which the effects of electric fields on seizure activity may be simulated.

Stimulus Competition by Inhibitory Interference

When two stimuli are present in the receptive field of a V4 neuron, the firing rate response is between the weakest and strongest response elicited by each of the stimuli when presented alone (Reynolds, Chelazzi, & Desimone, 1999). When attention is directed toward the stimulus eliciting the strongest response (the preferred stimulus), the response to the pair is increased, whereas the response decreases when attention is directed to the other stimulus (the poor stimulus). When attention is directed to either of the two stimuli presented alone, the firing rate remains the same or increases slightly, but the coherence between the neuron's spike train and the local field potential can increase (Fries, Reynolds, Rorie, & Desimone, 2001). These experimental results were reproduced in a model of a V4 neuron under the assumption that attention modulates the activity of local interneuron networks. The V4 model neuron received stimulus-specific excitation from V2 and synchronous inhibitory inputs from two local interneuron networks in V4. Each interneuron network was driven by stimulus-specific excitatory inputs from V2 and was modulated by the activity of the frontal eye fields. Stimulus competition was present because of a delay in arrival time of synchronous volleys from each interneuron network. For small delays, the firing rate was close to the rate elicited by the preferred stimulus alone, whereas for larger delays, it approached the firing rate of the poor stimulus. When either stimulus was presented alone, the neuron's response was not altered by the change in delay, but could change due to modulation of the degree of synchrony of the corresponding interneuron network. The model suggests that top-down attention biases the competition between V2 columns for control of V4 neurons primarily by changing the relative timing of inhibition, whereas changes in the degree of synchrony of interneuron networks modulate the response to a single stimulus. The new mechanism proposed here for attentional modulation of firing rate, gain modulation by inhibitory interference, is likely to have more general applicability to cortical information processing.

Multivariate linear discrimination of seizures

OBJECTIVE

To discriminate seizures from interictal dynamics based on multivariate synchrony measures, and to identify dynamics of a pre-seizure state.

METHODS

A linear discriminator was constructed from two different measures of synchronization: cross-correlation and phase synchronization. We applied this discriminator to a sequence of seizures recorded from the intracranial EEG of a patient monitored over 6 days.

RESULTS

Surprisingly, we found that this bivariate measure of synchronization was not a reliable seizure discriminator for 7 of 9 seizures. Furthermore, the method did not appear to reliably detect a pre-seizure state. An association between anti-convulsant dosage, frequency of clinical seizures, and discriminator performance was noted.

CONCLUSIONS

Using a bivariate measure of synchronization failed to reliably differentiate seizures from non-seizure periods in these data, nor did such methods show reliable detection of a synchronous pre-seizure state. The non-stationary variables of decreasing antiepileptic medication (without available serum concentration measurements), and concomitant increasing seizure frequency contributed to the difficulties in validating a seizure prediction tool on such data.

SIGNIFICANCE

The finding that these seizures were not a simple reflection of increasing synchronization in the EEG has important implications. The non-stationary characteristics of human post-implantation intracranial EEG is an inherent limitation of pre-resection data sets.

Rapid temporal modulation of synchrony by competition in cortical interneuron networks

The synchrony of neurons in extrastriate visual cortex is modulated by selective attention even when there are only small changes in firing rate (Fries, Reynolds, Rorie, & Desimone, 2001). We used Hodgkin-Huxley type models of cortical neurons to investigate the mechanism by which the degree of synchrony can be modulated independently of changes in firing rates. The synchrony of local networks of model cortical interneurons interacting through GABA(A) synapses was modulated on a fast timescale by selectively activating a fraction of the interneurons. The activated interneurons became rapidly synchronized and suppressed the activity of the other neurons in the network but only if the network was in a restricted range of balanced synaptic background activity. During stronger background activity, the network did not synchronize, and for weaker background activity, the network synchronized but did not return to an asynchronous state after synchronizing. The inhibitory output of the network blocked the activity of pyramidal neurons during asynchronous network activity, and during synchronous network activity, it enhanced the impact of the stimulus-related activity of pyramidal cells on receiving cortical areas (Salinas & Sejnowski, 2001). Synchrony by competition provides a mechanism for controlling synchrony with minor alterations in rate, which could be useful for information processing. Because traditional methods such as cross-correlation and the spike field coherence require several hundred milliseconds of recordings and cannot measure rapid changes in the degree of synchrony, we introduced a new method to detect rapid changes in the degree of coincidence and precision of spike timing.

Breaking synchrony by heterogeneity in complex networks

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in the presence of coupling heterogeneity precisely timed periodic firing patterns replace the state of global synchrony that exists in homogenous networks only. With increasing disorder, these firing patterns persist until a critical temporal extent is reached that is of the order of the interaction delay. For stronger disorder the periodic firing patterns cease to exist and only asynchronous, aperiodic states are observed. We derive self-consistency equations to predict the precise temporal structure of a pattern from a network of given connectivity and heterogeneity. Moreover, we show how to design heterogeneous coupling architectures to create an arbitrary prescribed pattern.

Electrical synapses and synchrony: the role of intrinsic currents

Electrical synapses are ubiquitous in the mammalian CNS. Particularly in the neocortex, electrical synapses have been shown to connect low-threshold spiking (LTS) as well as fast spiking (FS) interneurons. Experiments have highlighted the roles of electrical synapses in the dynamics of neuronal networks. Here we investigate theoretically how intrinsic cell properties affect the synchronization of neurons interacting by electrical synapses. Numerical simulations of a network of conductance-based neurons randomly connected with electrical synapses show that potassium currents promote synchrony, whereas the persistent sodium current impedes it. Furthermore, synchrony varies with the firing rate in qualitatively different ways depending on the intrinsic currents. We also study analytically a network of quadratic integrate-and-fire neurons. We relate the stability of the asynchronous state of this network to the phase-response function (PRF), which characterizes the effect of small perturbations on the firing timing of the neurons. In particular, we show that the greater the skew of the PRF toward the first half of the period, the more stable the asynchronous state. Combining our simulations with our analytical results, we establish general rules to predict the dynamic state of large networks of neurons coupled with electrical synapses. Our work provides a natural explanation for surprising experimental observations that blocking electrical synapses may increase the synchrony of neuronal activity. It also suggests different synchronization properties for LTS and FS cells. Finally, we propose to further test our predictions in experiments using dynamic clamp techniques.

What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance

When the local field potential of a cortical network displays coherent fast oscillations ( approximately 40-Hz gamma or approximately 200-Hz sharp-wave ripples), the spike trains of constituent neurons are typically irregular and sparse. The dichotomy between rhythmic local field and stochastic spike trains presents a challenge to the theory of brain rhythms in the framework of coupled oscillators. Previous studies have shown that when noise is large and recurrent inhibition is strong, a coherent network rhythm can be generated while single neurons fire intermittently at low rates compared to the frequency of the oscillation. However, these studies used too simplified synaptic kinetics to allow quantitative predictions of the population rhythmic frequency. Here we show how to derive quantitatively the coherent oscillation frequency for a randomly connected network of leaky integrate-and-fire neurons with realistic synaptic parameters. In a noise-dominated interneuronal network, the oscillation frequency depends much more on the shortest synaptic time constants (delay and rise time) than on the longer synaptic decay time, and approximately 200-Hz frequency can be realized with synaptic time constants taken from slice data. In a network composed of both interneurons and excitatory cells, the rhythmogenesis is a compromise between two scenarios: the fast purely interneuronal mechanism, and the slower feedback mechanism (relying on the excitatory-inhibitory loop). The properties of the rhythm are determined essentially by the ratio of time scales of excitatory and inhibitory currents and by the balance between the mean recurrent excitation and inhibition. Faster excitation than inhibition, or a higher excitation/inhibition ratio, favors the feedback loop and a much slower oscillation (typically in the gamma range).

Neuronal aggregate formation underlies spatiotemporal dynamics of nonsynaptic seizure initiation

High-frequency activity often precedes seizure onset. We found that electrographic seizures, induced in vitro using the low-Ca(2+) model, start with high-frequency (>150 Hz) activity that then decreases in frequency while increasing in amplitude. Multichannel and unit recordings showed that the mechanism of this transition was the progressive formation of larger neuronal aggregates. Thus the apparent high-frequency activity, at seizure onset, can reflect the simultaneous recording of several slower firing aggregates. Aggregate formation rate can be accelerated by reducing osmolarity. Because synaptic transmission is blocked when extracellular Ca(2+) is reduced, nonsynaptic mechanisms (gap junctions, field effects) must be sufficient for aggregate formation and recruitment.

Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity

In model networks of E-cells and I-cells (excitatory and inhibitory neurons, respectively), synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the E-cells synchronize the I-cells and vice versa. Under ideal conditions-homogeneity in relevant network parameters and all-to-all connectivity, for instance-this mechanism can yield perfect synchronization. We find that approximate, imperfect synchronization is possible even with very sparse, random connectivity. The crucial quantity is the expected number of inputs per cell. As long as it is large enough (more precisely, as long as the variance of the total number of synaptic inputs per cell is small enough), tight synchronization is possible. The desynchronizing effect of random connectivity can be reduced by strengthening the E --> I synapses. More surprising, it cannot be reduced by strengthening the I --> E synapses. However, the decay time constant of inhibition plays an important role. Faster decay yields tighter synchrony. In particular, in models in which the inhibitory synapses are assumed to be instantaneous, the effects of sparse, random connectivity cannot be seen.

Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons

We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized. We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur. We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.

Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multioperator problem exactly and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying regular dynamics, coexists with a balanced state exhibiting irregular dynamics. External signals may switch the network between qualitatively distinct states.

Fast convergence of spike sequences to periodic patterns in recurrent networks

The dynamical attractors are thought to underlie many biological functions of recurrent neural networks. Here we show that stable periodic spike sequences with precise timings are the attractors of the spiking dynamics of recurrent neural networks with global inhibition. Almost all spike sequences converge within a finite number of transient spikes to these attractors. The convergence is fast, especially when the global inhibition is strong. These results support the possibility that precise spatiotemporal sequences of spikes are useful for information encoding and processing in biological neural networks.

Decreased neuronal synchronization during experimental seizures

Synchronization between CA1 pyramidal neurons was studied using dual-cell patch-clamp techniques simultaneous with an extracellular measurement of network activity. We explored various linear and nonlinear methods to detect weak synchronization in this network, using cross-correlation, mutual information in one and two dimensions, and phase correlation in both broad and narrow band. The linear and nonlinear methods demonstrated different patterns of sensitivity to detect synchrony in this network, depending on the dynamical state of the network. Bursts in 4-amino-pyridine (4AP) were highly synchronous events. Unexpectedly, seizure-like events in 4AP were desynchronous events, both in comparison with interictal periods preceding the seizure without bursts (cut Schaffer collateral tract) and in comparison with bursts preceding the seizures (intact Schaffer collateral tract). The finding that seizure-like events are associated with desynchronization in such networks is consistent with recent theoretical work, suggesting that asynchrony is necessary to maintain a high level of activity in neuronal networks for sustained periods of time and that synchrony may disrupt such activity.

Computational model of carbachol-induced delta, theta, and gamma oscillations in the hippocampus

Field potential recordings from the rat hippocampus in vivo contain distinct frequency bands of activity, including delta (0.5-2 Hz), theta (4-12 Hz), and gamma (30-80 Hz), that are correlated with the behavioral state of the animal. The cholinergic agonist carbachol (CCH) induces oscillations in the delta (CCH-delta), theta (CCH-theta), and gamma (CCH-gamma) frequency ranges in the hippocampal slice preparation, eliciting asynchronous CCH-theta, synchronous CCH-delta, and synchronous CCH-theta with increasing CCH concentration (Fellous and Seinowski, Hippocampus 2000;1 0:187-197). In a network model of area CA3, the time scale for CCH-delta corresponded to the decay constant of the gating variable of the calcium-dependent potassium (K-AHP) current, that of CCH-theta to an intrinsic subthreshold membrane potential oscillation of the pyramidal cells, and that of CCH-gamma to the decay constant of GABAergic inhibitory synaptic potentials onto the pyramidal cells. In model simulations, the known physiological effects of carbachol on the muscarinic and K-AHP currents, and on the strengths of excitatory postsynaptic potentials, reproduced transitions from asynchronous CCH-theta to CCH-delta and from CCH-delta to synchronous CCH-theta. The simulations also exhibited the interspersed CCH-gamma/CCH-delta and CCH-gamma/CCH-theta that were observed in experiments. The model, in addition, predicted an oscillatory state with all three frequency bands present, which has not yet been observed experimentally.

The spatial dimensions of electrically coupled networks of interneurons in the neocortex

Inhibitory interneurons of the neocortex are electrically coupled to cells of the same type through gap junctions. We studied the spatial organization of two types of interneurons in the rat somatosensory cortex: fast-spiking (FS) parvalbumin-immunoreactive (PV+) cells, and low threshold-spiking (LTS) somatostatin-immunoreactive (SS+) cells. Paired recordings in layer 4 demonstrated that both the probability of coupling and the coupling coefficient drop steeply with intersomatic distance, reaching zero beyond 200 microm. The dendritic arbors of FS and LTS cells were reconstructed from electrophysiologically characterized, biocytin-filled cells; the two cell types had only minor differences in the number and span of their dendrites. However, there was a markedly higher density of PV+ cells than SS+ cells. PV+ cells were densest in layer 4, while SS+ cell density peaked in the subgranular layers. From these data we estimate that there is measurable electrical coupling (directly or indirectly via intermediary cells) between each interneuron and 20-50 others. The large number of electrical synapses implies that each interneuron participates in a large, continuous syncytium. To evaluate the functional significance of these findings, we examined several simple architectures of coupled networks analytically. We present a mathematical method to estimate the average summated coupling conductance that each cell receives from all of its neighbors, and the average leak conductance of individual cells, and we suggest that these have the same order of magnitude. These quantitative results have important implications for the effects of electrical coupling on the dynamic behavior of interneuron networks.

Relation between structure and size in social networks

In the context of complex network systems, we model social networks with the property that there is certain degradation of the information flowing through the network. We analyze different kinds of networks, from regular lattices to random graphs. We define an average coordination degree for the network, which can be associated with a certain notion of efficiency. Assuming that there is a limit to the information a person may handle, we show that there exists a close relationship between the structure of the network and its maximum size.

Pacemaker neurons for the theta rhythm and their synchronization in the septohippocampal reciprocal loop

Hippocampal theta (4-10 Hz) oscillation represents a well-known brain rhythm implicated in spatial cognition and memory processes. Its cellular mechanisms remain a matter of debate, and previous computational work has focused mostly on mechanisms intrinsic to the hippocampus. On the other hand, experimental data indicate that GABAergic cells in the medial septum play a pacemaker role for the theta rhythm. We have used biophysical modeling to address two major questions raised by the septal pacemaker hypothesis: what is the ion channel mechanism for the single-cell pacemaker behavior and how do these cells become synchronized? Our model predicts that theta oscillations of septal GABAergic cells depend critically on a low-threshold, slowly inactivating potassium current. Network simulations show that theta oscillations are not coherent in an isolated population of pacemaker cells. Robust synchronization emerges with the addition of a second GABAergic cell population. Such a reciprocally inhibitory circuit can be realized by the hippocampo-septal feedback loop.

Steady state of an inhibitory neural network

We investigate the dynamics of a neural network where each neuron evolves according to the combined effects of deterministic integrate-and-fire dynamics and purely inhibitory coupling with K randomly chosen "neighbors." The inhibition reduces the voltage of a given neuron by an amount Delta when one of its neighbors fires. The interplay between the integration and inhibition leads to a steady state that is determined by solving the rate equations for the neuronal voltage distribution. We also study the evolution of a single neuron and find that the mean lifetime between firing events equals 1+K delta and that the probability that a neuron has not yet fired decays exponentially with time.

Connection topology dependence of synchronization of neural assemblies on class 1 and 2 excitability

Two main classes of excitable neurons are analyzed in terms of connection topology and strength of the coupling in a network of neurons. In both cases, we measure the degree of synchronization and responsiveness of the neural assembly. Class 2 excitability presents a fast wave-like propagation of the activity pattern, strong frequency dependence on the connection topology and a good level of synchronization regardless of the topology. On the other hand, class 1 excitability shows a strong dependence of the wave propagation speed and the synchronization degree on the connection topology, in addition no frequency adaptation is observed. We conclude that both types of neural excitability endow the neural assembly with very different dynamical properties. Although, for simplicity reasons, no inhibition has been included in our study, the emergent properties described in this paper may help to determine the class of excitability underlying a neural assembly.