Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity

Exact solutions for signal propagation along an excitable transmission line

A simple transmission line composed of pulse-coupled units is presented. The model captures the basic properties of excitable media with, in particular, the robust transmission of information via traveling wave solutions. For rectified linear units with a cut-off threshold, the model is exactly solvable and analytical results on propagation are presented. The ability to convey a nontrivial message is studied in detail.

Computational models for state-dependent traveling waves in hippocampal formation

Hippocampal theta (4-10 Hz) oscillations have been identified as traveling waves in both rodents and humans. In freely foraging rodents, the theta traveling wave is a planar wave propagating from the dorsal to ventral hippocampus along the septotemporal axis. Motivated from experimental findings, we develop a spiking neural network of excitatory and inhibitory neurons to generate state-dependent hippocampal traveling waves to improve current mechanistic understanding of propagating waves. Model simulations demonstrate the necessary conditions for generating wave propagation and characterize the traveling wave properties with respect to model parameters, running speed and brain state of the animal. Networks with long-range inhibitory connections are more suitable than networks with long-range excitatory connections. We further generalize the spiking neural network to model traveling waves in the medial entorhinal cortex (MEC) and predict that traveling theta waves in the hippocampus and entorhinal cortex are in sink.

Rich dynamics and functional organization on topographically designed neuronal networks in vitro

Neuronal cultures are a prominent experimental tool to understand complex functional organization in neuronal assemblies. However, neurons grown on flat surfaces exhibit a strongly coherent bursting behavior with limited functionality. To approach the functional richness of naturally formed neuronal circuits, here we studied neuronal networks grown on polydimethylsiloxane (PDMS) topographical patterns shaped as either parallel tracks or square valleys. We followed the evolution of spontaneous activity in these cultures along 20 days in vitro using fluorescence calcium imaging. The networks were characterized by rich spatiotemporal activity patterns that comprised from small regions of the culture to its whole extent. Effective connectivity analysis revealed the emergence of spatially compact functional modules that were associated with both the underpinned topographical features and predominant spatiotemporal activity fronts. Our results show the capacity of spatial constraints to mold activity and functional organization, bringing new opportunities to comprehend the structure-function relationship in living neuronal circuits.

Traveling Waves in Quasi-One-Dimensional Neuronal Minicolumns

Traveling waves of neuronal activity in the cortex have been observed in vivo. These traveling waves have been correlated to various features of observed cortical dynamics, including spike timing variability and correlated fluctuations in neuron membrane potential. Although traveling waves are typically studied as either strictly one-dimensional or two-dimensional excitations, here we investigate the conditions for the existence of quasi-one-dimensional traveling waves that could be sustainable in parts of the brain containing cortical minicolumns. For that, we explore a quasi-one-dimensional network of heterogeneous neurons with a biologically influenced computational model of neuron dynamics and connectivity. We find that background stimulus reliably evokes traveling waves in networks with local connectivity between neurons. We also observe traveling waves in fully connected networks when a model for action potential propagation speed is incorporated. The biological properties of the neurons influence the generation and propagation of the traveling waves. Our quasi-one-dimensional model is not only useful for studying the basic properties of traveling waves in neuronal networks; it also provides a simplified representation of possible wave propagation in columnar or minicolumnar networks found in the cortex.

Spontaneous traveling waves naturally emerge from horizontal fiber time delays and travel through locally asynchronous-irregular states

Studies of sensory-evoked neuronal responses often focus on mean spike rates, with fluctuations treated as internally-generated noise. However, fluctuations of spontaneous activity, often organized as traveling waves, shape stimulus-evoked responses and perceptual sensitivity. The mechanisms underlying these waves are unknown. Further, it is unclear whether waves are consistent with the low rate and weakly correlated “asynchronous-irregular” dynamics observed in cortical recordings. Here, we describe a large-scale computational model with topographically-organized connectivity and conduction delays relevant to biological scales. We find that spontaneous traveling waves are a general property of these networks. The traveling waves that occur in the model are sparse, with only a small fraction of neurons participating in any individual wave. Consequently, they do not induce measurable spike correlations and remain consistent with locally asynchronous irregular states. Further, by modulating local network state, they can shape responses to incoming inputs as observed in vivo.

Spontaneous traveling cortical waves shape neural responses. Using a large-scale computational model, the authors show that transmission delays shape locally asynchronous spiking dynamics into traveling waves without inducing correlations and boost responses to external input, as observed in vivo.

The impact of a closed-loop thalamocortical model on the spatiotemporal dynamics of cortical and thalamic traveling waves

Propagation of activity in spatially structured neuronal networks has been observed in awake, anesthetized, and sleeping brains. How these wave patterns emerge and organize across brain structures, and how network connectivity affects spatiotemporal neural activity remains unclear. Here, we develop a computational model of a two-dimensional thalamocortical network, which gives rise to emergent traveling waves similar to those observed experimentally. We illustrate how spontaneous and evoked oscillatory activity in space and time emerge using a closed-loop thalamocortical architecture, sustaining smooth waves in the cortex and staggered waves in the thalamus. We further show that intracortical and thalamocortical network connectivity, cortical excitation/inhibition balance, and thalamocortical or corticothalamic delay can independently or jointly change the spatiotemporal patterns (radial, planar and rotating waves) and characteristics (speed, direction, and frequency) of cortical and thalamic traveling waves. Computer simulations predict that increased thalamic inhibition induces slower cortical frequencies and that enhanced cortical excitation increases traveling wave speed and frequency. Overall, our results provide insight into the genesis and sustainability of thalamocortical spatiotemporal patterns, showing how simple synaptic alterations cause varied spontaneous and evoked wave patterns. Our model and simulations highlight the need for spatially spread neural recordings to uncover critical circuit mechanisms for brain functions.

Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson (Phys Rev E 85(5):055,101(R), 2012), is known to support stationary and travelling bumps of localised activity. We pose the model on a ring and study the existence and stability of these patterns in various limits using a combination of analytical and numerical techniques. In a purely deterministic version of the model, posed on a continuum, we construct bumps and travelling waves analytically using standard interface methods from neural field theory. In a stochastic version with Heaviside firing rate, we construct approximate analytical probability mass functions associated with bumps and travelling waves. In the full stochastic model posed on a discrete lattice, where a coarse analytic description is unavailable, we compute patterns and their linear stability using equation-free methods. The lifting procedure used in the coarse time-stepper is informed by the analysis in the deterministic and stochastic limits. In all settings, we identify the synaptic profile as a mesoscopic variable, and the width of the corresponding activity set as a macroscopic variable. Stationary and travelling bumps have similar meso- and macroscopic profiles, but different microscopic structure, hence we propose lifting operators which use microscopic motifs to disambiguate them. We provide numerical evidence that waves are supported by a combination of high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times.

Traveling waves and breathers in an excitatory-inhibitory neural field

We study existence and stability of traveling activity bump solutions in an excitatory-inhibitory (E-I) neural field with Heaviside firing rate functions by deriving existence conditions for traveling bumps and an Evans function to analyze their spectral stability. Subsequently, we show that these existence and stability results reduce, in the limit of wave speed c→0, to the equivalent conditions developed for the stationary bump case. Using the results for the stationary bump case, we show that drift bifurcations of stationary bumps serve as a mechanism for generating traveling bump solutions in the E-I neural field as parameters are varied. Furthermore, we explore the interrelations between stationary and traveling types of bumps and breathers (time-periodic oscillatory bumps) by bridging together analytical and simulation results for stationary and traveling bumps and their bifurcations in a region of parameter space. Interestingly, we find evidence for a codimension-2 drift-Hopf bifurcation occurring as two parameters, inhibitory time constant τ and I-to-I synaptic connection strength w[over ¯]_{ii}, are varied and show that the codimension-2 point serves as an organizing center for the dynamics of these four types of spatially localized solutions. Additionally, we describe a case involving subcritical bifurcations that lead to traveling waves and breathers as τ is varied.

Feed-Forward Propagation of Temporal and Rate Information between Cortical Populations during Coherent Activation in Engineered In Vitro Networks

Transient propagation of information across neuronal assembles is thought to underlie many cognitive processes. However, the nature of the neural code that is embedded within these transmissions remains uncertain. Much of our understanding of how information is transmitted among these assemblies has been derived from computational models. While these models have been instrumental in understanding these processes they often make simplifying assumptions about the biophysical properties of neurons that may influence the nature and properties expressed. To address this issue we created an in vitro analog of a feed-forward network composed of two small populations (also referred to as assemblies or layers) of living dissociated rat cortical neurons. The populations were separated by, and communicated through, a microelectromechanical systems (MEMS) device containing a strip of microscale tunnels. Delayed culturing of one population in the first layer followed by the second a few days later induced the unidirectional growth of axons through the microtunnels resulting in a primarily feed-forward communication between these two small neural populations. In this study we systematically manipulated the number of tunnels that connected each layer and hence, the number of axons providing communication between those populations. We then assess the effect of reducing the number of tunnels has upon the properties of between-layer communication capacity and fidelity of neural transmission among spike trains transmitted across and within layers. We show evidence based on Victor-Purpura's and van Rossum's spike train similarity metrics supporting the presence of both rate and temporal information embedded within these transmissions whose fidelity increased during communication both between and within layers when the number of tunnels are increased. We also provide evidence reinforcing the role of synchronized activity upon transmission fidelity during the spontaneous synchronized network burst events that propagated between layers and highlight the potential applications of these MEMs devices as a tool for further investigation of structure and functional dynamics among neural populations.

An in vitro method to manipulate the direction and functional strength between neural populations

We report the design and application of a Micro Electro Mechanical Systems (MEMs) device that permits investigators to create arbitrary network topologies. With this device investigators can manipulate the degree of functional connectivity among distinct neural populations by systematically altering their geometric connectivity in vitro. Each polydimethylsilxane (PDMS) device was cast from molds and consisted of two wells each containing a small neural population of dissociated rat cortical neurons. Wells were separated by a series of parallel micrometer scale tunnels that permitted passage of axonal processes but not somata; with the device placed over an 8 × 8 microelectrode array, action potentials from somata in wells and axons in microtunnels can be recorded and stimulated. In our earlier report we showed that a one week delay in plating of neurons from one well to the other led to a filling and blocking of the microtunnels by axons from the older well resulting in strong directionality (older to younger) of both axon action potentials in tunnels and longer duration and more slowly propagating bursts of action potentials between wells. Here we show that changing the number of tunnels, and hence the number of axons, connecting the two wells leads to changes in connectivity and propagation of bursting activity. More specifically, the greater the number of tunnels the stronger the connectivity, the greater the probability of bursting propagating between wells, and shorter peak-to-peak delays between bursts and time to first spike measured in the opposing well. We estimate that a minimum of 100 axons are needed to reliably initiate a burst in the opposing well. This device provides a tool for researchers interested in understanding network dynamics who will profit from having the ability to design both the degree and directionality connectivity among multiple small neural populations.

Minimal conductance-based model of auditory coincidence detector neurons

Sound localization is a fundamental sensory function of a wide variety of animals. The interaural time difference (ITD), an important cue for sound localization, is computed in the auditory brainstem. In our previous modeling study, we introduced a two-compartment Hodgkin-Huxley type model to investigate how cellular and synaptic specializations may contribute to precise ITD computation of the barn owl's auditory coincidence detector neuron. Although our model successfully reproduced fundamental physiological properties observed in vivo, it was unsuitable for mathematical analyses and large scale simulations because of a number of nonlinear variables. In the present study, we reduce our former model into three types of conductance-based integrate-and-fire (IF) models. We test their electrophysiological properties using data from published in vivo and in vitro studies. Their robustness to parameter changes and computational efficiencies are also examined. Our numerical results suggest that the single-compartment active IF model is superior to other reduced models in terms of physiological reproducibility and computational performance. This model will allow future theoretical studies that use more rigorous mathematical analysis and network simulations.

Learning-induced modulation of the GABAB-mediated inhibitory synaptic transmission: mechanisms and functional significance

Complex olfactory-discrimination (OD) learning results in a series of intrinsic and excitatory synaptic modifications in piriform cortex pyramidal neurons that enhance the circuit excitability. Such overexcitation must be balanced to prevent runway activity while maintaining the efficient ability to store memories. We showed previously that OD learning is accompanied by enhancement of the GABAA-mediated inhibition. Here we show that GABAB-mediated inhibition is also enhanced after learning and study the mechanism underlying such enhancement and explore its functional role. We show that presynaptic, GABAB-mediated synaptic inhibition is enhanced after learning. In contrast, the population-average postsynaptic GABAB-mediated synaptic inhibition is unchanged, but its standard deviation is enhanced. Learning-induced reduction in paired pulse facilitation in the glutamatergic synapses interconnecting pyramidal neurons was abolished by application of the GABAB antagonist CGP55845 but not by blocking G protein-gated inwardly rectifying potassium channels only, indicating enhanced suppression of excitatory synaptic release via presynaptic GABAB-receptor activation. In addition, the correlation between the strengths of the early (GABAA-mediated) and late (GABAB-mediated) synaptic inhibition was much stronger for each particular neuron after learning. Consequently, GABAB-mediated inhibition was also more efficient in controlling epileptic-like activity induced by blocking GABAA receptors. We suggest that complex OD learning is accompanied by enhancement of the GABAB-mediated inhibition that enables the cortical network to store memories, while preventing uncontrolled activity.

Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-induced oscillations

Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG) activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.

Neural mechanism of activity spread in the cat motor cortex and its relation to the intrinsic connectivity

Motor cortical points are linked by intrinsic horizontal connections having a recurrent network topology. However, it is not known whether neural activity can propagate over the area covered by these intrinsic connections and whether there are spatial anisotropies of synaptic strength, as opposed to synaptic density. Moreover, the mechanisms by which activity spreads have yet to be determined. To address these issues, an 8 × 8 microelectrode array was inserted in the forelimb area of the cat motor cortex (MCx). The centre of the array had a laser etched hole ∼500 μm in diameter. A microiontophoretic pipette, with a tip diameter of 2-3 μm, containing bicuculline methiodide (BIC) was inserted in the hole and driven to a depth of 1200-1400 μm from the cortical surface. BIC was ejected for ∼2min from the tip of the micropipette with positive direct current ranging between 20 and 40 nA in different experiments. This produced spontaneous nearly periodic bursts (0.2-1.0 Hz) of multi-unit activity in a radius of about 400 μm from the tip of the micropipette. The bursts of neural activity spread at a velocity of 0.11-0.24 ms⁻¹ (mean=0.14 mm ms⁻¹, SD=0.05)with decreasing amplitude.The area activated was on average 7.22 mm² (SD=0.91 mm²), or ∼92% of the area covered by the recording array. The mode of propagation was determined to occur by progressive recruitment of cortical territory, driven by a central locus of activity of some 400 μm in radius. Thus, activity did not propagate as a wave. Transection of the connections between the thalamus and MCx did not significantly alter the propagation velocity or the size of the recruited area, demonstrating that the bursts spread along the routes of intrinsic cortical connectivity. These experiments demonstrate that neural activity initiated within a small motor cortical locus (≤ 400 μm in radius) can recruit a relatively large neighbourhood in which a variety of muscles acting at several forelimb joints are represented. These results support the hypothesis that the MCx controls the forelimb musculature in an integrated and anticipatory manner based on a recurrent network topology

Bifurcations of lurching waves in a thalamic neuronal network

We consider a two-layer, one-dimensional lattice of neurons; one layer consists of excitatory thalamocortical neurons, while the other is comprised of inhibitory reticular thalamic neurons. Such networks are known to support "lurching" waves, for which propagation does not appear smooth, but rather progresses in a saltatory fashion; these waves can be characterized by different spatial widths (different numbers of neurons active at the same time). We show that these lurching waves are fixed points of appropriately defined Poincaré maps, and follow these fixed points as parameters are varied. In this way, we are able to explain observed transitions in behavior, and, in particular, to show how branches with different spatial widths are linked with each other. Our computer-assisted analysis is quite general and could be applied to other spatially extended systems which exhibit this non-trivial form of wave propagation.

BDNF and NT-3 Increase Velocity of Activity Front Propagation in Unidimensional Hippocampal Cultures

Neurotrophins are known to promote synapse development as well as to regulate the efficacy of mature synapses. We have previously reported that in two-dimensional rat hippocampal cultures, brain-derived neurotrophic factor (BDNF) and neurotrophin-3 significantly increase the number of excitatory input connections. Here we measure the effect of these neurotrophic agents on propagating fronts that arise spontaneously in quasi-one-dimensional rat hippocampal cultures. We observe that chronic treatment with BDNF increased the velocity of the propagation front by about 30%. This change is attributed to an increase in the excitatory input connectivity. We analyze the experiment using the Feinerman–Golomb/Ermentrout–Jacobi/Moses–Osan model for the propagation of fronts in a one-dimensional neuronal network with synaptic delay and introduce the synaptic connection probability between adjacent neurons as a new parameter of the model. We conclude that BDNF increases the number of excitatory connections by favoring the probability to form connections between neurons, but without significantly modifying the range of the connections (connectivity footprint).

Leaders of neuronal cultures in a quorum percolation model

We present a theoretical framework using quorum percolation for describing the initiation of activity in a neural culture. The cultures are modeled as random graphs, whose nodes are excitatory neurons with k_in inputs and k_out outputs, and whose input degrees k_in = k obey given distribution functions p_k. We examine the firing activity of the population of neurons according to their input degree (k) classes and calculate for each class its firing probability Φ_k(t) as a function of t. The probability of a node to fire is found to be determined by its in-degree k, and the first-to-fire neurons are those that have a high k. A small minority of high-k-classes may be called “Leaders,” as they form an interconnected sub-network that consistently fires much before the rest of the culture. Once initiated, the activity spreads from the Leaders to the less connected majority of the culture. We then use the distribution of in-degree of the Leaders to study the growth rate of the number of neurons active in a burst, which was experimentally measured to be initially exponential. We find that this kind of growth rate is best described by a population that has an in-degree distribution that is a Gaussian centered around k = 75 with width σ = 31 for the majority of the neurons, but also has a power law tail with exponent −2 for 10% of the population. Neurons in the tail may have as many as k = 4,700 inputs. We explore and discuss the correspondence between the degree distribution and a dynamic neuronal threshold, showing that from the functional point of view, structure and elementary dynamics are interchangeable. We discuss possible geometric origins of this distribution, and comment on the importance of size, or of having a large number of neurons, in the culture.

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.

Analyzing Neuronal Networks Using Discrete-Time Dynamics

We develop mathematical techniques for analyzing detailed Hodgkin-Huxley like models for excitatory-inhibitory neuronal networks. Our strategy for studying a given network is to first reduce it to a discrete-time dynamical system. The discrete model is considerably easier to analyze, both mathematically and computationally, and parameters in the discrete model correspond directly to parameters in the original system of differential equations. While these networks arise in many important applications, a primary focus of this paper is to better understand mechanisms that underlie temporally dynamic responses in early processing of olfactory sensory information. The models presented here exhibit several properties that have been described for olfactory codes in an insect's Antennal Lobe. These include transient patterns of synchronization and decorrelation of sensory inputs. By reducing the model to a discrete system, we are able to systematically study how properties of the dynamics, including the complex structure of the transients and attractors, depend on factors related to connectivity and the intrinsic and synaptic properties of cells within the network.

Delays in activity-based neural networks

In this paper, we study the effect of two distinct discrete delays on the dynamics of a Wilson-Cowan neural network. This activity-based model describes the dynamics of synaptically interacting excitatory and inhibitory neuronal populations. We discuss the interpretation of the delays in the language of neurobiology and show how they can contribute to the generation of network rhythms. First, we focus on the use of linear stability theory to show how to destabilize a fixed point, leading to the onset of oscillatory behaviour. Next, we show for the choice of a Heaviside nonlinearity for the firing rate that such emergent oscillations can be either synchronous or anti-synchronous, depending on whether inhibition or excitation dominates the network architecture. To probe the behaviour of smooth (sigmoidal) nonlinear firing rates, we use a mixture of numerical bifurcation analysis and direct simulations, and uncover parameter windows that support chaotic behaviour. Finally, we comment on the role of delays in the generation of bursting oscillations, and discuss natural extensions of the work in this paper.

Variability and corresponding amplitude-velocity relation of activity propagating in one-dimensional neural cultures

We investigate the propagation of neural activity along one-dimensional rat hippocampal cultures patterned in lines over multielectrode arrays. Activity occurs spontaneously or is evoked by local electrical or chemical stimuli, with different resulting propagation velocities and firing rate amplitudes. A variability of an order of magnitude in velocity and amplitude is observed in spontaneous activity. A linear relation between velocity and amplitude is identified. We define a measure for neuron activation synchrony and find that it correlates with front velocity and is higher for electrically evoked fronts. We present a model that explains the linear relation between amplitude and velocity, which highlights the role of synchrony. The relation to current models for signal propagation in neural media is discussed.

An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons

It has been suggested that spontaneous synchronous neuronal activity is an essential step in the formation of functional networks in the central nervous system. The key features of this type of activity consist of bursts of action potentials with associated spikes of elevated cytoplasmic calcium. These features are also observed in networks of rat cortical neurons that have been formed in culture. Experimental studies of these cultured networks have led to several hypotheses for the mechanisms underlying the observed synchronized oscillations. In this paper, bursting integrate-and-fire type mathematical models for regular spiking (RS) and intrinsic bursting (IB) neurons are introduced and incorporated through a small-world connection scheme into a two-dimensional excitatory network similar to those in the cultured network. This computer model exhibits spontaneous synchronous activity through mechanisms similar to those hypothesized for the cultured experimental networks. Traces of the membrane potential and cytoplasmic calcium from the model closely match those obtained from experiments. We also consider the impact on network behavior of the IB neurons, the geometry and the small world connection scheme.

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

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 for the interaction of oscillations and pattern generation with real-time computing in generic neural microcircuit models

It is shown that real-time computations on spike patterns and temporal integration of information in neural microcircuit models are compatible with potentially descruptive additional inputs such as oscillations. A minor change in the connection statistics of such circuits (making synaptic connections to more distal target neurons more likely for excitatory than for inhibitory neurons) endows such generic neural microcircuit model with the ability to generate periodic patterns autonomously. We show that such pattern generation can also be multiplexed with pattern classification and temporal integration of information in the same neural circuit. These results can be interpreted as showing that periodic activity provides a second channel for communication in neural systems which can be used to synchronize or coordinate spatially separated processes, without encumbering local real-time computations on spike trains in diverse neural circuits.

Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks

We study the effect of delays on the dynamics of large networks of neurons. We show that delays give rise to a wealth of bifurcations and to a rich phase diagram, which includes oscillatory bumps, traveling waves, lurching waves, standing waves arising via a period-doubling bifurcation, aperiodic regimes, and regimes of multistability. We study the existence and the stability of the various dynamical patterns analytically and numerically in a simplified rate model as a function of the interaction parameters. The results derived in that framework allow us to understand the origin of the diversity of dynamical states observed in large networks of spiking neurons.

Travelling wave patterns in a model of the spinal pattern generator using spiking neurons

The aim of this study is to produce travelling waves in a planar net of artificial spiking neurons. Provided that the parameters of the waves--frequency, wavelength and orientation--can be sufficiently controlled, such a network can serve as a model of the spinal pattern generator for swimming and terrestrial quadruped locomotion. A previous implementation using non-spiking, sigmoid neurons lacked the physiological plausibility that can only be attained using more realistic spiking neurons. Simulations were conducted using three types of spiking neuronal models. First, leaky integrate-and-fire neurons were used. Second, we introduced a phenomenological bursting neuron. And third, a canonical model neuron was implemented which could reproduce the full dynamics of the Hodgkin-Huxley neuron. The conditions necessary to produce appropriate travelling waves corresponded largely to the known anatomy and physiology of the spinal cord. Especially important features for the generation of travelling waves were the topology of the local connections--so-called off-centre connectivity--the availability of dynamic synapses and, to some extent, the availability of bursting cell types. The latter were necessary to produce stable waves at the low frequencies observed in quadruped locomotion. In general, the phenomenon of travelling waves was very robust and largely independent of the network parameters and emulated cell types.

Pulse propagation in discrete excitatory networks of integrate-and-fire neurons

We study the propagation of solitary waves in a discrete excitatory network of integrate-and-fire neurons. We show the existence and the stability of a fast wave and a family of slow waves. Fast waves are similar to those already described in continuum networks. Stable slow waves have not been previously reported in purely excitatory networks and their propagation is particular to the discrete nature of the network. The robustness of our results is studied in the presence of noise.

Multiple-spike waves in a one-dimensional integrate-and-fire neural network

This paper builds on the past study of single-spike waves in one-dimensional integrate-and-fire networks to provide a framework for the study of waves with arbitrary (finite or countably infinite) collections of spike times. Based on this framework, we prove an existence theorem for single-spike traveling waves, and we combine analysis and numerics to study two-spike traveling waves, periodic traveling waves, and general infinite spike trains. For a fixed wave speed, finite-spike waves, periodic waves, and other infinite-spike waves may all occur, and we discuss the relationships among them. We also relate the waves considered analytically to waves generated in numerical simulations by the transient application of localized excitation.

Effects of quasiactive membrane on multiply periodic traveling waves in integrate-and-fire systems

We consider the dynamics of a one-dimensional continuum of synaptically interacting integrate-and-fire neurons with realistic forms of axodendritic interaction. The speed and stability of traveling waves are investigated as a function of discrete communication delays, distributed synaptic delays, and axodendritic delays arising from the spatially extended nature of the model neuron. In particular, dispersion curves for periodic traveling waves are constructed. Nonlinear ionic channels in the dendrite responsible for a so-called quasiactive bandpass response are shown to significantly influence the shape of dispersion curves. Moreover, a kinematic theory of spike train propagation suggests that period-doubling bifurcations of a singly periodic wave can occur in dendritic systems with a quasiactive membrane. The explicit construction of period-doubled solutions is used to confirm this prediction.

Dynamics of synaptically coupled integrate-and-fire-or-burst neurons

The minimal integrate-and-fire-or-burst (IFB) neuron model reproduces the salient features of experimentally observed thalamocortical (TC) relay neuron response properties, including the temporal tuning of both tonic spiking (i.e., conventional action potentials) and postinhibitory rebound bursting mediated by a low-threshold calcium current. In this paper we consider networks of IFB neurons with slow synaptic interactions and show how the dynamics may be described with a smooth firing-rate model. When the firing rate of the IFB model is dominated by a refractory process the equations of motion simplify and may be solved exactly. Numerical simulations are used to show that a pair of reciprocally interacting inhibitory spiking IFB TC neurons supports an alternating rhythm of the type predicted from the firing-rate theory. A change in a single parameter of the IFB neuron allows it to fire a burst of spikes in response to a depolarizing signal, so that it mimics the behavior of a reticular (RE) cell. Within a continuum model we show that a network of RE cells with on-center excitation can support a fast traveling pulse. In contrast, a network of inhibitory TC cells is found to support a slowly propagating lurching pulse.

Activity dynamics in nonlocal interacting neural fields

We study the activity of a synaptically coupled neuronal network consisting of an excitatory and an inhibitory layer with isotropic connections and nonlinear interactions. Using the mathematical model of Wilson and Cowan in two spatial dimensions, we first discuss a spatial hysteresis phenomenon. Then we analyze special traveling wave solutions with stationary shape. We establish existence conditions, derive analytic expressions of the particular solutions and their velocity, and finally present numerical simulations.

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.

Traveling waves of excitation in neural field models: equivalence of rate descriptions and integrate-and-fire dynamics

Field models provide an elegant mathematical framework to analyze large-scale patterns of neural activity. On the microscopic level, these models are usually based on either a firing-rate picture or integrate-and-fire dynamics. This article shows that in spite of the large conceptual differences between the two types of dynamics, both generate closely related plane-wave solutions. Furthermore, for a large group of models, estimates about the network connectivity derived from the speed of these plane waves only marginally depend on the assumed class of microscopic dynamics. We derive quantitative results about this phenomenon and discuss consequences for the interpretation of experimental data.

Slow excitation supports propagation of slow pulses in networks of excitatory and inhibitory populations

We study the propagation of traveling solitary pulses in one-dimensional networks of excitatory and inhibitory neurons. Each neuron is represented by the integrate-and-fire model, and is allowed to fire only one spike. Two types of propagating pulses are observed. During fast pulses, inhibitory neurons fire a short time before or after the excitatory neurons. During slow pulses, inhibitory cells fire well before neighboring excitatory cells, and potentials of excitatory cells become negative and then positive before they fire. There is a bistable parameter regime in which both fast and slow pulses can propagate. Fast pulses can propagate at low levels of inhibition, are affected by fast excitation but are almost unaffected by slow excitation, and are easily elicited by stimulating groups of neurons. In contrast, slow pulses can propagate at intermediate levels of inhibition, and are difficult to evoke. They can propagate without slow excitation, but slow excitation makes their propagation substantially more robust. Fast pulses can propagate in a wider parameter regime if inhibition decays slowly with time, whereas slow pulses can propagate in a wider parameter regime if the passive time constant of inhibitory cells is large. Strong inhibitory-to-inhibitory conductance eliminates the slow pulses and converts the fast traveling pulses into irregular pulses, in which the inhibitory neurons segregate into two groups that have different firing delays with respect to their neighboring excitatory cells. In general, the velocity of the fast pulse increases with the axonal conductance velocity c, but there are cases in which it decreases with c. We suggest that the fast and slow pulses observed in our model correspond to the fast and slow propagating activity observed in experiments on neocortical slices.

Nonlinear dynamics of direction-selective recurrent neural media

The direction selectivity of cortical neurons can be accounted for by asymmetric lateral connections. Such lateral connectivity leads to a network dynamics with characteristic properties that can be exploited for distinguishing in neurophysiological experiments this mechanism for direction selectivity from other possible mechanisms. We present a mathematical analysis for a class of direction-selective neural models with asymmetric lateral connections. Contrasting with earlier theoretical studies that have analyzed approximations of the network dynamics by neglecting nonlinearities using methods from linear systems theory, we study the network dynamics with nonlinearity taken into consideration. We show that asymmetrically coupled networks can stabilize stimulus-locked traveling pulse solutions that are appropriate for the modeling of the responses of direction-selective neurons. In addition, our analysis shows that outside a certain regime of stimulus speeds the stability of these solutions breaks down, giving rise to lurching activity waves with specific spatiotemporal periodicity. These solutions, and the bifurcation by which they arise, cannot be easily accounted for by classical models for direction selectivity.

Bistability in pulse propagation in networks of excitatory and inhibitory populations

We study the propagation of traveling solitary pulses in one-dimensional networks of excitatory and inhibitory integrate-and-fire neurons. Slow pulses, during which inhibitory cells fire well before neighboring excitatory cells, can propagate along the network at intermediate inhibition levels. At higher levels, they destabilize via a Hopf bifurcation. There is a bistable parameter regime in which both fast and slow pulses can propagate. Lurching pulses with spatiotemporal periodicity can propagate in regimes for which continuous pulses do not exist.

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.

Stability properties of solitary waves and periodic wave trains in a two-dimensional network of spiking neurons

We investigate solitary waves of excitation in a two-dimensional network of spiking neurons with distance-dependent couplings. A continuum description is developed that does not require spatial or temporal averaging. Using this description, propagation velocities and dispersion relations for solitary waves and periodic wave trains can be calculated analytically. We show that the stability properties of solitary waves and wave trains are dominated by form instabilities which are genuine to the two-dimensional nature of the system.

Spiking-bursting activity in the thalamic reticular nucleus initiates sequences of spindle oscillations in thalamic networks

Recent intracellular and local field potential recordings from thalamic reticular (RE) neurons in vivo as well as computational modeling of the isolated RE nucleus suggest that, at relatively hyperpolarized levels of membrane potentials, the inhibitory postsynaptic potentials (IPSPs) between RE cells can be reversed and gamma-aminobutyric acid-A (GABA(A)) -mediated depolarization can generate persistent spatio-temporal patterns in the RE nucleus. Here we investigate how this activity affects the spatio-temporal properties of spindle oscillations with computer models of interacting RE and thalamocortical (TC) cells. In a one-dimensional network of RE and TC cells, sequences of spindle oscillations alternated with localized patterns of spike-burst activity propagating inside the RE network. New sequences of spindle oscillations were initiated after removal of I(h)-mediated depolarization of the TC cells. The length of the interspindle lulls depended on the intrinsic and synaptic properties of RE and TC cells and was in the range of 3-20 s. In a two-dimensional model, GABA(A)-mediated 2-3 Hz oscillations persisted in the RE nucleus during interspindle lulls and initiated spindle sequences at many foci within the RE-TC network simultaneously. This model predicts that the intrinsic properties of the reticular thalamus may contribute to the synchrony of spindle oscillations observed in vivo.