Bistability in pulse propagation in networks of excitatory and inhibitory populations

The inhibitory control of traveling waves in cortical networks

Propagating waves of activity can be evoked and can occur spontaneously in vivo and in vitro in cerebral cortex. These waves are thought to be instrumental in the propagation of information across cortical regions and as a means to modulate the sensitivity of neurons to subsequent stimuli. In normal tissue, the waves are sparse and tightly controlled by inhibition and other negative feedback processes. However, alterations of this balance between excitation and inhibition can lead to pathological behavior such as seizure-type dynamics (with low inhibition) or failure to propagate (with high inhibition). We develop a spiking one-dimensional network of neurons to explore the reliability and control of evoked waves and compare this to a cortical slice preparation where the excitability can be pharmacologically manipulated. We show that the waves enhance sensitivity of the cortical network to stimuli in specific spatial and temporal ways. To gain further insight into the mechanisms of propagation and transitions to pathological behavior, we derive a mean-field model for the synaptic activity. We analyze the mean-field model and a piece-wise constant approximation of it and study the stability of the propagating waves as spatial and temporal properties of the inhibition are altered. We show that that the transition to seizure-like activity is gradual but that the loss of propagation is abrupt and can occur via either the loss of existence of the wave or through a loss of stability leading to complex patterns of propagation.

Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems

Bistable nonequilibrium systems are realized in catalytic reaction-diffusion processes, biological transport and regulation, spatial epidemics, etc. Behavior in spatially continuous formulations, described at the mean-field level by reaction-diffusion type equations (RDEs), often mimics that of classic equilibrium van der Waals type systems. When accounting for noise, similarities include a discontinuous phase transition at some value, p_{eq}, of a control parameter, p, with metastability and hysteresis around p_{eq}. For each p, there is a unique critical droplet of the more stable phase embedded in the less stable or metastable phase which is stationary (neither shrinking nor growing), and with size diverging as p→p_{eq}. Spatially discrete analogs of these mean-field formulations, described by lattice differential equations (LDEs), are more appropriate for some applications, but have received less attention. It is recognized that LDEs can exhibit richer behavior than RDEs, specifically propagation failure for planar interphases separating distinct phases. We show that this feature, together with an orientation dependence of planar interface propagation also deriving from spatial discreteness, results in the occurrence of entire families of stationary droplets. The extent of these families increases approaching the transition and can be infinite if propagation failure is realized. In addition, there can exist a regime of generic two-phase coexistence where arbitrarily large droplets of either phase always shrink. Such rich behavior is qualitatively distinct from that for classic nucleation in equilibrium and spatially continuous nonequilibrium systems.

Reaction-diffusion memory unit: Modeling of sensitization, habituation and dishabituation in the brain

We propose a novel approach to investigate the effects of sensitization, habituation and dishabituation in the brain using the analysis of the reaction-diffusion memory unit (RDMU). This unit consists of Morris-Lecar-type sensory, motor, interneuron and two input excitable cables, linked by four synapses with adjustable strength defined by Hebbian rules. Stimulation of the sensory neuron through the first input cable causes sensitization by activating two excitatory synapses, C₁ and C₂, connected to the interneuron and motor neuron, respectively. In turn, the stimulation of the interneuron causes habituation through the activation of inhibitory synapse C₃. Likewise, dishabituation is caused through the activation of another inhibitory synapse C₄. We have determined sensitization-habituation (BSH) and habituation-dishabituation (BHDH) boundaries as functions between synaptic strengths C₂ and C₃ at various strengths of C₁ and C₄. When BSH and BHDH curves shift towards larger values of C₂, the RDMU can be easily inhibited. On the contrary, the RDMU can be easily sensitized or dishabituated if BSH and BHDH curves shift towards smaller values of C₂. Our numerical simulations readily demonstrate that higher values of the Morris-Lecar relaxation parameter, greater leakage and potassium conductances, reduced length of the interneuron, and higher values of C₁ all result in easier habituation of the RDMU. In contrast, we found that at higher values of C₄ the RDMU becomes significantly more prone to dishabituation. Based on these simulations one can quantify BSH and BHDH curve shifts and relate them to particular neural outcomes.

Optimal Detection of a Localized Perturbation in Random Networks of Integrate-and-Fire Neurons

Experimental and theoretical studies suggest that cortical networks are chaotic and coding relies on averages over large populations. However, there is evidence that rats can respond to the short stimulation of a single cortical cell, a theoretically unexplained fact. We study effects of single-cell stimulation on a large recurrent network of integrate-and-fire neurons and propose a simple way to detect the perturbation. Detection rates obtained from simulations and analytical estimates are similar to experimental response rates if the readout is slightly biased towards specific neurons. Near-optimal detection is attained for a broad range of intermediate values of the mean coupling between neurons.

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.

Linking dynamics of the inhibitory network to the input structure

Networks of inhibitory interneurons are found in many distinct classes of biological systems. Inhibitory interneurons govern the dynamics of principal cells and are likely to be critically involved in the coding of information. In this theoretical study, we describe the dynamics of a generic inhibitory network in terms of low-dimensional, simplified rate models. We study the relationship between the structure of external input applied to the network and the patterns of activity arising in response to that stimulation. We found that even a minimal inhibitory network can generate a great diversity of spatio-temporal patterning including complex bursting regimes with non-trivial ratios of burst firing. Despite the complexity of these dynamics, the network's response patterns can be predicted from the rankings of the magnitudes of external inputs to the inhibitory neurons. This type of invariant dynamics is robust to noise and stable in densely connected networks with strong inhibitory coupling. Our study predicts that the response dynamics generated by an inhibitory network may provide critical insights about the temporal structure of the sensory input it receives.

Forecasting Bifurcations from Large Perturbation Recoveries in Feedback Ecosystems

Forecasting bifurcations such as critical transitions is an active research area of relevance to the management and preservation of ecological systems. In particular, anticipating the distance to critical transitions remains a challenge, together with predicting the state of the system after these transitions are breached. In this work, a new model-less method is presented that addresses both these issues based on monitoring recoveries from large perturbations. The approach uses data from recoveries of the system from at least two separate parameter values before the critical point, to predict both the bifurcation and the post-bifurcation dynamics. The proposed method is demonstrated, and its performance evaluated under different levels of measurement noise, with two ecological models that have been used extensively in previous studies of tipping points and alternative steady states. The first one considers the dynamics of vegetation under grazing; the second, those of macrophyte and phytoplankton in shallow lakes. Applications of the method to more complex situations are discussed together with the kinds of empirical data needed for its implementation.

Mechanisms of seizure propagation in 2-dimensional centre-surround recurrent networks

Understanding how seizures spread throughout the brain is an important problem in the treatment of epilepsy, especially for implantable devices that aim to avert focal seizures before they spread to, and overwhelm, the rest of the brain. This paper presents an analysis of the speed of propagation in a computational model of seizure-like activity in a 2-dimensional recurrent network of integrate-and-fire neurons containing both excitatory and inhibitory populations and having a difference of Gaussians connectivity structure, an approximation to that observed in cerebral cortex. In the same computational model network, alternative mechanisms are explored in order to simulate the range of seizure-like activity propagation speeds (0.1-100 mm/s) observed in two animal-slice-based models of epilepsy: (1) low extracellular [Formula: see text], which creates excess excitation and (2) introduction of gamma-aminobutyric acid (GABA) antagonists, which reduce inhibition. Moreover, two alternative connection topologies are considered: excitation broader than inhibition, and inhibition broader than excitation. It was found that the empirically observed range of propagation velocities can be obtained for both connection topologies. For the case of the GABA antagonist model simulation, consistent with other studies, it was found that there is an effective threshold in the degree of inhibition below which waves begin to propagate. For the case of the low extracellular [Formula: see text] model simulation, it was found that activity-dependent reductions in inhibition provide a potential explanation for the emergence of slowly propagating waves. This was simulated as a depression of inhibitory synapses, but it may also be achieved by other mechanisms. This work provides a localised network understanding of the propagation of seizures in 2-dimensional centre-surround networks that can be tested empirically.

Spatiotemporal alterations of cortical network activity by selective loss of NOS-expressing interneurons

Deciphering the role of GABAergic neurons in large neuronal networks such as the neocortex forms a particularly complex task as they comprise a highly diverse population. The neuronal isoform of the enzyme nitric oxide synthase (nNOS) is expressed in the neocortex by specific subsets of GABAergic neurons. These neurons can be identified in live brain slices by the nitric oxide (NO) fluorescent indicator diaminofluorescein-2 diacetate (DAF-2DA). However, this indicator was found to be highly toxic to the stained neurons. We used this feature to induce acute phototoxic damage to NO-producing neurons in cortical slices, and measured subsequent alterations in parameters of cellular and network activity. Neocortical slices were briefly incubated in DAF-2DA and then illuminated through the 4× objective. Histochemistry for NADPH-diaphorase (NADPH-d), a marker for nNOS activity, revealed elimination of staining in the illuminated areas following treatment. Whole cell recordings from several neuronal types before, during, and after illumination confirmed the selective damage to non-fast-spiking (FS) interneurons. Treated slices displayed mild disinhibition. The reversal potential of compound synaptic events on pyramidal neurons became more positive, and their decay time constant was elongated, substantiating the removal of an inhibitory conductance. The horizontal decay of local field potentials (LFPs) was significantly reduced at distances of 300-400 μm from the stimulation, but not when inhibition was non-selectively weakened with the GABA(A) blocker picrotoxin. Finally, whereas the depression of LFPs along short trains of 40 Hz stimuli was linearly reduced with distance or initial amplitude in control slices, this ordered relationship was disrupted in DAF-treated slices. These results reveal that NO-producing interneurons in the neocortex convey lateral inhibition to neighboring columns, and shape the spatiotemporal dynamics of the network's activity.

The mechanisms for compression and reflection of cortical waves

Waves are common in cortical networks and may be important for carrying information about a stimulus from one local circuit to another. In a recent study of visually evoked waves in rat cortex, compression and reflection of waves are observed as the activation passes from visual areas V1 to V2. The authors of this study apply bicuculline (BMI) and demonstrate that the reflection disappears. They conclude that inhibition plays a major role in compression and reflection. We present several models for propagating waves in heterogeneous media and show that the velocity and thus compression depends weakly on inhibition. We propose that the main site of action of BMI with respect to wave propagation is on the threshold for firing which we suggest is related to action on potassium channels. We combine numerical and analytic methods to explore both compression and reflection in an excitable system with synaptic coupling.

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling

This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.

Forecasting a class of bifurcations: theory and experiment

Forecasting bifurcations before they occur is a significant challenge and an important need in several fields. Existing approaches detect bifurcations before they occur by exploiting the critical slowing down phenomenon. However, the perturbations used in those approaches are limited to being very small and this represents a significant drawback. Large levels of perturbation have not been used mainly because of a lack of an adequate formulation that is robust to experimental noise. This paper provides such a formulation, and discusses how this approach to forecasting bifurcations is more accurate, especially when the dynamics are far from the bifurcation. Both numerical and experimental results are presented to demonstrate the technique and highlight its advantages over other prediction methods.

Neural adaptation facilitates oscillatory responses to static inputs in a recurrent network of ON and OFF cells

We investigate the role of adaptation in a neural field model, composed of ON and OFF cells, with delayed all-to-all recurrent connections. As external spatially profiled inputs drive the network, ON cells receive inputs directly, while OFF cells receive an inverted image of the original signals. Via global and delayed inhibitory connections, these signals can cause the system to enter states of sustained oscillatory activity. We perform a bifurcation analysis of our model to elucidate how neural adaptation influences the ability of the network to exhibit oscillatory activity. We show that slow adaptation encourages input-induced rhythmic states by decreasing the Andronov-Hopf bifurcation threshold. We further determine how the feedback and adaptation together shape the resonant properties of the ON and OFF cell network and how this affects the response to time-periodic input. By introducing an additional frequency in the system, adaptation alters the resonance frequency by shifting the peaks where the response is maximal. We support these results with numerical experiments of the neural field model. Although developed in the context of the circuitry of the electric sense, these results are applicable to any network of spontaneously firing cells with global inhibitory feedback to themselves, in which a fraction of these cells receive external input directly, while the remaining ones receive an inverted version of this input via feedforward di-synaptic inhibition. Thus the results are relevant beyond the many sensory systems where ON and OFF cells are usually identified, and provide the backbone for understanding dynamical network effects of lateral connections and various forms of ON/OFF responses.

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.

Oscillatory response in a sensory network of ON and OFF cells with instantaneous and delayed recurrent connections

A neural field model with multiple cell-to-cell feedback connections is investigated. Our model incorporates populations of ON and OFF cells, receiving sensory inputs with direct and inverted polarity, respectively. Oscillatory responses to spatially localized stimuli are found to occur via Andronov-Hopf bifurcations of stationary activity. We explore the impact of multiple delayed feedback components as well as additional excitatory and/or inhibitory non-delayed recurrent signals on the instability threshold. Paradoxically, instantaneous excitatory recurrent terms are found to enhance network responsiveness by reducing the oscillatory response threshold, allowing smaller inputs to trigger oscillatory activity. Instantaneous inhibitory components do the opposite. The frequency of these response oscillations is further shaped by the polarity of the non-delayed terms.

Dynamics of driven recurrent networks of ON and OFF cells

A globally coupled network of ON and OFF cells is studied using neural field theory. ON cells increase their activity when the amplitude of an external stimulus increases, while OFF cells do the opposite given the same stimulus. Theory predicts that, without input, multiple transitions to oscillations can occur depending on feedback delay and the difference between ON and OFF resting states. Static spatial stimuli can induce or suppress global oscillations via a Andronov-Hopf bifurcation. This is the case for either polarity of such stimuli. In contrast, only excitatory inputs can induce or suppress oscillations in an equivalent network built of ON cells only even though oscillations are more prevalent in such systems. Nonmonotonic responses to local stimuli occur where responses lateral to the stimulus switch from excitatory to inhibitory as the input amplitude increases. With local time-periodic forcing, the unforced cells oscillate at twice the driving frequency via full-wave rectification mediated by the feedback. Our results agree with simulations of the neural field model, and further, qualitative agreement is found with the behavior of a network of spiking stochastic integrate-and-fire model neurons.

Transient Turing patterns in a neural field model

We investigate Turing bifurcations in a neural field model with one spatial dimension. For some parameter values the resulting Turing patterns are stable, while for others the patterns appear transiently. We show that this difference is due to the relative position in parameter space of the saddle-node bifurcation of a spatially periodic pattern and the Turing bifurcation point. By varying parameters we are able to observe transient patterns whose duration scales in the same way as type-I intermittency. Similar behavior occurs in two spatial dimensions.

Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons

We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or "shortcuts", and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponentially distributed.

The importance of different timings of excitatory and inhibitory pathways in neural field models

In this paper, we consider a neural field model comprised of two distinct populations of neurons, excitatory and inhibitory, for which both the velocities of action potential propagation and the time courses of synaptic processing are different. Using recently-developed techniques, we construct the Evans function characterising the stability of both stationary and travelling wave solutions, under the assumption that the firing rate function is the Heaviside step. We find that these differences in timing for the two populations can cause instabilities of these solutions, leading to, for example, stationary breathers. We also analyse "anti-pulses", a novel type of pattern for which all but a small interval of the domain (in moving coordinates) is active. These results extend previous work on neural fields with space-dependent delays, and demonstrate the importance of considering the effects of the different time-courses of excitatory and inhibitory neural activity.

Enhanced neuronal response induced by fast inhibition

We report a facilitatory role of inhibitory synaptic input that can enhance a neuron's firing rate, in contrast to the conventional belief that inhibition suppresses firing. We study this phenomenon using the Hodgkin-Huxley model of spike generation with random Poisson trains of subthreshold excitatory and inhibitory inputs. Enhancement occurs when, by chance, brief inhibition leads excitation with a favorable timing and counterintuitively induces a reduction of the spike threshold. The basic mechanism is also illustrated with the phase-plane analysis of a two variable model.

Two cortical circuits control propagating waves in visual cortex

Visual stimuli produce waves of activity that propagate across the visual cortex of fresh water turtles. This study used a large-scale model of the cortex to examine the roles of specific types of cortical neurons in controlling the formation, speed and duration of these waves. The waves were divided into three components: initial depolarizations, primary propagating waves and secondary waves. The maximal conductances of each receptor type postsynaptic to each population of neurons in the model was systematically varied and the speed of primary waves, durations of primary waves and total wave durations were measured. The analyses indicate that wave formation and speed are controlled principally by feedforward excitation and inhibition, while wave duration is controlled principally by recurrent excitation and feedback inhibition.

Synchronized firings in the networks of class 1 excitable neurons with excitatory and inhibitory connections and their dependences on the forms of interactions

Synchronized firings in the networks of class 1 excitable neurons with excitatory and inhibitory connections are investigated, and their dependences on the forms of interactions are analyzed. As the forms of interactions, we treat the double exponential coupling and the interactions derived from it: pulse coupling, exponential coupling, and alpha coupling. It is found that the bifurcation structure of the networks depends mainly on the decay time of the synaptic interaction and the effect of the rise time is smaller than that of the decay time.

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.

On the dynamics of a pair of coupled neurons subject to alternating input rates

We present a statistical analysis of the firing activity of two coupled neuronal units that interact according to a 'sending-receiving' model. The membrane potential's behavior of both units is described by the Stein equations under the additional assumption that the spikes released by the sending neuron constitute an extra excitation for the receiving one. We also assume the presence of an alternating behavior for the rates of inputs to the sending neuron. By means of ad hoc simulations, we obtain, and then discuss, some statistical results concerning the spike production times of the units within the subintervals of the alternating inputs, as well as the reaction times of the receiving neuron.

An analysis of globally connected active rotators with excitatory and inhibitory connections having different time constants using the nonlinear Fokker-Planck equations

The globally connected active rotators with excitatory and inhibitory connections having different time constants under noise are analyzed using the nonlinear Fokker-Planck equation, and their oscillatory phenomena are investigated. Based on numerically calculated bifurcation diagrams, both periodic solutions and chaotic solutions are found. The periodic firings are classified based on the firing period, the coefficient of variation, and the correlation coefficient, and weakly synchronized periodic firings which are often observed in physiological experiments are found.

Oscillatory phase transition and pulse propagation in noisy integrate-and-fire neurons

We study nonlocally coupled noisy integrate-and-fire neurons with the Fokker-Planck equation. A propagating pulse state and a wavy state appear as a phase transition from an asynchronous state. We also find a solution in which traveling pulses are emitted periodically from a pacemaker region.

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.

Self-sustained activity in a small-world network of excitable neurons

We study the dynamics of excitable integrate-and-fire neurons in a small-world network. At low densities p of directed random connections, a localized transient stimulus results either in self-sustained persistent activity or in a brief transient followed by failure. Averages over the quenched ensemble reveal that the probability of failure changes from 0 to 1 over a narrow range in p; this failure transition can be described analytically through an extension of an existing mean-field result. Exceedingly long transients emerge at higher densities p; their activity patterns are disordered, in contrast to the mostly periodic persistent patterns observed at low p. The times at which such patterns die out follow a stretched-exponential distribution, which depends sensitively on the propagation velocity of the excitation.

Potassium model for slow (2-3 Hz) in vivo neocortical paroxysmal oscillations

In slow neocortical paroxysmal oscillations, the de- and hyperpolarizing envelopes in neocortical neurons are large compared with slow sleep oscillations. Increased local synchrony of membrane potential oscillations during seizure is reflected in larger electroencephalographic oscillations and the appearance of spike- or polyspike-wave complex recruitment at 2- to 3-Hz frequencies. The oscillatory mechanisms underlying this paroxysmal activity were investigated in computational models of cortical networks. The extracellular K(+) concentration ([K(+)](o)) was continuously computed based on neuronal K(+) currents and K(+) pumps as well as glial buffering. An increase of [K(+)](o) triggered a transition from normal awake-like oscillations to 2- to 3-Hz seizure-like activity. In this mode, the cells fired periodic bursts and nearby neurons oscillated highly synchronously; in some cells depolarization led to spike inactivation lasting 50-100 ms. A [K(+)](o) increase, sufficient to produce oscillations could result from excessive firing (e.g., induced by external stimulation) or inability of K(+) regulatory system (e.g., when glial buffering was blocked). A combination of currents including high-threshold Ca(2+), persistent Na(+) and hyperpolarization-activated depolarizing (I(h)) currents was sufficient to maintain 2- to 3-Hz activity. In a network model that included lateral K(+) diffusion between cells, increase of [K(+)](o) in a small region was generally sufficient to maintain paroxysmal oscillations in the whole network. Slow changes of [K(+)](o) modulated the frequency of bursting and, in some case, led to fast oscillations in the 10- to 15-Hz frequency range, similar to the fast runs observed during seizures in vivo. These results suggest that modifications of the intrinsic currents mediated by increase of [K(+)](o) can explain the range of neocortical paroxysmal oscillations in vivo.

Dynamical Consequences of Fast-Rising, Slow-Decaying Synapses in Neuronal Networks

Synapses that rise quickly but have long persistence are shown to have certain computational advantages. They have some unique mathematical properties as well and in some instances can make neurons behave as if they are weakly coupled oscillators. This property allows us to determine their synchronization properties. Furthermore, slowly decaying synapses allow recurrent networks to maintain excitation in the absence of inputs, whereas faster decaying synapses do not. There is an interaction between the synaptic strength and the persistence that allows recurrent networks to fire at low rates if the synapses are sufficiently slow. Waves and localized structures are constructed in spatially extended networks with slowly decaying synapses.

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.

Cellular and network mechanisms of slow oscillatory activity (<1 Hz) and wave propagations in a cortical network model

Slow oscillatory activity (<1 Hz) is observed in vivo in the cortex during slow-wave sleep or under anesthesia and in vitro when the bath solution is chosen to more closely mimic cerebrospinal fluid. Here we present a biophysical network model for the slow oscillations observed in vitro that reproduces the single neuron behaviors and collective network firing patterns in control as well as under pharmacological manipulations. The membrane potential of a neuron oscillates slowly (at <1 Hz) between a down state and an up state; the up state is maintained by strong recurrent excitation balanced by inhibition, and the transition to the down state is due to a slow adaptation current (Na(+)-dependent K(+) current). Consistent with in vivo data, the input resistance of a model neuron, on average, is the largest at the end of the down state and the smallest during the initial phase of the up state. An activity wave is initiated by spontaneous spike discharges in a minority of neurons, and propagates across the network at a speed of 3-8 mm/s in control and 20-50 mm/s with inhibition block. Our work suggests that long-range excitatory patchy connections contribute significantly to this wave propagation. Finally, we show with this model that various known physiological effects of neuromodulation can switch the network to tonic firing, thus simulating a transition to the waking state.

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.

Model of thalamocortical slow-wave sleep oscillations and transitions to activated States

During natural slow-wave sleep (SWS) in nonanesthetized cats, silent (down) states alternate with active (up) states; the down states are absent during rapid-eye-movement sleep and waking. Oscillations (<1 Hz) in SWS and transformation to an activated awake state were investigated with intracellular recordings in vivo and with computational models of the corticothalamic system. Occasional summation of the miniature EPSPs during the hyperpolarized (silent) phase of SWS oscillation activated the persistent sodium current and depolarized the membrane of cortical pyramidal (PY) cells sufficiently for spike generation. In the model, this triggered the active phase, which was maintained by lateral PY-PY excitation and persistent sodium current. Progressive depression of the excitatory interconnections and activation of Ca2+-dependent K+ current led to termination of the 20-25 Hz activity after 500-1000 msec. Including thalamocortical (TC) and thalamic reticular neurons in the model increased the duration of the active epochs up to 1-1.5 sec and introduced waning spindle sequences. An increase in acetylcholine activity, which is associated with activated states, was modeled by the reduction in the K+ leak current in PY and TC cells and by a decrease in intracortical PY-PY synaptic conductances. These changes eliminated the hyperpolarizing phases of network activity and transformed cortical neurons to tonic firing at 15-20 Hz. During the transition from SWS to the activated state, the input resistance of cortical neurons gradually increased and, in a fully activated state, reached the same or even higher values as during silent phases of SWS oscillations. The model describes many essential features of SWS and activated states in the thalamocortical system as well as the transition between them.

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.