Modelling Feedback Excitation,Pacemaker Properties and Sensory Switching of Electrically Coupled Brainstem Neurons Controlling Rhythmic Activity

What cellular and network properties allow reliable neuronal rhythm generation or firing that can be started and stopped by brief synaptic inputs? We investigate rhythmic activity in an electrically-coupled population of brainstem neurons driving swimming locomotion in young frog tadpoles, and how activity is switched on and off by brief sensory stimulation. We build a computational model of 30 electrically-coupled conditional pacemaker neurons on one side of the tadpole hindbrain and spinal cord. Based on experimental estimates for neuron properties, population sizes, synapse strengths and connections, we show that: long-lasting, mutual, glutamatergic excitation between the neurons allows the network to sustain rhythmic pacemaker firing at swimming frequencies following brief synaptic excitation; activity persists but rhythm breaks down without electrical coupling; NMDA voltage-dependency doubles the range of synaptic feedback strengths generating sustained rhythm. The network can be switched on and off at short latency by brief synaptic excitation and inhibition. We demonstrate that a population of generic Hodgkin-Huxley type neurons coupled by glutamatergic excitatory feedback can generate sustained asynchronous firing switched on and off synaptically. We conclude that networks of neurons with NMDAR mediated feedback excitation can generate self-sustained activity following brief synaptic excitation. The frequency of activity is limited by the kinetics of the neuron membrane channels and can be stopped by brief inhibitory input. Network activity can be rhythmic at lower frequencies if the neurons are electrically coupled. Our key finding is that excitatory synaptic feedback within a population of neurons can produce switchable, stable, sustained firing without synaptic inhibition.     

Small Modifications to Network Topology Can Induce Stochastic Bistable Spiking Dynamics in a Balanced Cortical Model

Directed random graph models frequently are used successfully in modeling the population dynamics of networks of cortical neurons connected by chemical synapses. Experimental results consistently reveal that neuronal network topology is complex, however, in the sense that it differs statistically from a random network, and differs for classes of neurons that are physiologically different. This suggests that complex network models whose subnetworks have distinct topological structure may be a useful, and more biologically realistic, alternative to random networks. Here we demonstrate that the balanced excitation and inhibition frequently observed in small cortical regions can transiently disappear in otherwise standard neuronal-scale models of fluctuation-driven dynamics, solely because the random network topology was replaced by a complex clustered one, whilst not changing the in-degree of any neurons. In this network, a small subset of cells whose inhibition comes only from outside their local cluster are the cause of bistable population dynamics, where different clusters of these cells irregularly switch back and forth from a sparsely firing state to a highly active state. Transitions to the highly active state occur when a cluster of these cells spikes sufficiently often to cause strong unbalanced positive feedback to each other. Transitions back to the sparsely firing state rely on occasional large fluctuations in the amount of non-local inhibition received. Neurons in the model are homogeneous in their intrinsic dynamics and in-degrees, but differ in the abundance of various directed feedback motifs in which they participate. Our findings suggest that (i) models and simulations should take into account complex structure that varies for neuron and synapse classes; (ii) differences in the dynamics of neurons with similar intrinsic properties may be caused by their membership in distinctive local networks; (iii) it is important to identify neurons that share physiological properties and location, but differ in their connectivity.     

To Burst or Not to Burst?

It is well known that some neurons tend to fire packets of action potentials followed by periods of quiescence (bursts) while others within the same stage of sensory processing fire in a tonic manner. However, the respective computational advantages of bursting and tonic neurons for encoding time varying signals largely remain a mystery. Weakly electric fish use cutaneous electroreceptors to convey information about sensory stimuli and it has been shown that some electroreceptors exhibit bursting dynamics while others do not. In this study, we compare the neural coding capabilities of tonically firing and bursting electroreceptor model neurons using information theoretic measures. We find that both bursting and tonically firing model neurons efficiently transmit information about the stimulus. However, the decoding mechanisms that must be used for each differ greatly: a non-linear decoder would be required to extract all the available information transmitted by the bursting model neuron whereas a linear one might suffice for the tonically firing model neuron. Further investigations using stimulus reconstruction techniques reveal that, unlike the tonically firing model neuron, the bursting model neuron does not encode the detailed time course of the stimulus. A novel measure of feature detection reveals that the bursting neuron signals certain stimulus features. Finally, we show that feature extraction and stimulus estimation are mutually exclusive computations occurring in bursting and tonically firing model neurons, respectively. Our results therefore suggest that stimulus estimation and feature extraction might be parallel computations in certain sensory systems rather than being sequential as has been previously proposed.     

Variable synaptic strengths controls the firing rate distribution in feedforward neural networks

Heterogeneity of firing rate statistics is known to have severe consequences on neural coding. Recent experimental recordings in weakly electric fish indicate that the distribution-width of superficial pyramidal cell firing rates (trial- and time-averaged) in the electrosensory lateral line lobe (ELL) depends on the stimulus, and also that network inputs can mediate changes in the firing rate distribution across the population. We previously developed theoretical methods to understand how two attributes (synaptic and intrinsic heterogeneity) interact and alter the firing rate distribution in a population of integrate-and-fire neurons with random recurrent coupling. Inspired by our experimental data, we extend these theoretical results to a delayed feedforward spiking network that qualitatively capture the changes of firing rate heterogeneity observed in in-vivo recordings. We demonstrate how heterogeneous neural attributes alter firing rate heterogeneity, accounting for the effect with various sensory stimuli. The model predicts how the strength of the effective network connectivity is related to intrinsic heterogeneity in such delayed feedforward networks: the strength of the feedforward input is positively correlated with excitability (threshold value for spiking) when firing rate heterogeneity is low and is negatively correlated with excitability with high firing rate heterogeneity. We also show how our theory can be used to predict effective neural architecture. We demonstrate that neural attributes do not interact in a simple manner but rather in a complex stimulus-dependent fashion to control neural heterogeneity and discuss how it can ultimately shape population codes.     

A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Analysis and an Application to Orientation Tuning

We explore a computationally efficient method of simulating realistic networks of neurons introduced by Knight, Manin, and Sirovich (1996) in which integrate-and-fire neurons are grouped into large populations of similar neurons. For each population, we form a probability density that represents the distribution of neurons over all possible states. The populations are coupled via stochastic synapses in which the conductance of a neuron is modulated according to the firing rates of its presynaptic populations. The evolution equation for each of these probability densities is a partial differential-integral equation, which we solve numerically. Results obtained for several example networks are tested against conventional computations for groups of individual neurons.We apply this approach to modeling orientation tuning in the visual cortex. Our population density model is based on the recurrent feedback model of a hypercolumn in cat visual cortex of Somers et al. (1995). We simulate the response to oriented flashed bars. As in the Somers model, a weak orientation bias provided by feed-forward lateral geniculate input is transformed by intracortical circuitry into sharper orientation tuning that is independent of stimulus contrast.The population density approach appears to be a viable method for simulating large neural networks. Its computational efficiency overcomes some of the restrictions imposed by computation time in individual neuron simulations, allowing one to build more complex networks and to explore parameter space more easily. The method produces smooth rate functions with one pass of the stimulus and does not require signal averaging. At the same time, this model captures the dynamics of single-neuron activity that are missed in simple firing-rate models.     

Spatial firing patterns of auditory neuron network modelling by computer simulation

This communication examines, in digital computer simulated network, input signals and response patterns established at excitatory neurons' level i.e. the membrane potential of neuron soma. It is restricted to spatial patterns of the auditory neuron networks and time factor for nervous conduction and transmission is neglected compared with long maintained membrane potentials of neuron somas. The model analyzes the change in the spatial patterns of the membrane potential in the two dimensional networks of the auditory system. In order to evaluate the contribution of the various parameters, it is started that the simplest model has only one parameter, lateral inhibition. The other parameters are then added, one at a time, to successive models. The lateral inhibition is a necessary condition in the auditory nervous system if any sharpening of the response areas in the single neurons is to occur. A necessary condition for the validity of the model is that it should be applicable to the other senses such as vision and chemical patterns, taste. The threshold feature of auditory neurons aids in producing a sharpening in the neuron of the auditory relay nuclei. It does this clipping the spatial response patterns in one dimensional arrays of excitatory neurons. Recurrent inhibition seems a necessary condition in the sensory nervous system that any kinds of input signals are to be preserved over a wide range of stimulus intensity. In other words, this network has a wide dynamic range against any kinds of input signals. A simple self-recurrent negative feedback does not contribute to the sharpening, but more complex socalled averaged type does. A neuron network is capable of responding stably to stimuli with a wide range of intensity and with any kind of spatial patterns if there is a simple negative feedback mechanism. When there is no negative feedback, input signals soon disappear or saturate in the neuron network. Therefore, recurrent inhibition is the most important mechanism. Spontaneous activity appears to aid in the sharpening by providing a kind of contrast, that is by reducting the amount of activity in neurons adjacent to the excitatory area. Moreover, the effect of spontaneous activity in the model seems to make repples around the excitatory area and suggests that an introduction of activity at any stage of the networks, from whatever source for example reticulum formation and thalamus, might appreciably alter the response patterns at subsequent neuron network. This suggests that the mechanism of the consciousness that might be controlled by the thalamus and or reticular formation. These two dimensional neuron networks may be expanded to three dimensional neuron networks. The former might simulate the auditory nervous system while the latter might simulate the visual system.     

Decorrelation of Neural-Network Activity by Inhibitory Feedback

Correlations in spike-train ensembles can seriously impair the encoding of information by their spatio-temporal structure. An inevitable source of correlation in finite neural networks is common presynaptic input to pairs of neurons. Recent studies demonstrate that spike correlations in recurrent neural networks are considerably smaller than expected based on the amount of shared presynaptic input. Here, we explain this observation by means of a linear network model and simulations of networks of leaky integrate-and-fire neurons. We show that inhibitory feedback efficiently suppresses pairwise correlations and, hence, population-rate fluctuations, thereby assigning inhibitory neurons the new role of active decorrelation. We quantify this decorrelation by comparing the responses of the intact recurrent network (feedback system) and systems where the statistics of the feedback channel is perturbed (feedforward system). Manipulations of the feedback statistics can lead to a significant increase in the power and coherence of the population response. In particular, neglecting correlations within the ensemble of feedback channels or between the external stimulus and the feedback amplifies population-rate fluctuations by orders of magnitude. The fluctuation suppression in homogeneous inhibitory networks is explained by a negative feedback loop in the one-dimensional dynamics of the compound activity. Similarly, a change of coordinates exposes an effective negative feedback loop in the compound dynamics of stable excitatory-inhibitory networks. The suppression of input correlations in finite networks is explained by the population averaged correlations in the linear network model: In purely inhibitory networks, shared-input correlations are canceled by negative spike-train correlations. In excitatory-inhibitory networks, spike-train correlations are typically positive. Here, the suppression of input correlations is not a result of the mere existence of correlations between excitatory (E) and inhibitory (I) neurons, but a consequence of a particular structure of correlations among the three possible pairings (EE, EI, II).     

Role of type-specific neuron properties in a spinal cord motor network

Recent recordings from spinal neurons in hatchling frog tadpoles allow their type-specific properties to be defined. Seven main types of neuron involved in the control of swimming have been characterized. To investigate the significance of type-specific properties, we build models of each neuron type and assemble them into a network using known connectivity between: sensory neurons, sensory pathway interneurons, central pattern generator (CPG) interneurons and motoneurons. A single stimulus to a sensory neuron initiates swimming where modelled neuronal and network activity parallels physiological activity. Substitution of firing properties between neuron types shows that those of excitatory CPG interneurons are critical for stable swimming. We suggest that type-specific neuronal properties can reflect the requirements for involvement in one particular network response (like swimming), but may also reflect the need to participate in more than one response (like swimming and slower struggling). Action Editor: Eberhard E. Fetz     

Distribution of Orientation Selectivity in Recurrent Networks of Spiking Neurons with Different Random Topologies

Neurons in the primary visual cortex are more or less selective for the orientation of a light bar used for stimulation. A broad distribution of individual grades of orientation selectivity has in fact been reported in all species. A possible reason for emergence of broad distributions is the recurrent network within which the stimulus is being processed. Here we compute the distribution of orientation selectivity in randomly connected model networks that are equipped with different spatial patterns of connectivity. We show that, for a wide variety of connectivity patterns, a linear theory based on firing rates accurately approximates the outcome of direct numerical simulations of networks of spiking neurons. Distance dependent connectivity in networks with a more biologically realistic structure does not compromise our linear analysis, as long as the linearized dynamics, and hence the uniform asynchronous irregular activity state, remain stable. We conclude that linear mechanisms of stimulus processing are indeed responsible for the emergence of orientation selectivity and its distribution in recurrent networks with functionally heterogeneous synaptic connectivity.     

Traveling Waves and the Processing of Weakly Tuned Inputs in a Cortical Network Module

Recent studies have shown that local cortical feedback can havean important effect on the response of neurons in primary visualcortex to the orientation of visual stimuli. In this work, westudy the role of the cortical feedback in shaping thespatiotemporal patterns of activity in cortex. Two questionsare addressed: one, what are the limitations on the ability ofcortical neurons to lock their activity to rotatingoriented stimuli within a single receptive field? Two, can thelocal architecture of visual cortex lead to the generation ofspontaneous traveling pulses of activity? We study theseissues analytically by a population-dynamic model of ahypercolumn in visual cortex. The order parameter thatdescribes the macroscopic behavior of the network is thetime-dependent population vector of the network. We firststudy the network dynamics under the influence of a weakly tunedinput that slowly rotates within the receptive field. We showthat if the cortical interactions have strong spatialmodulation, the network generates a sharply tuned activityprofile that propagates across the hypercolumn in a path thatis completely locked to the stimulus rotation. The resultantrotating population vector maintains a constant angular lagrelative to the stimulus, the magnitude of which grows with thestimulus rotation frequency. Beyond a critical frequency thepopulation vector does not lock to the stimulus but executes aquasi-periodic motion with an average frequency that is smallerthan that of the stimulus. In the second part we consider thestable intrinsic state of the cortex under the influence of isotropic stimulation. We show that if the local inhibitoryfeedback is sufficiently strong, the network does not settleinto a stationary state but develops spontaneous travelingpulses of activity. Unlike recent models of wave propagation incortical networks, the connectivity pattern in our model isspatially symmetric, hence the direction of propagation ofthese waves is arbitrary. The interaction of these waves withan external-oriented stimulus is studied. It is shown that thesystem can lock to a weakly tuned rotating stimulus if thestimulus frequency is close to the frequency of the intrinsic wave.     

Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

The regulation of biological networks relies significantly on convergent feedback signaling loops that render a global output locally accessible. Ideally, the recurrent connectivity within these systems is self-organized by a time-dependent phase-locking mechanism. This study analyzes recurrent fractal neural networks (RFNNs), which utilize a self-similar or fractal branching structure of dendrites and downstream networks for phase-locking of reciprocal feedback loops: output from outer branch nodes of the network tree enters inner branch nodes of the dendritic tree in single neurons. This structural organization enables RFNNs to amplify re-entrant input by over-the-threshold signal summation from feedback loops with equivalent signal traveling times. The columnar organization of pyramidal neurons in the neocortical layers V and III is discussed as the structural substrate for this network architecture. RFNNs self-organize spike trains and render the entire neural network output accessible to the dendritic tree of each neuron within this network. As the result of a contraction mapping operation, the local dendritic input pattern contains a downscaled version of the network output coding structure. RFNNs perform robust, fractal data compression, thus coping with a limited number of feedback loops for signal transport in convergent neural networks. This property is discussed as a significant step toward the solution of a fundamental problem in neuroscience: how is neuronal computation in separate neurons and remote brain areas unified as an instance of experience in consciousness? RFNNs are promising candidates for engaging neural networks into a coherent activity and provide a strategy for the exchange of global and local information processing in the human brain, thereby ensuring the completeness of a transformation from neuronal computation into conscious experience.     

Spike-based population coding and working memory

Compelling behavioral evidence suggests that humans can make optimal decisions despite the uncertainty inherent in perceptual or motor tasks. A key question in neuroscience is how populations of spiking neurons can implement such probabilistic computations. In this article, we develop a comprehensive framework for optimal, spike-based sensory integration and working memory in a dynamic environment. We propose that probability distributions are inferred spike-per-spike in recurrently connected networks of integrate-and-fire neurons. As a result, these networks can combine sensory cues optimally, track the state of a time-varying stimulus and memorize accumulated evidence over periods much longer than the time constant of single neurons. Importantly, we propose that population responses and persistent working memory states represent entire probability distributions and not only single stimulus values. These memories are reflected by sustained, asynchronous patterns of activity which make relevant information available to downstream neurons within their short time window of integration. Model neurons act as predictive encoders, only firing spikes which account for new information that has not yet been signaled. Thus, spike times signal deterministically a prediction error, contrary to rate codes in which spike times are considered to be random samples of an underlying firing rate. As a consequence of this coding scheme, a multitude of spike patterns can reliably encode the same information. This results in weakly correlated, Poisson-like spike trains that are sensitive to initial conditions but robust to even high levels of external neural noise. This spike train variability reproduces the one observed in cortical sensory spike trains, but cannot be equated to noise. On the contrary, it is a consequence of optimal spike-based inference. In contrast, we show that rate-based models perform poorly when implemented with stochastically spiking neurons.     

A network of spiking neurons that can represent interval timing: mean field analysis

Despite the vital importance of our ability to accurately process and encode temporal information, the underlying neural mechanisms are largely unknown. We have previously described a theoretical framework that explains how temporal representations, similar to those reported in the visual cortex, can form in locally recurrent cortical networks as a function of reward modulated synaptic plasticity. This framework allows networks of both linear and spiking neurons to learn the temporal interval between a stimulus and paired reward signal presented during training. Here we use a mean field approach to analyze the dynamics of non-linear stochastic spiking neurons in a network trained to encode specific time intervals. This analysis explains how recurrent excitatory feedback allows a network structure to encode temporal representations.     

Inhibitory Contribution to Suprathreshold Corticostriatal Responses: An Experimental and Modeling Study

Neostriatal neurons may undergo events of spontaneous synchronization as those observed in recurrent networks of excitatory neurons, even when cortical afferents are transected. It is necessary to explain these events because the neostriatum is a recurrent network of inhibitory neurons. Synchronization of neuronal activity may be caused by plateau-like depolarizations. Plateau-like orthodromic depolarizations that resemble up-states in medium spiny neostriatal neurons (MSNs) may be induced by a single corticostriatal suprathreshold stimulus. Slow synaptic depolarizations may last hundreds of milliseconds, decay slower than the monosynaptic glutamatergic synaptic potentials that induce them, and sustain repetitive firing. Because inhibitory inputs impinging onto MSNs have a reversal potential above the resting membrane potential but below the threshold for firing, they conform a type of “shunting inhibition”. This work asks if shunting GABAergic inputs onto MSNs arrive asynchronously enough as to help in sustaining the plateau-like corticostriatal response after a single cortical stimulus. This may help to begin explaining autonomous processing in the striatal micro-circuitry in the presence of a tonic excitatory drive and independently of spatio-temporally organized inputs. It is shown here that besides synaptic currents from AMPA/KA- and NMDA-receptors, as well as L-type intrinsic Ca²⁺- currents, inhibitory synapses help in maintaining the slow depolarization, although they accomplish the role of depressing firing at the beginning of the response. We then used a NEURON model of spiny cells to show that inhibitory synapses arriving asynchronously on the dendrites can help to simulate a plateau potential similar to that observed experimentally. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. II. Input selectivity—symmetry breaking

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network.     

A Model of Visuospatial Working Memory in Prefrontal Cortex: Recurrent Network and Cellular Bistability

We report a computer simulation of the visuospatial delayed-response experiments of Funahashi et al. (1989), using a firing-rate model that combines intrinsic cellular bistability with the recurrent local network architecture of the neocortex. In our model, the visuospatial working memory is stored in the form of a continuum of network activity profiles that coexist with a spontaneous activity state. These neuronal firing patterns provide a population code for the cue position in a graded manner. We show that neuronal persistent activity and tuning curves of delay-period activity (memory fields) can be generated by an excitatory feedback circuit and recurrent synaptic inhibition. However, if the memory fields are constructed solely by network mechanisms, noise may induce a random drift over time in the encoded cue position, so that the working memory storage becomes unreliable. Furthermore, a distraction stimulus presented during the delay period produces a systematic shift in the encoded cue position. We found that the working memory performance can be rendered robust against noise and distraction stimuli if single neurons are endowed with cellular bistability (presumably due to intrinsic ion channel mechanisms) that is conditional and realized only with sustained synaptic inputs from the recurrent network. We discuss how cellular bistability at the single cell level may be detected by analysis of spike trains recorded during delay-period activity and how local modulation of intrinsic cell properties and/or synaptic transmission can alter the memory fields of individual neurons in the prefrontal cortex.     

Event-based minimum-time control of oscillatory neuron models

We present an event-based feedback control method for randomizing the asymptotic phase of oscillatory neurons. Phase randomization is achieved by driving the neuron’s state to its phaseless set, a point at which its phase is undefined and is extremely sensitive to background noise. We consider the biologically relevant case of a fixed magnitude constraint on the stimulus signal, and show how the control objective can be accomplished in minimum time. The control synthesis problem is addressed using the minimum-time-optimal Hamilton–Jacobi–Bellman framework, which is quite general and can be applied to any spiking neuron model in the conductance-based Hodgkin–Huxley formalism. We also use this methodology to compute a feedback control protocol for optimal spike rate increase. This framework provides a straightforward means of visualizing isochrons, without actually calculating them in the traditional way. Finally, we present an extension of the phase randomizing control scheme that is applied at the population level, to a network of globally coupled neurons that are firing in synchrony. The applied control signal desynchronizes the population in a demand-controlled way.     

Noise in attractor networks in the brain produced by graded firing rate representations

Representations in the cortex are often distributed with graded firing rates in the neuronal populations. The firing rate probability distribution of each neuron to a set of stimuli is often exponential or gamma. In processes in the brain, such as decision-making, that are influenced by the noise produced by the close to random spike timings of each neuron for a given mean rate, the noise with this graded type of representation may be larger than with the binary firing rate distribution that is usually investigated. In integrate-and-fire simulations of an attractor decision-making network, we show that the noise is indeed greater for a given sparseness of the representation for graded, exponential, than for binary firing rate distributions. The greater noise was measured by faster escaping times from the spontaneous firing rate state when the decision cues are applied, and this corresponds to faster decision or reaction times. The greater noise was also evident as less stability of the spontaneous firing state before the decision cues are applied. The implication is that spiking-related noise will continue to be a factor that influences processes such as decision-making, signal detection, short-term memory, and memory recall even with the quite large networks found in the cerebral cortex. In these networks there are several thousand recurrent collateral synapses onto each neuron. The greater noise with graded firing rate distributions has the advantage that it can increase the speed of operation of cortical circuitry.     

Calcium Involvement in Regulation of Neuronal Bursting in Disinhibited Neuronal Networks: Insights from Calcium Studies in a Spherical Cell Model

Cytosolic calcium is involved in the regulation of many intracellular processes. Intracellular calcium may therefore potentially affect the behavior of both single neurons and synaptically connected neuronal assemblies. In computer model studies, we investigated calcium dynamics in spherical neurons during periods of recurrent neuronal bursting that were simulated in a disinhibited neuronal network. The model takes into account calcium influx via voltage-gated calcium channels, extrusion through the cell membrane, and binding to two different buffers representing fixed and mobile endogenous calcium buffers. Throughout the duration of the simulated recurrent neuronal bursting, the concentration of free fixed buffers shows a hyperbolic decrease in time at a rate that is not uniform inside a neuron. Recurrent calcium influxes associated with bursting lead to the formation of gradients in the concentration of the fixed buffer in the radial direction, and are accompanied by the redistribution of mobile buffers acting to compensate for these gradients. Simulated intracellular calcium transients have a slow component characterized by a gradual increase in the calcium baseline level that reaches a plateau 120-200 s after the onset of recurrent bursting. Using this model, we demonstrate what we believe is a novel mechanism of regulation of network excitability that occurs in conditions of prolonged and recurrent neuronal bursting in disinhibited networks. This mechanism is expressed via interaction of calcium clearance systems inside neurons with calcium-dependent potassium regulation of neuronal excitability in membranes. This is a network phenomenon because it arises largely by synaptic interactions. Therefore, it can serve as a network safety mechanism to prevent excessive and uncontrolled neuronal firing resulting from the lack of inhibition or after acute suppression of the inhibitory drive.