Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism

Generalization of the event-based Carnevale-Hines integration scheme for integrate-and-fire models

An event-based integration scheme for an integrate-and-fire neuron model with exponentially decaying excitatory synaptic currents and double exponential inhibitory synaptic currents has been introduced by Carnevale and Hines. However, the integration scheme imposes nonphysiological constraints on the time constants of the synaptic currents, which hamper its general applicability. This letter addresses this problem in two ways. First, we provide physical arguments demonstrating why these constraints on the time constants can be relaxed. Second, we give a formal proof showing which constraints can be abolished. As part of our formal proof, we introduce the generalized Carnevale-Hines lemma, a new tool for comparing double exponentials as they naturally occur in many cascaded decay systems, including receptor-neurotransmitter dissociation followed by channel closing. Through repeated application of the generalized lemma, we lift most of the original constraints on the time constants. Thus, we show that the Carnevale-Hines integration scheme for the integrate-and-fire model can be employed for simulating a much wider range of neuron and synapse types than was previously thought.

Analysis of intracerebral EEG recordings of epileptic spikes: insights from a neural network model

The pathophysiological interpretation of EEG signals recorded with depth electrodes [i.e., local field potentials (LFPs)] during interictal (between seizures) or ictal (during seizures) periods is fundamental in the presurgical evaluation of patients with drug-resistant epilepsy. Our objective was to explain specific shape features of interictal spikes in the hippocampus (observed in LFPs) in terms of cell- and network-related parameters of neuronal circuits that generate these events. We developed a neural network model based on "minimal" but biologically relevant neuron models interconnected through GABAergic and glutamatergic synapses that reproduce the main physiological features of the CA1 subfield. Simulated LFPs were obtained by solving the forward problem (dipole theory) from networks including a large number ( approximately 3000) of cells. Insertion of appropriate parameters allowed the model to simulate events that closely resemble actual epileptic spikes. Moreover, the shape of the early fast component ("spike'') and the late slow component ("negative wave'') was linked to the relative contribution of glutamatergic and GABAergic synaptic currents in pyramidal cells. In addition, the model provides insights about the sensitivity of electrode localization with respect to recorded tissue volume and about the relationship between the LFP and the intracellular activity of principal cells and interneurons represented in the network.

Generating oscillatory bursts from a network of regular spiking neurons without inhibition

Avian nucleus isthmi pars parvocellularis (Ipc) neurons are reciprocally connected with the layer 10 (L10) neurons in the optic tectum and respond with oscillatory bursts to visual stimulation. Our in vitro experiments show that both neuron types respond with regular spiking to somatic current injection and that the feedforward and feedback synaptic connections are excitatory, but of different strength and time course. To elucidate mechanisms of oscillatory bursting in this network of regularly spiking neurons, we investigated an experimentally constrained model of coupled leaky integrate-and-fire neurons with spike-rate adaptation. The model reproduces the observed Ipc oscillatory bursting in response to simulated visual stimulation. A scan through the model parameter volume reveals that Ipc oscillatory burst generation can be caused by strong and brief feedforward synaptic conductance changes. The mechanism is sensitive to the parameter values of spike-rate adaptation. In conclusion, we show that a network of regular-spiking neurons with feedforward excitation and spike-rate adaptation can generate oscillatory bursting in response to a constant input.

Brain stem feedback in a computational model of birdsong sequencing

Uncovering the roles of neural feedback in the brain is an active area of experimental research. In songbirds, the telencephalic premotor nucleus HVC receives neural feedback from both forebrain and brain stem areas. Here we present a computational model of birdsong sequencing that incorporates HVC and associated nuclei and builds on the model of sparse bursting presented in our preceding companion paper. Our model embodies the hypotheses that 1) different networks in HVC control different syllables or notes of birdsong, 2) interneurons in HVC not only participate in sparse bursting but also provide mutual inhibition between networks controlling syllables or notes, and 3) these syllable networks are sequentially excited by neural feedback via the brain stem and the afferent thalamic nucleus Uva, or a similar feedback pathway. We discuss the model's ability to unify physiological, behavioral, and lesion results and we use it to make novel predictions that can be tested experimentally. The model suggests a neural basis for sequence variations, shows that stimulation in the feedback pathway may have different effects depending on the balance of excitation and inhibition at the input to HVC from Uva, and predicts deviations from uniform expansion of syllables and gaps during HVC cooling.

Markov models for ion channels: versatility versus identifiability and speed

Markov models (MMs) represent a generalization of Hodgkin-Huxley models. They provide a versatile structure for modelling single channel data, gating currents, state-dependent drug interaction data, exchanger and pump dynamics, etc. This paper uses examples from cardiac electrophysiology to discuss aspects related to parameter estimation. (i) Parameter unidentifiability (found in 9 out of 13 of the considered models) results in an inability to determine the correct layout of a model, contradicting the idea that model structure and parameters provide insights into underlying molecular processes. (ii) The information content of experimental voltage step clamp data is discussed, and a short but sufficient protocol for parameter estimation is presented. (iii) MMs have been associated with high computational cost (owing to their large number of state variables), presenting an obstacle for multicellular whole organ simulations as well as parameter estimation. It is shown that the stiffness of models increases computation time more than the number of states. (iv) Algorithms and software programs are provided for steady-state analysis, analytical solutions for voltage steps and numerical derivation of parameter identifiability. The results provide a new standard for ion channel modelling to further the automation of model development, the validation process and the predictive power of these models.

Inhibition and recurrent excitation in a computational model of sparse bursting in song nucleus HVC

The telencephalic premotor nucleus HVC is situated at a critical point in the pattern-generating premotor circuitry of oscine songbirds. A striking feature of HVC's premotor activity is that its projection neurons burst extremely sparsely. Here we present a computational model of HVC embodying several central hypotheses: 1) sparse bursting is generated in bistable groups of recurrently connected robust nucleus of the arcopallium (RA)-projecting (HVCRA) neurons; 2) inhibitory interneurons terminate bursts in the HVCRA groups; and 3) sparse sequences of bursts are generated by the propagation of waves of bursting activity along networks of HVCRA neurons. Our model of sparse bursting places HVC in the context of central pattern generators and cortical networks using inhibition, recurrent excitation, and bistability. Importantly, the unintuitive result that inhibitory interneurons can precisely terminate the bursts of HVCRA groups while showing relatively sustained activity throughout the song is made possible by a specific constraint on their connectivity. We use the model to make novel predictions that can be tested experimentally.

Comparison of time-frequency responses and the event-related potential to auditory speech stimuli in human cortex

We recorded the electrocorticogram directly from the exposed cortical surface of awake neurosurgical patients during the presentation of auditory syllable stimuli. All patients were unanesthetized as part of a language-mapping procedure for subsequent left-hemisphere tumor resection. Time-frequency analyses showed significant high-gamma (gammahigh: 70-160 Hz) responses from the left superior temporal gyrus, but no reliable response from the left inferior frontal gyrus. Alpha suppression (alpha: 7-14 Hz) and event-related potential responses exhibited a more widespread topography. Across electrodes, the alpha suppression from 200 to 450 ms correlated with the preceding (50-200 ms) gammahigh increase. The results are discussed in terms of the different physiological origins of these electrocortical signals.

Dendritic encoding of sensory stimuli controlled by deep cortical interneurons

The computational power of single neurons is greatly enhanced by active dendritic conductances that have a large influence on their spike activity. In cortical output neurons such as the large pyramidal cells of layer 5 (L5), activation of apical dendritic calcium channels leads to plateau potentials that increase the gain of the input/output function and switch the cell to burst-firing mode. The apical dendrites are innervated by local excitatory and inhibitory inputs as well as thalamic and corticocortical projections, which makes it a formidable task to predict how these inputs influence active dendritic properties in vivo. Here we investigate activity in populations of L5 pyramidal dendrites of the somatosensory cortex in awake and anaesthetized rats following sensory stimulation using a new fibre-optic method for recording dendritic calcium changes. We show that the strength of sensory stimulation is encoded in the combined dendritic calcium response of a local population of L5 pyramidal cells in a graded manner. The slope of the stimulus-response function was under the control of a particular subset of inhibitory neurons activated by synaptic inputs predominantly in L5. Recordings from single apical tuft dendrites in vitro showed that activity in L5 pyramidal neurons disynaptically coupled via interneurons directly blocks the initiation of dendritic calcium spikes in neighbouring pyramidal neurons. The results constitute a functional description of a cortical microcircuit in awake animals that relies on the active properties of L5 pyramidal dendrites and their very high sensitivity to inhibition. The microcircuit is organized so that local populations of apical dendrites can adaptively encode bottom-up sensory stimuli linearly across their full dynamic range.

Spiking neural network simulation: numerical integration with the Parker-Sochacki method

Mathematical neuronal models are normally expressed using differential equations. The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models. Using this method, the solution order can be adapted according to the local conditions at each time step, enabling adaptive error control without changing the integration timestep. The method has been limited to polynomial equations, but we present division and power operations that expand its scope. We apply the Parker-Sochacki method to the Izhikevich ‘simple’ model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods. Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

Neuropathic pain memory is maintained by Rac1-regulated dendritic spine remodeling after spinal cord injury

Localized increases in synaptic strength constitute a synaptic basis for learning and memory in the CNS and may also contribute to the maintenance of neuropathic pain after spinal cord injury (SCI) through the de novo formation or elaboration of postsynaptic dendritic structures. To determine whether SCI-induced dendritic spine remodeling contributes to neuronal hyperexcitability and neuropathic pain, we analyzed spine morphometry, localization, and functional influence in dorsal horn (DH) neurons in adult rats 1 month after sham surgery, contusion SCI, and SCI treated with a selective inhibitor of Rac1 activation, NSC23766. After SCI, DH neurons located in lamina IV-V exhibited increased spine density, redistributed spines, and mature spines compared with control neurons, which was associated with enhancement of EPSCs in computer simulations and hyperexcitable responsiveness to innocuous and noxious peripheral stimuli in unit recordings in vivo. SCI animals also exhibited symptoms of tactile allodynia and thermal hyperalgesia. Inhibition of the small GTP-binding protein Rac1 ameliorated post-SCI changes in spine morphology, attenuated injury-induced hyperexcitability of wide-dynamic range neurons, and progressively increased pain thresholds over a 3 d period. This suggests that Rac1 is an important intracellular signaling molecule involved in a spinal dendritic spine pathology associated with chronic neuropathic pain after SCI. Our report provides robust evidence for a novel conceptual bridge between learning and memory on the one hand, and neuropathic pain on the other.

Evaluating statistical methods used to estimate the number of postsynaptic receptors

Calcium levels in spines play a significant role in determining the sign and magnitude of synaptic plasticity. The magnitude of calcium influx into spines is highly dependent on influx through N-methyl D-aspartate (NMDA) receptors, and therefore depends on the number of postsynaptic NMDA receptors in each spine. We have calculated previously how the number of postsynaptic NMDA receptors determines the mean and variance of calcium transients in the postsynaptic density, and how this alters the shape of plasticity curves. However, the number of postsynaptic NMDA receptors in the postsynaptic density is not well known. Anatomical methods for estimating the number of NMDA receptors produce estimates that are very different than those produced by physiological techniques. The physiological techniques are based on the statistics of synaptic transmission and it is difficult to experimentally estimate their precision. In this paper we use stochastic simulations in order to test the validity of a physiological estimation technique based on failure analysis. We find that the method is likely to underestimate the number of postsynaptic NMDA receptors, explain the source of the error, and re-derive a more precise estimation technique. We also show that the original failure analysis as well as our improved formulas are not robust to small estimation errors in key parameters.

A kinetic model unifying presynaptic short-term facilitation and depression

Short-term facilitation and depression refer to the increase and decrease of synaptic strength under repetitive stimuli within a timescale of milliseconds to seconds. This phenomenon has been attributed to primarily presynaptic mechanisms such as calcium-dependent transmitter release and presynaptic vesicle depletion. Previous modeling studies that aimed to integrate the complex short-term facilitation and short-term depression data derived from varying synapses have relied on computer simulation or abstract mathematical approaches. Here, we propose a unified theory of synaptic short-term plasticity based on realistic yet tractable and testable model descriptions of the underlying intracellular biochemical processes. Analysis of the model equations leads to a closed-form solution of the resonance frequency, a function of several critical biophysical parameters, as the single key indicator of the propensity for synaptic facilitation or depression under repetitive stimuli. This integrative model is supported by a broad range of transient and frequency response experimental data including those from facilitating, depressing or mixed-mode synapses. Specifically, the theory predicts that high calcium initial concentration and large gain of calcium action result in low resonance frequency and hence depressing behavior. In contrast, for synapses that are less sensitive to calcium or have higher recovery rate, resonance frequency becomes higher and thus facilitation prevails. The notion of resonance frequency therefore allows valuable quantitative parametric assessment of the contributions of various presynaptic mechanisms to the directionality of synaptic short-term plasticity. Thus, the model provides the reasons behind the switching behavior between facilitation and depression observed in experiments. New experiments are also suggested to control the short-term synaptic signal processing through adjusting the resonance frequency and bandwidth.

A quantitative model of cortical spreading depression due to purinergic and gap-junction transmission in astrocyte networks

Spreading depression (SD), a propagating wave of electrical silence in the cortex and archicortex, involves depolarization of neurons and astrocytes for approximately 1 min, due principally to a large increase in extracellular K+. SD is accompanied by large increases in extracellular ATP and is blocked by glutamate N-methyl-D-aspartate receptor antagonists. As a principal means of transmission between astrocytes is through their release of ATP, we have investigated if a model in which SD is driven by the effects of astrocyte waves of ATP interacting with waves of glutamate release from neurons and astrocytes can give a quantitative account of experimental observations on SD. We show that the characteristics of SD and the accompanying extracellular ionic changes can be accommodated by such a model-whether astrocyte transmission is principally through the release of ATP, as in archicortex (hippocampus) and spinal cord, or via gap junctions, as in the neocortex. Furthermore, these models give quantitative accounts of the effects on the characteristics of SD of agents toxic for astrocytes and of gap-junction blockers. Finally, an additional series of critical tests of the model is suggested.

Variety of synchronous regimes in neuronal ensembles

We consider a Hodgkin-Huxley-type model of oscillatory activity in neurons of the snail Helix pomatia. This model has a distinctive feature: It demonstrates multistability in oscillatory and silent modes that is typical for the thalamocortical neurons. A single neuron cell can demonstrate a variety of oscillatory activity: Regular and chaotic spiking and bursting behavior. We study collective phenomena in small and large arrays of nonidentical cells coupled by models of electrical and chemical synapses. Two single elements coupled by electrical coupling show different types of synchronous behavior, in particular in-phase and antiphase synchronous regimes. In an ensemble of three inhibitory synaptically coupled elements, the phenomenon of sequential synchronous dynamics is observed. We study the synchronization phenomena in the chain of nonidentical neurons at different oscillatory behavior coupled with electrical and chemical synapses. Various regimes of phase synchronization are observed: (i) Synchronous regular and chaotic spiking; (ii) synchronous regular and chaotic bursting; and (iii) synchronous regular and chaotic bursting with different numbers of spikes inside the bursts. We detect and study the effect of collective synchronous burst generation due to the cluster formation and the oscillatory death.

Focal generation of paroxysmal fast runs during electrographic seizures

PURPOSE

A cortically generated Lennox-Gastaut type seizure is associated with spike-wave/poly-spike-wave discharges at 1.0-2.5 Hz and fast runs at 7-16 Hz. Here we studied the patterns of synchronization during runs of paroxysmal fast spikes.

METHODS

Electrographic activities were recorded using multisite intracellular and field potential recordings in vivo from cats anesthetized with ketamine-xylazine. In different experiments, the recording electrodes were located either at short distances (<1 mm) or at longer distances (up to 12 mm). The main experimental findings were tested in computational models.

RESULTS

In the majority of cases, the onset and the offset of fast runs occurred almost simultaneously in different recording sites. The amplitude and duration of fast runs could vary by orders of magnitude. Within the fast runs, the patterns of synchronization recorded in different electrodes were as following: (1) synchronous, in phase, (2) synchronous, with phase shift, (3) patchy, repeated in phase/phase shift transitions, and (4) nonsynchronous, slightly different frequencies in different recording sites or absence of oscillatory activity in one of the recording sites; the synchronous patterns (in phase or with phase shifts) were most common. All these patterns could be recorded in the same pair of electrodes during different seizures, and they were reproduced in a computational network model. Intrinsically bursting (IB) neurons fired more spikes per cycle than any other neurons suggesting their leading role in the fast run generation.

CONCLUSIONS

Once started, the fast runs are generated locally with variable correlations between neighboring cortical foci.

Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier

A computational model is presented for the simulation of three-dimensional electrodiffusion of ions. Finite volume techniques were used to solve the Poisson-Nernst-Planck equation, and a dual Delaunay-Voronoi mesh was constructed to evaluate fluxes of ions, as well as resulting electric potentials. The algorithm has been validated and applied to a generalized node of Ranvier, where numerical results for computed action potentials agree well with cable model predictions for large clusters of voltage-gated ion channels. At smaller channel clusters, however, the three-dimensional electrodiffusion predictions diverge from the cable model predictions and show a broadening of the action potential, indicating a significant effect due to each channel's own local electric field. The node of Ranvier complex is an elaborate organization of membrane-bound aqueous compartments, and the model presented here represents what we believe is a significant first step in simulating electrophysiological events with combined realistic structural and physiological data.

A geometric comparison of single chain multi-state models of ion channel gating

Multi-state models of ion channel gating have been used extensively, but choosing optimally small yet sufficiently complex models to describe particular experimental data remains a difficult task. In order to provide some insight into appropriate model selection, this paper presents some basic results about the behavior of solutions of multi-state models, particularly those arranged in a chain formation. Some properties of the eigenvalues and eigenvectors of constant-rate multi-state models are presented. A geometric description of a three-state chain is given and, in particular, differences between a chain equivalent to an Hodgkin-Huxley model and a chain with identical rates are analyzed. One distinguishing feature between these two types of systems is that decay from the open state in the Hodgkin-Huxley model is dominated by the most negative eigenvalue while the identical rate chain displays a mix of modes over all eigenvalues.

Stochastic models for the in silico simulation of synaptic processes

Background

Research in life sciences is benefiting from a large availability of formal description techniques and analysis methodologies. These allow both the phenomena investigated to be precisely modeled and virtual experiments to be performed in silico. Such experiments may result in easier, faster, and satisfying approximations of their in vitro/vivo counterparts. A promising approach is represented by the study of biological phenomena as a collection of interactive entities through process calculi equipped with stochastic semantics. These exploit formal grounds developed in the theory of concurrency in computer science, account for the not continuous, nor discrete, nature of many phenomena, enjoy nice compositional properties and allow for simulations that have been demonstrated to be coherent with data in literature.

Results

Motivated by the need to address some aspects of the functioning of neural synapses, we have developed one such model for synaptic processes in the calyx of Held, which is a glutamatergic synapse in the auditory pathway of the mammalia. We have developed such a stochastic model starting from existing kinetic models based on ODEs of some sub-components of the synapse, integrating other data from literature and making some assumptions about non-fully understood processes. Experiments have confirmed the coherence of our model with known biological data, also validating the assumptions made. Our model overcomes some limitations of the kinetic ones and, to our knowledge, represents the first model of synaptic processes based on process calculi. The compositionality of the approach has permitted us to independently focus on tuning the models of the pre- and post- synaptic traits, and then to naturally connect them, by dealing with “interface” issues. Furthermore, we have improved the expressiveness of the model, e.g. by embedding easy control of element concentration time courses. Sensitivity analysis over several parameters of the model has provided results that may help clarify the dynamics of synaptic transmission, while experiments with the model of the complete synapse seem worth explaining short-term plasticity mechanisms.

Conclusions

Specific presynaptic and postsynaptic mechanisms can be further analysed under various conditions, for instance by studying the presynaptic behaviour under repeated activations. The level of details of the description can be refined, for instance by further specifying the neurotransmitter generation and release steps. Taking advantage of the compositionality of the approach, an enhanced model could then be composed with other neural models, designed within the same framework, in order to obtain a more detailed and comprehensive model. In the long term, we are interested, in particular, in addressing models of synaptic plasticity, i.e. activity dependent mechanisms, which are the bases of memory and learning processes.

More on the computer science side, we plan to follow some directions to improve the underlying computational model and the linguistic primitives it provides as suggested by the experiments carried out, e.g. by introducing a suitable notion of (spatial) locality.

Stochastic models for the in silico simulation of synaptic processes

BACKGROUND

Research in life sciences is benefiting from a large availability of formal description techniques and analysis methodologies. These allow both the phenomena investigated to be precisely modeled and virtual experiments to be performed in silico. Such experiments may result in easier, faster, and satisfying approximations of their in vitro/vivo counterparts. A promising approach is represented by the study of biological phenomena as a collection of interactive entities through process calculi equipped with stochastic semantics. These exploit formal grounds developed in the theory of concurrency in computer science, account for the not continuous, nor discrete, nature of many phenomena, enjoy nice compositional properties and allow for simulations that have been demonstrated to be coherent with data in literature.

RESULTS

Motivated by the need to address some aspects of the functioning of neural synapses, we have developed one such model for synaptic processes in the calyx of Held, which is a glutamatergic synapse in the auditory pathway of the mammalia. We have developed such a stochastic model starting from existing kinetic models based on ODEs of some sub-components of the synapse, integrating other data from literature and making some assumptions about non-fully understood processes. Experiments have confirmed the coherence of our model with known biological data, also validating the assumptions made. Our model overcomes some limitations of the kinetic ones and, to our knowledge, represents the first model of synaptic processes based on process calculi. The compositionality of the approach has permitted us to independently focus on tuning the models of the pre- and post- synaptic traits, and then to naturally connect them, by dealing with "interface" issues. Furthermore, we have improved the expressiveness of the model, e.g. by embedding easy control of element concentration time courses. Sensitivity analysis over several parameters of the model has provided results that may help clarify the dynamics of synaptic transmission, while experiments with the model of the complete synapse seem worth explaining short-term plasticity mechanisms.

CONCLUSIONS

Specific presynaptic and postsynaptic mechanisms can be further analysed under various conditions, for instance by studying the presynaptic behaviour under repeated activations. The level of details of the description can be refined, for instance by further specifying the neurotransmitter generation and release steps. Taking advantage of the compositionality of the approach, an enhanced model could then be composed with other neural models, designed within the same framework, in order to obtain a more detailed and comprehensive model. In the long term, we are interested, in particular, in addressing models of synaptic plasticity, i.e. activity dependent mechanisms, which are the bases of memory and learning processes. More on the computer science side, we plan to follow some directions to improve the underlying computational model and the linguistic primitives it provides as suggested by the experiments carried out, e.g. by introducing a suitable notion of (spatial) locality.

Fronto-parietal connection asymmetry regulates working memory distractibility

Recent functional magnetic resonance imaging studies demonstrate that increased task-related neural activity in parietal and frontal cortex during development and training is positively correlated with improved visuospatial working memory (vsWM) performance. Yet, the analysis of the corresponding underlying functional reorganization of the fronto-parietal network has received little attention. Here, we perform an integrative experimental and computational analysis to determine the effective balance between the superior frontal sulcus (SFS) and intraparietal sulcus (IPS) and their putative role(s) in protecting against distracters. To this end, we performed electroencephalographic (EEG) recordings during a vsWM task. We utilized a biophysically based computational cortical network model to analyze the effects of different neural changes in the underlying cortical networks on the directed transfer function (DTF) and spiking activity. Combining a DTF analysis of our EEG data with the DTF analysis of the computational model, a directed strong SFS --> IPS network was revealed. Such a configuration offers protection against distracters, whereas the opposite is true for strong IPS --> SFS connections. Our results therefore suggest that the previously demonstrated improvement of vsWM performance during development could be due to a shift in the control of the effective balance between the SFS-IPS networks.

Method to calculate the moments of the membrane voltage in a model neuron driven by multiplicative filtered shot noise

Neurons are subject to synaptic inputs from many other cells. These inputs consist of spikes changing the conductivity of the target cell, i.e., they enter the neural dynamics as multiplicative shot noise. Up to now, only for simplified models like current-based (additive-noise) point neurons or models with Gaussian white-noise input, exact solutions are available. We present a method to calculate the exact time-dependent moments for the voltage of a point neuron with conductance-based shot noise and a passive membrane. The exact solutions show features (for instance, maxima of the moments vs time) which are also confirmed by numerical simulations. The theoretical analysis of subthreshold membrane fluctuations may contribute to a better comprehension of neural noise in general. We also discuss how the analytical results may provide additional conditions for estimating parameters from experimental data.

Pathological effect of homeostatic synaptic scaling on network dynamics in diseases of the cortex

Slow periodic EEG discharges are common in CNS disorders. The pathophysiology of this aberrant rhythmic activity is poorly understood. We used a computational model of a neocortical network with a dynamic homeostatic scaling rule to show that loss of input (partial deafferentation) can trigger network reorganization that results in pathological periodic discharges. The decrease in average firing rate in the network by deafferentation was compensated by homeostatic synaptic scaling of recurrent excitation among pyramidal cells. Synaptic scaling succeeded in recovering the network target firing rate for all degrees of deafferentation (fraction of deafferented cells), but there was a critical degree of deafferentation for pathological network reorganization. For deafferentation degrees below this value, homeostatic upregulation of recurrent excitation had minimal effect on the macroscopic network dynamics. For deafferentation above this threshold, however, a slow periodic oscillation appeared, patterns of activity were less sparse, and bursting occurred in individual neurons. Also, comparison of spike-triggered afferent and recurrent excitatory conductances revealed that information transmission was strongly impaired. These results suggest that homeostatic plasticity can lead to secondary functional impairment in case of cortical disorders associated with cell loss.

Olfactory Representations by Drosophila Mushroom Body Neurons

Learning and memory has been studied extensively in Drosophila using behavioral, molecular, and genetic approaches. These studies have identified the mushroom body as essential for the formation and retrieval of olfactory memories. We investigated odor responses of the principal neurons of the mushroom body, the Kenyon cells (KCs), in Drosophila using whole cell recordings in vivo. KC responses to odors were highly selective and, thus sparse, compared with those of their direct inputs, the antennal lobe projection neurons (PNs). We examined the mechanisms that might underlie this transformation and identified at least three contributing factors: excitatory synaptic potentials (from PNs) decay rapidly, curtailing temporal integration, PN convergence onto individual KCs is low (∼10 PNs per KC on average), and KC firing thresholds are high. Sparse activity is thought to be useful in structures involved in memory in part because sparseness tends to reduce representation overlaps. By comparing activity patterns evoked by the same odors across olfactory receptor neurons and across KCs, we show that representations of different odors do indeed become less correlated as they progress through the olfactory system.

Artificial synaptic modification reveals a dynamical invariant in the pyloric CPG

The sequential firing of neurons in central pattern generators (CPGs) is generally thought to be a result of an interaction between intrinsic cellular and synaptic properties of the component neurons. Due to experimental limitations, it is usually difficult to address the role of each of these properties separately. We have done so by using the crustacean stomatogastric CPG and the dynamic clamp technique to measure how the network responds to the selective modification of an individual important synapse. Our results show that the burst periods and the phase lags between the constrictor (LP) and dilator (PD) neurons across preparations showed significant variability during equivalent experimental manipulations. Despite this variability, the ratio between the change in the burst period and the change in the phase lag between the same neurons was tightly preserved in all preparations, revealing a dynamical invariant in the system. This dynamical invariant was preserved despite the individual variability in the period and phase lag measurements, suggesting a tightly regulated constraint between the parameters of the network.

Effects of an exogenous noise on a realistic network model: encoding of an EM signal

Endogenous noise has been shown to play a central role in the detection of an electromagnetic signal in the nervous system. In this work, following a biomedical perspective, an exogenous noise applied to a realistic feedforward network model has been considered. It will be shown that, if the exogenous noise is properly filtered and its level is adjusted, a clear optimization of network encoding of an electromagnetic signal, representative of an external stimulation, is obtained through the stochastic resonance paradigm.

NeuReal: an interactive simulation system for implementing artificial dendrites and large hybrid networks

The dynamic clamp is a technique which allows the introduction of artificial conductances into living cells. Up to now, this technique has been mainly used to add small numbers of 'virtual' ion channels to real cells or to construct small hybrid neuronal circuits. In this paper we describe a prototype computer system, NeuReal, that extends the dynamic clamp technique to include (i) the attachment of artificial dendritic structures consisting of multiple compartments and (ii) the construction of large hybrid networks comprising several hundred biophysically realistic modelled neurons. NeuReal is a fully interactive system that runs on Windows XP, is written in a combination of C++ and assembler, and uses the Microsoft DirectX application programming interface (API) to achieve high-performance graphics. By using the sampling hardware-based representation of membrane potential at all stages of computation and by employing simple look-up tables, NeuReal can simulate over 1000 independent Hodgkin and Huxley type conductances in real-time on a modern personal computer (PC). In addition, whilst not being a hard real-time system, NeuReal still offers reliable performance and tolerable jitter levels up to an update rate of 50kHz. A key feature of NeuReal is that rather than being a simple dedicated dynamic clamp, it operates as a fast simulation system within which neurons can be specified as either real or simulated. We demonstrate the power of NeuReal with several example experiments and argue that it provides an effective tool for examining various aspects of neuronal function.

A simulation method for the firing sequences of motor units

The firing sequences of motoneurons contain important information with regard to the underlying neural processes. Several methods have been proposed in the literature to simulate these sequences, however, one of the limitations is that they are not capable of simulating the complex neural dynamics of motor neurons, especially those of concurrently active ones, such as motor unit synchrony and motor unit common drive. In this paper, a novel model based on the Hodgkin-Huxley (HH) system is proposed, which has the ability to simulate the complex neurodynamics of the firing sequences of motor neurons. The model is presented at the cellular level and network level, and some simulation results from a simple 3-neuron network are presented to demonstrate its applications.

The properties of reticular thalamic neuron GABA(A) IPSCs of absence epilepsy rats lead to enhanced network excitability

Both human investigations and studies in animal models have suggested that abnormalities in GABA(A) receptor function have a potential role in the pathophysiology of absence seizures. Recently we showed that, prior to seizure onset, GABA(A) IPSCs in thalamic reticular (NRT) neurons of genetic absence epilepsy rats from Strasbourg (GAERS) had a 25% larger amplitude, a 40% faster decay and a 45% smaller paired-pulse depression than those of nonepileptic control (NEC) rats. By means of a novel mathematical description, the properties of both GAERS and NEC GABAergic synapses can be mimicked. These model synapses were then used in an NRT network model in order to investigate their potential impact on the neuronal firing patterns. Compared to NEC, GAERS NRT neurons show an overall increase in excitability and a higher frequency and regularity of firing in response to periodic input signals. Moreover, in response to randomly distributed stimuli, the GAERS but not the NEC model produces resonance between 7 and 9 Hz, the frequency range of spike-wave discharges in GAERS. The implications of these results for the epileptogenesis of absence seizures are discussed.

Adaptive regulation of sparseness by feedforward inhibition

In the mushroom body of insects, odors are represented by very few spikes in a small number of neurons, a highly efficient strategy known as sparse coding. Physiological studies of these neurons have shown that sparseness is maintained across thousand-fold changes in odor concentration. Using a realistic computational model, we propose that sparseness in the olfactory system is regulated by adaptive feedforward inhibition. When odor concentration changes, feedforward inhibition modulates the duration of the temporal window over which the mushroom body neurons may integrate excitatory presynaptic input. This simple adaptive mechanism could maintain the sparseness of sensory representations across wide ranges of stimulus conditions.

Simulation of networks of spiking neurons: a review of tools and strategies

We review different aspects of the simulation of spiking neural networks. We start by reviewing the different types of simulation strategies and algorithms that are currently implemented. We next review the precision of those simulation strategies, in particular in cases where plasticity depends on the exact timing of the spikes. We overview different simulators and simulation environments presently available (restricted to those freely available, open source and documented). For each simulation tool, its advantages and pitfalls are reviewed, with an aim to allow the reader to identify which simulator is appropriate for a given task. Finally, we provide a series of benchmark simulations of different types of networks of spiking neurons, including Hodgkin-Huxley type, integrate-and-fire models, interacting with current-based or conductance-based synapses, using clock-driven or event-driven integration strategies. The same set of models are implemented on the different simulators, and the codes are made available. The ultimate goal of this review is to provide a resource to facilitate identifying the appropriate integration strategy and simulation tool to use for a given modeling problem related to spiking neural networks.

The impact of input fluctuations on the frequency-current relationships of layer 5 pyramidal neurons in the rat medial prefrontal cortex

The role of irregular cortical firing in neuronal computation is still debated, and it is unclear how signals carried by fluctuating synaptic potentials are decoded by downstream neurons. We examined in vitro frequency versus current (f-I) relationships of layer 5 (L5) pyramidal cells of the rat medial prefrontal cortex (mPFC) using fluctuating stimuli. Studies in the somatosensory cortex show that L5 neurons become insensitive to input fluctuations as input mean increases and that their f-I response becomes linear. In contrast, our results show that mPFC L5 pyramidal neurons retain an increased sensitivity to input fluctuations, whereas their sensitivity to the input mean diminishes to near zero. This implies that the discharge properties of L5 mPFC neurons are well suited to encode input fluctuations rather than input mean in their firing rates, with important consequences for information processing and stability of persistent activity at the network level.

Network mechanism for burst generation

We report on the mechanism of burst generation by populations of intrinsically spiking neurons, when a certain threshold in coupling strength is exceeded. These ensembles synchronize at relatively low coupling strength and lose synchronization at stronger coupling via spatiotemporal intermittency. The latter transition triggers fast repetitive spiking, which results in synchronized bursting. We present evidence that this mechanism is generic for various network topologies from regular to small-world and scale-free ones, different types of coupling and neuronal model.

Computation of spiking activity for a stochastic spatial neuron model: effects of spatial distribution of input on bimodality and CV of the ISI distribution

We obtain computational results for a new extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation and the synaptic input current is also a function of space and time, obeying a first order linear partial differential equation driven by a two-parameter random process. The model is first described explicitly with the inclusion of all biophysical parameters. Simplified equations are obtained with dimensionless space and time variables. A standard parameter set is described, based mainly on values appropriate for cortical pyramidal cells. When the noise is small and the mean voltage crosses threshold, a formula is derived for the expected time to spike. A simulation algorithm, involving one-dimensional random processes is given and used to obtain moments and distributions of the interspike interval (ISI). The parameters used are those for a near balanced state and there is great sensitivity of the firing rate around the balance point. This sensitivity may be related to genetically induced pathological brain properties (Rett's syndrome). The simulation procedure is employed to find the ISI distribution for some simple patterns of synaptic input with various relative strengths for excitation and inhibition. With excitation only, the ISI distribution is unimodal of exponential type and with a large coefficient of variation. As inhibition near the soma grows, two striking effects emerge. The ISI distribution shifts first to bimodal and then to unimodal with an approximately Gaussian shape with a concentration at large intervals. At the same time the coefficient of variation of the ISI drops dramatically to less than 1/5 of its value without inhibition.

Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study

Basal dendrites receive the majority of synapses that contact neocortical pyramidal neurons, yet our knowledge of synaptic processing in these dendrites has been hampered by their inaccessibility for electrical recordings. A new approach to patch-clamp recordings enabled us to characterize the integrative properties of these cells. Despite the short physical length of rat basal dendrites, synaptic inputs were electrotonically remote from the soma (>30-fold excitatory postsynaptic potential (EPSP) attenuation) and back-propagating action potentials were significantly attenuated. Unitary EPSPs were location dependent, reaching large amplitudes distally (>8 mV), yet their somatic contribution was relatively location independent. Basal dendrites support sodium and NMDA spikes, but not calcium spikes, for 75% of their length. This suggests that basal dendrites, despite their proximity to the site of action potential initiation, do not form a single basal-somatic region but rather should be considered as a separate integrative compartment favoring two integration modes: subthreshold, location-independent summation versus local amplification of incoming spatiotemporally clustered information.

Analytical integrate-and-fire neuron models with conductance-based dynamics for event-driven simulation strategies

Event-driven simulation strategies were proposed recently to simulate integrate-and-fire (IF) type neuronal models. These strategies can lead to computationally efficient algorithms for simulating large-scale networks of neurons; most important, such approaches are more precise than traditional clock-driven numerical integration approaches because the timing of spikes is treated exactly. The drawback of such event-driven methods is that in order to be efficient, the membrane equations must be solvable analytically, or at least provide simple analytic approximations for the state variables describing the system. This requirement prevents, in general, the use of conductance-based synaptic interactions within the framework of event-driven simulations and, thus, the investigation of network paradigms where synaptic conductances are important. We propose here a number of extensions of the classical leaky IF neuron model involving approximations of the membrane equation with conductance-based synaptic current, which lead to simple analytic expressions for the membrane state, and therefore can be used in the event-driven framework. These conductance-based IF (gIF) models are compared to commonly used models, such as the leaky IF model or biophysical models in which conductances are explicitly integrated. All models are compared with respect to various spiking response properties in the presence of synaptic activity, such as the spontaneous discharge statistics, the temporal precision in resolving synaptic inputs, and gain modulation under in vivo-like synaptic bombardment. Being based on the passive membrane equation with fixed-threshold spike generation, the proposed gIF models are situated in between leaky IF and biophysical models but are much closer to the latter with respect to their dynamic behavior and response characteristics, while still being nearly as computationally efficient as simple IF neuron models. gIF models should therefore provide a useful tool for efficient and precise simulation of large-scale neuronal networks with realistic, conductance-based synaptic interactions.

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the "canard structure," in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking ("canard explosion") is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models.

Slow state transitions of sustained neural oscillations by activity-dependent modulation of intrinsic excitability

Little is known about the dynamics and mechanisms of transitions between tonic firing and bursting in cortical networks. Here, we use a computational model of a neocortical circuit with extracellular potassium dynamics to show that activity-dependent modulation of intrinsic excitability can lead to sustained oscillations with slow transitions between two distinct firing modes: fast run (tonic spiking or fast bursts with few spikes) and slow bursting. These transitions are caused by a bistability with hysteresis in a pyramidal cell model. Balanced excitation and inhibition stabilizes a network of pyramidal cells and inhibitory interneurons in the bistable region and causes sustained periodic alternations between distinct oscillatory states. During spike-wave seizures, neocortical paroxysmal activity exhibits qualitatively similar slow transitions between fast run and bursting. We therefore predict that extracellular potassium dynamics can cause alternating episodes of fast and slow oscillatory states in both normal and epileptic neocortical networks.

Computational analysis of subthalamic nucleus and lenticular fasciculus activation during therapeutic deep brain stimulation

The subthalamic nucleus (STN) is the most common target for the treatment of Parkinson's disease (PD) with deep brain stimulation (DBS). DBS of the globus pallidus internus (GPi) is also effective in the treatment of PD. The output fibers of the GPi that form the lenticular fasciculus pass in close proximity to STN DBS electrodes. In turn, both STN projection neurons and GPi fibers of passage represent possible therapeutic targets of DBS in the STN region. We built a comprehensive computational model of STN DBS in parkinsonian macaques to study the effects of stimulation in a controlled environment. The model consisted of three fundamental components: 1) a three-dimensional (3D) anatomical model of the macaque basal ganglia, 2) a finite element model of the DBS electrode and electric field transmitted to the tissue medium, and 3) multicompartment biophysical models of STN projection neurons, GPi fibers of passage, and internal capsule fibers of passage. Populations of neurons were positioned within the 3D anatomical model. Neurons were stimulated with electrode positions and stimulation parameters defined as clinically effective in two parkinsonian monkeys. The model predicted axonal activation of STN neurons and GPi fibers during STN DBS. Model predictions regarding the degree of GPi fiber activation matched well with experimental recordings in both monkeys. Only axonal activation of the STN neurons showed a statistically significant increase in both monkeys when comparing clinically effective and ineffective stimulation. Nonetheless, both neural targets may play important roles in the therapeutic mechanisms of STN DBS.

A computational model of how an interaction between the thalamocortical and thalamic reticular neurons transforms the low-frequency oscillations of the globus pallidus

In Parkinson's disease, neurons of the internal segment of the globus pallidus (GPi) display the low-frequency tremor-related oscillations. These oscillatory activities are transmitted to the thalamic relay nuclei. Computer models of the interacting thalamocortical (TC) and thalamic reticular (RE) neurons were used to explore how the TC-RE network processes the low-frequency oscillations of the GPi neurons. The simulation results show that, by an interaction between the TC and RE neurons, the TC-RE network transforms a low-frequency oscillatory activity of the GPi neurons to a higher frequency of oscillatory activity of the TC neurons (the superharmonic frequency transformation). In addition to the interaction between the TC and RE neurons, the low-threshold calcium current in the RE and TC neurons and the hyperpolarization-activated cation current (I (h)) in the TC neurons have significant roles in the superharmonic frequency transformation property of the TC-RE network. The external globus pallidus (GPe) oscillatory activity, which is directly transmitted to the RE nucleus also displays a significant modulatory effect on the superharmonic frequency transformation property of the TC-RE network.

Travelling waves and EEG patterns during epileptic seizure: analysis with an integrate-and-fire neural network

Epilepsy is characterized by paradoxical patterns of neural activity. They may cause different types of electroencephalogram (EEG), which dynamically change in shape and frequency content during the temporal evolution of seizure. It is generally assumed that these epileptic patterns may originate in a network of strongly interconnected neurons, when excitation dominates over inhibition. The aim of this work is to use a neural network composed of 50 x 50 integrate-and-fire neurons to analyse which parameter alterations, at the level of synapse topology, may induce network instability and epileptic-like discharges, and to study the corresponding spatio-temporal characteristics of electrical activity in the network. We assume that a small group of central neurons is stimulated by a depolarizing current (epileptic focus) and that neurons are connected via a Mexican-hat topology of synapses. A signal representative of cortical EEG (ECoG) is simulated by summing the membrane potential changes of all neurons. A sensitivity analysis on the parameters describing the synapse topology shows that an increase in the strength and in spatial extension of excitatory vs. inhibitory synapses may cause the occurrence of travelling waves, which propagate along the network. These propagating waves may cause EEG patterns with different shape and frequency, depending on the particular parameter set used during the simulations. The resulting model EEG signals include irregular rhythms with large amplitude and a wide frequency content, low-amplitude high-frequency rapid discharges, isolated or repeated bursts, and low-frequency quasi-sinusoidal patterns. A slow progressive temporal variation in a single parameter may cause the transition from one pattern to another, thus generating a highly non-stationary signal which resembles that observed during ECoG measurements. These results may help to elucidate the mechanisms at the basis of some epileptic discharges, and to relate rapid changes in EEG patterns with the underlying alterations at the network level.

Role of inhibitory feedback for information processing in thalamocortical circuits

The information transfer in the thalamus is blocked dynamically during sleep, in conjunction with the occurrence of spindle waves. In order to describe the dynamic mechanisms which control the sensory transfer of information, it is necessary to have a qualitative model for the response properties of thalamic neurons. As the theoretical understanding of the mechanism remains incomplete, we analyze two modeling approaches for a recent experiment by Le Masson et al. [Nature (London) 417, 854 (2002)] on the thalamocortical loop. We use a conductance based model in order to motivate an extension of the Hindmarsh-Rose model, which mimics experimental observations of Le Masson et al. Typically, thalamic neurons possess two different firing modes, depending on their membrane potential. At depolarized potentials, the cells fire in a single spike mode and relay synaptic inputs in a one-to-one manner to the cortex. If the cell gets hyperpolarized, T-type calcium currents generate burst-mode firing which leads to a decrease in the spike transfer. In thalamocortical circuits, the cell membrane gets hyperpolarized by recurrent inhibitory feedback loops. In the case of reciprocally coupled excitatory and inhibitory neurons, inhibitory feedback leads to metastable self-sustained oscillations, which mask the incoming input, and thereby reduce the information transfer significantly.

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.

Inferring network activity from synaptic noise

During intense network activity in vivo, cortical neurons are in a high-conductance state, in which the membrane potential (V(m)) is subject to a tremendous fluctuating activity. Clearly, this "synaptic noise" contains information about the activity of the network, but there are presently no methods available to extract this information. We focus here on this problem from a computational neuroscience perspective, with the aim of drawing methods to analyze experimental data. We start from models of cortical neurons, in which high-conductance states stem from the random release of thousands of excitatory and inhibitory synapses. This highly complex system can be simplified by using global synaptic conductances described by effective stochastic processes. The advantage of this approach is that one can derive analytically a number of properties from the statistics of resulting V(m) fluctuations. For example, the global excitatory and inhibitory conductances can be extracted from synaptic noise, and can be related to the mean activity of presynaptic neurons. We show here that extracting the variances of excitatory and inhibitory synaptic conductances can provide estimates of the mean temporal correlation-or level of synchrony-among thousands of neurons in the network. Thus, "probing the network" through intracellular V(m) activity is possible and constitutes a promising approach, but it will require a continuous effort combining theory, computational models and intracellular physiology.

Exploring spike transfer through the thalamus using hybrid artificial-biological neuronal networks

We use dynamic clamp to construct "hybrid" thalamic circuits by connecting a biological neuron in situ to silicon- or software-generated "neurons" through artificial synapses. The purpose is to explore cellular sensory gating mechanisms that regulate the transfer efficiency of signals during different sleep-wake states. Hybrid technology is applied in vitro to different paradigms such as: (1) simulating interactions between biological thalamocortical neurons, artificial reticular thalamic inhibitory interneurons and a simulated sensory input, (2) grafting an artificial sensory input to a wholly biological thalamic network that generates spontaneous sleep-like oscillations, (3) injecting in thalamocortical neurons a background synaptic bombardment mimicking the activity of corticothalamic inputs. We show that the graded control of the strength of intrathalamic inhibition, combined with the membrane polarization and the fluctuating synaptic noise in thalamocortical neurons, is able to govern functional shifts between different input/output transmission states of the thalamic gate.