We designed a model-based analysis to predict the occurrence of population patterns in distributed spiking activity. Using a maximum entropy principle with a Markovian assumption, we obtain a model ...that accounts for both spatial and temporal pairwise correlations among neurons. This model is tested on data generated with a Glauber spin-glass system and is shown to correctly predict the occurrence probabilities of spatiotemporal patterns significantly better than Ising models only based on spatial correlations. This increase of predictability was also observed on experimental data recorded in parietal cortex during slow-wave sleep. This approach can also be used to generate surrogates that reproduce the spatial and temporal correlations of a given data set.
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Many complex systems display self-organized critical states characterized by 1/f frequency scaling of power spectra. Global variables such as the electroencephalogram, scale as 1/f, which could be ...the sign of self-organized critical states in neuronal activity. By analyzing simultaneous recordings of global and neuronal activities, we confirm the 1/f scaling of global variables for selected frequency bands, but show that neuronal activity is not consistent with critical states. We propose a model of 1/f scaling which does not rely on critical states, and which is testable experimentally.
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We present a mean-field formalism able to predict the collective dynamics of large networks of conductance-based interacting spiking neurons. We apply this formalism to several neuronal models, from ...the simplest Adaptive Exponential Integrate-and-Fire model to the more complex Hodgkin-Huxley and Morris-Lecar models. We show that the resulting mean-field models are capable of predicting the correct spontaneous activity of both excitatory and inhibitory neurons in asynchronous irregular regimes, typical of cortical dynamics. Moreover, it is possible to quantitatively predict the population response to external stimuli in the form of external spike trains. This mean-field formalism therefore provides a paradigm to bridge the scale between population dynamics and the microscopic complexity of the individual cells physiology.
Population models are a powerful mathematical tool to study the dynamics of neuronal networks and to simulate the brain at macroscopic scales. We present a mean-field model capable of quantitatively predicting the temporal dynamics of a network of complex spiking neuronal models, from Integrate-and-Fire to Hodgkin-Huxley, thus linking population models to neurons electrophysiology. This opens a perspective on generating biologically realistic mean-field models from electrophysiological recordings.
Key points
We recreated in vitro the fluctuation‐driven regime observed at the soma during asynchronous network activity in vivo and we studied the firing rate response as a function of the ...properties of the membrane potential fluctuations.
We provide a simple analytical template that captures the firing response of both pyramidal neurons and various theoretical models.
We found a strong heterogeneity in the firing rate response of layer V pyramidal neurons: in particular, individual neurons differ not only in their mean excitability level, but also in their sensitivity to fluctuations.
Theoretical modelling suggest that this observed heterogeneity might arise from various expression levels of the following biophysical properties: sodium inactivation, density of sodium channels and spike frequency adaptation.
Characterizing the input–output properties of neocortical neurons is of crucial importance for understanding the properties emerging at the network level. In the regime of low‐rate irregular firing (such as in the awake state), determining those properties for neocortical cells remains, however, both experimentally and theoretically challenging. Here, we studied this problem using a combination of theoretical modelling and in vitro experiments. We first identified, theoretically, three somatic variables that describe the dynamical state at the soma in this fluctuation‐driven regime: the mean, standard deviation and time constant of the membrane potential fluctuations. Next, we characterized the firing rate response of individual layer V pyramidal cells in this three‐dimensional space by means of perforated‐patch recordings and dynamic clamp in the visual cortex of juvenile mice in vitro. We found that individual neurons strongly differ not only in terms of their excitability, but also, and unexpectedly, in their sensitivities to fluctuations. Finally, using theoretical modelling, we attempted to reproduce these results. The model predicts that heterogeneous levels of biophysical properties such as sodium inactivation, sharpness of sodium activation and spike frequency adaptation account for the observed diversity of firing rate responses. Because the firing rate response will determine population rate dynamics during asynchronous neocortical activity, our results show that cortical populations are functionally strongly inhomogeneous in young mouse visual cortex, which should have important consequences on the strategies of cortical computation at early stages of sensory processing.
Key points
We recreated in vitro the fluctuation‐driven regime observed at the soma during asynchronous network activity in vivo and we studied the firing rate response as a function of the properties of the membrane potential fluctuations.
We provide a simple analytical template that captures the firing response of both pyramidal neurons and various theoretical models.
We found a strong heterogeneity in the firing rate response of layer V pyramidal neurons: in particular, individual neurons differ not only in their mean excitability level, but also in their sensitivity to fluctuations.
Theoretical modelling suggest that this observed heterogeneity might arise from various expression levels of the following biophysical properties: sodium inactivation, density of sodium channels and spike frequency adaptation.
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To investigate the basis of the fluctuating activity present in neocortical neurons
in vivo, we have combined computational models with whole-cell recordings using the dynamic-clamp technique. A ...simplified ‘point-conductance’ model was used to represent the currents generated by thousands of stochastically releasing synapses. Synaptic activity was represented by two independent fast glutamatergic and GABAergic conductances described by stochastic random-walk processes. An advantage of this approach is that all the model parameters can be determined from voltage-clamp experiments. We show that the point-conductance model captures the amplitude and spectral characteristics of the synaptic conductances during background activity. To determine if it can recreate
in vivo-like activity, we injected this point-conductance model into a single-compartment model, or in rat prefrontal cortical neurons
in vitro using dynamic clamp. This procedure successfully recreated several properties of neurons intracellularly recorded
in vivo, such as a depolarized membrane potential, the presence of high-amplitude membrane potential fluctuations, a low-input resistance and irregular spontaneous firing activity. In addition, the point-conductance model could simulate the enhancement of responsiveness due to background activity.
We conclude that many of the characteristics of cortical neurons
in vivo can be explained by fast glutamatergic and GABAergic conductances varying stochastically.
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Local field potentials (LFPs) are routinely measured experimentally in brain tissue, and exhibit strong low-pass frequency filtering properties, with high frequencies (such as action potentials) ...being visible only at very short distances (approximately 10 microm) from the recording electrode. Understanding this filtering is crucial to relate LFP signals with neuronal activity, but not much is known about the exact mechanisms underlying this low-pass filtering. In this paper, we investigate a possible biophysical mechanism for the low-pass filtering properties of LFPs. We investigate the propagation of electric fields and its frequency dependence close to the current source, i.e., at length scales in the order of average interneuronal distances. We take into account the presence of a high density of cellular membranes around current sources, such as glial cells. By considering them as passive cells, we show that under the influence of the electric source field, they respond by polarization. Because of the finite velocity of ionic charge movements, this polarization will not be instantaneous. Consequently, the induced electric field will be frequency-dependent, and much reduced for high frequencies. Our model establishes that this situation is analogous to an equivalent RC circuit, or better yet a system of coupled RC circuits. We present a number of numerical simulations of an induced electric field for biologically realistic values of parameters, and show the frequency filtering effect as well as the attenuation of extracellular potentials with distance. We suggest that induced electric fields in passive cells surrounding neurons are the physical origin of frequency filtering properties of LFPs. Experimentally testable predictions are provided allowing us to verify the validity of this model.
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In vivo,
in vitro and computational studies were used to investigate the impact of the synaptic background activity observed in neocortical neurons
in vivo. We simulated background activity
in vitro ...using two stochastic Ornstein-Uhlenbeck processes describing glutamatergic and GABAergic synaptic conductances, which were injected into a cell in real time using the dynamic clamp technique. With parameters chosen to mimic
in vivo conditions, layer 5 rat prefrontal cortex cells recorded
in vitro were depolarized by about 15 mV, their membrane fluctuated with a S.D. of about 4 mV, their input resistances decreased five-fold, their spontaneous firing had a high coefficient of variation and an average firing rate of about 5–10 Hz. Brief changes in the variance of the α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) synaptic conductance fluctuations induced time-locked spiking without significantly changing the average membrane potential of the cell. These transients mimicked increases in the correlation of excitatory inputs. Background activity was highly effective in modulating the firing-rate/current curve of the cell: the variance of the simulated γ-aminobutyric acid (GABA) and AMPA conductances individually set the input/output gain, the mean excitatory and inhibitory conductances set the working point, and the mean inhibitory conductance controlled the input resistance. An average ratio of inhibitory to excitatory mean conductances close to 4 was optimal in generating membrane potential fluctuations with high coefficients of variation. We conclude that background synaptic activity can dynamically modulate the input/output properties of individual neocortical neurons
in vivo.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
Unité de Neurosciences Intégratives et Computationnelles, Centre National de la Recherche Scientifique, Gif-sur-Yvette, France; Howard Hughes Medical Institute and the Salk Institute; Department of ...Biology, University of California at San Diego, La Jolla, California
Destexhe, A., and T. J. Sejnowski. Interactions Between Membrane Conductances Underlying Thalamocortical Slow-Wave Oscillations. Physiol Rev 83: 1401-1453, 2003; 10.1152/physrev.00012.2003.Neurons of the central nervous system display a broad spectrum of intrinsic electrophysiological properties that are absent in the traditional "integrate-and-fire" model. A network of neurons with these properties interacting through synaptic receptors with many time scales can produce complex patterns of activity that cannot be intuitively predicted. Computational methods, tightly linked to experimental data, provide insights into the dynamics of neural networks. We review this approach for the case of bursting neurons of the thalamus, with a focus on thalamic and thalamocortical slow-wave oscillations. At the single-cell level, intrinsic bursting or oscillations can be explained by interactions between calcium- and voltage-dependent channels. At the network level, the genesis of oscillations, their initiation, propagation, termination, and large-scale synchrony can be explained by interactions between neurons with a variety of intrinsic cellular properties through different types of synaptic receptors. These interactions can be altered by neuromodulators, which can dramatically shift the large-scale behavior of the network, and can also be disrupted in many ways, resulting in pathological patterns of activity, such as seizures. We suggest a coherent framework that accounts for a large body of experimental data at the ion-channel, single-cell, and network levels. This framework suggests physiological roles for the highly synchronized oscillations of slow-wave sleep.
1 All the conclusions of the model with high uniform density of I T in dendrites (1.7 x 10 -5 cm/s in soma and 8.5 x 10 -5 cm/s in dendrites; Ref. 106) could be obtained using a nonuniform distribution of T channels (10.3 x 10 -5 cm/s in soma, 20.6 x 10 -5 cm/s in proximal dendrites <40 µm from soma, and 2.5 x 10 -5 cm/s elsewhere), similar to the pattern estimated by Williams and Stuart (361).
2 Computer-generated animations of the membrane voltage are available at http://cns.iaf.cnrs-gif.fr or http://www.salk.edu/~alain .
3 Heterogeneity was created by randomizing the values of the I h conductance, such that the majority of TC cells was resting around -60 mV, while only a small minority were spontaneous oscillators, similar to the proportion found in vitro (191). This minority served as "initiators" of the oscillation in the entire network.
4 It is also conceivable that RE cells coupled through gap junctions could induce bursts in each other if their resting level is hyperpolarized enough to deinactivate the I T . In this case, oscillations should be observed in slices of the RE nucleus, where the level of RE cells is typically very hyperpolarized (see, for example, Ref. 337). Such oscillations have, however, never been reported.
5 Bicuculline was later shown to also block the apamin-sensitive calcium-dependent current in RE cells (76). Because this current is important for controlling burst generation in RE cells (17), bicuculline therefore does not exert specific effects on GABA A receptors. Other antagonists are used, such as picrotoxin, that also induce slow thalamic oscillations, showing that this oscillation is generated through antagonist actions on GABA A receptors in the RE nucleus (266).
6 This may also be described as an afterdepolarization (ADP) following the spindle wave, which is actually the terminology used in the in vitro experiments (19).
7 Inhibitory dominance was not by itself a prediction, given the large body of experimental evidence showing that cortical stimulation primarily evoke IPSPs in TC cells (4, 45, 66, 82, 196, 263, 315, 330, 359).
8 This is converse to the claims that the low-threshold spike in TC cells is not involved in generating seizures in GAERS rats (251).
9 This is possible in slices from the visual thalamus, in which the corticothalamic and retinal fibers are both accessible (18, 334).
Address for reprint requests and other correspondence: A. Destexhe, Unité de Neurosciences Intégratives et Computation-nelles, CNRS, UPR-2191, Avenue de la Terrasse, Bat. 33, 91198 Gif-sur-Yvette, France (E-mail: Destexhe{at}iaf.cnrs-gif.fr ).
Synaptic noise due to intense network activity can have a significant impact on the electrophysiological properties of individual neurons. This is the case for the cerebral cortex, where ongoing ...activity leads to strong barrages of synaptic inputs, which act as the main source of synaptic noise affecting on neuronal dynamics. Here, we characterize the sub-threshold behavior of neuronal models in which synaptic noise is represented by either additive or multiplicative noise, described by Ornstein-Uhlenbeck processes. We derive and solve the Fokker-Planck equation for this system, which describes the time evolution of the probability density function for the membrane potential. We obtain an analytic expression for the membrane potential distribution at steady state and compare this expression with the subthreshold activity obtained in Hodgkin-Huxley-type models with stochastic synaptic inputs. The differences between multiplicative and additive noise models suggest that multiplicative noise is adequate to describe the high-conductance states similar to in vivo conditions. Because the steady-state membrane potential distribution is easily obtained experimentally, this approach provides a possible method to estimate the mean and variance of synaptic conduct ancesinreal neurons.
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Synaptically generated subthreshold membrane potential (V
) fluctuations can be characterized within the framework of stochastic calculus. It is possible to obtain analytic expressions for the ...steady-state V
distribution, even in the case of conductance-based synaptic currents. However, as we show here, the analytic expressions obtained may substantially deviate from numerical solutions if the stochastic membrane equations are solved exclusively based on expectation values of differentials of the stochastic variables, hence neglecting the spectral properties of the underlying stochastic processes. We suggest a simple solution that corrects these deviations, leading to extended analytic expressions of the V
distribution valid for a parameter regime that covers several orders of magnitude around physiologically realistic values. These extended expressions should enable finer characterization of the stochasticity of synaptic currents by analyzing experimentally recorded V
distributions and may be applicable to other classes of stochastic processes as well.
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